diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzciee" "b/data_all_eng_slimpj/shuffled/split2/finalzzciee" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzciee" @@ -0,0 +1,5 @@ +{"text":"\\subsection{Calibration}\nThe instrumental magnitudes for reference images were converted to\nthe standard ones using linear transformations based on 817 and 570\nlocal Stetson's standards in fields W and E, respectively (Stetson\n2000; 2009 on-line version). The following transformations were\nderived:\n\\begin{eqnarray}\n{\\rm v=V -0.086(3)\\times (B-V) -1.591(3)}\\nonumber \\\\\n{\\rm b=B -0.146(4)\\times (B-V) -2.054(4)} \\\\\n{\\rm b-v=0.938(3)\\times (B-V) -0.462(3)}\\nonumber\n\\end{eqnarray}\n(field E) and\n\\begin{eqnarray}\n{\\rm v=V -0.074(3)\\times (B-V) -1.602(3)}\\nonumber \\\\\n{\\rm b=B -0.139(4)\\times (B-V) -2.038(4)} \\\\\n{\\rm b-v=0.936(3)\\times (B-V) -0.437(3)~}\\nonumber\n\\end{eqnarray}\n(field W), where capital letters denote the standard magnitudes.\n\nAs most of the monitoring was conducted in $V$ only, the standard magnitudes\nwere derived using average values of $B-V$. This introduced some systematic\nerrors in the case of color-changing objects. However, for most of the\nvariables the amplitude of color variations did nor exceed 0.1 mag,\nand the systematic error of $V$ magnitude was smaller than 0.01 mag.\nFor the main goal of our survey, which is is to detect\nnew variables, inaccuracies of this size are unimportant.\n\nThe astrometric solutions for the reference images in $V$ were found\nbased on positions of 2997 UCAC3 stars (Zacharias et al. 2010). The\naverage residuals in RA and DEC between cataloged and recovered\ncoordinates amount to 0.14 and 0.14 arcsec, respectively. For the\ndetection of variables we used methods described in Kaluzny et al.\n(2013). A total of 65 variables were found (among them 19 new ones).\nThey are listed in Table 1 along with their equatorial coordinates\nand cross-identifications for catalogs published by Weldrake et al.\n(2004), Kaluzny et al. (1998) and Samus et al. (2009). We detected\nvariability of all but one object from Weldrake et al. (2004)\npresent in the surveyed field (our photometry shows no brightness\nvariations for their star V93. \n\nFinding charts for newly detected variables are shown in Fig. 2.\n\n\\section{Variables}\n\nThe basic properties of variables detected in our survey are listed\nin Table 2. The periods in column 2 were derived with the method\nemploying periodic orthogonal polynomials to fit the observations\nand the analysis of variance statistic to evaluate the quality of\nthe fit (Schwarzenberg-Czerny 1996), as implemented in the TATRY\ncode kindly made available by the author.\n\nThe parameter $\\Delta V$ listed in column 4 gives the full range of\nthe measured $V$-magnitude, including seasonal changes of light curves. The last\ncolumn of Table 2 contains the proposed classification of variables,\nwith ``new'' indicating newly detected objects. EW, EB and EA denote\neclipsing binaries with light curves of W~UMa, $\\beta$ Lyrae and\nAlgol type, respectively; Ell stands for ellipsoidal variables, and\nstars located along or near the red giant branch on the cluster\ncolor-magnitude diagram (CMD) are labeled with RGB \/ RS CVn. The variability\nof the latter is likely caused by ellipsoidal effect and\/or\nchromospheric activity related to binarity. Among the new variables\nthere are five likely optical counterparts to X-ray sources detected\nwith the Chandra telescope by Heinke et al. (2005).\n\nFigure 3 presents the CMD of a small section of the monitored field\nwith marked locations of all but two variables listed in Tables 1\nand 2. Not included are W26 and E20 - very red Miras with $B-V>3$\nmag which were originally detected by the OGLE group (Soszynski et al.\n2011). The group of variables located to the blue of the cluster's\nmain sequence at $V\\approx19$ is composed of RR~Lyr pulsators from the\nSMC. The sequence of red long period variables starting at\n($V\\approx19$, $B-V\\approx1.1$) and extending to $B-V\\approx2.0$\nalso consists of SMC stars. Fig. 4 shows the positions in the \ncluster CMD of a selection of\nour variables, including all eclipsing binaries and three\nparticularly interesting objects which will be discussed below.\nPhased light curves for 14 new variables are\ndisplayed in Fig.~5 (not shown are RR-Lyrae stars, long period\nvariables, and E32 whose period is not known). Light curves for all\n65 variables detected within the present survey can be found in the\nelectronic version of this paper available from the Acta Astronomica \narchive or from CASE archive at http:\/\/case.camk.edu.pl. \n\n\\subsection{Eclipsing binaries}\nThe main result of our survey is the detection of four new detached\neclipsing binaries. All these objects are located beyond the core region of\nthe cluster and are suitable for spectroscopic follow-up studies\nwith ground-based telescopes.\n\nThe light curve of W12 shows a shallow secondary eclipse with $\\Delta V\n\\approx 0.1$~mag and a primary eclipse with $\\Delta V \\approx 0.4$~mag.\nA preliminary solution of the $V$-light curve indicates a high luminosity\nratio of the components: $L_{p}\/L_{s}=9$. The expected luminosity ratio\nfor the $I$ band equals to 6, so that the near-IR spectroscopy may allow\nfor the determination of radial velocity curves of both components.\n\nThe available light curve of E32 shows only one clear eclipse\nwith $\\Delta V=0.6$ mag. This means that the orbital period has to\nbe longer than 10 days (and may be significantly longer). We\nobtained a single spectrum of E32 with the MIKE Echelle spectrograph\non the 6.5 m Magellan Clay telescope, and we found that the\nbinary is an SB2 system. Several additional spectra are needed to\nestablish the ephemeris with confidence. This in turn will enable\nfurther photometric observations in eclipses, and finally the\ndetermination of absolute parameters of the system. The location of\nE32 on the CMD indicates that at least one of its components is an\nevolved star at or past the turnoff. The analysis of E32 together\nwith V69 (Thompson et al. 2010) may allow interesting limits \nto be placed on the helium abundance of 47~Tuc.\n\\begin{table}\n\\caption{Equatorial coordinates of variable stars identified\nwithin the present survey \\label{tab:coords}}\n{\\scriptsize\n\\setlength{\\tabcolsep}{0.48em}\n\\begin{tabular}{|l|c|c|c|c|c|l|c|c|c|c|c|}\n\\hline\nID & RA$_{J2000}$& Dec$_{J2000}$ & ID-W$^{a}$ & ID-K$^{b}$ & ID-S$^{c}$& ID &\n RA$_{J2000}$& Dec$_{J2000}$ & ID-W$^{a}$ & ID-K$^{b}$ & ID-S$^{c}$\\\\\n & [deg] & [deg] & & & & & [deg] & [deg] & & \\\\\n\\hline\n\\hline\nW1 &6.04262 &-72.06708& & & &E8 &6.51136 &-71.97276& &231 & \\\\\nW2 &5.91532 &-72.02788& & & &E9 &6.45689&-71.97410& V99 &226& \\\\ \nW3 &6.05276 &-72.11105& & &V001 &E10 &6.55163&-72.03701& V27 &230& \\\\ \nW4 &6.02083 &-72.08958& & & &E11 &6.51274&-72.05086& V28 &229& \\\\\nW5 &5.99704 &-72.11350& & & &E12 &6.38434&-72.03098& V100&225 & V044 \\\\\nW6 &5.91897 &-72.09997& &217 &V009 &E13 &6.53692&-72.11706& V7 &228 & V046 \\\\\nW7 &5.95557 &-72.21893& & & &E14 &6.42508&-72.10040& V10 &223 & \\\\\nW8 &5.69983 &-71.94696& & & &E15 &6.54387&-72.18549& V6 &227 & V045\\\\\nW9 &5.68282 &-71.95575& V71&234 & &E16 &6.53915&-72.20798& V5 &255 & \\\\\nW10 &5.77913 &-72.02260& &218& &E17 &6.39210&-72.16616& V9 &222 & \\\\\nW11 &5.84984 &-72.05125& & & &E18 &6.51177&-72.24155& & & \\\\\nW12 &5.73147 &-72.08756& & & &E19 &6.40987&-72.23572& & & \\\\\nW13 &5.72354 &-72.06296& V69& & &E20 &6.40448&-72.25761& & 252 & \\\\\nW14 &5.65323 &-72.11997& & & &E21 &6.31753&-71.93490& V30 & 238 &V047\\\\\nW15 &5.81761 &-72.18639& & & &E22 &6.24824&-71.89658& & 241 & \\\\\nW16 &5.77493 &-72.15847& V97& & &E23 &6.20544&-71.93885& V34 & & \\\\\nW17 &5.67023 &-72.15625& V96 &215& &E24 &6.25194&-72.00076& V32 &221&V043\\\\\nW18 &5.69786 &-72.22143& V95&245&V050 &E25 &6.17727&-72.10601& & & \\\\\nW19 &5.63677 &-71.99193& V70&216& &E26 &6.17207&-72.10331& & & \\\\\nW20 &5.54779 &-71.98831& V62 & & &E27 &6.29647&-72.20400& V14&250 &V052\\\\\nW21 &5.50268 &-72.03445& V61 &214&V042 &E28 &6.27575&-72.17319& V15&219 & \\\\\nW22 &5.46433 &-72.18055& V91 & & &E29 &6.15076&-71.95535& V33&239 \\\\\nW23 &5.63907 &-72.23045& V94 & 246& &E30 &6.17717&-71.98994& V31&220 & \\\\\nW24 &5.27715 &-71.94372& V64& & &E31 &6.05647&-71.96737& & & \\\\\nW25 &5.34974 &-72.25939& V92&243& &E32 &6.15633&-72.06476& & & \\\\\nW26 &5.27031 &-72.25646& &242& &E33 &6.07805&-72.08192& & & \\\\\nE1 &6.75007 &-71.91918& V26 & & &E34 &6.10767&-72.11765& & & \\\\\nE2 &6.69230 &-71.99538& & & &E35 &6.09773&-72.14155& V17 & & \\\\\nE3 &6.73688 &-72.17036& V13 & 232 & &E36 &6.09675&-72.12296& & & \\\\\nE4 &6.69681 &-72.27622& & & &E37 &6.07745&-72.13304& & &V002\\\\\nE5 &6.67914 &-72.25578& V12 & 253&V053 &E38 &6.13241&-72.15819& V16&251 & \\\\\nE6 &6.58064 &-72.23394& V4& 254 & &E39 &6.10552&-72.22732& & & \\\\\nE7 &6.40631 &-71.93440& V29 & & & & & & &\\\\\n\\hline\n\\end{tabular}\n}\n{\\footnotesize$^a$ Weldrake et al. 2004;\n$^b$ Kaluzny et al. 1988; $^c$ Samus et al. 2009}\n\\end{table}\n\nFor W7 we have three MIKE\/Magellan spectra. They show that the\nvariable is an SB1 binary with measured velocities of\n$-37.41\\pm0.32$, $-74.27\\pm0.22$ and $+16.38\\pm0.16$ km\/s at orbital\nphases 0.49 (HJD 245 5769.866), 0.353 (HJD 245 5770.839) and 0.692 \n(HJD 55836.642), respectively. \nThese measurements indicate that the binary is a\nlikely member of the cluster. The systemic velocity of 47 Tuc is\nequal to -18.0 km\/s, and the velocity dispersion at the location \nof W7 amounts to $\\sim$11 km\/s (Harris 1996). This can be\ncompared with the binary's velocity near conjunction at orbital phase 0.49.\nThe fourth detached system, E39, is placed rather far\nfrom the cluster main sequence and may be a foreground halo object.\nIts low luminosity and short orbital period make it a\ndifficult target for spectroscopy.\n\n\n\\begin{center}\n\\footnotesize{\n\\begin{longtable}{|l|l|c|c|c|c|}\n\\caption[]{Properties of variable stars identified\nwithin the present survey}\\label{tab:properties}\\\\\n\\hline\nID & P & $V_{max}$ & $\\Delta V$ & $$ & Type of variability \\\\\n & [d] & [mag] & [mag] & & Remarks \\\\\n\\hline \\hline\n\\endfirsthead\n\\multicolumn{6}{l}\n{{ \\tablename\\ \\thetable{} -- continued from previous page}} \\\\\n\\hline\nID & P & $V_{max}$ & $\\Delta V$ & $$ & Type of variability \\\\\n & [d] & [mag] & [mag] & [mag] & Remarks \\\\\n\\hline\n\\hline\n\\endhead\n\\hline \\multicolumn{6}{|r|}{Continued on next page} \\\\ \\hline\n\\endfoot\n\n\\hline\n\\endlastfoot\nW1 & 0.36115544(1) & 15.607& 0.163 & 0.600 & EW, YS, X$^{a}$, {\\bf new}, {\\scriptsize Ch-0024410.2-720401}\\\\\nW2 & 0.28287568(1) & 17.565 & 0.317 & 0.543 & EW, {\\bf new} \\\\\nW3 & - & 11.753 & 2.872 & 1.516 & LP \\\\\nW4 & 0.35298992(1) & 15.838 & 0.163 & 0.271 & EW, BS, X, {\\bf new}, {\\scriptsize Ch-002404.9-720522}\\\\\nW5 & 0.28036018(8) & 17.143 & 0.225 & 0.420 & EW, BS, X, {\\bf new}, {\\scriptsize Ch-002359.3-720648}\\\\\nW6 & 0.73703372(1) & 13.063 & 1.024 & 0.261 & RR, {\\scriptsize OGLE-SMC-RRLYR-0051}$^{~b}$\\\\\nW7 & 3.88255(2) & 19.091 & 0.384 & 0.815 & EA, {\\bf new} \\\\\nW8 & 12.78227(2) & 14.150 & 0.032 & 1.173 & RS CVn, RGB, {\\bf new} \\\\\nW9 & 0.6158898(1) & 19.084 & 0.811 & 0.439 & RR, {\\scriptsize OGLE-SMC-RRLYR-0030} \\\\\nW10& - & 15.976 & 0.384 & 1.652 & LP, {\\scriptsize OGLE-SMC-LPV-00128}$^{~c}$ \\\\\nW11& 0.63228616(9) & 18.948 & 0.65 & 0.18 & RR, {\\scriptsize OGLE-SMC-RRLYR-0044} \\\\\nW12& 3.732001(2) & 17.462 & 0.411 & 0.634 & EA, {\\bf new} \\\\\nW13& 29.53975(1) & 16.799 & 0.616 & 0.537 & EA \\\\\nW14& 0.04595852(3) & 14.679 & 0.037 & 0.145 & SX, BS, {\\bf new}$^d$ \\\\\nW15& 71.4156(5) & 17.138 & 0.141 & 1.557 & LP, {\\scriptsize OGLE-SMC-LPV-00143} \\\\\nW16& 0.39708061(4) & 18.859 & 0.274 & 0.834 & EB \\\\\nW17& 8.4278094(1) & 16.638 & 0.203 & 0.931 & BY~Dra \\\\\nW18& 0.27890017(1) & 15.459 & 0.424 & 0.565 & EW, YS \\\\\nW19& 0.36164588(6) & 19.584 & 0.624 & 0.276 & RR, {\\scriptsize OGLE-SMC-RRLYR-0026} \\\\\nW20& 66.6 & 16.991 & 0.611 & 1.792 & LP, {\\scriptsize OGLE-SMC-LPV-00066} \\\\\nW21& 0.273741797(2)& 17.919 & 0.360 & 0.610 & EW \\\\\nW22& 0.65390427(7) & 18.738 & 0.264 & 0.582 & RR (blend?), {\\scriptsize OGLE-SMC-RRLYR-0016} \\\\\nW23& 0.57218584(4) & 19.017 & 0.980 & 0.428 & RR, {\\scriptsize OGLE-SMC-RRLYR-0027} \\\\\nW24& 0.595771(2) & 19.323 & 0.839 & 0.301 & RR, {\\scriptsize OGLE-SMC-RRLYR-0011} \\\\\nW25& 0.6256305(15) & 19.364 & 0.752 & 0.404 & RR, {\\scriptsize OGLE-SMC-RRLYR-0012} \\\\\nW26& 269.3 & 16.879 & 1.406 & 3.347 & LP, {\\scriptsize OGLE-SMC-LPV-00015} \\\\\nE1 & 0.347489131(4)& 17.025 & 0.485 & 0.493 & EW, BS \\\\\nE2 & 0.8102273(2) & 18.306 & 0.168 & 0.734 & RR, {\\bf new} \\\\\nE3 & 0.36349524(3) & 19.129 & 0.733 & 0.240 & RR, {\\scriptsize OGLE-SMC-RRLYR-0122} \\\\\nE4 & 79.92 & 16.850 & 0.297 & 2.053 & LP, {\\scriptsize OGLE-SMC-LPV-00446} \\\\\nE5 & 0.44625922(1) & 16.659 & 0.398 & 0.434 & EB, BS \\\\\nE6 & - & 16.539 & 0.254 & 1.783 & LP, {\\scriptsize OGLE-SMC-LPV-00398} \\\\\nE7 & 4.463003(6) & 18.792 & 0.210 & 1.273 & ? \\\\\nE8 & 6.371513(8) & 14.141 & 0.152 & 0.760 & RS CVn, RGB-clump \\\\\nE9 & 0.64752185(2) & 18.959 & 0.852 & 0.446 & RR, {\\scriptsize OGLE-SMC-RRLYR-0098} \\\\\nE10& 4.78667(2) & 17.474 & 0.603 & 0.917 & BY~Dra \\\\\nE11& 8.3711839(8) & 14.916 & 0.157 & 0.911 & RS CVn, RGB \\\\\nE12& 0.234635701(3)& 19.441 & 0.787 & 0.981 & EW \\\\\nE13& 1.150686034(3)& 15.797 & 0.417 & 0.233 & EB, BS \\\\\nE14& 0.29712114(1) & 17.306 & 0.505 & 0.238 & RR, {\\scriptsize OGLE-SMC-RRLYR-0098} \\\\\nE15& 0.378854028(1)& 16.397 & 0.311 & 0.349 & EW, BS \\\\\nE16& 0.52515175(1) & 18.977 & 1.254 & 0.422 & RR, {\\scriptsize OGLE-SMC-RRLYR-0104} \\\\\nE17& 20.791(1) & 16.576 & 0.207 & 0.733 & RS CVn, RGB \\\\\nE18& 14.09425(2) & 17.176 & 0.070 & 1.405 & ?, {\\bf new} \\\\\nE19& - & 14.344 & 0.194 & 0.989 & LP, {\\bf new} \\\\\nE20& 266.4 & 17.911 & 2.254 & 3.607 & LP, {\\scriptsize OGLE-SMC-LPV-00329} \\\\\nE21& 0.250546155(3)& 18.397 & 0.457 & 0.669 & EW \\\\\nE22& 35.3519(1) & 16.747 & 0.111 & 1.680 & LP, {\\scriptsize OGLE-SMC-LPV-00271} \\\\\nE23& 0.24169575(1) & 18.551 & 0.337 & 0.693 & EW \\\\\nE24& 0.313434596(1)& 17.650 & 0.519 & 0.511 & EW \\\\\nE25& 3.33587(4) & 17.849 & 0.143 & 0.589 & ?, {\\bf new} \\\\\nE26& 1.4043996(8) & 18.528 & 0.282 & 1.276 & ?, {\\bf new} \\\\\nE27& 0.351384786(1)& 16.244 & 0.246 & 0.318 & EW, BS \\\\\nE28& 36.8914(12) & 15.252 & 0.148 & 0.900 & RS CVn, RGB \\\\\nE29& 40.39(16) & 16.613 & 0.111 & 1.608 & LP, {\\scriptsize OGLE-SMC-LPV-00235} \\\\\nE30& 10.77526(2) & 16.007 & 0.320 & 0.814 & RS CVn, RGB \\\\\nE31& 20.43463(6) & 16.564 & 0.457 & 0.901 & RS CVn, RGB, {\\bf new} \\\\\nE32& - & 17.109 & 0.362 & 0.569 & EA, {\\bf new} \\\\\nE33& 0.25151879(1) & 16.391 & 0.107 & 1.044 & Ell?, X, {\\bf new}, {\\scriptsize Ch-002418.6-720455}\\\\\nE34& 9.87379(4) & 15.495 & 0.294 & 0.878 & RGB, X, {\\bf new}, {\\scriptsize Ch-002425.8-720703 }\\\\\nE35& 0.300361921(7)& 18.021 & 0.345 & 0.648 & EW \\\\\nE36& 0.279869331(2)& 18.053 & 0.641 & 0.703 & EW, {\\bf new} \\\\\nE37& 135.43559(2) & 13.180 & 2.480 & 1.301 & LP \\\\\nE38& 3.480885(3) & 16.468 & 0.313 & 0.843 & RS CVn, RGB \\\\\nE39& 0.987519(2) & 18.792 & 0.507 & 0.937 & EA, {\\bf new} \\\\\n\\end{longtable}\n}\n\\end{center}\n\\vskip -1cm\n{\\footnotesize\n$^a$ X - likely counterpart of an X-ray source \nCh-nnnnnnn.n-nnnnnn cataloged by Heinke et al. (2005)\\\\\n$^b$ OGLE-SMC-RRLYR-nnnn = stars cataloged by Soszynski et al. (2010)\\\\\n$^c$ OGLE-SMC-LPV-nnnnn = stars catalogud by Soszynski et al. (2011)\\\\\n$^d$ independently discovered by Poleski (2012)}\n\nIn addition to the five detached binaries our sample includes one\nsemi-detached system (E13; Kaluzny et al. 2007) and 15 contact or\nnearly-contact binaries. Six of them are blue stragglers, and two\n(W1 and W18) are located in the region occupied by yellow stragglers\n(see e.g. Stetson 1994). Variable yellow stragglers are rare objects\nand therefore we examined HST\/ACS images of W1 and W18 to check \nwhether or not \nthe ground based photometry is affected by blending. We found\nW18 to be an isolated object with no indication for unresolved\nvisual companions on ACS images. Still, we cannot rule out the\npossibility that it is a triple system like many (perhaps most) W\nUMa stars (Rucinski et al 2007). Spectroscopic observations can\nclarify this issue. As for W1, ACS images show three visual\ncomponents forming a blend which cannot be resolved on our ground\nbased images. Two closest components of the blend are separated by\n0.3 arcsec. Thus, based on the available evidence, W1 cannot be\nregarded as a yellow straggler.\n\n\\subsection{Notes on individual objects}\nSome of the newly detected variables deserve a short comment on\ntheir properties. The red straggler candidate E33 is located only 61 arcsec\naway from the cluster centre; however it is an isolated object whose\nground based photometry is free from blending-related problems. This\nwas confirmed by the examination of HST\/ACS\/F625W images. The star\nhas a sine-like light curve with $\\Delta V=0.11$ mag and a period\nof 0.25~d which was coherent and stable during several observing\nseasons between 1998 and 2010. This indicates that the observed\nvariability is very likely related to the binarity of E33. We have\ntentatively classified E33 as an ellipsoidal variable. Heinke et al.\n(2005) analyzed two sets of Chandra data for 47 Tuc. The X-ray\ncounterpart of E33 was detected in 2002 with an X-ray luminosity of\n$0.7$E30 erg\/s in 0.5-2.5~keV band, but no X-rays were detected at\nE33 position in 2000. This implies a seasonal variability of the\nX-ray source connected with the star. The variable is too red to be\nan ordinary contact binary regardless of its membership status in 47\nTuc. This is evident by comparing its color with colors of contact\nsystems E21 and E23 whose orbital periods are similar to the period\nof E33 (see Fig. 4). As for the membership status of E33, the\nexamination of stacked subtracted images from seasons 1998, 1999,\n2009, and 2010 does not indicate any proper motion with respect to\ncluster stars. The qualitative method we used to estimate the proper\nmotion is based on Eyer and Wozniak (2001). Spectroscopic\ndata are needed to clarify the evolutionary and membership status of\nE33. \n\nThe variable E8 is located on the red horizontal branch on\nthe cluster CMD. The light curve shows coherent changes with $\\Delta\nV= 0.15$~mag and $P=6.4$~d. The amplitude of the light curve\nexhibits small but easily visible seasonal changes. Similar\nvariations with the same period were detected by the OGLE group in\n1993 (Kaluzny et al. 1998). Given the coherence of the variability\nit is likely that the star is a binary. If so, it would be a rare\nexample of a photometrically variable binary from the red horizontal\nbranch. With $P=6.4$~d and a radius of the red giant component of\nabout 10~$R_{\\odot}$, E8 would have to be a rather compact system.\nObviously, if the variability was induced mainly by the ellipsoidal\neffect, the actual orbital period of a binary would double to\n12.8~d.\nThe bright blue straggler W14 turned out to be an SX~Phe-type\npulsator. It is one of a few such variables detected in 47 Tuc\n(Gilliland et al. 1998; Poleski 2012). \nSX Phe stars are\ncommon among blue stragglers in metal-poor globular clusters, but\nrather rare in metal rich ones.\n\nFinally, we note the presence of several variables located on or\nslightly to the red of the red giant branch of the cluster, newly \ndetected of which are E31 and E34 (see Figs. 3 and 4). They are \ngood candidates for binaries and are easy targets for a spectroscopic \nfollow-up.\n\\section{Summary}\nWe performed a photometric study of the globular\ncluster 47 Tuc with a time baseline of more than ten years. \nBased on over 6500 $BV$ frames we have identified 65\nvariable stars, 19 of which are new detections. We provide celestial\ncoordinates of all variables, and cross-identifications of the\nvariables discovered earlier by other authors. Finding charts for\nthe new variables are also provided. Five of the new variables are\nlikely optical counterparts of X-ray sources, and another four ones\nare detached eclipsing binaries. Two detached eclipsing systems are\nlocated close to the main-sequence turnoff on the CMD of 47 Tuc, and\nwe argue that they are promising targets for detailed photometric\nand spectroscopic studies: when combined with the results of an\nearlier study of W13 (Thompson et al. 2010; their variable V69) they\nmay allow an improved constraint on the helium content of the cluster.\nThe yellow straggler W18, the red straggler E33 and the red\nhorizontal branch object E8 are another systems deserving further\nstudy which would clarify their membership and evolutionary status.\n\n\\Acknow{We are grateful to Igor Soszynski for a very detailed and helpful\nreferee report, and to Radek Poleski for pointing out the references to \nSX Phe stars in 47 Tuc. JK, MR, WP and WN were partly \nsupported by the grant NCN 2012\/05\/B\/ST9\/03931 from the Polish Ministry of Science.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nAt present, the continuum limit of antiferromagnetic Lie superalgebra spin chains is poorly understood. High degeneracies or continuous spectra~\\cite{Essler:2005ag, Saleur:2006tf, 2006cond.mat.12037I} and singular IR scattering amplitude behavior~\\cite{Essler:2005ag, Saleur:2009bf} emerge in the continuum large volume limit. We interpret this as a hint of new physical phenomena.\nStrikingly, when $q$-deformed, connections with long known puzzling theories with no obvious supergroup symmetry appear~\\cite{Saleur:2009bf}.\n\n\nPrevious, above mentioned, research has mostly dealt with $\\osp(2|2)\\simeq \\sgl(2|1)$ spin chains.\nWe adopt a general approach, taking as a model~\\cite{Essler:2005ag}, and hoping that the greater $\\gl(M|N)$ symmetry and the relatively new cohomology techniques developed in~\\cite{Candu:2010yg} will allow us to better understand the nature of these new phenomena, thus, suggesting how to correctly interpret, treat and, eventually, apply them.\nThe goal is to ultimately formulate a meaningful factorizable scattering theory for the continuum limit of superspin chains.\nThis is an even harder task then it sounds, because there are few massive relativistic field theories with supergroup symmetry that have been understood, at least partially~\\cite{Saleur:2001cw, Saleur:2009bf, Bassi:1999ua, 2000NuPhB.583..475G}.\n\n\nSo, clarifying the continuum limit of superspin chains must ultimately result in a better understanding of formally simple field theories such as the $\\gl(M|N)$ and $\\osp(R|2S)$ Gross-Neveu models, which play an important role in disordered electronic system~\\cite{Bernard:1995as, 2000NuPhB.583..475G, LeClair:2007aj}, provide instances of continuous families of conformal field theories~\\cite{Candu:2008yw, Candu:2009ep} and appear in strong-weak coupling dualities with sigma models~\\cite{Candu:2008yw, 2000NuPhB.583..475G, Creutzig:2008an}.\nDue to strong violation of unitarity, it is not obvious how to treat these super Gross-Neveu models directly in the continuum by standard bootstrap methods~\\cite{Zamolodchikov:1978xm, ORW}.\nInsight from coordinate Bethe ansatz for chiral Gross-Neveu models~\\cite{Andrei:1979un, Andrei:1979wy, Andrei:1979sq, Destri:1987ug} has led us to consider the spin chain of $\\gl(M|N)$ alternating fundamental and dual representations as the most natural candidate for an integrable discretization of the $\\gl(M|N)$ Gross-Neveu model.\nThe spin chain of $\\gl(M|N)$ fundamental representations is discarded because it cannot lead to a relativistic field theory~\\cite{Saleur:1999cx, 1994IJMPB...8.3243E}.\nThis can be immediately seen from the fact that the dual of any $\\gl(M|N)$ representation appearing in the tensor product of $\\gl(M|N)$ fundamental representations does not belong to this tensor product if $N>0$. \n\n\n\nThe article is organized as follows.\nIn sec.~\\ref{sec:formalism} we define the spin chain and its integrable dynamics.\nIn sec.~\\ref{sec:num_diag} we explain how to perform numerical calculations for spin chains of modest size, but arbitrary $\\gl(M|N)$ symmetry, using the walled Brauer algebra. Then we present numerical results on the spectrum.\nSec.~\\ref{sec:rl} is of primary technical importance.\nWe discover a class of solutions to the $\\gl(M|N)$ Bethe ansatz equations (BAE) which can be fully characterized in terms of solutions to the $\\gl(M-1|N-1)$ BAE. This provides an explicit embedding of the $\\gl(M-1|N-1)$ spectrum into the $\\gl(M|N)$ spectrum.\nWe give an algebraic explanation to this relationship.\nFurther, in sec.~\\ref{sec:BAE} we identify the vacuum solution of $\\gl(M|N)$ BAE and write it down explicitly in terms of the $\\gl(M-N)$ vacuum solution. We then classify $\\gl(M|N)$ excitations that lead to a $\\gl(M-N)$ like spectrum and the simplest ones that do not. For the latter, we analyze numerically the form of the solutions to the BAE.\nFinally, in sec.~\\ref{sec:cl} we consider the continuum limit of spin chains with alternating inhomogeneities.\nExcitations which can be characterized in terms of solutions to $\\gl(M-N)$ BAE lead to a spectrum of massive particles with $\\gl(M-N)$ Gross-Neveu mass ratios and $S$-matrix eigenvalues.\nWe argue that the continuum limit of the spin chain is the $\\gl(M|N)$ Gross-Neveu model.\nWe end with analyzing some low lying excitations of the $\\gl(2m|1)$ spin chain which are not of $\\gl(2m-1)$ type and lead to new massive particles.\n\n\n\n\\section{Transfer matrices, Hamiltonians and their spectra}\n\\label{sec:formalism}\n\nLet $V$ be the fundamental module of $\\gl(M|N)$, $V^*$ denote its dual and $\\rho:\\gl(M|N)\\mapsto \\gl(V)$ together with $\\bar{\\rho}:\\gl(M|N)\\mapsto \\gl(V^*)$ be the corresponding representations.\nIf $\\{e_\\alpha\\}_{\\alpha=1}^{M+N}$ is a graded basis of $V$ with grading $|\\alpha|:=|e_\\alpha|\\in \\mathbb{Z}\/2\\mathbb{Z}$ and $\\{e^\\alpha\\}_{\\alpha=1}^{M+N}$ is the dual basis $e^\\alpha(e_\\beta)=\\delta_{\\alpha\\beta}$, then $E_{\\alpha\\beta}$ are the standard generators of $\\gl(M|N)$ acting in $V$ as $E_{\\alpha\\beta}\\cdot e_\\gamma=\\delta_{\\beta\\gamma}e_\\alpha$ and in $V^*$ as $E_{\\alpha\\beta}\\cdot e^\\gamma=-(-1)^{(|\\alpha|+|\\beta|)|\\alpha|}\\delta_{\\alpha\\gamma}e^\\beta$.\nLet us label the $V$ factors of the spin chain $\\mathcal{C}(L)=(V\\otimes V^*)^{\\otimes L}$ from left to right by a subscript $1,\\dots,L$. Similarly, we label the $V^*$ factors by a subscript $\\bar{1},\\dots,\\bar{L}$.\nFor $\\mathcal{E}\\in \\End V$ we denote by $\\mathcal{E}_k\\in\\End\\mathcal{C}(L)$ the endomorphisms acting as $\\mathcal{E}$ on $V_k$ and trivially, up to grading signs, everywhere else in the chain. Similarly, for $\\mathcal{E}\\in \\End V^*$, $\\mathcal{E}_{\\bar{k}}\\in\\End\\mathcal{C}(L)$ will act as $\\mathcal{E}$ on $V^*_{\\bar{k}}$ and trivially, up to grading signs, everywhere else. \n\nThe $\\mathcal{C}(L)$--endomorphisms $P_{kl}=(-1)^{|\\beta|}\\rho_k(E_{\\alpha\\beta}) \\rho_l(E_{\\beta\\alpha})$ provide a representation for the symmetric group acting on the $V$ factors of $\\mathcal{C}(L)$. Similarly, $P_{\\bar{k}\\bar{l}}=(-1)^{|\\beta|}\\bar{\\rho}_{\\bar{k}}(E_{\\alpha\\beta}) \\bar{\\rho}_{\\bar{l}}(E_{\\beta\\alpha})$\ngenerate a representation for the symmetric group acting on the $V^*$ factors of $\\mathcal{C}(L)$.\nOn the other hand, $Q_{k\\bar{l}}= - (-1)^{|\\beta|}\\rho_k(E_{\\alpha\\beta}) \\bar{\\rho}_{\\bar{l}}(E_{\\beta\\alpha})$ generate a representation of the Temperley-Lieb algebra $T_{2L}(n)$ with loop weight $n=M-N$.\nTogether, the $P$'s and $Q$'s generate the $\\gl(M|N)$--centralizer algebra of $\\mathcal{C}(L)$, that is the set of all endomorphisms of the spin chain that commute with the $\\gl(M|N)$ action~\\cite{Sergeev}.\nThis centralizer algebra is a representation of the walled Brauer algebra $B_{L,L}(n)$, which can be viewed as a subalgebra of the Brauer algebra $B_{2L}(n)$. We shall discuss in detail the algebra $B_{L,L}(n)$ and its representations in sec.~\\ref{sec:wb_sub}.\n\nIn terms of walled Brauer algebra generators we introduce the $R$--matrices\n\\begin{align*}\nR_{ij}(u)&= u+P_{ij}& R_{\\bar{\\imath}j}(u)&= u-Q_{\\bar{\\imath}j}\\\\\nR_{i\\bar{\\jmath}}(u)&=u-Q_{i\\bar{\\jmath}}& R_{\\bar{\\imath}\\bar{\\jmath}}(u)&=u+P_{\\bar{\\imath}\\bar{\\jmath}}\\ ,\n\\end{align*}\nwhich satisfy the Yang-Baxter algebra\n\\begin{align}\\label{eq:YBE}\n R_{ij}(u-v)R_{ik}(u)R_{jk}(v)&=R_{jk}(v)R_{ik}(u)R_{ij}(u-v) \\\\ \\notag R_{ij}(u-v)R_{i\\bar{k}}(u)R_{j\\bar{k}}(v)& = R_{j\\bar{k}}(v)R_{i\\bar{k}}(u)R_{ij}(u-v)\\\\ \\notag\n R_{i\\bar{\\jmath}}(u-v+n)R_{ik}(u)R_{\\bar{\\jmath}k}(v)&=R_{\\bar{\\jmath}k}(v)R_{ik}(u)R_{i\\bar{\\jmath}}(u-v+n)\n\\end{align}\nand similar ``dual'' relations, that is the above relations with all barred indices replaced with unbarred ones and all unbarred indices with barred ones.\nWe define two one-parameter families of monodromies\n\\begin{align}\\label{eq:mon_mat_a}\n T_a(u)&=R_{a\\bar{L}}(u +n\/2)R_{aL}(u)\\dots R_{a\\bar{1}}(u+n\/2)R_{a1}(u)\\\\\n \\bar{T}_{\\bar{a}}(u)&= R_{\\bar{a}\\bar{L}}(u)R_{\\bar{a}L}(u+n\/2) \\dots R_{\\bar{a}\\bar{1}}(u) R_{\\bar{a}1}(u+n\/2) \\label{eq:mon_mat_ab}\n\\end{align}\nacting on $V_a\\otimes\\mathcal{C}(L)$ and $V_{\\bar{a}}\\otimes \\mathcal{C}(L)$, respectively,\nand corresponding transfer matrices\n\\begin{equation}\\label{eq:tmat}\n t(u)=\\str_a T_a(u),\\qquad \\bar{t}(u)=\\str_{\\bar{a}}\\bar{T}_{\\bar{a}}(u)\\ .\n\\end{equation}\nYang-Baxter relations~\\eqref{eq:YBE} imply a Yangian structure given by the two relations\n\\begin{align}\\label{eq:comm1}\n R_{ab}(u-v)T_a(u)T_b(v)&=T_b(v)T_a(u)R_{ab}(u-v)\\\\\nR_{a\\bar{b}}(u-v+n\/2)T_a(u)\\bar{T}_{\\bar{b}}(v)&=\\bar{T}_{\\bar{b}}(v)T_a(u)R_{a\\bar{b}}(u-v+n\/2)\\label{eq:comm2}\n\\end{align}\nand their duals.\nThe following commutation relations immediately follow\n\\begin{equation*}\n [t(u),t(v)]=[t(u),\\bar{t}(v)]=[\\bar{t}(u),\\bar{t}(v)]=0\\ .\n\\end{equation*}\n\n\n\nThe nested algebraic Bethe ansatz for the most general $\\gl(M|N)$ spin chain was considered in \\cite{Belliard:2008di}. The Bethe ansatz equations and the spectrum of $t(u)$ formally depend on the nesting order, that is an ordering of a basis of $V$ and the induced ordering of the dual basis of $V^*$.\nIf the basis $\\{e_\\alpha\\}_{\\alpha=1}^{M+N}$ of $V$ diagonalizes the Cartan subalgebra, then, without loss of generality, we label the basis vectors so that the total ordering reads\n\\begin{equation}\\label{eq:tot_ordering_bas_vecs}\n e_1>e_2>\\dots>e_{M+N}\\ ,\n\\end{equation}\nwhere, however, we keep unspecified the grading of the basis vectors.\nLet $\\wt(e_\\alpha)=\\epsilon_\\alpha$ denote the weights of basis vectors called \\emph{fundamental weights}. The ordering~\\eqref{eq:tot_ordering_bas_vecs} induces a weight space ordering $\\epsilon_1>\\dots>\\epsilon_{M+N}$\nwhich fixes the simple root system $\\Delta_0=\\{\\alpha_i:=\\epsilon_i-\\epsilon_{i+1}\\}_{i=1}^{M+N-1}$.\nThe choice of grading, which we denote by $\\Sigma=\\{\\sigma_\\alpha = |e_\\alpha|=|\\alpha|\\}_{\\alpha=1}^{M+N}$, is equivalent to the choice of a Cartan matrix, or a Dynkin diagram.\nChanging the grading can be equivalently seen as changing the total ordering~\\eqref{eq:tot_ordering_bas_vecs}. As a result, the simple root system, the positive root system and the Borel subalgebra changes with $\\Sigma$.\nSo, keep in mind that the notion of highest weight always depends on the grading choice and changes when $\\Sigma$ changes.\n\nDefine the operator matrix elements of the monodromies~(\\ref{eq:mon_mat_a}, \\ref{eq:mon_mat_ab}) as\n$T = E_{ij}\\otimes T_{ij}$, where $T_{ij}\\in \\End\\mathcal{C}(L)$ and $T=T_{a},T_{\\bar{a}}$. \nChoosing the reference state $\\Omega$ to be the highest weight state of $\\mathcal{C}(L)$, the eigenvalues of $t(u)$ can be written in terms of polynomials $(T_a)_{ii}(u)\\Omega=\\Lambda_i(u)\\Omega$\n\\begin{equation*}\n \\Lambda_i(u)=\\begin{cases}\n(u+(-1)^{|1|})^L(u+n\/2)^L, & i=1\\\\\n u^L(u+n\/2)^L, & 2\\leq i\\leq r\\\\\nu^L(u+n\/2-(-1)^{|M+N|})^L,& i=M+N\n\\end{cases}\n\\end{equation*}\nand simple root $Q$-polynomials\n\\begin{equation*}\n Q_k(u)=\\prod_{j=1}^{\\nu^{(k)}}(u-u^{(k)}_j)\n\\end{equation*}\nas follows\n\\begin{equation}\\label{eq:spec_trmat}\n \\Lambda(u)=\\sum_{k=1}^{M+N} (-1)^{|k|}\\Lambda_k(u) \\frac{Q_{k-1}(u+(-1)^{|k|})Q_k(u-(-1)^{|k|})}{Q_{k-1}(u)Q_{k}(u)}\n\\end{equation}\nwhere the $u^{(k)}_j$, $k=1,\\dots,r=M+N-1$ are the Bethe roots appearing at the $k$--th step of the nesting. There are solutions of the following system of nested Bethe ansatz equations\n\\begin{equation}\\label{eq:BAE_Qform}\n\\frac{\\Lambda_{k}(u^{(k)}_j)}{\\Lambda_{k+1}(u^{(k)}_j)}=-(-1)^{|k|+|k+1|}\\frac{Q_{k-1}(u^{(k)}_j)Q_k(u^{(k)}_j+(-1)^{|k+1|})Q_{k+1}(u^{(k)}_j-(-1)^{|k+1|})}{Q_{k-1}(u^{(k)}_j+(-1)^{|k|})Q_k(u^{(k)}_j-(-1)^{|k|})Q_{k+1}(u^{(k)}_j)}\\ ,\n\\end{equation}\nensuring the analyticity of eigenvalues~\\eqref{eq:spec_trmat}, where $k=1,\\dots,r$ and $j=1,\\dots,\\nu^{(k)}$ and it has to be understood that $Q_0(u)=Q_{M+N}(u)=1$. We stress that the BAE~\\eqref{eq:BAE_Qform} are equivalent to the analyticity requirement of the transfer matrix~\\eqref{eq:spec_trmat} if and only if none of the Bethe roots of the same type coincide, which is an essential requirement in the algebraic Bethe ansatz construction. \n\n\nThe BAE take a more familiar look~\\cite{Ogievetsky:1986hu}\n\\begin{equation}\\label{eq:BAE_short}\n \\prod_{k=1}^{L} e_{\\langle \\Lambda_{k},\\alpha \\rangle }(x_j^{(\\alpha)}-y_k)e_{\\langle \\Lambda_{\\bar{k}},\\alpha \\rangle }(x_j^{(\\alpha)}-y_{\\bar{k}})=-(-1)^{|\\alpha|}\\prod_{\\beta\\in\\Delta_0}\\prod_{i=1}^{\\nu^{\\alpha}}e_{\\langle\\alpha,\\beta\\rangle}(x_j^{(\\alpha)}-x_i^{(\\beta)})\\ ,\n\\end{equation}\nwhen written in terms of new variables\n\\begin{equation}\\label{eq:shifts}\n iu^{(k)}_{j}= x^{(k)}_j - \\frac{i}{2}\\sum_{l=1}^k(-1)^{|l|}\\ ,\n\\end{equation}\nthe Takahashi functions\n\\begin{equation*}\n e_t(x)=\\frac{x+it\/2}{x-it\/2}\\, ,\n\\end{equation*}\nthe highest weights $\\Lambda_k=\\epsilon_1$ and $\\Lambda_{\\bar{k}}=-\\epsilon_{M+N}$ of modules $V_k$ and $V^*_{\\bar{k}}$ in $\\mathcal{C}(L)$, \nand the weight space scalar product $\\langle \\epsilon_i, \\epsilon_j\\rangle = \\delta_{ij}(-1)^{|i|}$.\nThe degree of a root $\\alpha_k=\\epsilon_k-\\epsilon_{k+1}$ is defined as $|\\alpha_k|=|k|+|k+1|$.\nWe shall be mostly considering the homogeneous case~(\\ref{eq:mon_mat_a}, \\ref{eq:mon_mat_ab}) corresponding to \n$y_k = y_{\\bar{k}}= 0$, although the inhomogeneous deformation $y_k,y_{\\bar{k}}\\neq 0$ shall also be required.\nThere are $j=1,\\dots,\\nu^{\\alpha}$ equations for every $\\alpha\\in \\Delta_0$.\n\nThe weight of the reference state is $\\wt(\\Omega)=\\sum_{k=1}^L\\Lambda_k+\\sum_{\\bar{k}=1}^L\\Lambda_{\\bar{k}}=L(\\epsilon_1-\\epsilon_{M+N})$. Bethe vectors $\\omega$ described by the Bethe roots~\\eqref{eq:BAE_short} are highest weight vectors of weight\n\\begin{equation}\n \\wt(\\omega) = \\wt(\\Omega) - \\sum_{k=1}^r\\alpha_k \\nu^{(k)}\\ .\n\\label{eq:weight_BV1}\n\\end{equation}\n\nWe define the dynamics of the spin chain by the momentum and Hamiltonian operators\n\\begin{equation}\\label{eq:first_charges}\n H = \\frac{d}{du}\\bigg\\vert_{u=0}\\log \\frac{t(u)\\bar{t}(u)}{\\Lambda_1(u)\\bar{\\Lambda}_{M+N}(u)}\\ ,\\qquad \\exp{i P} = (-1)^{|1|+|M+N|}\\frac{t(0)\\bar{t}(0)}{\\Lambda_1(0)\\bar{\\Lambda}_{M+N}(0)}\\ .\n\\end{equation}\nIn order to have explicit expression for the spectrum of these operators, the eigenvalues~\\eqref{eq:spec_trmat} of $t(u)$ are not enough. One also needs to evaluate the eigenvalue of $\\bar{t}(u)$ on a given Bethe eigenvector of $t(u)$.\nA \\emph{fundamental difference} w.r.t. $\\gl(N)$ spin chains is that one cannot solve this problem by fusion. This is because tensor products of the fundamental representation $V$ of $\\sgl(M|N)$ will never generate the dual representation $V^*$ as a direct summand, nor even as a subquotient.\nTo develop a clear idea about how this should be done, let us recall that a set of Bethe vectors for $t(u)$ can be constructed in the framework of the algebraic Bethe ansatz (ABA) by using only the commutation relations~\\eqref{eq:comm1}. Then, the eigenvalue of $\\bar{t}(u)$ on such a Bethe vector can be calculated, in principle, by using the second type of commutation relations~\\eqref{eq:comm2}.\nWe shall not pursue this rather tedious route. Instead, we guess the eigenvalue of $\\bar{t}(u)$ on a given Bethe vector of $t(u)$ as follows.\n\nFirst, notice that a different set of Bethe vectors can be obtained by performing the ABA for $\\bar{t}(u)$, that is by using the commutation relations dual to eq.~\\eqref{eq:comm1}.\nWe perform the nesting by ordering the dual basis vectors $\\{e^\\alpha\\}_{\\alpha=1}^{M+N}$ of the auxiliary space $V^*$ according to their weights $\\wt(e^\\alpha)=-\\epsilon_\\alpha$\n\\begin{equation}\n e^{M+N}>\\dots >e^2>e^1\\ .\n\\label{eq:dual_ord}\n\\end{equation}\nWe denote the Bethe roots appearing at step $k$ of the nested ABA by $\\bar{u}^{(M+N-k)}_j$, because the simple root which must be associated to them is $\\wt(e^{M+N-k+1})-\\wt(e^{M+N-k})=\\alpha_{M+N-k}$.\nThen, the eigenvalues of $\\bar{t}(u)$ can be written in term of polynomials \n$(T_{\\bar{a}})_{ii}(u)\\Omega=\\bar{\\Lambda}_i(u)\\Omega$\n\\begin{equation*}\n \\bar{\\Lambda}_i(u)=\\begin{cases}\n(u-(-1)^{|1|}+n\/2)^L u^L, & i=1\\\\\n (u+n\/2)^Lu^L, & 2\\leq i\\leq r\\\\\n(u+n\/2)^L(u+(-1)^{|M+N|})^L,& i=M+N\n\\end{cases}\n\\end{equation*}\nand simple root polynomials $\\bar{Q}_k(u)=\\prod_{j=1}^{\\bar{\\nu}^{(k)}}(u-\\bar{u}^{(k)}_j)$\nin complete analogy with~\\eqref{eq:spec_trmat}\n\\begin{equation}\n\\bar{\\Lambda}(u)=\\sum_{k=1}^{M+N}(-1)^{|k|}\\bar{\\Lambda}_k(u)\\frac{\\bar{Q}_{k-1}(u-(-1)^{|k|})\\bar{Q}_k(u+(-1)^{|k|})}{\\bar{Q}_{k-1}(u)\\bar{Q}_k(u)}\\ .\n\\label{eq:eg_trmat_bar}\n\\end{equation}\nAnalyticity conditions for $\\bar{\\Lambda}(u)$ take the same form as eqs.~\\eqref{eq:BAE_short} in terms of variables\n\\begin{equation}\n \\bar{u}^{(k)}_j = \\bar{x}^{(k)}_j-\\frac{i}{2}\\sum_{l=k+1}^{M+N}(-1)^{|l|}\n\\label{eq:bshifts}\n\\end{equation}\nand parameters $\\bar{\\nu}^{(k)}$.\nA Bethe vector constructed in this way, which we denote by $\\bar{\\omega}$, has weight \n\\begin{equation*}\n\\wt(\\bar{\\omega})=\\wt(\\Omega)-\\sum_{k=1}^r \\alpha_k \\bar{\\nu^{(k)}}\\ .\n\\end{equation*}\n\nAt this point, the eigenvalues of $t(u)$ w.r.t. the second set of Bethe vectors $\\bar{\\omega}$ is not known. If we can match Bethe vectors $\\omega$ with Bethe vectors $\\bar{\\omega}$ then the problem is solved.\nDue to the subtle completeness issue of ABA for super spin chains, it is not clear at all if the matching can actually be performed.\nWe assume it can and we do it as follows: a Bethe vector $\\omega$ of $t(u)$ is identified with a Bethe vector $\\bar{\\omega}$ of $\\bar{t}(u)$ if\n\\begin{align}\\label{eq:indent_BV}\n&&\\omega =& \\bar{\\omega} & {}&\\Leftrightarrow&\n\\begin{cases}\n \\nu^{(k)}=\\bar{\\nu}^{(k)}\\\\ \\{x^{(k)}_j\\}_{j=1}^{\\nu^{(k)}}=\\{\\bar{x}^{(k)}_j\\}_{j=1}^{\\bar{\\nu}^{(k)}} &\n \\end{cases} & k=1,\\dots r \\ .\n\\end{align}\nThe first condition in the braces means $\\wt(\\omega)=\\wt(\\bar{\\omega})$.\nEqs.~(\\ref{eq:spec_trmat}, \\ref{eq:eg_trmat_bar}, \\ref{eq:shifts}, \\ref{eq:bshifts}) allow us to compute both the eigenvalues of $t(u)$ and $\\bar{t}(u)$ on a Bethe vector $\\omega=\\bar{\\omega}$ in~\\eqref{eq:indent_BV}.\n\nAfter this long detour, we come back to the spectrum of the operators~\\eqref{eq:first_charges}, which can now be explicitly computed\n\\begin{align}\\label{eq:spec_ham_mom}\n E &= -\\sum_{j=1}^{\\nu^{(1)}}\\frac{(-1)^{|1|}}{x^{(1)}_j {}^2+1\/4}-\\sum_{j=1}^{\\nu^{(r)}}\\frac{(-1)^{|M+N|}}{x^{(r)}_j{}^2+1\/4}\\\\\nP &\\equiv \\sum_{i=1}^{\\nu^{(1)}}(-1)^{|1|}\\theta_1(x^{(1)}_j)+ \\sum_{i=1}^{\\nu^{(r)}}(-1)^{|M+N|}\\theta_1(x^{(r)}_j)\\mod 2\\pi \\ ,\n\\end{align}\nwhere $\\theta_t(u)=i \\log e_t(u)+\\pi=2\\tan^{-1}\\tfrac{2u}{t}$ with some fixed branch.\nFirst thing to be noticed is the explicit dependence of the definitions~\\eqref{eq:first_charges} on the grading.\nTherefore, it looks like the type of the chain -- ferromagnetic or antiferromagnetic -- also depends on it, which is also suggested by the grading signs in \\eqref{eq:spec_ham_mom}.\nHowever, the charges $H$ and $P$ can be written explicitly as a representation of an element of the periodic walled Brauer algebra\n\\begin{align}\\label{eq:ham}\n H &= \\sum_{i=1}^L -(-1)^{|1|} -(-1)^{|M+N|}+ P_{ii+1}+ P_{\\overline{i-1i}} -\\frac{2}{n}\\left(\\{ Q_{\\bar{i}i+1},Q_{i\\bar{i}}\\} + \\{Q_{\\overline{i-1}i},Q_{i\\bar{i}}\\}\\right)\n\\end{align}\nwhere all (un)barred indices are defined modulo $L$ and the affine generators are expressed in terms of non-periodic walled Brauer algebra elements $P_{L1}:=P_{1L}=P_{12}\\dots P_{L-1L}\\dots P_{12}$, $P_{\\overline{L1}}:=P_{\\overline{1L}}=P_{\\overline{12}}\\dots P_{\\overline{L-1L}}\\dots P_{\\overline{12}}$ and $Q_{\\bar{L}1}:=Q_{1\\bar{L}} = P_{1L}Q_{L\\bar{L}}P_{1L}$.\nThe following general formulas have been used\n\\begin{align*}\n \\frac{d\\log t(0)}{du}&=\\sum_{i=1}^L R^{-1}_{i\\bar{i}}\\left(\\frac{n}{2}\\right)\\dot{R}_{i\\bar{i}}\\left(\\frac{n}{2}\\right)+\\sum_{i=1}^{L} R^{-1}_{i\\bar{\\imath}}\\left(\\frac{n}{2}\\right)\\dot{\\check{R}}_{ii+1}(0)R_{i\\bar{\\imath}}\\left(\\frac{n}{2}\\right)\\\\\n \\frac{d\\log \\bar{t}(0)}{du}&=\\sum_{i=1}^L\\dot{R}_{i\\bar{i}}\\left(\\frac{n}{2}\\right)R^{-1}_{i\\bar{i}}\\left(\\frac{n}{2}\\right)+\\sum_{i=1}^{L}R_{i\\bar{\\imath}}\\left(\\frac{n}{2}\\right)\\dot{\\check{R}}_{\\overline{i\\imath-1}}(0)R^{-1}_{i\\bar{i}}\\left(\\frac{n}{2}\\right)\\ .\n\\end{align*}\nThe latter can be derived using only the cyclicity of the supertrace, which holds for a graded tensor product when even endomorphisms are considered, relations of the form $P_{ij}R_{jk}(u)= R_{ik}(u)P_{ij}$, $P_{ij}R_{j\\bar{k}}(u)= R_{i\\bar{k}}(u)P_{ij}$, $\\str_a P_{aj} = \\id$ and their duals.\nWe see from eq.~\\eqref{eq:ham} that the grading enters in the definition of the Hamiltonian only as a shift, therefore having nothing to do with ferro or antiferromagnetism.\nThe equivalence of the solutions of BAE in different gradings, and therefore of $\\spec H$, is at present somewhat understood in terms of particle hole transformations~\\cite{Tsuboi:1998ne}, although the relationship between Bethe vectors in different gradings not at all.\n\nA relation between the spectrum of $H$ and $-H$ can be constructed, but one has to consider different chains. Let $H_{M|N}$ denote the integrable Hamiltonian of the $\\gl(M|N)$ spin chain. Then on has the following relation\n\\begin{equation}\\label{eq:aut}\n H_{M|N} = - H_{N|M} \\ .\n\\end{equation}\nTo prove it one uses the algebra homomorphism between the walled Brauer algebras $B_{L,L}(n)$ and $B_{L,L}(-n)$ provided by\n$P\\mapsto -P$ and $Q\\mapsto -Q$. This automorphism is realized in the spin chain representation by the shift of the grading function $|i|\\mapsto |i|+1$, which maps $\\gl(M|N)\\mapsto \\gl(N|M)$.\nAs we shall see in the next section, the sign in front of our Hamiltonian~\\eqref{eq:ham} ensures that we are dealing with antiferromagnetic spin chains for $n=M-N>0$.\nEq.~\\eqref{eq:aut} allows to fold back the $\\gl(N|M)$ spin chains with $N0$ by changing the sign of the Hamiltonian~\\eqref{eq:ham}, but now they will be ferromagnetic.\nAs the $\\gl(N|N)$ chain in eq.~\\eqref{eq:ham} is poorly defined, we shall discard it from the main discussion and come back to it only in the conclusions.\nFrom now on we restrict to $\\gl(M|N)$ antiferromagnetic spin chains~\\eqref{eq:ham} for which $n=M-N>0$.\n\n\\section{Numerical diagonalization of the Hamiltonian}\n\\label{sec:num_diag}\n\n\n\n\nDiagonalizing the matrix~\\eqref{eq:ham} is not the smartest thing to do if one is solely interested in its spectrum, especially if one intends to consider on the same footing all the $\\gl(M|N)$--spin chains. This is because many eigenvalues have multiplicities corresponding to the dimension of irreducible $\\gl(M|N)$ representations appearing (generically as quotients of submodules) in the spin chain. These multiplicities quickly grow with the rank and the computing power per eigenvalue increases respectively.\nAn approach which allows to select the eigenvalues corresponding to a given $\\gl(M|N)$--irreducible representation and eliminate the corresponding degeneracy would be considerably more efficient.\nTo implement this approach one interprets the Hamiltonian $H$ in eq.~\\eqref{eq:ham} as an element $\\mathsf{H}$ of some algebra, namely the walled Brauer algebra $B_{L,L}(n=M-N)$, in a particular representation provided by the $\\gl(M|N)$--centralizer of the $\\mathcal{C}(L)$ spin chain.\nAs we shall explain shortly, the algebra $B_{L,L}(n)$ can be abstractly defined independently of the $\\gl(n+N|N)$ spin chains and, in particular, it does not depend on $N$. Centralizers of\n$\\gl(n+N|N)$ spin chains provide $N$ dependent representations for $B_{L,L}(n)$.\nThe important thing is that the representation theory of $B_{L,L}(n)$ is understood well enough, so that one can find the spectrum of the algebraic Hamiltonian $\\mathsf{H}$ by working in other representations where its spectrum is much less\ndegenerate compared to that in the $\\gl(n+N|N)$ spin chain, namely it does not depend on $N$.\n\n\\subsection{Walled Brauer algebra and its standard modules}\n\\label{sec:wb_sub}\n\n\nThe walled Brauer algebra $B_{L,L}(n)$ can be conveniently viewed as a subalgebra of the Brauer algebra $B_{2L}(n)$.\nThe defining relations of $B_{2L}(n)$ can be found in \\cite{ram}.\nThe words of $B_{2L}(n)$ admit a representation as graphs\non $4L$ labeled vertices with $2L$ edges connecting the vertices\npairwise in all $(4L-1)!!$ possible ways (crossings are allowed).\nThe identity $\\mathsf{I}$ of the Brauer algebra and the generators $\\mathsf{E}_i,\\mathsf{P}_i$\nare represented by the graphs on the left in\nfig.~\\ref{fig:gen_br_alg}.\n\n\\begin{figure}[t]%\n\\psfrag{1}{$1$}\n\\psfrag{l}{$L$}\n\\psfrag{d}{$\\cdots$}\n\\psfrag{i}{$i$}\n\\psfrag{i}{$i$}\n\\psfrag{P}{$\\mathsf{P}_i$}\n\\psfrag{I}{$\\mathsf{I}$}\n\\psfrag{E}{$\\mathsf{E}_i$}\n\\psfrag{PPP}{$\\mathsf{P}_{i-1}\\mathsf{P}_{i}\\mathsf{P}_{i+1}$}\n\\centerline{\\includegraphics[scale=0.8]{brauer_diags.eps}}\\caption{The identity $\\mathsf{I}$ and the generators $\\mathsf{E}_i,\\mathsf{P}_i$ of the Brauer algebra of\ndimension\n$(2L-1)!!$ are represented on the left; the walled Brauer algebra generator\n$\\mathsf{P}_{i-1} \\mathsf{P}_{}\\mathsf{P}_{i+1}$\nis represented on the right.}%\n\\label{fig:gen_br_alg}%\n\\end{figure}\n\nIn order to multiply the diagrams one arranges the first $2L$\nvertices horizontally with the remaining $2L$ vertices on top of\nthe first ones. The product of a diagram $d_1$ with a diagram\n$d_2$ is the diagram $d_1 d_2$ obtained by i) placing the diagram\n$d_1$ on top of the diagram $d_2$, ii) identifying the top of the\ndiagram $d_2$ with the bottom of the diagram $d_1$ and iii)\nreplacing every loop generated in this process by $n$.\nThe walled Brauer algebra $B_{L,L}(n)$ is the subalgebra of $B_{2L}(n)$ generated by the elements $\\mathsf{Q}_{i\\bar{\\imath}}:=\\mathsf{E}_{2i-1}$, $\\mathsf{Q}_{\\bar{\\imath}i+1}:=\\mathsf{E}_{2i}$, $\\mathsf{P}_{i,i+1}:= \\mathsf{P}_{2i-1} \\mathsf{P}_{2i}\\mathsf{P}_{2i-1}$, $\\mathsf{P}_{\\overline{\\imath\\imath+1}}:= \\mathsf{P}_{2i} \\mathsf{P}_{2i+1}\\mathsf{P}_{2i}$. The generators\n$\\mathsf{P}_i\\mathsf{P}_{i+1}\\mathsf{P}_i$ are represented on the right in\nfig.~\\ref{fig:gen_br_alg}.\nFor every diagram spanning $B_{2L}(n)$ with vertices labeled as in fig.~\\ref{fig:gen_br_alg}, imagine moving all odd vertices to the left of a wall, all the even ones to the right, while keeping the connectivity unchanged. Then $B_{L,L}(n)$ is spanned by the set of $B_{2L}(n)$ diagrams, such that only strictly horizontal edge cross the wall.\nIn this representation of $B_{L,L}(n)$ we label the $L$ up and $L$ down vertices on the left of the wall by the set $1,\\dots,L$ from left to right and similarly those on the right by the set $\\bar{1},\\dots,\\bar{L}$.\nThe $\\mathsf{P}_{ii+1}$ generators act on the left of the wall, the $\\mathsf{P}_{\\overline{\\imath\\imath+1}}$ act on the right, while the generators $\\mathsf{Q}_{i\\bar{\\imath}}$ and $\\mathsf{Q}_{\\bar{\\imath}i+1}$ act across the wall.\n\n\nNext we give a brief description of a set of modules of $B_{L,L}(n)$ over which we actually numerically diagonalize the algebraic Hamiltonian $\\mathsf{H}$.\nThese modules shall be related in the following to $\\gl(n+N|N)$ \\emph{traceless tensors} of fixed co-- and contravariant shapes.\nFor $\\lambda$ and $\\mu$ partitions of the non-negative integers $|\\lambda|,|\\mu|=f\\leq L$, we denote by $\\Delta_{L,L}(\\lambda,\\mu)$ the \\emph{standard modules} of $B_{L,L}(n)$.\nThere are constructed in the following way.\nLet $\\Sym(f)$ denote the symmetric group on $f$ objects and $S(\\lambda)$, $S(\\mu)$ denote its simple modules labeled by the corresponding partitions.\nThen, $\\Delta_{L,L}(\\lambda,\\mu)$ has as basis the tensor products between the set of some diagrams on $2L$ points, with $L$ of them on the either side of the wall, and some basis of $S(\\lambda)$ and $S(\\mu)$.\nThe diagrams are such that every point is either free or belongs to a horizontal edge crossing the wall. The number of edges is fixed to $L-f$.\nWe give a rough idea about how the action of the walled Brauer algebra can be constructed by diagrammatic multiplication in in fig.~\\ref{fig:mult_diag}: i) in a first step, before the diagrammatic multiplication, assign labels to the free points of the diagrams on each side of the wall; ii) in a second step, after the diagrammatic multiplication, apply a surjective homomorphism from labeled diagrams to the tensor product of unlabeled diagrams with $S(\\lambda)$ and $S(\\mu)$, that is to $\\Delta_{L,L}(\\lambda,\\mu)$. All labeled diagrams with more then $L-f$ horizontal edges that can appear as a result of the diagrammatic multiplication belong to the kernel of this homomorphism.\nIn particular, $\\Delta_{f,f}(\\lambda,\\mu)\\simeq S(\\lambda)\\times S(\\mu)$.\nThe detailed definitions can be found in \\cite{martin}.\n\\begin{figure}%\n\\psfrag{x}{$\\times$}\n\\psfrag{o}{$\\otimes$}\n\\psfrag{v}{$\\mathbf{v}$}\n\\psfrag{u}{$((2,1), (1,2))\\cdot \\mathbf{v}$}\n\\psfrag{1}{$1$}\n\\psfrag{2}{$2$}\n\\psfrag{f}{$\\phantom{fffff}$}\n\\centerline{\\includegraphics[scale=0.7]{mult_diag.eps}}\n\\caption{Example of diagrammatic multiplication of a basis element of $\\Delta_{3,3}(\\lambda,\\mu)$, where $\\lambda,\\mu\\vdash 2$ and $\\mathbf{v}$ is an element of the module $S(\\lambda)\\times S(\\mu)$ of $\\Sym(2)\\times\\Sym(2)$.}\n\\label{fig:mult_diag}\n\\end{figure}\n\n\n\nWe implement the action of $B_{L,L}(n)$ in $\\Delta_{L,L}(\\lambda,\\mu)$ on a computer and investigate the spectrum of the algebraic Hamiltonian $\\mathsf{H}$. Before presenting and discussing the numerical results of sec.~\\ref{sec:spec} one should explain how to extract from these data the spectrum of the original spin chain Hamiltonian $H$.\n\n\n\\subsection{Traceless tensors}\\label{sec:tens}\n\nConsider the $\\gl(M|N)$ tensor $(V\\otimes V^*)^{\\otimes f}$.\nThere is an obvious action of $\\Sym(f)\\times\\Sym(f)$ on the $V$ and $V^*$ factors.\nLet $\\lambda,\\mu$ be partitions of $f$, which we symbolically write as $\\lambda,\\mu\\vdash f$.\nOne can apply a Young symmetrizer of shape $\\lambda$ to the $V$ factors and a Young symmetrizer of shape $\\mu$ to the $V^*$ factors to get a tensor of shape $t(\\lambda,\\mu)$ \\cite{Weyl}.\nWe say that $t(\\lambda,\\mu)$ has rank $(f,f)$, covariant shape $\\lambda$ and contravariant shape $\\mu$ or, shortly, shape $(\\lambda,\\mu)$.\nThe number of Young symmetrizers of shape $\\lambda$ is equal to the \nnumber of standard Young tableau of shape $\\lambda$, which is also equal to $\\dim S(\\lambda)$.\nTherefore, the number of tensors of shape $(\\lambda,\\mu)$ is $\\dim S(\\lambda)\\times \\dim S(\\mu)$.\nEvery tensor $t(\\lambda,\\mu)$ is a $\\gl(M|N)$ module, which appears as a direct summand in $(V\\otimes V^*)^{\\otimes f}$.\nThe symmetric group $\\Sym(f)\\times\\Sym(f)$ acts in the space of tensors of the same shape $(\\lambda,\\mu)$, transforming them into each other. The latter subspace is isomorphic to a direct sum of $S(\\lambda)\\otimes S(\\mu)$ modules of $\\Sym(f)\\times\\Sym(f)$. \nThese statements can be compactly written as follows\n\\begin{equation*}\n(V\\otimes V^*)^{\\otimes f}\\mathop{\\simeq}_{\\Sym(f)\\times \\Sym(f)}\\bigoplus_{\\lambda,\\mu \\vdash f}\\dim t(\\lambda,\\mu)S(\\lambda)\\otimes S(\\mu)\\ .\n\\end{equation*}\n\n\n\nConsider now the subspace $t_0(\\lambda,\\mu)\\subset t(\\lambda,\\mu)$ of traceless tensors. Notice that this is a $\\gl(M|N)$\nsubmodule, which will \\emph{not} necessarily be \\emph{a direct summand} of $t(\\lambda,\\mu)$.\nMore generally, one can consider the $\\gl(M|N)$ submodules $t_{n-1}(\\lambda,\\mu)\\subset t(\\lambda,\\mu)$ composed of tensors whose all contractions of $n$ covariant indices with $n$ contravariant indices vanish.\nThese provide a filtration of the tensor $t(\\lambda,\\mu)$\n\\begin{equation}\n t_0(\\lambda,\\mu)\\subset t_1(\\lambda,\\mu)\\subset \\cdots \\subset t_f(\\lambda,\\mu)=t(\\lambda,\\mu)\\ .\n\\label{eq:filtr}\n\\end{equation}\nIt is very important to observe that the subquotients $t_{n}(\\lambda,\\mu)\/t_{n-1}(\\lambda,\\mu)$ of this filtration are isomorphic to traceless tensors of lower rank $(f-n,f-n)$.\nFor instance, taking a trace of $t_1(\\lambda,\\mu)$ one gets a traceless tensor of rank $f-1$ because, by definition, all double contractions of $t_1(\\lambda,\\mu)$ must vanish. Now, the preimage of single traces of $t_1(\\lambda,\\mu)$ modulo the kernel $t_0(\\lambda,\\mu)$ of the single trace homomorphisms is precisely to $t_1(\\lambda,\\mu)\/t_0(\\lambda,\\mu)$. Therefore, $t_1(\\lambda,\\mu)\/t_0(\\lambda,\\mu)$ is isomorphic to traceless tensors of rank $(f-1,f-1)$.\n\n\nWe see that traceless tensors $t_0(\\lambda,\\mu)$ have a fundamental role --- all direct summands of the spin chain $(V\\otimes V^*)^{\\otimes L}$ can be built out of them.\nAs a consequence, the full spectrum of the spin chain Hamiltonian can be reconstructed from the spectra in the subspaces of traceless tensors of shape $\\lambda,\\mu\\vdash f$, $f=0,1,\\dots,L$.\nIn fact, not all of these shapes are possible. We shall determine the class of shapes for which the traceless tensors do not vanish later.\n\n\nThe traceless tensors are not necessarily irreducible. One way to build submodules of a traceless tensor $t_0(\\lambda,\\mu)$ is by embedding into it quotients of traceless tensors of lower rank as follows. Let $t_0(\\lambda',\\mu')$ be a traceless tensor, $\\lambda',\\mu'\\vdash f-k$, $\\lambda'\\subset \\lambda$, $\\mu'\\subset \\mu$ and $\\mathsf{e}_{\\lambda},\\mathsf{e}_\\mu$ denote some Young symmetrizers of shape $\\lambda$, $\\mu$. The tensor \n\\begin{equation*}\n\\mathsf{e}_{\\lambda}\\mathsf{e}_\\mu t_0(\\lambda',\\mu')\\otimes \\left((V\\otimes V^*)^{\\otimes k}\\right)^{\\gl(M|N)} \\subset t(\\lambda,\\mu)\n\\end{equation*}\nmight have an intersection with a non-trivial submodule of $t_0(\\lambda,\\mu)$. The latter will generally be isomorphic to only a quotient of $t_0(\\lambda',\\mu')$, because the Young symmetrizers $\\mathsf{e}_{\\lambda},\\mathsf{e}_\\mu$ are projectors.\nAn illustrative example is the indecomposable $\\gl(N|N)$ tensor $t(1,1)= V\\otimes V^*$. The traceless subspace $t_0(1,1)$, isomorphic to the adjoint representation, is spanned by elements of the form $G^i_j e_i\\otimes e^j$ subject to the constraint $\\str G = G^i_i (-1)^{|i|}=0$, where $\\{e_i\\}_{i=1}^{2N}$ is a basis of $V$ and $\\{e^i\\}_{i=1}^{2N}$ is the dual basis. The quotient $t_1(1,1)\/t_0(1,1)$ is one dimensional. A coset representative for this quotient is, for instance, $(-1)^{|i|}e_i\\otimes e^i$.\nThe traceless tensor $t_0(1,1)$ is also indecomposable, but reducible.\nIt has a unique proper non-trivial submodule spanned by the $\\gl(N|N)$ traceless invariant $e_i\\otimes e^i$.\n\n\nRepresent now the covariant part of every tensor $t_0(\\lambda,\\mu)$ of fixed shape $(\\lambda,\\mu)$ and ranks $(f,f)$ by $f$ dots on the left of an imaginary wall and the contravariant part by $f$ dots on the right of that wall.\nThen the walled Brauer algebra generators $\\mathsf{P}_{ii+1}$ will act on the left of the wall as in $S(\\lambda)$ and the $\\mathsf{P}_{\\overline{\\imath\\imath+1}}$ generators will act on the right as in $S(\\mu)$ by transforming $\\dim S(\\lambda)\\times \\dim S(\\mu)$ different traceless tensors of shape $(\\lambda,\\mu)$ into each other. The generators $\\mathsf{Q}_{i\\bar{\\imath}}$ and $\\mathsf{Q}_{\\bar{\\imath}i+1}$ will act across the wall by contracting a covariant index with a contravariant one. In view of the tracelessness condition this action is trivial.\nThus, the space of all traceless tensors $t_0(\\lambda,\\mu)$ of the same shape $(\\lambda,\\mu)$ is isomorphic to a direct sum of $\\dim t_0(\\lambda,\\mu)$ modules $\\Delta_{f,f}(\\lambda,\\mu)\\simeq S(\\lambda)\\times S(\\mu)$ of the walled Brauer algebra.\nA very important observation is the \\emph{triviality of the centralizer} of a traceless tensor\n\\begin{equation}\n \\End_{\\gl(M|N)} t_0(\\lambda,\\mu)\\simeq \\mathbb{C}\\ .\n\\label{eq:centralizer_simple}\n\\end{equation}\nThis is a generalization of the Schur lemma for $\\gl(n)$ traceless tensors, which are irreducible. The statement~\\eqref{eq:centralizer_simple} follows immediately from the action of the walled Brauer algebra in the space of traceless tensors of shape $(\\lambda,\\mu)$ that we have just described. \nIt means that traceless tensors are a very special type of indecomposables, namely any $\\gl(M|N)$ Casimir is diagonalizable and proportional to the identity in a traceless tensor. \nIt should be noticed that this is typical of highest weight or Kac modules \\cite{Kac77a, Kac77b}. \n\\begin{assumption}{1}\\label{ass:hw}\n Traceless tensors are highest weight modules.\n\\end{assumption}\n\\noindent This means that there is a Borel subalgebra $\\mathfrak{b}$ of $\\gl(M|N)\\simeq \\mathfrak{b}\\oplus \\mathfrak{n}^-$ and a $\\mathfrak{b}$--highest weight vector $\\mathbf{v}\\in t_0(\\lambda,\\mu)$ such that the full tensor $t_0(\\lambda,\\mu)$ can be generated from $\\mathbf{v}$ by repeated action of $\\mathfrak{n}^-$. We shall see later how to choose $\\mathfrak{b}$ for given $t_0(\\lambda,\\mu)$.\n\nConsider now the vector space $\\delta_{L,L}(\\lambda,\\mu)$ of all possible embeddings of traceless tensors of shape $(\\lambda,\\mu)$ and ranks $(f,f)$ into the spin chain $(V\\otimes V^*)^{\\otimes L}$. It consists of tensor products of tensors $t_0(\\lambda,\\mu)$ with $\\gl(M|N)$--invariants of $(V\\otimes V^*)^{\\otimes (L-f)}$.\nNotice that there is a unique $\\gl(M|N)$ invariant in $V\\otimes V^*$, which can be written as $e_i\\otimes e^i$.\nRepresenting this invariant by an edge connecting two vertices across the wall and the traceless tensors $t_0(\\lambda,\\mu)$ as we did before,\nwe can visualize $\\delta_{L,L}(\\lambda,\\mu)$ as a diagram with $L$ vertices on each side of the wall and $L-f$ edges connecting pairs of vertices across the wall.\nThus, we reconstruct the same diagrammatic representation of $\\delta_{L,L}(\\lambda,\\mu)$ as for $\\Delta_{L,L}(\\lambda,\\mu)$. \nThis proves that all the relations between the generators of $B_{L,L}(n)$ satisfied in $\\Delta_{L,L}(\\lambda,\\mu)$\nwill be satisfied in $\\delta_{L,L}(\\lambda,\\mu)$ as well.\nThe converse is generally not true, meaning that $\\delta_{L,L}(\\lambda,\\mu)$ is generally only a quotient of ($\\dim t_0(\\lambda,\\mu)$ direct sums of) $\\Delta_{L,L}(\\lambda,\\mu)$.\nWe stress that neither $\\delta_{L,L}(\\lambda,\\mu)$ nor $\\Delta_{L,L}(\\lambda,\\mu)$ are necessarily simple $B_{L,L}(n)$ modules and, therefore, the quotient can be non-trivial.\n\n\n\nThe ABA in some grading $\\Sigma$ provides $\\gl(M|N)$ Bethe eigenvectors of highest weight with respect to the Borel subalgebra $\\mathfrak{b}_\\Sigma$ determined by $\\Sigma$ and the ordering~\\eqref{eq:tot_ordering_bas_vecs}.\nTherefore, in order to match the numerical spectrum of $\\mathsf{H}$ in $\\Delta_{L,L}(\\lambda,\\mu)$ with the exact spectrum of $H$ by the ABA we need to know the highest weight of a traceless tensor $t_0(\\lambda,\\mu)$ at least in one grading $\\Sigma$. We explain below how to evaluate it.\n\nConsider the Young diagrams corresponding to the shape $(\\lambda,\\mu)$ of a \\emph{full} tensor $t(\\lambda,\\mu)$.\nThe basis vectors in the tensor subspace of co(ntra)variant shape $\\lambda\\, (\\mu)$ can be represented by co(ntra)variant Young supertableaux of shape $\\lambda\\, (\\mu)$, that is Young diagrams of shape $\\lambda\\, (\\mu)$ with a fundamental weight $\\epsilon_i\\, (-\\epsilon_j)$ inscribed in every box.\nThe pattern of weights within the Young diagrams must satisfy the supersymmetrization rules, that is i) in the same row the bosonic (fermionic) weights are weakly (strongly) ordered w.r.t. each other, ii) in the same column the bosonic (fermionic) weights are strongly (weakly) ordered w.r.t. each other\nand iii) bosonic weights are weakly (strongly) ordered w.r.t. fermionic weight in the same row (column).\nThe weight of a supertableau is the sum of all weights it carries in its boxes.\n\nBoth $\\lambda$ and $\\mu$ must fit into a hook whose horizontal (vertical) arm is of width $M\\, (N)$, otherwise $t(\\lambda,\\mu)$ vanishes identically because it is not possible to fill in the Young diagrams and get Young supertableaux compatible with the supersymmetrization rules.\nThe highest weight of a supertableau depends on the grading.\nThe choice of grading is a splitting of the set of basis vectors into two sets $B\\, (F)=\\{\\epsilon_i\\mid |i|\\equiv 0\\, (1)\\}^{<}$ which are ordered w.r.t. the total ordering~\\eqref{eq:tot_ordering_bas_vecs}.\nEquivalently, it can be represented by paths connecting the two corners of the $(M,N)$--hooks as represented in fig.~\\ref{fig:young}.\nFix these paths and consider Young diagrams $\\lambda$, $\\mu$\nwith rows $\\lambda_i$, $\\mu_i$ and columns $\\lambda'_i$, $\\mu'_i$.\nThen, the highest weight of $t(\\lambda,\\mu)$ can be written in the following form\n\\begin{equation}\\label{eq:weight_tensor}\n \\Lambda_\\Sigma(\\lambda,\\mu) = \\sum_{i=1}^M [r_i\\epsilon_{b(i)}-\\bar{r}_i\\epsilon_{\\bar{b}(i)}]+\\sum_{i=1}^N [c_i\\epsilon_{f(i)} -\\bar{c}_i\\epsilon_{\\bar{f}(i)}]\n\\end{equation}\nwhere $b(i)$ and $f(i)$ are the elements at position $i$ in the ordered sets $B$ and $F$,\n $\\bar{b}(i)$ and $\\bar{f}(i)$ are the elements at position $i$ in the ordered sets $-B$ and $-F$, \n $r_i=\\max(0,\\, \\lambda_i - \\sum_{j=1}^{b(i)}(1-(-1)^{|i|})\/2)$, $c_i=\\max(0,\\, \\lambda'_i- \\sum_{j=1}^i (1-(-1)^{|i|})\/2$,\n$\\bar{r}_i = \\max(0,\\, \\mu_i-\\sum_{i=\\bar{b}(i)}^M(1-(-1)^{|i|})\/2 )$ and $\\bar{c}_i = \\max(0,\\, \\mu'_i-\\sum_{i=\\bar{f}(i)}^N(1+(-1)^{|i|})\/2 )$\nare number of boxes in a row or column of $\\lambda$ or $\\mu$ overpassing the grading paths as represented in fig.~\\ref{fig:young}.\n\\begin{figure}%\n\\psfrag{mu}{$\\mu$}\n\\psfrag{lam}{$\\lambda$}\n\\psfrag{r1}{$r_1$}\n\\psfrag{r2}{$r_2$}\n\\psfrag{r3}{$r_3$}\n\\psfrag{r4}{$r_4$}\n\\psfrag{r5}{$r_5$}\n\\psfrag{x1}{$\\bar{r}_1$}\n\\psfrag{x2}{$\\bar{r}_2$}\n\\psfrag{x3}{$\\bar{r}_3$}\n\\psfrag{x4}{$\\bar{r}_4$}\n\\psfrag{x5}{$\\bar{r}_5$}\n\\psfrag{c1}{$c_1$}\n\\psfrag{c2}{$c_2$}\n\\psfrag{c3}{$c_3$}\n\\psfrag{c4}{$c_4$}\n\\psfrag{y1}{$\\bar{c}_1$}\n\\psfrag{y2}{$\\bar{c}_2$}\n\\psfrag{y3}{$\\bar{c}_3$}\n\\psfrag{y4}{$\\bar{c}_4$}\n\\psfrag{e1}{$\\epsilon_1$}\n\\psfrag{e2}{$\\epsilon_2$}\n\\psfrag{e3}{$\\epsilon_3$}\n\\psfrag{e4}{$\\epsilon_4$}\n\\psfrag{e5}{$\\epsilon_5$}\n\\psfrag{e6}{$\\epsilon_6$}\n\\psfrag{e7}{$\\epsilon_7$}\n\\psfrag{e8}{$\\epsilon_8$}\n\\psfrag{e9}{$\\epsilon_9$}\n\\psfrag{d1}{$-\\epsilon_1$}\n\\psfrag{d2}{$-\\epsilon_2$}\n\\psfrag{d3}{$-\\epsilon_3$}\n\\psfrag{d4}{$-\\epsilon_4$}\n\\psfrag{d5}{$-\\epsilon_5$}\n\\psfrag{d6}{$-\\epsilon_6$}\n\\psfrag{d7}{$-\\epsilon_7$}\n\\psfrag{d8}{$-\\epsilon_8$}\n\\psfrag{d9}{$-\\epsilon_9$}\n\\psfrag{s}{$\\Sigma$}\n\\psfrag{v}{$v_\\Sigma(\\lambda)$}\n\\psfrag{bv}{$\\bar{v}_\\Sigma(\\mu)$}\n\\psfrag{bh}{$\\bar{h}_\\Sigma(\\mu)$}\n\\psfrag{h}{$h_\\Sigma(\\lambda)$}\n\\centerline{\\includegraphics[scale=1.3]{SAVE.eps}}\n\\caption{Covariant Young tableau $\\lambda$ and contravariant Young tableau $\\mu$ of $(5,4)$--hook shape for $\\gl(5|4)$. Bosonic fundamental weights belong to $B=\\{\\epsilon_2,\\epsilon_3,\\epsilon_5,\\epsilon_6,\\epsilon_9\\}$, while fermionic weights to $F=\\{\\epsilon_1,\\epsilon_4,\\epsilon_7,\\epsilon_8\\}$. Correspondingly $-B=\\{-\\epsilon_9,-\\epsilon_6,-\\epsilon_5,-\\epsilon_3,-\\epsilon_2\\}$ and $-F=\\{-\\epsilon_8,-\\epsilon_7,-\\epsilon_4,-\\epsilon_1\\}$. The values of $r_4$, $r_5$, $c_3$, $c_4$, $\\bar{r}_2,\\dots,\\bar{r}_5$, $\\bar{c}_3$, $\\bar{c}_4$ are zero.}\n\\label{fig:young}\n\\end{figure}\n\nThe highest weight component of a tensor $t(\\lambda,\\mu)$ w.r.t. the grading $\\Sigma$ will belong to the traceless subspace $t_0(\\lambda,\\mu)$ if the highest weight Young supertableau of shape $(\\lambda,\\mu)$ does not contain some fundamental weight $\\pm \\epsilon_i$ in both $\\lambda$ and $\\mu$.\nOtherwise, the corresponding highest weight component of $t(\\lambda,\\mu)$ w.r.t. the grading $\\Sigma$ will either i) not belong to $t_0(\\lambda,\\mu)$ or ii) generate a submodule of $t_0(\\lambda,\\mu)$ isomorphic to the embedding of a quotient of a lower rank tensor $t_0(\\lambda',\\mu')$. If the latter case holds, then the possible Young diagrams $(\\lambda',\\mu')$ are obtained from the Young diagrams $(\\lambda,\\mu)$ by removing pairs of boxes from the highest weight Young supertableau of shape $(\\lambda,\\mu)$: a box of $\\lambda$ carrying some weights $\\epsilon_i$ together with a box of $\\mu$ carrying the opposite weight $-\\epsilon_i$.\nMoreover, in the case ii) the highest weight of the top~\\footnote{The top of a module is the quotient by the intersection of all maximal ideals. The top of a Kac module is the irreducible representation of highest weight.} of $t_0(\\lambda,\\mu)$ will be smaller then the highest weight of the submodule $t_0(\\lambda',\\mu')$.\nTherefore, $t_0(\\lambda,\\mu)$ cannot be a Kac module w.r.t. $\\mathfrak{b}_\\Sigma$.\n\nWe call $\\gl(M|N)$--\\emph{admissible} the shapes $(\\lambda,\\mu)$ for which the $\\gl(M|N)$ traceless tensors $t_0(\\lambda,\\mu)$ neither vanish nor are isomorphic to lower rank traceless tensors.\nWe are now ready to answer the very important question: what are the admissible shapes of traceless tensors?\nAccording to the previous discussion on the highest weight component of a tensor $t(\\lambda,\\mu)$, a shape $(\\lambda,\\mu)$ is admissible if there \\emph{exists} a grading $\\Sigma$ such that \nthe highest weight Young supertableau of shape $(\\lambda,\\mu)$ does not contain any fundamental weight $\\pm \\epsilon_i$ both in $\\lambda$ and in $\\mu$.\nConsequently, for a traceless tensor $t_0(\\lambda,\\mu)$ of admissible shape there is a grading $\\Sigma$ and a corresponding highest weight~\\eqref{eq:weight_tensor} such that no cancellation between the $r_i$ and $\\bar{r}_i$ or $c_i$ and $\\bar{c}_i$ terms can occur.\nIf $v_\\Sigma(\\lambda)$, $\\bar{v}_\\Sigma(\\mu)$ denote the number of nonzero ``reduced rows'' $r_i$, $\\bar{r}_i$ and $h_\\Sigma(\\lambda)$, $\\bar{h}_\\Sigma(\\mu)$ denote the number of non-zero ``reduced'' columns\n$c_i$, $\\bar{c}_i$, then one must have\n\\begin{equation}\n\\exists \\Sigma \\quad \\text{such that} \\quad v_\\Sigma(\\lambda)+\\bar{v}_\\Sigma(\\mu)\\leq M,\\qquad h_\\Sigma(\\lambda)+\\bar{h}_\\Sigma(\\mu)\\leq N\n\\label{eq:restr_form}\n\\end{equation}\nfor a $\\gl(M|N)$--admissible shape $(\\lambda,\\mu)$, as represented in fig.~\\ref{fig:young}\nThese admissible shapes nicely reduce to hook shapes for purely covariant or contravariant $\\gl(M|N)$ tensors and to staircases for $\\gl(n)$ traceless tensors \\cite{staircase}.\n\nWe say that a shape $(\\lambda,\\mu)$ of a traceless tensor $t_0(\\lambda,\\mu)$ is $\\Sigma$--admissible if the inequalities in eq.~\\eqref{eq:restr_form} are satisfied.\nLet $K_\\Sigma(\\Lambda)$ be the Kac module of highest weight $\\Lambda$ w.r.t. $\\mathfrak{b}_\\Sigma$. \nWe can make now assumption~\\ref{ass:hw} more precise.\n\\begin{assumption}{1$'$}\\label{ass:hw2}\n The following isomorphism holds for $\\Sigma$--admissible shapes $(\\lambda,\\mu)$\n\\begin{equation*}\n t_0(\\lambda,\\mu)\\simeq K_\\Sigma(\\Lambda_\\Sigma(\\lambda,\\mu))\\ .\n\\end{equation*}\n\\end{assumption}\n\\noindent \nThis assumption in combination with the general theory of Kac modules \\cite{Kac77a, Kac77b} is very useful for counting highest weight vectors. Namely, if $(\\lambda,\\mu)$ is $\\Sigma$--admissible then the number of highest weight vectors in $t_0(\\lambda,\\mu)$ w.r.t. $\\mathfrak{b}_\\Sigma$ is equal to the number of irreducible subquotients.\nNoticing that a highest weight vector cannot generate more then a highest weight module, we prove the following claim in app.~\\ref{sec:proofs}.\n\\begin{claim}\\label{claim:cv}\nHighest weight vectors of $(V\\otimes V^*)^{\\otimes L}$ w.r.t. any Borel subalgebra belong to submodules isomorphic to traceless tensors $t_0(\\lambda,\\mu)$, $\\lambda,\\mu\\vdash f=0,1,\\dots,L$ of admissible shape.\n\\end{claim}\n\nTo sum up, we have explained the connexion between traceless tensors $t_0(\\lambda,\\mu)$ of admissible shapes $(\\lambda,\\mu)$ and standard modules $\\Delta_{L,L}(\\lambda,\\mu)$. Secondly, if $(\\lambda,\\mu)$ is $\\Sigma$-admissible, then eq.~\\eqref{eq:weight_tensor} allows to compute the highest weight of $t_0(\\lambda,\\mu)$ w.r.t. the Borel subalgebra $\\mathfrak{b}_\\Sigma$. Evaluating the highest weight of $t_0(\\lambda,\\mu)$ w.r.t. arbitrary Borel subalgebras is more delicate, mainly because of indecomposability issues.\nFinally, claim~\\ref{claim:cv} indicates a natural relationship between traceless tensors and Bethe vectors, which have the highest weight property, constructed in the framework of ABA.\n\n\n\\subsection{Spectrum}\n\\label{sec:spec}\n \n\nWe present the spectrum of $\\mathsf{H}$ in various $\\Delta_{L,L}(\\lambda,\\mu)$ standard modules of $B_{L,L}(n)$ in tab.~\\ref{tab:1}.\n\\begin{table}[t]\\label{tab:1}\n\\begin{center}\n\\caption{Lowest eigenvalue of $\\mathsf{H}$ in $\\Delta_L(\\lambda,\\mu)$ with opposite sign.}\n\\begin{tabular}{|c|c|ccccccc|}\n\\hline\n $L$& $n$ & $(1^2,2)$ & $(2,2)$ & $(1^3,21)$ & $(1^3,3)$ & $(21,21)$ & $(21,3)$ & $(3,3)$\\\\ \\hline\n \\multirow{8}{*}{5} & 1 &\n$32.130$ & $29.797$ & $29.461$ & $27.453$ & $27.156$ & $24.101$ & $21.446$ \\\\ \n& &\n$\\pm0.591 i$& & & & & & \\\\ \n& 2 &\n$23.237$ & $21.904$ & $22.626$ & $20.498$ & $20.506$ & $17.999$ & $15.548$ \\\\ \n& &\n$\\pm 0.339i$& & & & & & \\\\\n& 3 &\n$20.763$ & $20.130$ & $20.554$ & $18.530$ & $18.762$ & $16.572$ & $14.472$ \\\\\n& &\n & & & & & & \\\\\n& 4 &\n$19.968$ & $19.487$ & $19.588$ & $17.682$ & $18.139$ &$16.180$ & $14.153$ \\\\\n& &\n & & & & & & \\\\ \\hline\n \\multirow{8}{*}{6} & 1 &\n41.558 & 39.639 & 38.640 & 36.835 & 36.559 & 33.762& 31.264 \\\\ \n& &\n & & $\\pm 0.355 i$ & & $\\pm 0.513 i$& $\\pm 0.127i$ & \\\\ \n& 2 & 29.537 & 28.282 & 28.802 & 26.976 & 26.984 & 24.784& 22.607 \\\\ \n& & & & $\\pm 0.267 i$ & & $\\pm 0.426 i$ & $\\pm 0.148i$ & \\\\\n& 3 & 26.123 & 25.600 & 25.884 & 24.210 & 24.365& & 20.713\\\\\n& & & & $\\pm 0.194i$ & & $\\pm0.258i$& 22.550 & \\\\\n& 4 & 24.917 & 24.557 & 24.566 & 23.031 &23.416& 21.800 & 20.021\\\\\n& & & & $\\pm 0.129i$ & & & & \\\\ \\hline\n\n\\end{tabular}\n\\end{center}\n\\end{table}\nAt a first glance, it appears that the vacuum always lies in $\\Delta_{L,L}(\\emptyset,\\emptyset)$.\nTo check more thoroughly this vacuum hypothesis we need an additional assumption on the spectrum.\nConsider the spectral sets of $\\mathsf{H}$ defined as\n\\begin{equation*}\n\\spec f=\\bigcup_{\\lambda,\\mu\\vdash f } \\spec\\Delta_{L,L}(\\lambda,\\mu)\\ .\n\\end{equation*}\nNotice from tab.~\\ref{tab:1} that the lowest eigenvalue in $\\spec f$ always lies in $\\Delta_{L,L}(1^f,1^f)$, where $1^f$ denotes the Young diagram with a single column of length $f$. We have checked this observation extensively \\textbf{(more details)}.\n\\begin{assumption}{2}\\label{ass:ord}\n The lowest eigenvalues of $\\mathsf{H}$ in $\\spec f$ always lies in $\\Delta_{L,L}(1^f,1^f)$.\n\\end{assumption}\n\\noindent Comparing only the lowest eigenvalues in $\\Delta_{L,L}(1^f,1^f)$ allows us to gain several spin chain length units and check the vacuum hypothesis further, see tab.~\\ref{tab:2}.\n\n\n\n\n\n\n\n\\begin{table}\\label{tab:2}\n\\begin{center}\n\\caption{Lowest eigenvalue of $\\mathsf{H}$ in $\\Delta_L(1^k,1^k)$ with opposite sign.}\n\\begin{tabular}{|c|c|ccccccccc|}\n\\hline\n $L$& $n$ & $k=0$ & $k=1$ & $k=2$ & $k=3$ & $k=4$ & $k=5$ & $k=6$ & $k=7$ & $k=8$\\\\ \\hline\n \\multirow{4}{*}{5} & 1 & 40.000 & 38.846 & 36.564 & 32.710 & 26.385& 20.000 & & & \\\\ \n& 2 & 28.062 & 26.369 & 26.156 & 25.082 & 22.606& 20.000 & & &\\\\ \n& 3 & 24.625 & 22.950 & 22.944 & 22.640 & 21.407& 20.000 & & & \\\\\n& 4 & 23.123 & 21.631 & 21.456 & 21.456 & 20.828& 20.000 & & & \\\\ \\hline\n\\multirow{4}{*}{6} & 1 & 48.000 & 46.723 & 45.626 & 41.538 & 36.782& 30.397 & 24.000 & & \\\\ \n& 2 & 33.550 & 32.126 & 32.126 & 30.902 & 29.253& 26.647 & 24.000 & & \\\\ \n& 3 & 29.388 & 27.989 & 28.024 & 27.617 & 26.885& 25.476 & 24.000 & & \\\\ \n& 4 & 27.574 & 26.339 & 26.146 & 26.086 & 25.754& 24.921 & 24.000 & & \\\\ \\hline\n \\multirow{4}{*}{7} & 1 & 56.000 & 55.134 & 53.631 & 50.894 & 46.015 & 40.796 & 34.399 & 28.000 &\\\\ \n& 2 & 39.054 & 37.826 & 37.699 & 36.992 & 35.260 & 33.307 & 30.660 & 28.000 & \\\\ \n& 3 & 34.172 & 32.971 & 32.939 & 32.748 & 31.954& 30.981 & 29.504 & 28.000 & \\\\\n& 4 & 32.046 & 30.991 & 30.800 & 30.800 & 30.420 & 29.885 & 28.962 & 28.000 & \\\\ \\hline\n\\multirow{4}{*}{8} & 1 & 64.000 & 63.035 & 63.296 & 59.127 & 55.589 & 50.296 & 44.799 & 38.400 & 32.000\\\\ \n& 2 & 44.569 & 43.488 & 43.488 & 42.533 & 41.457 & 39.453 & 37.325 & 34.664 & 32.000\\\\ \n& 3 & 38.970 & 37.917 & 39.480 & 37.652 & 37.146& 36.134 & 35.019 & 33.515 & 32.000\\\\ \n& 4 & 36.530 & 35.611 & 35.428 & 35.391 & 35.169& 34.603 & 33.944 & 32.982 & 32.000\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nTo extract the spectrum of the $\\gl(n+N|N)$ spin chain Hamiltonian~\\eqref{eq:ham} from the spectrum of the algebraic Hamiltonian $\\mathsf{H}$ we do the following.\nFor every pair of Young diagrams $(\\lambda,\\mu)$ with $f=0,1,\\dots, L$ boxes and of admissible shape we pick a grading $\\Sigma$ such that the inequalities in eq.~\\eqref{eq:restr_form} are satisfied.\nThen we try to reproduce the spectrum of $\\mathsf{H}$ in $\\Delta_{L,L}(\\lambda,\\mu)$ by means of eq.~\\eqref{eq:spec_ham_mom} from numerical solutions of BAE~\\eqref{eq:BAE_short} in the the form determined by the grading $\\Sigma$ and for root numbers corresponding\nto the highest weight $\\Lambda_\\Sigma(\\lambda,\\mu)$ in eqs.~(\\ref{eq:weight_BV1}, \\ref{eq:weight_tensor}).~\\footnote{One could work with a single form of BAE, say, that corresponding to the distinguished gradation $\\Sigma_0$.\nHowever, the weight of Bethe vectors reproducing eigenvalues of $\\mathsf{H}$ in $\\Delta_{L,L}(\\lambda,\\mu)$ will no longer be given by eq.~\\eqref{eq:weight_tensor} if the shapes of $(\\lambda,\\mu)$ do not satisfy the inequalities~\\eqref{eq:restr_form} w.r.t. $\\Sigma_0$.}\nNot all eigenvalues can be reproduced in this way. This happens because, as we have explained in sec.~\\ref{sec:tens}, \nthe vector space $\\delta_{L,L}(\\lambda,\\mu)$ of all possible embeddings of $\\gl(n+N|N)$ traceless tensors $t_0(\\lambda,\\mu)$ into $(V\\otimes V^*)^{\\otimes L}$ can be identified with only a quotient of the standard module $\\Delta_{L,L}(\\lambda,\\mu)$ of $B_{L,L}(n)$.\nWhen an eigenvalue of $\\mathsf{H}$ can be reproduced this way, we assume that a corresponding non-vanishing Bethe vector exist. Otherwise, we assume that there is no eigenstate of $H$ corresponding to that eigenvalue. So, we rely on the completeness of the ABA, at least as far as the spectrum is concerned.\n\n\nDenote the spectrum of the $\\gl(M|N)$ spin chain Hamiltonian~\\eqref{eq:ham} by $\\spec H_{M|N}$.\nThe central idea of this section was the existence of an abstract algebra $B_{L,L}(n)$ such that the centralizers of the series $N=0,1,2,\\dots$ of $\\gl(n+N|N)$ spin chains $\\mathcal{C}(L)$ provide different, $N$ dependent, representations of $B_{L,L}(n)$.\nThis suggests that the intersection $\\cap_{N\\in \\mathbb{Z}^+}\\spec H_{n+N|N}$ might be non-trivial.\nIn fact, with the cohomological techniques developed in \\cite{Candu:2010yg} one can prove the following relationship between the spectral sets $\\spec H_{N+n|N}$ with $n$ fixed\n\\begin{equation}\n\\spec H_{n|0}\\subset \\spec H_{n+1|1}\\subset \\spec H_{n+2|2}\\subset \\cdots\\subset \\mathsf{H}\\ . \n\\label{eq:embed_spec}\n\\end{equation}\nThis ``embedding of spectra'' is a very interesting and general feature of supergroup spin chains and one might wonder how does it carry on to the field theory description of the continuum limit.\nIn this respect, two scenarios are possible.\nThe first possibility is that $\\spec H_{n+N'|N'}$ becomes a very excited subset within $\\spec H_{n+N''|N''}$, where $N'0 &\\longrightarrow w^{j\\phantom{+1}}>0\\ ,& w^{j+1}=0&\\longrightarrow w^{j\\phantom{+1}} \\geq 0 \\ .\n\\end{align}\nThe main reason for introducing this restrictions and working with BAE in multiple gradings is the bounds on the number of Bethe roots resulting from eqs.~\\eqref{eq:cond_TL}.\n\n\\subsection{Restriction}\n\nWe wish to consider the BAE in the form~\\eqref{eq:BAE_Qform} corresponding to a simple root $\\alpha_k$ such that the following assumptions hold\n\\begin{description}\n\t\\item[A1] $\\alpha_k$ is odd, that is $\\sigma_k \\sigma_{k+1}=-1$\n\t\\item[A2] $\\alpha_k$ has no source terms or, equivalently, $k\\neq 1, r$\n\\end{description}\nWith these assumptions, we have $\\nu^{(k)}$ BAE for $\\alpha_{k-1}$ of the form\n\\begin{equation}\\label{eq:red1} \\frac{\\sigma_{k-1}\\Lambda_{k-1}(u^{(k-1)}_j)}{\\sigma_{k}\\Lambda_{k}(u^{(k-1)}_j)}=- \\frac{Q_{k-2}(u^{(k-1)}_j)Q_{k-1}(u^{(k-1)}_j+\\sigma_k)Q_{k}(u^{(k-1)}_j-\\sigma_k)}{Q_{k-2}(u^{(k-1)}_j+\\sigma_{k-1})Q_{k-1}(u^{(k-1)}_j-\\sigma_{k-1})Q_{k}(u^{(k-1)}_j)}\\ ,\n\\end{equation}\n$\\nu^{(k)}$ equations for $\\alpha_k$\n\\begin{equation}\\label{eq:red2} 1=\\frac{Q_{k-1}(u^{(k)}_j)Q_{k+1}(u^{(k)}_j+\\sigma_k)}{Q_{k-1}(u^{(k)}_j+\\sigma_k)Q_{k+1}(u^{(k)}_j)}\n\\end{equation}\nand $\\nu^{(k+1)}$ equations for $\\alpha_{k+1}$\n\\begin{equation}\\label{eq:red3} \\frac{\\sigma_k \\Lambda_{k+1}(u^{(k+1)}_j)}{\\sigma_{k+2}\\Lambda_{k+2}(u^{(k+1)}_j)}= \\frac{Q_{k}(u^{(k+1)}_j)Q_{k+1}(u^{(k+1)}_j+\\sigma_{k+2})Q_{k+2}(u^{(k+1)}_j-\\sigma_{k+2})}{Q_{k}(u^{(k+1)}_j-\\sigma_k)Q_{k+1}(u^{(k+1)}_j+\\sigma_k)Q_{k+2}(u^{(k+1)}_j)}\\ .\n\\end{equation}\nNotice that if $\\nu^{(k-1)}=\\nu^{(k+1)}$ then one can reduce the\nBAE~\\eqref{eq:BAE_Qform} for the $\\gl(M|N)$ spin chain $(V\\otimes V^*)^{\\otimes L}$ to the BAE for the $\\gl(M-1|N-1)$ spin chain of the same type $(V\\otimes V^*)^{\\otimes L}$ by\n\\begin{description}\n\t\\item[R1] identifying the Bethe roots corresponding to $\\alpha_{k-1}$ and $\\alpha_{k+1}$\n\\begin{equation} \\{u^{(k-1)}_j\\}_{j=1}^{\\nu^{(k-1)}}=\\{u^{(k+1)}_j\\}_{j=1}^{\\nu^{(k+1)}}\n\\end{equation}\n\\item[R2] multiplying the BAE for $\\alpha_{k-1}$ and $\\alpha_{k+1}$ corresponding to, say $u^{(k-1)}_j=u^{(k+1)}_j$\n\\begin{equation}\\label{eq:red}\n \\frac{\\sigma_{k-1}\\Lambda_{k-1}(u^{(k- 1)}_j)}{\\sigma_{k+2}\\Lambda_{k+2}(u^{(k- 1)}_j)}=-\n \\frac{Q_{k-2}(u^{(k- 1)}_j)Q_{k- 1}(u^{(k- 1)}_j+\\sigma_{k+2})Q_{k+2}(u^{(k- 1)}_j-\\sigma_{k+2})}{Q_{k-2}(u^{(k-1)}_j+\\sigma_{k-1})Q_{k- 1}(u^{(k- 1)}_j-\\sigma_{k-1})Q_{k+2}(u^{(k- 1)}_j)}\\ .\n\\end{equation}\n\\end{description}\nIndeed, eq.~\\eqref{eq:red2} is trivially satisfied because according to \\textbf{R1} one has $Q_{k-1}(u)=Q_{k+1}(u)$. \nFurthermore, multiplying BAE according to \\textbf{R2}, the Bethe roots $\\{u^{(k)}_j\\}_{j=1}^{\\nu^{(k)}}$ drop off yielding a BAE of the form~\\eqref{eq:red1} with $k$ replaced by $k+2$.\nWe call a \\emph{restriction} of BAE the procedure \\textbf{R1}--\\textbf{R2}.\n\nThe restriction procedure can be given an algebraic meaning and, therefore, partially explained as follows. \nAs described in detail in~\\cite{Candu:2010yg}, choose\n\\begin{equation}\\label{eq:q_choice}\nQ=E_{k+1k} \n\\end{equation}\nto be the odd $\\gl(M|N)$ element that squares to zero and defines the $\\gl(M-1|N-1)$ chain $(V_{M-1|N-1}\\otimes V^*_{M-1|N-1})^{\\otimes L}$ as the $Q$--cohomology of the $\\gl(M|N)$ chain $(V_{M|N}\\otimes V^*_{M|N})^{\\otimes L}$.\nA necessary condition for a highest weight vector $\\omega\\in (V_{M|N}\\otimes V^*_{M|N})^{\\otimes L}$ to yield a non-trivial $Q$--cohomology is for it to be in the kernel of $Q$ and, therefore, one must have\n\\begin{equation}\\label{eq:wt_ker}\n[E_{kk+1},E_{k+1k}]\\omega =0\\quad \\Rightarrow\\quad \\langle \\wt(\\omega),\\alpha_k\\rangle=0\\ .\n\\end{equation}\nApplying this constraint to a Bethe vector of weight~\\eqref{eq:weight_BV1} one recovers the condition $\\nu^{(k-1)}=\\nu^{(k+1)}$ necessary for \\textbf{R1} to hold.\nThe form of the reduced $\\gl(M-1|N-1)$ BAE can also be easily understood.\nThe subquotient $ \\big(\\ker_{\\langle\\alpha_k,-\\rangle} \\sum_{i=1}^{M+N}\\mathbb{C}\\epsilon_i\\big)\/\\mathbb{C}\\alpha_k$ of the $\\gl(M|N)$ weight space can be straightforwardly identified with the $\\gl(M-1|N-1)$ weight space $\\sum_{i\\neq k,k+1}^{M+N}\\mathbb{C}\\epsilon_i$.\nTherefore, the subquotient $ \\big(\\ker_{\\langle\\alpha_k,-\\rangle} \\Delta_0^{M|N}\\big) \/\\mathbb{C}\\alpha_k$ of the $\\gl(M|N)$ simple root system $\\Delta^{M|N}_0$ induced by cohomological reduction can be identified with a $\\gl(M-1|N-1)$ simple root system $\\Delta_0^{M-1|N-1}= \\{\\alpha_1,\\dots,\\alpha_{k-2},\\epsilon_{k-1}-\\epsilon_{k+2},\\alpha_{k+2},\\dots,\\alpha_r\\}$.\nThe reduced BAE have the form~\\eqref{eq:BAE_short} corresponding to precisely the simple root system $\\Delta_0^{M-1|N-1}$.\n\n\n\\subsection{Lift}\n\n\nTo resume, assuming \\textbf{A1}--\\textbf{A2} holds for the BAE of the spin chain $(V_{M|N}\\otimes V^*_{M|N})^{\\otimes L}$ written w.r.t. a simple root system $\\Delta^{M|N}_0$, we showed that one can restrict them to the system of BAE for the spin chain $(V_{M-1|N-1}\\otimes V^*_{M-1|N-1})^{\\otimes L}$ written w.r.t. the simple root system $\\Delta_0^{M-1|N-1}=\\{\\alpha_1,\\dots,\\alpha_{k-2},\\epsilon_{k-1}-\\epsilon_{k+2}, \\alpha_{k+2},\\dots,\\alpha_r\\}$ induced by cohomological reduction.\nThis restriction is represented at the level of Dynkin diagrams in fig.~\\ref{fig:col}.\n\\begin{figure}%\n\\psfrag{+}{$+$}\n\\psfrag{-}{$-$}\n\\psfrag{a1}{$\\alpha_{k-1}$}\n\\psfrag{a2}{$\\alpha_{k}$}\n\\psfrag{a3}{$\\alpha_{k+1}$}\n\\psfrag{a4}{$\\alpha_{k-1}\\simeq\\alpha_{k+1}$}\n\\psfrag{b1}{$\\gl(M|N)$}\n\\psfrag{b2}{$\\gl(M-1|N-1)$}\n\\psfrag{P}{$P$}\n\\psfrag{p}{$p$}\n\\centerline{\\includegraphics[scale=1]{col.eps}}\n\\caption{Restriction of BAE for supergroups. The dashed lines indicate possible roots on the left or on the right. For a root $\\alpha_j=\\epsilon_j-\\epsilon_{j+1}$ with $j=k,k\\pm 1$, we have indicated with $\\pm$ signs the gradings of fundamental weight $\\epsilon_j$ determining it.}\n\\label{fig:col}\n\\end{figure}\n\nNotice that we have not imposed any condition on the roots $u_j^{(k)}$.\nTherefore, it is legitimate to ask if the BAE satisfying \\textbf{A1}--\\textbf{A2} actually admit solutions of type \\textbf{R1}.\nSo, we are given a simple root system $\\Delta_0^{M|N}$ and a solution $\\{u_j^{(l)}\\}_{j=1}^{\\nu^{(l)}}$, $l=1,\\dots,k-1,k+2,\\dots,r$ of the \n$(V_{M-1|N-1}\\otimes V^*_{M-1|N-1})^{\\otimes L}$ BAE w.r.t the simple root system $\\Delta_0^{M-1|N-1}$ induced by cohomological reduction from $\\Delta_0^{M|N}$ with $Q$ as in eq.~\\eqref{eq:q_choice}.\nWe must show that there is a solution $\\{u_j^{(l)}\\}_{j=1}^{\\nu^{(l)}}$, $l=1,\\dots,r$ of the $(V_{M|N}\\otimes V^*_{M|N})^{\\otimes L}$ BAE w.r.t. $\\Delta_0^{M|N}$ which restricts to the given one.\nFirst of all, eq.~\\eqref{eq:wt_ker} implies $\\nu^{(k+1)}=\\nu^{(k-1)}=P$. After defining $u^{(k+1)}_j=u^{(k-1)}_j$ for $j=1,\\dots,P$ the task is reduced to finding a solution $\\{u^{(k)}_j\\}_{j=1}^p$ to either eq.~\\eqref{eq:red1} or eq.~\\eqref{eq:red3}, where we have set $p=\\nu^{(k)}$.\nIf $p\\geq P$, such a solution obviously exists. More then that, for $p>P$ there is a continuum of such solutions!\nHowever, recall that for a fixed root system and corresponding BAE we have restricted to Bethe vectors such that their weights satisfy the constraints~\\eqref{eq:cond_TL}.\nIf the solution we are looking for exists then the weight of the corresponding Bethe vector $\\omega$ can be written as $\\wt(\\omega)=\\dots+(P-p)(\\epsilon_k-\\epsilon_{k+1})+\\dots$.\nThe constraints~\\eqref{eq:cond_TL} imply $p\\leq P$.\nSo, for $p=P$ the solution \\emph{always exists}.\nIn our numerical investigations we have observed that solutions might exist even for $p0$ and $\\Img x_j^{(k)} < 0$, exactly as we did for the strange $\\pm$-strings~\\eqref{eq:ideal_sol_1}.\nHowever, a finite volume treatment will be required, because the imaginary parts of even roots vanish in the thermodynamic limit.\n\nThe need for a finite volume approach can also be seen from eq.~\\eqref{eq:sing_anti}.\nSolving for the Fourier transform of $\\hat{\\rho}_a(p)=\\int dp\\,\\exp(-2\\pi i p x)\\rho_a(x)$ one gets\n\\begin{equation}\\label{eq:dist_anti_four}\n\\hat{\\rho}_a(p\\mid\\xi^{(+)},\\xi^{(-)})=\\frac{i}{2L}e^{\\pi |p|-\\pi i p (\\xi^{(+)}+\\xi^{(-)})}\\frac{\\sin \\pi p (\\xi^{(+)}-\\xi^{(-)})}{\\sinh^2 \\tfrac{\\pi p }{2}}\\ ,\n\\end{equation}\nwhere we have considered only a single pair of $\\pm$-holes.\nThis is no loss of generality, because from~\\eqref{eq:lieb_deg1} the numbers of $\\pm$-holes is always equal\n\\begin{equation*}\nn^{\\pm}=2(\\nu^{(m-1)}-N_+-N_-) = 2(\\nu^{(m-1)}-\\nu^{(m)})\\ .\n\\end{equation*}\nand, therefore, they always come in pairs.\nThe Fourier transform~\\eqref{eq:dist_anti_four} is singular and clearly must be regularized, because one has to satisfy the constraint\n\\begin{equation}\\label{eq:charge_finally_found}\nb:=\\lim_{L\\to \\infty} \\frac{N_+-N_-}{2L} = L \\int_{-\\infty}^{\\infty} dx\\, \\rho_a(x)\t=\\hat{\\rho}_a(0)\\ ,\n\\end{equation}\nwhere $b$ is a fixed real number parametrizing the state.\nOn a lattice of length $L$ the the momentum can take a minimal value of $p_\\sim 1\/L$. Using this value as a regulator one gets for a pair of $\\pm$-holes from eqs.~(\\ref{eq:dist_anti_four} \\ref{eq:charge_finally_found}) \n\\begin{equation*}\n\\xi^{(+)}-\\xi^{(-)}=\\frac{\\pi b}{2i}\\ ,\n\\end{equation*}\nwhich clearly shows that for $b\\neq 0$ the deviations of strange $\\pm$-strings from the form~\\eqref{eq:ideal_sol_1} of the solution in the thermodynamic limit have to be taken into account.\nNotice that the energy does not depend on the continuous parameter~\\eqref{eq:charge_finally_found} parametrizing the state.\nTherefore, it is tempting to conclude that there is a \\emph{continuum} of new particles~\\eqref{eq:new_particles}.\n\n\n\n\\section{Conclusions and Outlook}\n\n\n\nWe have put on firm grounds the relationship between $\\gl(n+N|N)$ integrable spin chains with $n$ fixed.\nThis allowed us to prove that all $\\gl(n+N|N)$ spin chains $(V\\otimes V^*)^{\\otimes L}$ with $n,N>0$ possess in the continuum limit $2n-2$ multiplets of massive particles which scatter with $\\gl(n)$ Gross-Neveu like $S$-matrices, namely their eigenvalues do not depend on $N$.\nWe concluded that the continuum theory is the $\\gl(M|N)$ Gross-Neveu model.\nEvidence that the massive spectrum is much richer, possibly continuous, was established on the example of $\\gl(2m|1)$ chain.\nFinally, our analysis of the thermodynamic limit strongly suggests that understanding the nature of new particles requires a finite volume treatment.\n\nThe question that begs the quickest answer is how to close the fusion\nof $S$-matrices~\\eqref{eq:smat_11} of the $\\gl(M|N)$ Gross-Neveu model starting with just the vector multiplet and its antiparticles.\n\nThe $\\gl(N|N)$ spin chains require a separate treatment, which we hope to report on later.\\\\\n\n\\noindent\\textbf{Acknowledgements.} I would like to greatly thank Hubert Saleur, Volker Schomerus, Sergei Lukyanov and J\\\"org Techner for helpful discussions, important guidance and critical feedback. I am also grateful to Fabian E{\\ss}ler, Holger Frahm and Nikolay Gromov for sharing some of their expertise in solving numerically BAE.\nThe author thanks the Rutgers NHET center for their hospitality, where an important part of this paper was written, and SFB676 for partial financial support. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLet $G=(V(G),E(G))$ be an undirected simple graph.\nThe \\textit{adjacency matrix} of $G$ is the $n \\times n$ matrix $A(G)=(a_{ij})$, where $a_{ij}=1$\nif $v_{i}$ is adjacent to $v_{j}$, and $0$ otherwise.\nThe \\textit{$Q$-matrix} (or \\textit{signless Laplacian matrix}) of $G$ is defined as $Q(G)=D(G)+A(G)$,\nwhere $D(G)$ is the diagonal matrix of vertex degrees of $G$. The largest eigenvalue of $Q(G)$,\ndenoted by $q(G)$, is called the \\textit{$Q$-index} (or \\textit{signless Laplacian spectral radius}) of $G$.\nFor two vertex disjoint graphs $G$ and $H$, we denote by $G \\cup H$ the union of $G$ and $H$, and $G \\vee H$ the join of $G$ and $H$, i.e., joining every vertex of $G$ to every vertex of $H$.\nDenote by $kG$ the union of $k$ disjoint copies of $G$.\nAs usual, denote by $K_{n}$ a \\textit{complete graph} of order $n$, and $K_{m, n}$ a \\textit{complete bipartite graph} on $m + n$ vertices.\nLet $F_{s, t}(n):\\cong K_{s-1} \\vee(p K_{t}\\cup K_{r})$, where $2 \\leq s \\leq t, n-s+1=p t+r$ and $1\\leq r\\leq t$.\nIt is easy to check that $F_{s, t}(n)$ is a $K_{s, t}$-minor free graph of order $n$. \\\nFor a graph $G$, let $\\overline{G}$ be its complement. Denote by $S^{1}(G)$ a graph obtained from a graph $G$ by subdividing once of an edge $uv$ with minimum degree sum $d_{G}(u)+d_{G}(v)$.\nDenote by $H^{\\star}$ the Petersen graph.\nLet $H_{s, t}:\\cong(\\beta-1)K_{1,s} \\cup K_{1,\\alpha}$, where $1\\leq s\\leq t$, $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor$ and $\\alpha=t-(\\beta-1)(s+1)\\geq s$.\nGiven two graphs $G$ and $H$, $H$ is a \\textit{minor} of $G$\nif $H$ can be obtained from a subgraph of $G$ by contracting edges.\nA graph is said to be \\textit{$H$-minor free} if it does not contain $H$ as a minor.\n\nMinors play a key role in graph theory, and extremal problems on forbidding\nminors have attracted appreciable amount of interests.\nFirstly, it is very useful for\nstudying the structures and properties of graphs. For example,\nevery planar graph is $\\{K_{3,3}, K_5\\}$-minor free and every outer-planar graph is $\\{K_{2,3}, K_4\\}$-minor free.\nSecondly, one of the problems in extremal graph theory is to concern about the maximum\nnumber of edges for graphs that do not contain a given $H$ as a minor. It is known that a\nplanar graph has at most $3n-6$ edges and an outer planar graph has at most $2n-3$ edges, see \\cite{A.B}.\n\nIn 2017, Nikiforov \\cite{Nikiforov2} provided a unified extension of both the adjacency spectral\nradius and the signless Laplacian spectral radius. For a graph $G$, it was proposed by Nikiforov \\cite{Nikiforov2}\nto study the family of matrices $A_{\\alpha}(G)$ defined as\n$$A_{\\alpha}(G) = \\alpha D(G) + (1 - \\alpha)A(G),$$\nwhere $0\\leq\\alpha \\leq1$.\nThe \\textit{$A_\\alpha$-spectral radius} (or \\textbf{$\\alpha$-index} ) of $G$, denoted by $\\rho_{\\alpha}(G)$, is the largest eigenvalue of $A_{\\alpha}(G)$.\nNikiforov \\cite{Nikiforov2} posed the $A_{\\alpha}$-spectral extrema problem:\n\\begin{prob}\\label{prob-0}\nGiven a graph $F$, what is the maximum $A_\\alpha$-spectral radius of a graph $G$ of order $n$, with no\nsubgraph isomorphic to $F$?\n\\end{prob}\nAt present, for $0 \\leq \\alpha<1$, the $A_{\\alpha}$-spectral extrema problem is done when $F$ is a $K_r$ (see \\cite{Nikiforov2}), $F$ is a $K_r$-minor (see \\cite{M.T} and \\cite{C.M}), and $F$ is a star forest (see \\cite{C.M2} and \\cite{C.M}). In this paper, we pay our attention on Problem \\ref{prob-0}\nwhen $F$ is a $K_{s,t}$-minor. For special value $\\alpha\\in \\left\\lbrace 0, \\frac{1}{2}\\right\\rbrace$, there are plentiful results.\n\n\n\n\n\n\n\nFor the special value $\\alpha=0$, then $A_\\alpha(G) = A(G)$ of $G$.\nIn 2007,\nNikiforov \\cite{Nikiforov1} (resp. Zhai and Wang \\cite{M.Q1} ) obtained a sharp upper bound of adjacency spectral radius over all $K_{2,2}$-minor free graphs of odd (resp. even) order, and determined the extremal graph.\nIn 2017, Nikiforov \\cite{Nikiforov}\nestablished a sharp upper bound of adjacency spectral radius over all $K_{2,t}$-minor free graphs of large\norder $n$ for $t\\ge3$, and determined the extremal graph when $t \\mid n-1$. In addition, Nikiforov determined the extremal graph when $t=3$ for all $n$. In 2019,\nTait \\cite{M.T}\nobtained an upper bound of adjacency spectral radius\nover all $K_{s,t}$-minor free graphs of large order $n$ for $2\\leq s\\leq t$, and determined the extremal graph when\n$t \\mid n-s+1$. In 2021, Wang, Chen and Fang \\cite{B.W} determined the extremal graph with maximum adjacency spectral radius\nover all $K_{3,3}$ (resp. $K_{2,4}$)-minor free graphs of large order.\nIn 2022, Zhai and Lin \\cite{M.Q} improved Tait's result \\cite{M.T} by removing the condition $t \\mid n-s+1$.\nMeanwhile, they\ndetermined the extremal graph with maximum adjacency spectral radius over all $K_{1,t}$-minor free graphs.\nThus the adjacency spectral extremal problem on $K_{s,t}$-minor free graphs\nis completely solved for large order.\n\n\nFor the special values $\\alpha=\\frac{1}{2}$, then $A_\\alpha(G) = Q(G)$ of $G$.\nIn 2022, Zhang and Lou \\cite{Y.T} determined the unique extremal graph with maximum $Q$-index among all $n$-vertex connected $K_{1,t}$-minor free graphs.\nFor $2 \\leq s \\leq t$, Chen and Zhang proposed the following conjecture in \\cite{C.M1}.\n\\begin{conj}(\\cite{C.M1})\\label{conj::1}\nLet $2 \\leq s \\leq t$ and $G$ be a $K_{s,t}$-minor free graph of sufficiently large order $n$. Then\n$$q(G) \\leq q(F_{s,t}(n))$$\nwith equality if and only if $G \\cong F_{s,t}(n)$.\n\\end{conj}\nThe previous results showed that the Conjecture \\ref{conj::1} is true in several cases. In 2013, Freitas, Nikiforov and Patuzzi \\cite {M.F} showed that $F_{2,2}(n)$ is the extremal graph with maximum $Q$-index among all $K_{2,2}$-minor free graphs for $n \\geq 4$.\nIn 2021, Chen and Zhang \\cite {C.M1} showed that $F_{2,3}(n)$ (resp. $F_{3,3}(n)$) is the extremal graph with maximum $Q$-index among all $K_{2,3}$ (resp. $K_{3,3}$)-minor free graphs for $n \\geq 22$ (resp. $n \\geq 1186$).\nThey also obtained an upper bound of $Q$-index\nover all $K_{2,t}$-minor free graphs of order $n\\geq t^{2} +4t+1$ with $t\\ge 3$, and proved the extremal graph is $F_{2,t}(n)$ when $t \\mid n-1$.\n\nHowever, for general $\\alpha\\in(0,1)$,\nthe related results are few. In 2022, Chen, Liu and Zhang \\cite{C.M}\nobtained an upper bound of $A_\\alpha$-spectral radius over all $K_{s,t}$-minor free graphs of large order $n$ for $2\\leq s\\leq t$, and proved that $F_{s,t}(n)$ is the extremal graph when\n$t \\mid n-s+1$.\nTherefore,\nProblem \\ref{prob-0} is still open when $F$ is a $K_{s,t}$-minor for $t \\nmid n-s+1$.\nNaturally, we want to overcome the following problem:\n\\begin{prob}\\label{1.1}\nFor $2\\leq s\\leq t$, what is the extremal graph with maximum $A_\\alpha$-spectral radius among all $K_{s, t}$-minor free graphs of sufficiently large order for $0 < \\alpha <1$? Whether the extremal graph is consistent for $0 \\leq \\alpha <1$?\n\\end{prob}\n\n\n\nIn this paper, the \\emph{$A_\\alpha$-spectral extremal graph} is defined by a graph with maximum $A_\\alpha$-spectral radius among all $K_{s, t}$-minor free graphs of sufficiently order $n$ for $0 < \\alpha<1$ and $2\\leq s\\leq t$.\nIn order to characterize the structure of the {$A_\\alpha$-spectral extremal graph,\nour first challenge is to show that\nthe $A_\\alpha$-spectral extremal graph contains a $(s-1)$-clique dominating set, the way of which is different from the adjacency spectra. In addition, we prove that the $A_\\alpha$-spectral extremal graph has a property of\nlocal edge maximality. Meanwhile, we apply the double eigenvectors transformation technique to the $A_{\\alpha}(G)$-matrix and get that\nthe $A_\\alpha$-spectral extremal graph has a property of local degree sequence majorization.\nFinally, we completely determined the $A_\\alpha$-spectral extremal graph as follows.\n\n\\begin{thm}\\label{thm::1.1}\nLet $0<\\alpha<1$, $2 \\leq s \\leq t$, $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor$ and $G^{*}$ be a $K_{s,t}$-minor free graph of sufficiently large order $n$ with the maximum $A_\\alpha$-spectral radius, where $n-s+1=pt+r$ and $1\\leq r\\leq t$. Then\n$$\nG^{*} \\cong \\begin{cases}K_{s-1} \\vee \\left((p-1) K_{t} \\cup \\overline{H^{\\star}}\\right) & \\text { if } r=2, t=8 \\text { and } \\beta=1 ; \\\\ K_{s-1} \\vee\\left( (p-1) K_{t} \\cup S^{1}\\left(\\overline{H_{s, t}}\\right)\\right) & \\text { if } r=\\beta=2; \\\\ K_{s-1} \\vee\\left((p-r) K_{t} \\cup r \\overline{H_{s, t}}\\right) & \\text { if } r \\leq 2(\\beta-1) \\text { except } r=\\beta=2 ; \\\\ K_{s-1} \\vee\\left(p K_{t} \\cup K_{r}\\right) & \\text { otherwise. }\\end{cases}\n$$\n\\end{thm}\n\nTaking $\\alpha=\\frac{1}{2}$ in Theorem \\ref{thm::1.1}, we completely solve Conjecture \\ref{conj::1}.\nOn the other hand, we completely answer the front part of Problem \\ref{1.1}. Combining the result of the extremal graph with maximum adjacency spectral radius among all $K_{s,t}$-minor free graphs in \\cite{M.Q},\nwe give a positive answer to the last part of Problem \\ref{1.1}.\n\n\n\nOur proofs are based on the structural analysis of the $A_\\alpha$-spectral extremal graph $G^{*}$, which is motivated by the work of Zhai and Lin in \\cite{M.Q}. Some special notations, terminologies and lemmas will be presented in Section 2. We also prove that $G^{*}$ has a $(s-1)$-clique dominating set in Section 2. The proofs of $G^{*}$ has the structural properties of local edge maximality and local degree sequence majorization are shown in Section 3. The proof of Theorem \\ref{thm::1.1} will be shown in Section 4.\n\n\n\\section{Preliminaries}\nIn this section, we will list some symbols and useful lemmas. Let $\\pi(G)=\\left( d_{1}, d_{2},\\dots , d_{n}\\right) $ be the non-increasing degree sequence of an $n$-vertex graph $G$.\nFor $A, B \\subseteq V(G)$, $e(A, B)$ is denoted the number of the edges of $G$ with one end vertex in $A$ and the other in $B$.\nFor a vertex $v \\in V(G)$, we write $N_{G}(v)$ for the set of neighbors of $v$ in $G$. Let $d_{G}(v)$ be the degree of a vertex $v$ in $G$. Let $H$ be a subgraph of $G$, we write $N_{H}(v)$ for the set of neighbors of $v$ in $V(H)$, and $d_{H}(v)$ for the number of neighbors of $v$ in $V(H)$, that is, $d_{H}(v)=\\left|N_{H}(v)\\right|=|N_{G}(v) \\cap V(H)|$. Let $\\Delta(G)$ and $\\delta(G)$ denote the maximum degree and minimum degree of $G$, respectively.\nFor $A \\subseteq V(G)$, the graph $G[A]$ is the \\textit{induced subgraph} by $A$. $G[A]$ is called a clique if it is a complete subgraph of $G$. Let $G$ be an $n$-vertex graph and $A\\subseteq V(G)$, then $A$ is called a \\textit{clique dominating set} of $G$, if $d_{G}(v)=n-1$ for any $v\\in A$.\nLet $K_{n}-e$ be a graph obtained by deleting one edge from a complete graph $K_{n}$.\nFor graph notations and concepts undefined here, readers are referred to \\cite {A.B}.\n\n\n\\begin{lem}(\\cite{Nikiforov2})\\label{lem::2.2}\nLet $0\\leq\\alpha<1$, then the $\\alpha$-index of any proper subgraph of a connected graph is smaller than the $\\alpha$-index of the original graph.\n\\end{lem}\n\nThe following result is from the proof of Theorem 1.2 in \\cite{C.M}.\n\n\\begin{lem}(\\cite{C.M})\\label{lem::2.1}\nLet $G$ be a $K_{s,t}$-minor free graph of sufficiently large order $n$ with maximum $\\alpha$-index, where $2 \\leq s \\leq t$ and $0 < \\alpha < 1$. Then $G$ contains a vertex set $K=\\left\\{v_{1}, v_{2}, \\ldots, v_{s-1}\\right\\}$ such that all of $v_{i}$ have common neighborhood of size $n-s+1$ in $G$. That is, $d_{G-K}\\left(v_{i}\\right)=n-s+1$ for $i \\in\\{1,2, \\ldots, s-1\\}$.\n\\end{lem}\n\n\n\\begin{lem}(Lemma 2.1, \\cite{C.M})\\label{lem::2.1'}\nLet $0<\\alpha<1, s \\geq 2$, and $n \\geq s-1$. If $G=K_{s-1} \\vee \\overline{K}_{n-s+1}$, then $\\rho_\\alpha(G) \\geq \\alpha(n-1)+(1-\\alpha)(s-2)$.\n\\end{lem}\n\n\nRecall that $F_{s, t}(n):\\cong K_{s-1} \\vee(p K_{t}\\cup K_{r})$, where $2 \\leq s \\leq t, n-s+1=p t+r$ and $1\\leq r\\leq t$.\n\n\\begin{lem}\\label{lem::2.3}\nLet $0 < \\alpha <1$, $2 \\leq s \\leq t$ and $n \\geq s-1+\\frac{t^{2}-1}{\\alpha}$. Then\n$\\rho_{\\alpha}(F_{s, t}(n))$ is no more than the largest root of $g(x)=0$\nand $\\rho_{\\alpha}(F_{s, t}(n))$ is larger than the largest root of $h(x)=0$, where\n\\begin{equation*}\n\\begin{aligned}\nh(x)=&x^2-(\\alpha n+s+t-3) x+\n(\\alpha(n-s+1)+s-2)(\\alpha(s-1)+t-1)\\\\&-(1-\\alpha)^2(s-1)(n-s),\n\\end{aligned}\n\\end{equation*}\nand\n$\ng(x)=h(x)-(1-\\alpha)^2(s-1).\n$\n\\end{lem}\n\n\\begin{proof}\nLet $\\rho_{\\alpha}=\\rho_{\\alpha}(F_{s, t}(n))$ and\n$\\mathbf{x}$ be a positive eigenvector of $A_{\\alpha}(F_{s, t}(n))$ corresponding to $\\rho_{\\alpha}$. By symmetry and the Perron-Frobenius theorem, all vertices of subgraphs $K_{s-1}$, $p \\cdot K_{t}$, or $K_{r}$ in $F_{s, t}(n):=K_{s-1} \\vee(p \\cdot K_{t} \\cup K_{r} )$ have the same eigenvector components respectively, which are denoted by $x_{1}$, $x_{2}$, $x_{3}$, respectively. We consider the following two cases.\n\t\n{\\flushleft\\bf Case 1. $r=t$.} By $A_{\\alpha}(F_{s, t}(n)) \\mathbf{x}=\\rho_{\\alpha} \\mathbf{x}$, it is easy to see that\n$$\n\\begin{aligned}\n&(\\rho_{\\alpha}-\\alpha(n-1)-(1-\\alpha)(s-2)) x_{1} =(1-\\alpha)(n-s+1) x_{2}, \\\\\n&(\\rho_{\\alpha}-\\alpha(s+t-2)-(1-\\alpha)(t-1))x_{2} =(1-\\alpha)(s-1) x_{1}.\n\\end{aligned}\n$$\nThen $\\rho_{\\alpha}$ is the largest root of $g(x)=0$, where\n$\ng(x)=h(x)-(1-\\alpha)^2(s-1).\n$\nSince $0 <\\alpha <1$ and $s \\geq2$, we find that $\\rho_{\\alpha}$ is larger than the largest root of $h(x)=0$.\n\t\n\t\n{\\flushleft\\bf Case 2. $1 \\leq r\\rho_{\\alpha}\\left(K_{s+r-1}\\right)=s+r-2$, $0 < \\alpha <1$ and $s\\geq2$, we have\n$$\\rho_{\\alpha}-\\alpha(s+r-2)-(1-\\alpha)(r-1)>(s+r-2)-\\alpha(s+r-2)-(1-\\alpha)(r-1)=(1-\\alpha)(s-1)>0.$$\nMoreover, since $1 \\leq r&\\rho_{\\alpha}\\left(K_{s-1} \\vee \\overline{K}_{n-s+1}\\right)\\geq \\alpha(n-1)+(1-\\alpha)(s-2).\n\\end{aligned}\n\\end{equation*}\nRecall that $0 < \\alpha <1$, $s\\geq2$, $1\\leq r< t$ and $n \\geq s-1+\\frac{t^{2}-1}{\\alpha}$. We obtain\n\\begin{equation*}\n\\begin{aligned}\n\\rho_{\\alpha}+1-\\alpha (s-1)-r(1-r+t)>&\\alpha(n-1)+(1-\\alpha)(s-2)+1-\\alpha (s-1)-(t-1)(1+t)\\\\\n=&\\alpha (n-1) + s-1-\\alpha(2s-3)-t^{2}+1\\\\\n\\geq&\\alpha (n-1) -\\alpha(s-2)-t^{2}+1\\\\\n\\geq&\\alpha \\left( \\left( s-1+\\frac{t^{2}-1}{\\alpha}\\right) -1\\right) -\\alpha(s-2)-t^{2}+1\\\\\n=&0.\n\\end{aligned}\n\\end{equation*}\nHence, $h(\\rho_{\\alpha})>0$. Furthermore, since\n\\begin{equation*}\n\\begin{aligned}\nn&\\geq s-1+\\frac{t^{2}-1}{\\alpha}=2(s-1)+\\frac{t^{2}-1-\\alpha(s-1)}{\\alpha}\\geq2(s-1)+\\frac{t^{2}-1-(s-1)}{\\alpha}\\\\\n&\\geq2(s-1)+\\frac{t-(s-1)}{\\alpha}=2(s-1)+\\frac{t-s+1}{\\alpha},\n\\end{aligned}\n\\end{equation*}\nwe have\n\\begin{equation*}\n\\begin{aligned}\n\\rho_{\\alpha}>&\\alpha(n-1)+(1-\\alpha)(s-2)\n=\\frac{\\alpha n}{2} +\\frac{\\alpha n}{2}-\\alpha (s-1)+s-2\\\\\n\\geq&\\frac{\\alpha n}{2} +\\frac{\\alpha}{2}\\left(2(s-1)+\\frac{t-s+1}{\\alpha} \\right) -\\alpha (s-1)+s-2=\\dfrac{\\alpha n+s+t-3}{2}.\n\\end{aligned}\n\\end{equation*}\nIt follows that $\\rho_{\\alpha}$ is larger than the largest root of $h(x)=0$.\n\\end{proof}\n\n\\begin{lem}\\label{lem::2.5}\nLet $0<\\alpha<1$, $s\\geq4$, $n\\geq s$ and $G\\cong (K_{s-1}-e) \\vee \\overline{K}_{n-s+1}$. Then $\\rho_{\\alpha}(G)$ is the largest root of $f(x) = 0$, where\n\\begin{equation*}\n\\begin{aligned}\nf(x)=&x^3 -(2\\alpha n + s - 4)x^2 +(\\alpha^2n^2 +3\\alpha ns - \\alpha s^2 - 6\\alpha n +\\alpha s- ns + s^2 - 2\\alpha + n -\n4s +\\\\& 7)x -2\\alpha^2 n^2 s + \\alpha^2 ns^2 + 2 \\alpha^2 n^2 - \\alpha^2 n s + \\alpha n^2 s - \\alpha ns^2 + 2\\alpha^2 n - \\alpha n^2 + 6\\alpha ns -\n2\\alpha s^2 \\\\&-9\\alpha n + 4\\alpha s -2ns + 2s^2 - 2\\alpha + 4n - 6s + 4.\n\\end{aligned}\n\\end{equation*}\t\nMoreover, $\\rho_{\\alpha}(G) > \\alpha(n-1)+(1-\\alpha)(s-4).$\n\\end{lem}\n\n\\begin{proof}\nFirstly, denote by $\\{v_{1}, v_{2}, \\cdots, v_{s-3}, w_{1}, w_{2}\\}$ the vertex set of $K_{s-1}-e$ in the representation $G:\\cong (K_{s-1}-e) \\vee \\overline{K}_{n-s+1}$, where $d_{G}(v_{i})=n-1$ for $i\\in \\{1,2,\\dots,s-3\\}$ and $d_{G}(w_{1})=d_{G}(w_{2})=n-2$. Set for short $\\rho_{\\alpha}=\\rho_{\\alpha}(G)$ and let $\\mathbf{x}=\\left(x_{v}\\right)_{v \\in V(G)}$ be the Perron vector of $A_{\\alpha}(G)$ with respect to $\\rho_{\\alpha}$. By symmetry, we have $x_{v_{1}}=\\cdots=x_{v_{s-3}}$ and $x_{w_{1}}=x_{w_{2}}$. Additionally, $x_{z}=x_{z_{1}}$ for any two vertices\n$z, z_1\\in V(G)\\backslash \\{v_{1}, v_{2}, \\cdots, v_{s-3}, w_{1}, w_{2}\\}$. By eigen-equations of $A_{\\alpha}(G)$ on $v_{1}$, $w_{1}$ and $z_{1}$, we have\n$$\n\\begin{aligned}\n&(\\rho_{\\alpha}-\\alpha(n-1)-(1-\\alpha)(s-4)) x_{v_{1}} =2(1-\\alpha)x_{w_{1}}+(1-\\alpha)(n-s+1)x_{z_{1}}, \\\\\n&(\\rho_{\\alpha}-\\alpha(n-2))x_{w_{1}} =(1-\\alpha)(s-3) x_{v_{1}}+(1-\\alpha)(n-s+1)x_{z_{1}},\\\\\n&(\\rho_{\\alpha}-\\alpha(s-1))x_{z_{1}}=(1-\\alpha)(s-3) x_{v_{1}}+2(1-\\alpha)x_{w_{1}}.\n\\end{aligned}\n$$\nThen $\\rho_{\\alpha}$ is the largest real root of $f(x) = 0$, where\n\\begin{equation*}\n\\begin{aligned}\nf(x)=&x^3 -(2\\alpha n + s - 4)x^2 +(\\alpha^2n^2 + 3\\alpha ns - \\alpha s^2 - 6\\alpha n +\\alpha s - ns + s^2\n- 2\\alpha + n - 4s \\\\\n&+ 7)x -2\\alpha^2 n^2 s + \\alpha^2 ns^2 + 2 \\alpha^2 n^2 - \\alpha^2 n s + \\alpha n^2 s - \\alpha ns^2 + 2\\alpha^2 n - \\alpha n^2 + 6\\alpha ns -\\\\\n&2\\alpha s^2 - 9\\alpha n + 4\\alpha s -2ns + 2s^2 - 2\\alpha + 4n - 6s + 4.\n\\end{aligned}\n\\end{equation*}\nSince $0 < \\alpha < 1$, $s\\geq4$ and $n \\geq s$, we have\n\\begin{equation*}\n\\begin{aligned}\n(\\alpha(s-3)-s)(n-s+3) + 8\\leq&((s-3)-s)(n-s+3) + 8\n=-3(n-s+3) + 8\\\\\n\\leq&-3(s-s+3) + 8\n<0,\n\\end{aligned}\n\\end{equation*}\nand so\n$f(\\alpha(n-1)+(1-\\alpha)(s-4))=((\\alpha(s-3)-s)(n-s+3) + 8)(s - 3)(\\alpha - 1)^2<0,$\nwhich implies that $\\rho_{\\alpha} >\\alpha(n-1)+(1-\\alpha)(s-4).$\n\nThis completes the proof.\n\\end{proof}\n\n\\begin{lem}\\label{lem::2.6}\nLet $0< \\alpha <1$, $t\\geq 2$, $s\\geq3$, $C\\geq\\frac{1}{\\alpha}$ and\n$$n \\geq \\max\\{ 2s-3+\\frac{t-s+4}{\\alpha}, \\frac{(1-\\alpha)(s-1)(C(s+t)+2)}{2}+s+t \\}.$$ Suppose that $H$ is a graph of order $n-s+1$ and $G\\cong (K_{s-1}-e) \\vee H$, particularly, defined by $G^{\\prime}=G$ when $H$ is $(t-1)$-regular. If $\\Delta(H) \\leq t-1$, then $\\rho_{\\alpha}(G) \\leq \\rho_{\\alpha}(G^{\\prime})$, eauality holds if and only if $G\\cong G^{\\prime}$,\nmoreover, $\\rho_{\\alpha}(G^{\\prime})$ is less than the largest root of $h(x)=0$, where\n\\begin{equation*}\n\\begin{aligned}\nh(x)=&x^2-(\\alpha n+s+t-3) x+\n(\\alpha(n-s+1)+s-2)(\\alpha(s-1)+t-1)\\\\&-(1-\\alpha)^2(s-1)(n-s).\n\\end{aligned}\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nDenote by $\\{v_{1}, v_{2}, \\cdots, v_{s-3}, w_{1}, w_{2}\\}$ the vertex set of $K_{s-1}-e$ in the representation $G:\\cong (K_{s-1}-e) \\vee H$, where $d_{G}(v_{i})=n-1$ for $i\\in \\{1,2,\\dots,s-3\\}$ and $d_{G}(w_{1})=d_{G}(w_{2})=n-2$. Set for short $\\rho_{\\alpha}=\\rho_{\\alpha}(G)$ and let\n$\\mathbf{x}=\\left(x_{v}\\right)_{v \\in V(G)}$ be the Perron vector of $A_{\\alpha}(G)$ with respect to $\\rho_{\\alpha}$. Clearly, $x_{v_{1}}=\\cdots=x_{v_{s-3}}$ and $x_{w_{1}}=x_{w_{2}}$ by the symmetry. Choose a vertex $z_{1} \\in V(H)$ such that\n$$x_{z_{1}}=\\max _{v \\in V(H)} x_{v}.$$\nSince $\\Delta(H) \\leq t-1$, we have $d_{G}(z_{1})=d_{H}(z_{1})+s-1\\leq s+t-2$.\n\t\n{\\flushleft\\bf Case 1. $s=3$.}\nThen\n$\nh(x)=x^2-(\\alpha n+t) x+(\\alpha(n-2)+1)(2\\alpha+t-1)-2(1-\\alpha)^2(n-3)$.\nBy eigenequations of $A_{\\alpha}(G)$ on $w_{1}$ and $z_{1}$, we have\n$$\n\\begin{aligned}\n\\rho_{\\alpha}x_{w_{1}}&=\\alpha(n-2)x_{w_{1}}+(1-\\alpha)\\sum_{v \\in V(H)} x_{v}\\leq\\alpha(n-2)x_{w_{1}}+(1-\\alpha)(n-2)x_{z_{1}},\\\\\n\\rho_{\\alpha}x_{z_{1}}&=\\alpha d_{G}(z_{1})x_{z_{1}}+2(1-\\alpha)x_{w_{1}}+(1-\\alpha)\\sum_{v \\in N_{H}(z_{1})} x_{v}\n\\\\&\\leq ((t+1)\\alpha+(1-\\alpha)(t-1) )x_{z_{1}}+2(1-\\alpha)x_{w_{1}},\n\\end{aligned}\n$$\nand thus\n\\begin{equation}\\label{equ::2_}\n(\\rho_{\\alpha}-\\alpha(n-2)) x_{w_{1}} \\leq (1-\\alpha)(n-2)x_{z_{1}},\n\\end{equation}\n\\begin{equation}\\label{equ::3_}\n(\\rho_{\\alpha}-(t+1)\\alpha-(1-\\alpha)(t-1) ) x_{z_{1}} \\leq 2(1-\\alpha)x_{w_{1}}.\n\\end{equation}\nNotice that $n \\geq 2s-3+\\frac{t-s+4}{\\alpha}=3+\\frac{t+1}{\\alpha}$. Then we obtain\n\\begin{equation}\\label{equ::2-}\n\\begin{aligned}\n\\rho_{\\alpha}&\\geq\\rho_{\\alpha}(K_{2,n-2})=\\dfrac{\\alpha n+\\sqrt{(\\alpha n)^2+8(n-2)(1-2\\alpha)}}{2}\n=\\dfrac{\\alpha n+\\sqrt{(\\alpha (n-4))^2+8(\\alpha - 1)^2 (n - 2)}}{2}\\\\&\n>\\dfrac{\\alpha n+\\alpha (n-4)}{2}\n=\\alpha (n-2)\n\\geq\\alpha \\left( \\left( 3+\\frac{t+1}{\\alpha}\\right) -2\\right)\n>(t+1)\\alpha+(1-\\alpha)(t-1).\n\\end{aligned}\n\\end{equation}\nNow, we can multiply (\\ref{equ::2_})-(\\ref{equ::3_}), obtaining\n$$\n\\rho_{\\alpha}^2 -(\\alpha n + t - 1)\\rho_{\\alpha} +\\alpha nt + 3 \\alpha n - 2 \\alpha t - 6\\alpha - 2n + 4 \\leq 0.\n$$\nThis implies that $\\rho_{\\alpha}$ is no more than the largest root of $g(x)=0$, where\n$$g(x)=x^2 -(\\alpha n + t - 1)x +\\alpha nt + 3 \\alpha n - 2 \\alpha t - 6\\alpha - 2n + 4 .$$\nIf $\\rho_{\\alpha}$ is equal to the largest real root of $g(x)=0$, then equalities in (\\ref{equ::2_})-(\\ref{equ::3_}) hold. Therefore, for any vertex $z \\in V(H)$, we have $x_{z}=x_{z_{1}}$ and\n$$\n\\begin{aligned}\n\\rho_{\\alpha}-\\alpha d_{G}(z)x_{z}\n&=2(1-\\alpha)x_{w_{1}}+(1-\\alpha)\\sum_{v \\in N_{H}(z)} x_{v}\\\\\n&\\leq 2(1-\\alpha)x_{w_{1}}+(1-\\alpha)(t-1)x_{z_{1}}=\\rho_{\\alpha}-\\alpha d_{G}(z_{1})x_{z_{1}},\n\\end{aligned}\n$$\nwhich implies $d_{G}(z)=s+t-2$, that is, $d_{H}(z)=t-1$. Hence, $H$ is a $(t-1)$-regular graph, and so $G\\cong G^{\\prime}$.\nFurthermore, we see that $g(x)=h(x)+x-(t+1-2\\alpha(1-\\alpha))$. By (\\ref{equ::2-}), we have\n$\\rho_{\\alpha}>\\alpha \\left( \\left( 3+\\frac{t+1}{\\alpha}\\right) -2\\right)=t+1+\\alpha.$\nThus,\n$h(\\rho_{\\alpha})=-(\\rho_{\\alpha}-(t+1-2\\alpha(1-\\alpha)))\\leq-(\\rho_{\\alpha}-(t+1))<0$.\nTherefore, $\\rho_{\\alpha}$ is less than the largest root of $h(x)=0$.\n\t\n{\\flushleft\\bf Case 2. $s\\geq4$.}\nBy eigenequations of $A_{\\alpha}(G)$ on $v_{1}$, $w_{1}$ and $z_{1}$, we have\n\\begin{equation*}\\begin{array}{ll}\n(\\rho_{\\alpha}-\\alpha(n-1)-(1-\\alpha)(s-4))x_{v_{1}}\n&=2(1-\\alpha)x_{w_{1}}+(1-\\alpha)\\sum_{v \\in V(H)} x_{v}\\\\\n&\\leq2(1-\\alpha)x_{w_{1}}+(1-\\alpha)(n-s+1)x_{z_{1}},\n\\end{array}\n\\end{equation*}\n\\begin{equation*}\\begin{array}{ll}\n(\\rho_{\\alpha}-\\alpha(n-2))x_{w_{1}}\n&=(1-\\alpha)(s-3) x_{v_{1}}+(1-\\alpha)\\sum_{v \\in V(H)} x_{v}\\\\\n&\\leq(1-\\alpha)(s-3) x_{v_{1}}+(1-\\alpha)(n-s+1)x_{z_{1}},\n\\end{array}\n\\end{equation*}\n\\begin{equation*}\\begin{array}{ll}\n(\\rho_{\\alpha}-\\alpha(s+t-2))x_{z_{1}}\n&\\leq (\\rho_{\\alpha}-\\alpha d_{G}(z_{1})x_{z_{1}}\\\\\n&=(1-\\alpha)(s-3) x_{v_{1}}+2(1-\\alpha)x_{w_{1}}+(1-\\alpha)\\sum_{v \\in N_{H}(z_{1})} x_{v}\\\\\n&\\leq(1-\\alpha)(s-3) x_{v_{1}}+2(1-\\alpha)x_{w_{1}}+(1-\\alpha)(t-1)x_{z_{1}},\n\\end{array}\n\\end{equation*}\n\nthat is,\n\\begin{equation}\\label{equ::1}\n(\\rho_{\\alpha}-\\alpha(n-1)-(1-\\alpha)(s-4)) x_{v_{1}} \\leq2(1-\\alpha)x_{w_{1}}+(1-\\alpha)(n-s+1)x_{z_{1}},\n\\end{equation}\n\\begin{equation}\\label{equ::2}\n(\\rho_{\\alpha}-\\alpha(n-2)) x_{w_{1}} \\leq (1-\\alpha)(s-3) x_{v_{1}}+(1-\\alpha)(n-s+1)x_{z_{1}},\n\\end{equation}\n\\begin{equation}\\label{equ::3}\n(\\rho_{\\alpha}-\\alpha(s+t-2)-(1-\\alpha)(t-1)) x_{z_{1}} \\leq (1-\\alpha)(s-3) x_{v_{1}}+2(1-\\alpha)x_{w_{1}}.\n\\end{equation}\nNotice that and $G$ contains $(K_{s-1}-e) \\vee \\overline{K}_{n-s+1}$ as a subgraph. By Lemma \\ref{lem::2.5}, we have\n$$\n\\rho_{\\alpha}\\geq \\rho_{\\alpha}\\left((K_{s-1}-e) \\vee \\overline{K}_{n-s+1} \\right)>\\alpha(n-1)+(1-\\alpha)(s-4).\n$$\nSince $0<\\alpha<1$ and $s\\geq4$, we have\n\\begin{equation}\\label{equ::4-}\n\\begin{aligned}\n\\rho_{\\alpha}\\geq \\alpha(n-1)+(1-\\alpha)(s-4)>\\alpha(n-2).\n\\end{aligned}\n\\end{equation}\nRecall that $n \\geq 2s-3+\\frac{t-s+4}{\\alpha}$. Hence, we get that\n\\begin{equation}\\label{equ::4--}\n\\begin{aligned}\n\\rho_{\\alpha}\\geq \\alpha(n-1)+(1-\\alpha)(s-4)>\\alpha(s+t-2)+(1-\\alpha)(t-1).\n\\end{aligned}\n\\end{equation}\nLet $A=1$; $B=-(2\\alpha n + s + t - 5)$; $C=a^2 n^2 + 3\\alpha ns + 2\\alpha nt - \\alpha s^2 - \\alpha st - 8\\alpha n + 2\\alpha s + \\alpha t - ns + s^2 + st - 3\\alpha + n - 5s - 4t + 11$;\n$D= 10 + 4n - 6t - 8s - 6\\alpha + 2s^2 + 7\\alpha ns - 2\\alpha^2 ns - 2\\alpha^2 n^2 s + 4\\alpha nt - 2\\alpha st -\\alpha n^2+ 4\\alpha t + 3\\alpha^2 n + 3\\alpha^2 n^2 - 2\\alpha s^2 + 6\\alpha s - 2ns - 13\\alpha n + 2st + \\alpha n^2s + \\alpha^2 ns^2 - \\alpha ns^2 - \\alpha^2 n^2 t + \\alpha^2 nst - \\alpha nst - \\alpha^2 nt$.\nNow, we can multiply (\\ref{equ::1})-(\\ref{equ::3}), obtaining\n$A\\rho_{\\alpha}^{3}+B\\rho_{\\alpha}^{2}+C\\rho_{\\alpha}+D \\leq 0$.\nThis implies that $\\rho_{\\alpha}$ is no more than the largest root of $f(x)=0$, where\n$$\nf(x)=Ax^{3}+Bx^{2}+Cx+D.\n$$\nIf $\\rho_{\\alpha}$ is equal to the largest real root of $f(x)=0$, then equalities in (\\ref{equ::1})-(\\ref{equ::3}) hold. Therefore, for any vertex $z \\in V(H)$, we have $x_{z}=x_{z_{1}}$ and\n$$\n\\begin{aligned}\n\\rho_{\\alpha}-\\alpha d_{G}(z)x_{z}\n&=(1-\\alpha)(s-3) x_{v_{1}}+2(1-\\alpha)x_{w_{1}}+(1-\\alpha)\\sum_{v \\in N_{H}(z)} x_{v}\\\\\n&\\leq(1-\\alpha)(s-3) x_{v_{1}}+2(1-\\alpha)x_{w_{1}}+(1-\\alpha)(t-1)x_{z_{1}}=\\rho_{\\alpha}-\\alpha d_{G}(z_{1})x_{z_{1}},\n\\end{aligned}\n$$\nwhich implies $d_{G}(z)=s+t-2$, that is, $d_{H}(z)=t-1$. Hence, $H$ is a $(t-1)$-regular graph, and so $G\\cong G^{\\prime}$. Moreover, let\n\\begin{equation*}\n\\begin{aligned}\ng_{1}(x)=&x^2-(\\alpha n+s+t-3) x+ (\\alpha(n-s+1)+s-2)(\\alpha(s-1)+t-1)\\\\&-(1-\\alpha)^2(s-1)(n-s+1).\n\\end{aligned}\n\\end{equation*}\nwe find that\n$$\nf(x)=(x-(\\alpha n-2))g_{1}(x)+2(1-\\alpha)(x-((\\alpha-1) n + s + t - 2)).\n$$\nNote that $0<\\alpha<1$. By (\\ref{equ::4-}) we have $\\rho_{\\alpha}>\\alpha(n-2)\\geq\\alpha n-2,$\nand by (\\ref{equ::4--}) we get that\n$$\n\\begin{aligned}\n\\rho_{\\alpha}-((\\alpha-1) n + s + t - 2)\n&>\\alpha(s+t-2)+(1-\\alpha)(t-1)-((\\alpha-1) n + s + t - 2)\\\\\n&=(1-\\alpha)(n-s+1)>0.\n\\end{aligned}\n$$\nIt follows that\n$$\nh(\\rho_{\\alpha})=-\\dfrac{2(1-\\alpha)(\\rho_{\\alpha}-((\\alpha-1) n + s + t - 2))}{\\rho_{\\alpha}-(\\alpha n-2)}<0,\n$$\nTherefore, $\\rho_{\\alpha}$ is less than the largest root of $g_{1}(x)=0$. That is,\n$\n\\rho_{\\alpha}<\\dfrac{\\alpha n+s+t-3+\\sqrt{R}}{2},\n$\nwhere\n$$\n\\begin{aligned}\nR=&(\\alpha n+s+t-3)^{2}-4\\left( (\\alpha(n-s+1)+s-2)(\\alpha(s-1)+t-1)-(1-\\alpha)^2(s-1)(n-s+1)\\right)\\\\\n=&(\\alpha n +(2C-1)(s + t) + 3)^2-4(C(C-1)(s^2 + 2st+t^2)+(C\\alpha-1)ns + C\\alpha nt +\\alpha ns +\\\\\n&3s(C-1) +(3C-2)t + \\alpha (2s + t) +(1-\\alpha)(t+s)s + n +(3-\\alpha))\\\\\n<&(\\alpha n +(2C-1)(s + t) + 3)^2,\n\\end{aligned}\n$$\nsince $0<\\alpha<1$ and $C\\geq\\frac{1}{\\alpha}$.\nHence, we have\n$$\n\\begin{aligned}\n\\rho_{\\alpha}<\\dfrac{\\alpha n+s+t-3+\\alpha n +(2C-1)(s + t) + 3}{2}=\\alpha n+C(s+t).\n\\end{aligned}\n$$\nNow we see that\n$$\nf(x)=(x-(\\alpha n-2))h(x)+2(1-\\alpha)(x-((\\alpha-1) n + s + t - 2))-(x-(\\alpha n-2))(1-\\alpha)^{2}(s-1).\n$$\nNote that $n\\geq\\frac{(1-\\alpha)(s-1)(C(s+t)+2)}{2}+s+t$. Then we obtain\n$$\n\\begin{aligned}\n&2(1-\\alpha)(\\rho_{\\alpha}-((\\alpha-1) n + s + t - 2))-(\\rho_{\\alpha}-(\\alpha n-2))(1-\\alpha)^{2}(s-1)\\\\\n=&(1-\\alpha)\\left( 2(n-s-t)+(\\rho_{\\alpha}-(\\alpha n-2))\\left( -(1-\\alpha)(s-1)+2\\right) \\right) \\\\\n>&(1-\\alpha)\\left( 2(n-s-t)-(1-\\alpha)(s-1)(\\rho_{\\alpha}-(\\alpha n-2)) \\right) \\\\\n\\geq&(1-\\alpha)\\left( 2(n-s-t)-(1-\\alpha)(s-1)((\\alpha n+C(s+t))-(\\alpha n-2)) \\right) \\\\\n=&(1-\\alpha)\\left( 2(n-s-t)-(1-\\alpha)(s-1)(C(s+t)+2) \\right) \\\\\n\\geq&(1-\\alpha)\\left( 2\\left( \\frac{(1-\\alpha)(s-1)(C(s+t)+2)}{2}+s+t-s-t\\right) -(1-\\alpha)(s-1)(C(s+t)+2) \\right) \\\\\n=&0.\n\\end{aligned}\n$$\nIt follows that\n$$\nh(\\rho_{\\alpha})=-\\dfrac{2(1-\\alpha)(x-((\\alpha-1) n + s + t - 2))-(x-(\\alpha n-2))(1-\\alpha)^{2}(s-1)}{\\rho_{\\alpha}-(\\alpha n-2)}<0,\n$$\nTherefore, $\\rho_{\\alpha}$ is less than the largest root of $h(x)=0$.\n\\end{proof}\n\n\\begin{lem}\\label{lem::2.7}\nLet $0<\\alpha <1$, $2 \\leq s \\leq t$ and $G$ be a $K_{s,t}$-minor free graph of sufficiently large order $n$ with maximum $A_\\alpha$-spectral radius. Then $G$ contains a vertex set $K=\\left\\{v_{1}, v_{2}, \\ldots, v_{s-1}\\right\\}$ such that $d_{G}\\left(v_{i}\\right)=n-1$ for $i \\in\\{1,2, \\ldots, s-1\\}$.\n\\end{lem}\n\n\\begin{proof}\nLet\n\\begin{equation*}\n\\begin{aligned}\nh(x)=&x^2-(\\alpha n+s+t-3) x+\n(\\alpha(n-s+1)+s-2)(\\alpha(s-1)+t-1)\\\\&-(1-\\alpha)^2(s-1)(n-s).\n\\end{aligned}\n\\end{equation*}\nNote that $F_{s, t}(n)$ is a $K_{s, t}$-minor free graph. Then we have $\\rho_{\\alpha}(G) \\geq \\rho_{\\alpha}(F_{s, t}(n))$. Furthermore, by Lemma \\ref{lem::2.3}, we get that $\\rho_{\\alpha}(G)$ is larger than the largest root of $h(x)=0$. By Lemma \\ref{lem::2.1}, $G$ contains a vertex set $K$ of size $s-1$ such that $d_{G-K}\\left(v\\right)=n-s+1$ for any vertex $v \\in K$. Now, we need to show that $K$ induces a clique. Otherwise, we have $s\\geq3$ and $G[K] \\subseteq K_{s-1}-e$. Let $H=G-K$, i.e., $G=G[K] \\vee H$. Since $G$ is a $K_{s, t}$-minor free graph, we have $\\Delta(H)\\leq t-1$. If $G[K] \\cong K_{s-1}-e$, by Lemma \\ref{lem::2.6}, we obtain $\\rho_{\\alpha}(G)$ is less than the largest root of $h(x)=0$, a contradiction. Therefore, $G[K]$ is a proper subgraph of $K_{s-1}-e$. Let $G^{\\prime}$ be the graph obtained from $G$\nby adding edges to $G[K]$ to make it a graph $K_{s-1}-e$. Then we have $\\rho_{\\alpha}(G)<\\rho_{\\alpha}(G^{\\prime})$ by Lemma \\ref{lem::2.2}. However, from Lemma \\ref{lem::2.6}, $\\rho_{\\alpha}(G^{\\prime})$ is\nless than the largest root of $h(x)=0$, which is also a contradiction.\n\\end{proof}\n\n\\begin{lem}(\\cite{D.L})\\label{lem::2.4}\nLet $t \\geq 3$ and $n \\geq t+2$. If $G$ is an $n$-vertex connected graph with no $K_{1, t}$-minor, then $e(G) \\leq \\binom{t}{2} +n-t$, and this is best possible for all $n, t$.\n\\end{lem}\n\n\n\nFor $2\\leq s\\leq t$, we say a graph $G$ has the $(s, t)$-property, if $G$ is $K_{a, b}$-minor free for any two positive integers $a, b$ with $a+b=t+1$ and $a \\leq \\min \\left\\{s,\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor\\right\\}$.\nThe following Lemma gives a equivalent\ncondition whether a graph $G$ has $K_{s, t}$-minor or not.\n\n\\begin{lem}(Lemma 2.3, \\cite{M.Q})\\label{lem::2.10}\nLet $2\\leq s\\leq t$ and $G$ be a graph with a clique dominating set $K$ of size $s-1$. Then $G$ is $K_{s, t}$-minor free if and only if $G-K$ has the $(s, t)$-property.\n\\end{lem}\n\nRecall that $H_{s, t}:\\cong(\\beta-1)K_{1,s} \\cup K_{1,\\alpha}$, where $1\\leq s\\leq t$, $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor$ and $\\alpha=t-(\\beta-1)(s+1)\\geq s$,\n$S^{1}\\left(\\overline{H_{s, t}}\\right)$ is a graph obtained from a graph $\\overline{H_{s, t}}$ by subdividing once of an edge $uv$ with minimum degree sum $d_{G}(u)+d_{G}(v)$ and $H^{\\star}$ is the Petersen graph.\nThe following result is from Theorem 3.1 and the proofs of Claims $3.8$-$3.9$ in \\cite{M.Q}.\n\\begin{lem}(\\cite{M.Q})\\label{lem::2.16}\nLet $2\\leq s\\leq t$ and $t\\geq4$. Then\n\t\n(i) $\\overline{H_{s,t}}$ and $S^{1}\\left(\\overline{H_{s, t}}\\right)$ have the $(s, t)$-property. Moreover, if $\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor=2$, then $\\pi\\left(S^{1}\\left(\\overline{H_{s, t}}\\right)\\right)=(t-1, \\ldots, t-1, t-s, s+1,2)$.\n\n(ii) $\\overline{H^{\\star}}$ has the $(s, t)$-property for $t=8$.\n\\end{lem}\n\\begin{lem}(Lemma 3.1, \\cite{M.Q})\\label{lem::2.11}\nLet $2\\leq s\\leq t$, $t\\geq4$, $\\gamma = \\min \\left\\{s,\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor\\right\\}$ and $G$ be a connected graph with $|G|=t+1$. Then $G$ has the $(s, t)$-property if and only if each component of $\\overline{G}$ has at least $\\gamma+1$ vertices.\n\\end{lem}\n\nThe following result comes from the proof of Lemma 3.2 in \\cite{M.Q}.\n\\begin{lem}(\\cite{M.Q})\\label{lem::2.12}\nLet $2\\leq s\\leq t$, $t\\geq4$, $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor$ and $G$ be a connected graph with $|G|=t+1$. If $G$ is an edge-maximal graph with the $(s, t)$-property, then $e(G)=\\binom{t}{2}+\\beta-1$ and $\\overline{G}$ is a forest with $\\beta$ components.\n\\end{lem}\n\n\\begin{lem}(Lemma 3.3, \\cite{M.Q})\\label{lem::2.13}\nLet $G$ be a graph with $vw \\in E(G)$ and $uw \\notin E(G)$. If $d_{G}(u) \\geq d_{G}(v)$, then $\\pi(G) \\prec \\pi(G-\\{v w\\}+\\{u w\\})$ and $\\pi(G) \\neq \\pi(G-\\{v w\\}+\\{u w\\})$.\n\\end{lem}\n\n\\begin{lem}(Lemma 3.4, \\cite{M.Q})\\label{lem::2.15}\nLet $2\\leq s\\leq t$, $t\\geq4$, $\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor \\leq 2$ and $G$ be a connected graph with $|G|=t+2$. If $G$ has the $(s, t)$-property, then $e(G) \\leq \\binom{t}{2} +2$, and if equality holds, then $\\overline{G}$ is isomorphic to either the Petersen graph $H^{\\star}$ or some $H_{a, b, c}$ (see Fig. \\ref{fig-1}), where $a+b+c=t-1$.\n\\end{lem}\n\n\\begin{figure}[hbtp]\n\\centering\n\\begin{tikzpicture}[x=0.9mm, y=0.9mm, inner xsep=0pt, inner ysep=0pt, outer xsep=0pt, outer ysep=0pt]\n\\path[line width=0mm] (25.74,33.08) rectangle +(155.86,74.28);\n\\definecolor{L}{rgb}{0,0,0}\n\\path[line width=0.30mm, draw=L] (60.31,92.82) circle (12.55mm);\n\\path[line width=0.30mm, draw=L] (80.50,57.74) circle (12.55mm);\n\\path[line width=0.30mm, draw=L] (40.29,56.89) circle (12.55mm);\n\\definecolor{F}{rgb}{0,0,0}\n\\path[line width=0.30mm, draw=L, fill=F] (57.23,71.77) circle (1.00mm);\n\\path[line width=0.30mm, draw=L, fill=F] (63.22,71.09) circle (1.00mm);\n\\path[line width=0.30mm, draw=L, fill=F] (59.28,66.98) circle (1.00mm);\n\\path[line width=0.30mm, draw=L] (63.22,71.77) -- 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width=0.30mm, draw=L] (139.00,85.80) -- (130.10,67.15);\n\\path[line width=0.30mm, draw=L] (155.59,89.74) -- (169.28,67.32);\n\\path[line width=0.30mm, draw=L] (153.71,85.63) -- (165.00,67.15);\n\\path[line width=0.30mm, draw=L] (130.44,50.56) -- (162.26,50.21);\n\\path[line width=0.30mm, draw=L] (133.52,54.83) -- (158.67,55.01);\n\\path[line width=0.30mm, draw=L] (145.84,72.29) -- (142.08,84.43);\n\\path[line width=0.30mm, draw=L] (146.01,72.46) -- (148.75,84.43);\n\\path[line width=0.30mm, draw=L] (142.25,66.47) -- (131.81,64.42);\n\\path[line width=0.30mm, draw=L] (142.42,65.78) -- (134.89,58.08);\n\\path[line width=0.30mm, draw=L] (150.80,66.98) -- (161.24,65.10);\n\\path[line width=0.30mm, draw=L] (150.80,66.64) -- (159.01,59.80);\n\\path[line width=0.30mm, draw=L] (146.01,72.11) -- (142.59,67.15);\n\\path[line width=0.30mm, draw=L] (146.01,72.11) -- (150.29,66.98);\n\\draw(142.57,91.87) node[anchor=base west]{\\fontsize{14.23}{17.07}\\selectfont $K_a$};\n\\draw(121.06,54.28) 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and $2\\leq s\\leq t$.\nIn the following, we always assume that $G^*$ is the $A_\\alpha$-spectral extremal graph.\nIn this section, we will prove that $G^*$ has a property of\nlocal edge maximality. Meanwhile, we will apply the double eigenvectors transformation technique to the $A_{\\alpha}(G)$-matrix and prove that\n$G^*$ has a property of local degree sequence majorization.\n\n\nLet $\\mathbf{x}=\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right)^{T}$ and $\\mathbf{y}=\\left(y_{1}, y_{2}, \\ldots, y_{n}\\right)^{T}$ be two non-increasing real vectors. We say $\\mathbf{x}$ is weakly majorized by $\\mathbf{y}$, denoted by $\\mathbf{x} \\prec_{w} \\mathbf{y}$, if and only if $\\sum_{i=1}^{k} x_{i} \\leq \\sum_{i=1}^{k} y_{i}$ for $k=1,2, \\ldots, n$. We say $\\mathbf{x}$ is majorized by $\\mathbf{y}$, denoted $\\mathbf{x} \\prec \\mathbf{y}$, if and only if $\\mathbf{x} \\prec_{w} \\mathbf{y}$ and $\\sum_{i=1}^{n} x_{i}=\\sum_{i=1}^{n} y_{i}$.\n\n\\begin{lem}(\\cite{Lin})\\label{lem::2.8}\nLet $\\mathbf{x}, \\mathbf{y} \\in R^{n}$ be two non-negative and non-increasing vectors. If $\\mathbf{x} \\prec_{w} \\mathbf{y}$, then $\\|\\mathbf{x}\\|_{k} \\leq\\|\\mathbf{y}\\|_{k}$ for $k>1$, with equality if and only if $\\mathbf{x}=\\mathbf{y}$.\n\\end{lem}\n\nThe following lemma is from Exercises $5(i)$ on page $74$ of \\cite{Zhan}.\n\\begin{lem}(\\cite{Zhan})\\label{lem::2.9}\nLet $\\mathbf{x}, \\mathbf{y}, \\mathbf{z} \\in R^{n}$ be three non-increasing vectors. If $\\mathbf{x} \\prec \\mathbf{y}$, then $\\mathbf{x}^{T} \\cdot \\mathbf{z} \\leq \\mathbf{y}^{T} \\cdot \\mathbf{z}$.\n\\end{lem}\n\nSet for short $\\rho_{\\alpha} = \\rho_{\\alpha} (G^{*}) $ and let $\\mathbf{X}=(x_{v})_{v \\in V(G^{*})} \\in R^{n}$ be a Perron eigenvector of $A_{\\alpha}(G^{*})$ corresponding to $\\rho_{\\alpha}$. By Lemma \\ref{lem::2.7},\n$G^*$ contains a clique dominating set $K$ of size $s-1$. We immediately get that $G^{*}-K$ has the $(s, t)$-property from Lemma \\ref{lem::2.10}. So, $\\Delta(G^{*}-K)b.$ Then $a x_{2}>b x_{1}$ and $a x_{2}^{2}>b x_{1}^{2}$.\n\\end{lem}\n\n\\begin{proof}\nSince $H\\subseteq G^{*}-K$, we have $\\Delta(H)\\leq\\Delta(G^{*}-K) < t$.\nFor each $v \\in V(H)$, we see that\n\\begin{equation}\\label{equ::a}\ns-1\\leq d_{G^{*}}(v)=d_{H}(v)+s-1 < t+s-1.\n\\end{equation}\nNote that $x_{0}=\\sum\\limits_{v\\in K}x_{v}$. Therefore, we obtain $\\rho_{\\alpha} x_{1}<(\\alpha(t+s-1)+(1-\\alpha)t)x_{1}+(1-\\alpha)x_{0}$ and $\\rho_{\\alpha} x_{2} \\geq \\alpha(s-1)x_{2}+(1-\\alpha)x_{0}$, and thus\n\\begin{equation}\\label{equ::4}\nx_{1}<\\frac{(1-\\alpha)x_{0}}{\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)} \\quad \\text { and } \\quad x_{2} \\geq \\frac{(1-\\alpha)x_{0}}{\\rho_{\\alpha}-\\alpha(s-1)}.\n\\end{equation}\nSince $\\rho_{\\alpha}\\geq \\alpha(n-1)+(1-\\alpha)(s-2)$, $0<\\alpha<1$, $n$ is sufficiently large and $a, b, s, t$ are constants, we can easily get that\n$$\nax_{2}-bx_{1}>(1-\\alpha)x_{0}(\\frac{a}{\\rho_{\\alpha}-\\alpha(s-1)}-\\frac{b}{\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)})>0.\n$$\nSimilarly, we can obtain $a x_{2}^{2}>b x_{1}^{2}$.\n\\end{proof}\n\nThe following lemma implies that $G^{*}$ has a property of local edge maximality.\n\\begin{lem}\\label{lem::3.2}\nLet $H$ consist of some components of $G^{*}-K$ with $|H| \\leq N$ (a constant) and $H^{\\prime}$ be a graph with $V(H^{\\prime})=V(H)$. If $H^{\\prime}$ has the $(s, t)$-property, then $e(H^{\\prime}) \\leq e(H)$.\n\\end{lem}\n\n\\begin{proof}\nSuppose to the contrary that $e(H^{\\prime})>e(H)$. Let $G^{\\prime}=G^{*}-E(H)+E\\left( H^{\\prime}\\right) $. By Lemma \\ref{lem::2.10}, $G^{\\prime}$ is $K_{s,t}$-minor free since $H^{\\prime}$ has the $(s, t)$-property. Now, by Lemma \\ref{lem::3.1} we obtain\n$$\n\\begin{aligned}\n\\rho_{\\alpha}(G^{\\prime})-\\rho_{\\alpha} & \\geq \\mathbf{X}^{T}(A_{\\alpha}(G^{\\prime})-A_{\\alpha}(G^{*})) \\mathbf{X}\\\\\n&=\\sum_{u v \\in E(H^{\\prime})}(\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{v}+\\alpha x_{v}^{2})-\\sum_{u v \\in E(H)} (\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{v}+\\alpha x_{v}^{2}) \\\\\n& \\geq 2 e(H^{\\prime}) x_{2}^{2}-2 e(H) x_{1}^{2}>0,\n\\end{aligned}\n$$\nwhich contradicts the maximality of $\\rho_{\\alpha}$.\n\\end{proof}\n\n\\begin{lem}\\label{lem::3.3}\nLet $H$ consist of some connected components of $G^{*}-K$ with $|H| \\leq N$ (a constant), $H^{\\prime}$ be a graph with $V\\left(H^{\\prime}\\right)=V(H)$ and $e\\left(H^{\\prime}\\right)=e\\left( H\\right)$. If $H^{\\prime}$ has the $(s, t)$-property, then\n\\begin{equation}\\label{equ::5}\n\\sum_{u v \\in E\\left(H^{\\prime}\\right)} (d_{H}(u)+ d_{H}(v)) \\leq \\sum_{u v \\in E(H)} (d_{H}(u)+ d_{H}(v)).\n\\end{equation}\nMoreover, if (\\ref{equ::5}) holds in equality, then\n\\begin{equation}\\label{equ::6}\n\\sum_{u v \\in E\\left(H^{\\prime}\\right)}(d_{H^{\\prime}}(u)+ d_{H^{\\prime}}(v)) \\leq \\sum_{u v \\in E(H)}(d_{H}(u)+ d_{H}(v)).\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nLet $G^{\\prime}=G^{*}-E(H)+E\\left(H^{\\prime}\\right)$ and $\\rho_{\\alpha}^{\\prime}=\\rho_{\\alpha}(G^{\\prime})$.\nClearly, $G^{\\prime}$ is also $K_{s, t}$-minor free.\nLet $x_{1}=\\max\\limits_{v \\in V(H) }x_{v}$ and $ x_{2}=\\min\\limits_{v \\in V(H) }x_{v}$. Note that $x_{0}=\\sum\\limits_{v\\in K} x_{v}$.\nThen by (\\ref{equ::a}) and the eigen-equations of $A_{\\alpha}(G^{*})$, we have\n\\begin{equation}\\label{equ::8}\n\\frac{(1-\\alpha)(d_{H}(v)x_{2} +x_{0})}{\\rho_{\\alpha}-\\alpha(s-1)}\\leq\\frac{(1-\\alpha)(d_{H}(v)x_{2} +x_{0})}{\\rho_{\\alpha}-\\alpha d_{G^{*}}(v)}\\leq x_{v},\n\\end{equation}\nand\n\\begin{equation}\\label{equ::8''}\n x_{v}\\leq \\frac{(1-\\alpha)(d_{H}(v)x_{1} +x_{0})}{\\rho_{\\alpha}-\\alpha d_{G^{*}}(v)}< \\frac{(1-\\alpha)(d_{H}(v)x_{1} +x_{0})}{\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)},\n\\end{equation}\nfor each vertex $v \\in V(H)$.\nLet\n$a=\\sum\\limits_{u v \\in E(H^{\\prime})} (d_{H}(u)+ d_{H}(v))$,\n$b=\\sum\\limits_{u v \\in E(H)} (d_{H}(u)+ d_{H}(v))$,\n$c_{1}=\\sum\\limits_{u v \\in E(H)} \\left( d_{H}^{2}(u)+ d_{H}^{2}(v)\\right) $,\n$c_{2}=\\sum\\limits_{u v \\in E(H)} d_{H}(u)d_{H}(v)$.\nFirstly, we will show that $a\\leq b$. Suppose to the contrary that $a\\geq b+1$.\nBy Lemma \\ref{lem::3.1}, we obtain $\\left(a-\\frac{1}{2}\\right) x_{2}>b x_{1}$. Moreover, by (\\ref{equ::4}) we have\n$$x_{0} x_{2}-c_{1} x_{1}^{2}>(1-\\alpha)x_{0}^{2}\\left(\\frac{1}{\\rho_{\\alpha}-\\alpha(s-1)}-\\frac{(1-\\alpha)c_{1}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))^{2}} \\right) >0,$$\nsince $\\rho_{\\alpha}\\geq\\alpha(n-1)+(1-\\alpha)(s-2)$, $0<\\alpha<1$, $n$ is sufficiently large, and $s, t, c_{1}$ are constants.\nSimilarly, we can obtain $\\frac{1}{2}x_{0} x_{2}-c_{2} x_{1}^{2}>0.$\n\nNow, we denote\n$$A_{1}=\\sum\\limits_{uv\\in E(H^{\\prime})}( (d_{H}(u)x_{2}+x_{0})^{2}+(d_{H}(v)x_{2}+x_{0})^{2}),\\\\\nA_{2}=\\sum\\limits_{uv\\in E(H)}\\left( (d_{H}(u)x_{1} +x_{0})^{2}+(d_{H}(v)x_{1} +x_{0})^{2}\\right),$$\n$$A_{3}=\\sum\\limits_{uv\\in E(H^{\\prime})}( (d_{H}(u)x_{2}+x_{0})(d_{H}(v)x_{2}+x_{0})) \\ \\ \\mbox{ and} \\ \\ A_{4}=\\sum\\limits_{uv\\in E(H)}\\left( (d_{H}(u)x_{1} +x_{0})(d_{H}(v)x_{1} +x_{0})\\right).\n$$\nBy simple scaling, we see that\n$A_{1}\\geq 2ax_{2}x_{0}+2e(H^{\\prime})x_{0}^{2}$, $A_{2}= 2bx_{1}x_{0}+2e(H)x_{0}^{2}+c_{1}x_{1}^{2}$, $A_{3}\\geq ax_{2}x_{0}+e(H^{\\prime})x_{0}^{2}$ and $A_{4}= bx_{1}x_{0}+e(H)x_{0}^{2}+c_{2}x_{1}^{2}$.\nSince $e\\left(H^{\\prime}\\right)=e\\left( H\\right) $, we get that\n\\begin{align*}\nA_{1}-A_{2}\n&\\geq (2ax_{2}x_{0}+2e(H^{\\prime})x_{0}^{2})-(2bx_{1}x_{0}+2e(H)x_{0}^{2}+c_{1}x_{1}^{2})\\notag\\\\\n&=2ax_{2}x_{0}-2bx_{1}x_{0}-c_{1}x_{1}^{2}=2x_{0}\\left(\\left(a-\\frac{1}{2}\\right) x_{2}-b x_{1}\\right)+(x_{0} x_{2}-c_{1} x_{1}^{2})>0,\n\\end{align*}\nand\n\\begin{align*}\nA_{3}-A_{4}\n&\\geq (ax_{2}x_{0}+e(H^{\\prime})x_{0}^{2})-(bx_{1}x_{0}+e(H)x_{0}^{2}+c_{2}x_{1}^{2})\\notag\\\\\n&=ax_{2}x_{0}-bx_{1}x_{0}-c_{2}x_{1}^{2}=x_{0}\\left(\\left(a-\\frac{1}{2}\\right) x_{2}-b x_{1}\\right)+(\\frac{1}{2}x_{0} x_{2}-c_{2} x_{1}^{2})>0,\n\\end{align*}\nwhich include that $A_{1}>A_{2}$ and $A_{3}>A_{4}$.\nFurthermore, let\n$$B_{1}= \\sum_{uv\\in E(H^{\\prime})}\\left( \\frac{(d_{H}(u)x_{2}+x_{0})^{2}+(d_{H}(v)x_{2}+x_{0})^{2}}{(\\rho_{\\alpha}-\\alpha(s-1))^{2}}\\right)-\\sum_{uv\\in E(H)}\\left(\\frac{(d_{H}(u)x_{1} +x_{0})^{2}+(d_{H}(v)x_{1} +x_{0})^{2}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))^{2}}\\right),$$\nand\n$$\nB_{2}=\\sum_{u v \\in E(H^{\\prime})}\\frac{(d_{H}(u)x_{2}+x_{0})(d_{H}(v)x_{2}+x_{0})}{(\\rho_{\\alpha}-\\alpha(s-1))^{2}}-\\sum_{u v \\in E(H)} \\frac{(d_{H}(u)x_{1} +x_{0})(d_{H}(v)x_{1} +x_{0})}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))^{2}}.$$\nNotice that $\\rho_{\\alpha}\\geq\\alpha(n-1)+(1-\\alpha)(s-2)$, $0<\\alpha<1$, $n$ is sufficiently large. Then from (\\ref{equ::8})-(\\ref{equ::8''}), we have\n\\begin{align*}\n\\rho_{\\alpha}^{\\prime}-\\rho_{\\alpha}\\geq& \\mathbf{X}^{T}(A_{\\alpha}(G^{\\prime})-A_{\\alpha}(G^{*}))\\mathbf{X}\\\\\n=&\\sum_{u v \\in E(H^{\\prime})}(\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{v}+\\alpha x_{v}^{2})-\\sum_{u v \\in E(H)} (\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{v}+\\alpha x_{v}^{2})\\notag\\\\\n=&\\alpha\\left( \\sum_{u v \\in E(H^{\\prime})}( x_{u}^{2}+ x_{v}^{2})-\\sum_{u v \\in E(H)} (x_{u}^{2}+x_{v}^{2})\\right)+2(1-\\alpha)\\left(\\sum_{u v \\in E(H^{\\prime})}x_{u}x_{v}-\\sum_{u v \\in E(H)} x_{u}x_{v} \\right) \\\\\n\\geq&\\alpha(1-\\alpha)^{2}B_{1}+2(1-\\alpha)^{3}B_{2}\\\\\n=&\\alpha(1-\\alpha)^{2}\\left( \\frac{A_{1}}{(\\rho_{\\alpha}-\\alpha(s-1))^{2}}-\\frac{A_{2}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))^{2}}\\right) \\\\\n&+2(1-\\alpha)^{3}\\left( \\frac{A_{3}}{(\\rho_{\\alpha}-\\alpha(s-1))^{2}}-\\frac{A_{4}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))^{2}}\\right) \\\\\n>&0,\\label{equ::6'}\n\\end{align*}\nwhich contradicts the maximality of $\\rho_{\\alpha}$.\nSo (\\ref{equ::5}) holds.\n\t\nWe next show that the inequality (\\ref{equ::6}) holds when $a=b$.\nLet $\\mathbf{Y}=(y_{v})_{v \\in V(G^{*})} \\in R^{n}$ be a Perron vector of $A_{\\alpha}(G^{\\prime})$ corresponding to $\\rho_{\\alpha}^{\\prime}$.\nAssume that $y_{0}=\\sum\\limits_{v \\in K} y_{v}$, $y_{1}=\\max\\limits_{v \\in V\\left(H^{\\prime}\\right)} y_{v}$\n and $y_{2}=\\min\\limits_{v \\in V\\left(H^{\\prime}\\right)} y_{v}$. Similar as (\\ref{equ::4}), we obtain\n\\begin{equation}\\label{equ::14}\ny_{1}<\\frac{(1-\\alpha)y_{0}}{\\rho_{\\alpha}^{\\prime}-(\\alpha(t+s-1)+(1-\\alpha)t)} \\quad \\text { and } \\quad y_{2} \\geq \\frac{(1-\\alpha)y_{0}}{\\rho^{\\prime}_{\\alpha}-\\alpha(s-1)}.\n\\end{equation}\nMoreover, for any $v\\in V(H^{\\prime})$ we have\n\\begin{equation}\\label{equ::8'}\n\\frac{(1-\\alpha)(d_{H^{\\prime}}(v)y_{2} +y_{0})}{\\rho^{\\prime}_{\\alpha}-\\alpha(s-1)}\\leq\\frac{(1-\\alpha)(d_{H^{\\prime}}(v)y_{2} +y_{0})}{\\rho^{\\prime}_{\\alpha}-\\alpha d_{G^{\\prime}}(v)}\\leq y_{v},\n\\end{equation}\nand\n\\begin{equation}\\label{equ::8'''}\ny_{v}\\leq \\frac{(1-\\alpha)(d_{H^{\\prime}}(v)y_{1} +y_{0})}{\\rho^{\\prime}_{\\alpha}-\\alpha d_{G^{\\prime}}(v)}<\\frac{(1-\\alpha)(d_{H^{\\prime}}(v)y_{1} +y_{0})}{\\rho^{\\prime}_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)}.\n\\end{equation}\nNow, we let\n\\begin{align*}\na^{\\prime}&=\\sum\\limits_{u v \\in E(H^{\\prime})} (d_{H^{\\prime}}(u)+d_{H^{\\prime}}(v)),&\nb^{\\prime}&=\\sum\\limits_{u v \\in E(H)} (d_{H^{\\prime}}(u)+d_{H^{\\prime}}(v)),\\\\\nc^{\\prime}_{1}&=\\sum\\limits_{u v \\in E(H)}(d_{H}(u) d_{H^{\\prime}}(v)+d_{H}(v) d_{H^{\\prime}}(u)),&\nc^{\\prime}_{2}&=\\sum\\limits_{u\\in V(H)} d_{H}^{2}(u)d_{H^{\\prime}}(u).\n\\end{align*}\nSince $V\\left(H^{\\prime}\\right)=V(H)$, we have\n\\begin{align*}a&=\\sum_{uv \\in E\\left(H^{\\prime}\\right)} (d_{H}(u)+d_{H}(v))=\\sum_{u \\in V\\left(H^{\\prime}\\right)} d_{H}(u) d_{H^{\\prime}}(u)=\\sum_{u \\in V(H)} d_{H}(u) d_{H^{\\prime}}(u)\\\\\n&=\\sum_{u v \\in E(H)} (d_{H^{\\prime}}(u)+d_{H^{\\prime}}(v))=b^{\\prime}.\n\\end{align*}\nFurthermore, we see that\n$\\sum\\limits_{u \\in V\\left(H^{\\prime}\\right)} d_{H^{\\prime}}^{2}(u)=\\sum\\limits_{uv \\in E\\left(H^{\\prime}\\right)} (d_{H^{\\prime}}(u)+d_{H^{\\prime}}(v))=a^{\\prime}$\nand $\\sum\\limits_{u \\in V\\left(H\\right)} d_{H}^{2}(u)= \\sum\\limits_{uv \\in E\\left(H\\right)} (d_{H}(u)+d_{H}(v))=b$.\nNow we denote\n\\begin{equation*}\n\\begin{aligned}\nA^{\\prime}_{1}=&\\sum_{u v \\in E(H^{\\prime})}((x_{0}+d_{H}(u) x_{2})(y_{0}+d_{H^{\\prime}}(v) y_{2})+(x_{0}+d_{H}(v) x_{2})(y_{0}+d_{H^{\\prime}}(u) y_{2}))\\\\\nA^{\\prime}_{2}=&\\sum_{u\\in V(H^{\\prime})} d_{H^{\\prime}}(u) \\cdot (d_{H}(u)x_{2} +x_{0}) \\cdot(d_{H^{\\prime}}(u)y_{2} +y_{0})\\\\\nA^{\\prime}_{3}=&\\sum_{u v \\in E(H)}((x_{0}+d_{H}(u) x_{1})\\cdot( y_{0}+d_{H^{\\prime}}(v) y_{1})+(x_{0}+d_{H}(v) x_{1})(y_{0}+d_{H^{\\prime}}(u) y_{1}))\\\\\nA^{\\prime}_{4}=&\\sum_{u\\in V(H)} d_{H}(u) \\cdot (d_{H}(u)x_{1} +x_{0}) \\cdot(d_{H^{\\prime}}(u)y_{1} +y_{0})\n\\end{aligned}\n\\end{equation*}\nBy simple scaling, we see that\n\\begin{align*}\nA^{\\prime}_1\\geq&2e(H^{\\prime})x_{0}y_{0} + ax_{2}y_{0}+a^{\\prime}x_{0}y_{2}, \\ \\ A^{\\prime}_3=2e(H)x_{0}y_{0} + ax_{1}y_{0}+ax_{0}y_{1}+c^{\\prime}_{1}x_{1}y_{1},\\\\\nA^{\\prime}_2\\geq &a^{\\prime}x_{0}y_{2}+ax_{2}y_{0} +2e(H^{\\prime})x_{0}y_{0},\\ \\ A^{\\prime}_4= c^{\\prime}_{2}x_{1}y_{1}+ax_{0}y_{1}+ax_{1}y_{0}+2e(H)x_{0}y_{0}.\n\\end{align*}\nSuppose that (\\ref{equ::6}) does not hold, then $a^{\\prime} \\geq b+1=a+1$.\nFrom Lemma \\ref{lem::3.1}, we find that $\\left(a+\\frac{1}{2}\\right) y_{2}>a y_{1}$.\nAdditionally, by (\\ref{equ::4}) and (\\ref{equ::14}) we have\n\\begin{equation*}\\label{equ::22}\n\\begin{array}{ll}\n&\\frac{1}{2}x_{0} y_{2}+a y_{0}\\left(x_{2}-x_{1}\\right)-c^{\\prime}_{1} x_{1} y_{1}\\\\\n>&(1-\\alpha)x_{0} y_{0}\\left(\\frac{1}{ 2(\\rho^{\\prime}_{\\alpha}-\\alpha(s-1))}+\\frac{a}{\\rho_{\\alpha}-\\alpha(s-1)}-\\frac{a}{\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)}-\\frac{(1-\\alpha)c^{\\prime}_{1}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))\\left(\\rho^{\\prime}_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)\\right)}\\right)\\\\\n>&0,\n\\end{array}\n\\end{equation*}\nsince $\\rho_{\\alpha}\\geq\\alpha(n-1)+(1-\\alpha)(s-2)$, $\\rho^{\\prime}_{\\alpha}\\geq \\rho_{\\alpha}\\left( K_{s-1} \\vee \\overline{K}_{n-s+1}\\right)\\geq \\alpha(n-1)+(1-\\alpha)(s-2)$, $0<\\alpha<1$, and $n$ is sufficiently large. Similarly, we have $\\frac{1}{2}x_{0} y_{2}+a y_{0}\\left(x_{2}-x_{1}\\right)-c^{\\prime}_{2} x_{1} y_{1}>0,$\nFurthermore, since $e\\left(H^{\\prime}\\right)=e\\left( H\\right)$, we have\n\\begin{align*}\nA^{\\prime}_{1}-A^{\\prime}_{3}\\geq&(2e(H^{\\prime})x_{0}y_{0} + ax_{2}y_{0}+a^{\\prime}x_{0}y_{2})-(2e(H)x_{0}y_{0} +ax_{1}y_{0}+ax_{0}y_{1}+c^{\\prime}_{1}x_{1}y_{1})\\\\\n=& a^{\\prime} x_{0} y_{2}+a x_{2} y_{0}-a x_{0} y_{1}-a x_{1} y_{0}-c^{\\prime}_{1} x_{1} y_{1}\\\\\n\\geq&(a+1) x_{0} y_{2}+a x_{2} y_{0}-a x_{0} y_{1}-a x_{1} y_{0}-c^{\\prime}_{1} x_{1} y_{1} \\\\\n=&x_{0}\\left(\\left(a+\\frac{1}{2}\\right) y_{2}-a y_{1}\\right)+\\left(\\frac{1}{2}x_{0} y_{2}+a\\left(x_{2}-x_{1}\\right) y_{0}-c^{\\prime}_{1} x_{1} y_{1}\\right)\\\\\n>&0,\n\\end{align*}\nand similarly, we obtain $A^{\\prime}_{2}-A^{\\prime}_{4}>0$.\nBy (\\ref{equ::8})-(\\ref{equ::8''}) and (\\ref{equ::8'})-(\\ref{equ::8'''}), we find that\n\\begin{align*}\n\\left(\\rho_{\\alpha}^{\\prime}-\\rho_{\\alpha}\\right) \\mathbf{Y}^{T}\\mathbf{X} =&\\left(A_{\\alpha}\\left(G^{\\prime}\\right) \\mathbf{Y}\\right)^{T}\\mathbf{X}-\\mathbf{Y}^{T}\\left(A_{\\alpha}\\left(G^{*}\\right) \\mathbf{X}\\right)\\\\\n=&\\left( \\sum_{u v \\in E\\left(H^{\\prime}\\right)}(1-\\alpha)\\left( x_{u} y_{v}+x_{v} y_{u}\\right)+\\sum_{u\\in V\\left(H^{\\prime}\\right)} \\alpha d_{H^{\\prime}}(u) x_{u} y_{u}\\right) \\\\\n&-( \\sum_{u v \\in E(H)}(1-\\alpha)\\left( x_{u} y_{v}+x_{v}y_{u}\\right)+\\sum_{u\\in V\\left(H\\right)} \\alpha d_{H}(u) x_{u} y_{u})\\\\\n=&(1-\\alpha)\\left( \\sum_{u v \\in E\\left(H^{\\prime}\\right)}\\left( x_{u} y_{v}+x_{v} y_{u}\\right)-\\sum_{u v \\in E(H)}\\left( x_{u} y_{v}+x_{v}y_{u}\\right)\\right)\\\\\n&+\\alpha\\left( \\sum_{u\\in V\\left(H^{\\prime}\\right)} d_{H^{\\prime}}(u) x_{u} y_{u} - \\sum_{u\\in V\\left(H\\right)}d_{H}(u) x_{u} y_{u})\\right) \\\\\n\\geq&(1-\\alpha)^{3}\\left( \\frac{A^{\\prime}_{1}}{(\\rho_{\\alpha}-\\alpha(s-1))(\\rho^{\\prime}_{\\alpha}-\\alpha(s-1))}\\right.\\\\\n&\\left.-\\frac{A^{\\prime}_{3}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))(\\rho^{\\prime}_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))}\\right) \\\\\n&+\\alpha(1-\\alpha)^{2}\\left( \\frac{A^{\\prime}_{2}}{(\\rho_{\\alpha}-\\alpha(s-1))(\\rho^{\\prime}_{\\alpha}-\\alpha(s-1))}\\right.\\\\\n&\\left.-\\frac{A^{\\prime}_{4}}{(\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))(\\rho^{\\prime}_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t))}\\right) \\\\\n>&0,\n\\end{align*}\nwhich contradicts the maximality of $\\rho_{\\alpha}$. Therefore, (\\ref{equ::6}) holds.\n\\end{proof}\n\nThe following lemma implies that $G^{*}$ has a property of local degree sequence majorization.\n\\begin{lem}\\label{lem::3.4}\nLet $H$ consist of some components of $G^{*}-K$ with $|H| \\leq N$ (a constant), $H^{\\prime}$ be a graph with $V\\left(H^{\\prime}\\right)=V(H)$, $e\\left(H^{\\prime}\\right)=e\\left( H\\right)$ and $H^{\\prime}$ have the $(s, t)$-property. If $\\pi(H) \\prec \\pi\\left(H^{\\prime}\\right)$, then $\\pi(H)=\\pi\\left(H^{\\prime}\\right)$.\n\\end{lem}\n\n\\begin{proof}\nSuppose to the contrary that $\\pi(H) \\neq \\pi\\left(H^{\\prime}\\right)$. Since $V(H)=V\\left(H^{\\prime}\\right)$, we have $\\pi(H)=\\left(d_{1}, d_{2}, \\ldots, d_{|H|}\\right)$ and $\\pi\\left(H^{\\prime}\\right)=\\left(d_{1}^{\\prime}, d_{2}^{\\prime}, \\ldots, d_{|H|}^{\\prime}\\right)$. Let $k=2$ in Lemma \\ref{lem::2.8}, we have\n\\begin{equation}\\label{equ::25}\n\\sum_{i=1}^{|H|} d_{i}^{2}<\\sum_{i=1}^{|H|} d_{i}^{\\prime 2}.\n\\end{equation}\nFurthermore, let $\\mathbf{x}=\\mathbf{z}=\\pi(H)$ and $\\mathbf{y}=\\pi\\left(H^{\\prime}\\right)$ in Lemma \\ref{lem::2.9}, we obtain\n\\begin{equation}\\label{equ::26}\n\\sum_{i=1}^{|H|} d_{i}^{2} \\leq \\sum_{i=1}^{|H|} d_{i} d_{i}^{\\prime}.\n\\end{equation}\nNotice that $\\sum\\limits_{i=1}^{|H|} d_{i}^{2}=\\sum\\limits_{u v \\in E(H)}\\left(d_{H}(u)+d_{H}(v)\\right)$ and\n$$\\sum_{i=1}^{|H|} d_{i} d_{i}^{\\prime}=\\sum_{v \\in V\\left(H^{\\prime}\\right)} d_{H}(v) d_{H^{\\prime}}(v)=\\sum_{u v \\in E\\left(H^{\\prime}\\right)}\\left(d_{H}(u)+d_{H}(v)\\right).$$\nThen by (\\ref{equ::5}) we have $\\sum\\limits_{i=1}^{|H|} d_{i} d_{i}^{\\prime} \\leq \\sum\\limits_{i=1}^{|H|} d_{i}^{2}$. Together with (\\ref{equ::26}), we obtain $\\sum\\limits_{i=1}^{|H|} d_{i} d_{i}^{\\prime}=\\sum\\limits_{i=1}^{|H|} d_{i}^{2}$. Therefore, (\\ref{equ::6}) holds, that is, $\\sum\\limits_{i=1}^{|H|} d_{i}^{\\prime 2}\\leq \\sum\\limits_{i=1}^{|H|} d_{i}^{2}$, which contradicts (\\ref{equ::25}).\n\\end{proof}\n\n\n\n\n\\section{The characterization of the $A_\\alpha$-spectral extremal graph}\nRecall that $G^*$ is a graph with maximum $A_\\alpha$-spectral radius among all $K_{s,t}$-minor free graphs of sufficiently large order $n$, where $0 < \\alpha<1$ and $2\\leq s\\leq t$. Moreover, $\\rho_{\\alpha} = \\rho_{\\alpha}(G^{*})$, $\\mathbf{X}=(x_{v})_{v \\in V(G^{*})} \\in R^{n}$ is a Perron eigenvector of $A_{\\alpha}(G^{*})$ corresponding to $\\rho_{\\alpha}$, $x_{0}=\\sum\\limits_{v\\in K}x_{v}$. In addition,\n$G^*$ contains a clique dominating set $K$ of size $s-1$, $G^{*}-K$ has the $(s, t)$-property and $\\Delta(G^{*}-K)\\rho_{\\alpha}$ by Lemma \\ref{lem::2.2}, a contradiction.\n\n\n{\\flushleft\\bf Case 2. $d=4$.} We have $x_{v_1}=x_{v_4}$ and $x_{v_2}=x_{v_3}$ by symmetry. Let $G^{\\prime}=G^{*}-v_1 v_2+v_2 v_4$. Clearly, $G^{\\prime}$ is $K_{2,3}$-minor free and\n$$\n\\rho_{\\alpha}\\left(G^{\\prime}\\right)-\\rho_{\\alpha} \\geq\\left(\\alpha x^{2}_{v_2}+2(1-\\alpha)x_{v_2} x_{v_4}+\\alpha x^{2}_{v_4}\\right)-\\left(\\alpha x^{2}_{v_1}+2(1-\\alpha)x_{v_1} x_{v_2}+\\alpha x^{2}_{v_2}\\right)=0.\n$$\nIf $\\rho_{\\alpha}\\left(G^{\\prime}\\right)=\\rho_{\\alpha}$, then $\\mathbf{X}$ is also an unit eigenvector corresponding to $\\rho_{\\alpha}\\left(G^{\\prime}\\right)$. Since $v_2$, $v_3$, and $v_4$ are symmetric in $G^{\\prime}$, we have $x_{v_2}=x_{v_3}=x_{v_4}$, which implies that $x_{v_1}=x_{v_2}=x_{v_3}=x_{v_4}$. By eigenequations of $A_{\\alpha}(G^{*})$ on $v_1$ and $v_2$, we have\n$$\n\\rho_{\\alpha} x_{v_1}=2\\alpha x_{v_1}+(1-\\alpha)(x_{v_2}+x_v) \\quad \\text { and } \\quad \\rho_{\\alpha} x_{v_2}=3\\alpha x_{v_2}+(1-\\alpha)(x_{v_1}+x_{v_3}+x_v),\n$$\na contradiction. Thus, $\\rho_{\\alpha}\\left(G^{\\prime}\\right)>\\rho_{\\alpha}$, which is also a contradiction.\n\n\n{\\flushleft\\bf Case 3. $d\\geq5$.} We firstly assume that $d$ is odd, let $d=2j+1$ with $j \\geq 2$. Then we have $x_{v_k}=x_{v_{2j+2-k}}$ for $1 \\leq k \\leq j$ by symmetry. Let $G^{\\prime}=G^{*}-\\left\\{v_{j-1} v_j, v_{j+2} v_{j+3}\\right\\}+\\left\\{v_j v_{j+2}, v_{j-1} v_{j+3}\\right\\}$. Clearly, $G^{\\prime}$ is a $K_{2,3}$-minor free graph and\n$$\n\\begin{aligned}\n\\rho_{\\alpha}\\left(G^{\\prime}\\right)-\\rho_{\\alpha}\\geq& \\sum_{u w \\in E(G^{\\prime})}(\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{w}+\\alpha x_{w}^{2})-\\sum_{u w \\in E(G^{*})} (\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{w}+\\alpha x_{w}^{2}) \\\\\n=&2(1-\\alpha) (x_{v_j} x_{v_{j+2}}+x_{v_{j-1}} x_{v_{j+3}}-x_{v_{j-1}} x_{v_{j}}-x_{v_{j+2}} x_{v_{j+3}}) \\\\\n=& 2(1-\\alpha)( x_{v_{j}}^2+x_{v_{j-1}}^2-2x_{v_{j-1}} x_{v_{j}}) \\\\\n=& 2(1-\\alpha)(x_{v_{j-1}}-x_{v_{j}})^2 \\\\\n\\geq & 0.\n\\end{aligned}\n$$\nNow, suppose that $d$ is even, let $d=2 j$ with $j \\geq 3$. Then we have $x_{v_k}=x_{v_{2j+1-k}}$ for $1 \\leq k \\leq j$ by symmetry. Let $G^{\\prime}=G^{*}-\\left\\{v_{j-1} v_j, v_{j+2} v_{j+3}\\right\\}+\\left\\{v_j v_{j+2}, v_{j-1} v_{j+3}\\right\\}$. Clearly, $G^{\\prime}$ is $K_{2,3}$-minor free and\n$$\n\\begin{aligned}\n\\rho_{\\alpha}\\left(G^{\\prime}\\right)-\\rho_{\\alpha}\n\\geq& \\sum_{u w \\in E(G^{\\prime})}(\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{w}+\\alpha x_{w}^{2})-\\sum_{u w \\in E(G^{*})} (\\alpha x_{u}^{2}+2(1-\\alpha)x_{u}x_{w}+\\alpha x_{w}^{2}) \\\\\n=&2(1-\\alpha) (x_{v_{j}} x_{v_{j+2}}+ x_{v_{j-1}} x_{v_{j+3}}-x_{v_{j-1}} x_{v_{j}}-x_{v_{j+2}} x_{v_{j+3}})\\\\\n=&2(1-\\alpha) ( x_{v_{j}}(x_{v_{j+2}}-x_{v_{j-1}})- x_{v_{j+3}}(x_{v_{j+2}}-x_{v_{j-1}})) \\\\\n=&2(1-\\alpha) (x_{v_{j}}-x_{v_{j+3}})(x_{v_{j+2}}-x_{v_{j-1}})\\\\\n=&0.\n\\end{aligned}\n$$\nThus, whether $d$ is odd or even, if $\\rho_{\\alpha}\\left(G^{\\prime}\\right)=\\rho_{\\alpha}$, then $\\mathbf{X}$ is also an unit eigenvector corresponding to $\\rho_{\\alpha}\\left(G^{\\prime}\\right)$. Since $v_{v_{j}}$, $ v_{v_{j+1}}$, and $v_{v_{j+2}}$ are symmetric in $G^{\\prime}$, we have $x_{v_{j-1}}=x_{v_{j}}=x_{v_{j+1}}$. By the eigenequations of $A_{\\alpha}(G^{*})$, we have $x_{v_{1}}=\\cdots=x_{v_{d}}$. From the eigenequations of $A_{\\alpha}(G^{*})$ on $v_1$ and $v_2$, we have\n$$\n\\rho_{\\alpha} x_{v_1}=2\\alpha x_{v_1}+(1-\\alpha)(x_{v_2}+x_v) \\quad \\text { and } \\rho_{\\alpha} x_{v_2}=3 \\alpha x_{v_2}+(1-\\alpha)(x_{v_1}+x_{v_3}+x_v),\n$$\na contradiction. So, $\\rho_{\\alpha}\\left(G^{\\prime}\\right)>\\rho_{\\alpha}$, which is also a contradiction.\n\nTherefore, we get that $1 \\leq d \\leq 2$, and so $G^{*}-v$ consists of disjoint copies of triangles and at most a path of order $1$ or $2$, that is, $G^{*}\\cong K_{1}\\vee(pK_{2}\\cup K_{r}).$\n\\end{proof}\n\n\\begin{lem}\\label{lem::3.1'''}\nIf $s=3$, $t=3$, $n-1=pt+r$ and $1\\leq r \\leq3$. Then $G^{*}\\cong K_{2}\\vee(pK_{3}\\cup K_{r}).$\n\\end{lem}\n\\begin{proof}\nNote that $|K|=s-1=2$. Let $K=\\{v, v^{\\prime}\\}$. Thus, $d_{G^{*}}(v)=d_{G^{*}}(v^{\\prime})=n-1$. Since $G^{*}$ is $K_{3,3}$-minor free, we see that $G^{*}-\\{v, v^{\\prime}\\}$ does not contain any cycle of length at least $4$ as a subgraph. Moreover, $G^{*}$ contains no $K_{3,3}$ as a subgraph, which implies that $d_{G^{*}-\\{v, v^{\\prime}\\}}(u)\\leq2$ for any vertex $u \\in V(G^{*}-\\{v, v^{\\prime}\\})$.\nTherefore, each component of $G^{*}-\\{v, v^{\\prime}\\}$ is either a triangle or a path of order at least $1$. In fact, there is at most one component of $G^{*}-\\{v, v^{\\prime}\\}$ is a path. Otherwise, adding an edge to two pendant vertices in two different components which are paths leads to a $K_{3,3}$-minor free graph with larger $A_\\alpha$-spectral radius, a contradiction. Let $P$ be the unique component of $G^{*}-\\{v, v^{\\prime}\\}$ which is not triangle, that is, $P$ is a path of order $d \\geq 1$.\nSimilar as Cases 1-3 in the proof of Lemma \\ref{lem::3.1''}, we can get that $1 \\leq d \\leq 2$.\nThus $G^{*}-\\{v, v^{\\prime}\\}$ consists of disjoint copies of triangles and at most a path of order $1$ or $2$, and so $G^{*}\\cong K_{2}\\vee(pK_{3}\\cup K_{r}).$\n\\end{proof}\n\nLemmas \\ref{lem::3.1'}-\\ref{lem::3.1'''} imply that the cases $t=2$ and $t=3$ in Theorem \\ref{thm::1.1} hold. Thus, it remains to prove the Theorem \\ref{thm::1.1} for $t\\geq4$. We now let $t\\geq4$, $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor$, $\\gamma = \\min \\left\\{s,\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor\\right\\}$. In addition,\nwe use $H_{i}$, $H_{>i}$ and $H_{t+3}=\\varnothing$.\n\\end{lem}\n\n\\begin{proof}\nSuppose to the contrary that there exists a component $H \\in H_{>t+3}$. Let $|H|=pt+r$, where $p \\geq 1$ and $1 \\leq r \\leq t$. Since $H$ is $K_{1,t}$-minor free, by Lemma \\ref{lem::2.4} we have\n\\begin{equation}\\label{equ::27}\ne(H) \\leq \\binom{t}{2}+|H|-t= \\binom{t}{2} +(p-1) t+r.\n\\end{equation}\nNow, let $H^{\\prime} \\cong p K_{t} \\cup K_{r}$ with $V\\left(H^{\\prime}\\right)=V(H)$. Clearly, $H^{\\prime}$ also has the $(s, t)$-property. Since $|H|>t+3$ and $t \\geq 4$, then by (\\ref{equ::27}) we obtain\n$\ne(H)

t+3}=\\varnothing$. Furthermore, there exists at most one component in $H_{2 \\beta\\left(\\binom{t}{2}+\\beta-1\\right)=e(H^{\\prime}_{t+1}),\n$$\nwhich contradicts Lemma \\ref{lem::3.2}. Hence, there exists at most $2\\beta-1$ components in $H_{t+1}$.\nIf there are exactly $2\\beta-1$ components in $H_{t+1}$. Then $H_{t+1}\\cong(2 \\beta-1)\\overline{H_{s, t}}$. Let $H^{\\prime\\prime\\prime}_{t+1} \\cong(2 \\beta-1) K_{t} \\cup K_{2 \\beta-1}$. Thus, $e\\left(H^{\\prime\\prime\\prime}_{t+1}\\right)=e(H_{t+1})$. Note that $H_{s, t} \\cong (\\beta-1) K_{1, s} \\cup K_{1, \\alpha} $, where $\\alpha=t-(s+1)(\\beta-1) \\geq s$. Thus we obtain\n\\begin{equation}\\label{equ::30}\n\\delta\\left(\\overline{H_{s, t}}\\right)=(s+1)(\\beta-1)>2(\\beta-1),\n\\end{equation}\nwhich implies that $\\delta(H_{t+1})>2 \\beta-2$. Clearly, $\\pi\\left(H^{\\prime\\prime\\prime}_{t+1}\\right)=(t-1, \\ldots, t-1,2 \\beta-2, \\ldots, 2 \\beta-2)$. Then $\\pi(H_{t+1}) \\prec \\pi\\left(H^{\\prime\\prime\\prime}_{t+1}\\right)$, which contradicts Lemma \\ref{lem::3.4}. Thus, there exists at most $2(\\beta-1)$ components in $H_{t+1}$.\n\\end{proof}\n\n\\begin{lem}\\label{lem::3.9}\n$H_{t+3}=\\varnothing$.\n\\end{lem}\n\n\\begin{proof}\nSuppose to the contrary that there exists a component $H$ of $G^{*}-K$ with $|H|=t+3$. Then we have $e(H) \\leq\\binom{t }{2}+3$ by Lemma \\ref{lem::2.4}. Clearly, $K_{t} \\cup K_{3}$ also has the $(s, t)$-property. Then by Lemma \\ref{lem::3.2}, we have $e(H) \\geq e\\left(K_{t} \\cup K_{3}\\right)=\\binom{t}{2}+3$. Hence, $e(H)=\\binom{t}{2}+3$.\nSuppose that $\\beta \\geq 3$. Since $|\\overline{H_{s, t}}|=t+1$, $\\overline{H_{s,t}}$ has the $(s, t)$-property by Lemma \\ref{lem::2.16}(i) and $e\\left(\\overline{H_{s, t}}\\right)=\\binom{t}{2}+\\beta-1$ by Lemma \\ref{lem::3.5}. Moreover, from Lemma \\ref{lem::3.7}, there exists two components which are isomorphic to $K_{t}$ in $G^{*}-K$. Then\n$\ne\\left(H \\cup 2K_{t}\\right)=3\\binom{t}{2}+3<3\\binom{t}{2}+3(\\beta-1)=e\\left(3\\overline{H_{s, t}}\\right),\n$\nwhich contradicts Lemma \\ref{lem::3.2}. Hence $\\beta \\leq 2$.\n\t\n\\begin{clm}\\label{clm::3.5}\n$2 \\leq d_{H}(v) \\leq t-1$ for each $v \\in V(H)$.\n\\end{clm}\n\t\n\\begin{proof}\nClearly, $\\Delta(H)\\leq\\Delta(G^{*}-K)\\leq t-1$. Thus, it remains to show that $\\delta(H) \\geq 2$. Suppose to the contrary that $\\delta(H)=1$. Take an arbitrary vertex $v \\in V(H)$ with $d_{H}(v)=1$. Then $H-\\{v\\}$ is connected and $e(H-v)=\\binom{t}{2}+2$.\nBy Lemmas \\ref{lem::2.15} and \\ref{lem::3.2},\n$H-v$ is isomorphic to either $\\overline{H^{\\star}}$ or some $\\overline{H_{a, b, c}}$ with $a+b+c=t-1$. If $H-v$ is isomorphic to some $\\overline{H_{a, b, c}}$, by contracting the edges $u_{1} u_{2}$ and $u_{2} w$ from Fig. \\ref{fig-1}, we find that $\\overline{H_{a, b, c}}$ contains a $K_{t}$-minor, this implies that $H$ contains a $K_{1, t}$-minor, a contradiction. Hence, $H-v \\cong \\overline{H^{\\star}}$. Now, let $vuw$ be a path of length $2$ in $H$. Note that any two non-adjacent vertices in the Petersen graph $H^{\\star}$ have exactly one common neighbor. So, if we contract $uw$ in $H-\\{v\\}$, then the new vertex is of degree $t-1$ in the resulting graph. Consequently, if we contract $uw$ in $H$, then the new vertex is of degree $t$, which implies that $H$ contains a $K_{1, t}$-minor, a contradiction.\n\\end{proof}\nBy $e(H)=\\binom{t}{2}+3$ and Claim \\ref{clm::3.5}, we have $\\pi(H) \\prec (t-1, \\ldots, t-1,2,2,2)=\\pi\\left(K_{t} \\cup K_{3}\\right)$. Then by Lemma \\ref{lem::3.4}, we obtain $\\pi(H)=\\pi\\left(K_{t} \\cup K_{3}\\right)$.\nLet $M_{1}=\\{v \\in V(H) \\mid d_{H}(v)=2\\}$ and $M_{2}=\\cup_{v \\in M_{1}} N_{H}(v) \\backslash M_{1}$. Clearly, $1 \\leq d_{G[M_{1}]}(u) \\leq 3$ for any $u \\in M_{2}$.\n\t\n\\begin{clm}\\label{clm::3.6}\n$d_{G[M_{1}]}(u)=1$ for any $u \\in M_{2}$.\n\\end{clm}\n\t\n\\begin{proof}\nLet $L$ be the set of non-adjacent vertex-pairs in $M_{2}$. Since $|V(H) \\backslash M_{1}|=t$ and $d_{H}(u)=t-1$ for each $u \\in V(H) \\backslash M_{1}$, we have $|L|=\\frac{1}{2} e(M_{1}, M_{2})$, and so $1 \\leq|L| \\leq 3$. Suppose that $d_{G[M_{1}]}\\left(u_{0}\\right)=c \\in\\{2,3\\}$ for some $u_{0} \\in M_{2}$. Then $u_{0}$ has exactly $c$ non-neighbors, say $\\left\\{u_{1}, \\ldots, u_{c}\\right\\}$, in $M_{2}$. If $c=3$, then $L=\\left\\{\\left(u_{0}, u_{i}\\right) \\mid i=1,2,3\\right\\}$ and there are three paths $u_{0} v_{i} u_{i}$ in $H$, where $v_{i} \\in M_{1}$ and $i \\in\\{1,2,3\\}$. Now, if we contract two of the three paths into edges, then the resulting graph is isomorphic to $S^{1}(K_{t})$. However, $S^{1}\\left(K_{t}\\right)$ contains $K_{2, t-1}$ as a subgraph, which contradicts the fact that $H$ has the $(s,t)$-property. Therefore, $c=2$. Since $t \\geq 4$, we can find a vertex $u_{3} \\in N_{H}(u_{0}) \\backslash M_{1}$. Note that $|L| \\leq 3$. Hence, $\\{u_{1}, u_{2}\\} \\cap N_{H}(u_{3}) \\neq \\varnothing$. Moreover, if $u_{1}, u_{2} \\in N_{H}\\left(u_{3}\\right)$, then $H$ contains a double star with a non-pendant edge $u_{0} u_{3}$ and $t$ leaves, which implies that $H$ has a $K_{1, t}$-minor, a contradiction. So we may assume that $u_{1} u_{3} \\in E(H)$ and $u_{2} u_{3} \\notin E(H)$. Since $|L| \\leq 3$, we have $u_{1} u_{2} \\in E(H)$. It follows that $P=u_{0} u_{3} u_{1} u_{2}$ is an induced path in $H$. Now, $|L|=3$ and $M_{2}=\\left\\{u_{0}, u_{1}, u_{2}, u_{3}\\right\\}$. Furthermore, $d_{G[M_{1}]}\\left(u_{i}\\right)=2$ for $i \\in\\{0,2\\}$ and $d_{G[M_{1}]}(u_{i})=1$ for $i \\in\\{1,3\\}$. Then there exists a double star with a non-pendant edge in $E(P)$ and $t$ leaves, which implies that $H$ has a $K_{1, t}$-minor, a contradiction.\n\\end{proof}\n\t\n\\begin{clm}\\label{clm::3.7}\n$x_{u}>x_{v}$ for any two vertices $u, v$ with $u \\in M_{2}$ and $v \\in M_{1}$.\n\\end{clm}\n\\begin{proof}\nLet $x_{1}=\\max\\limits_{v \\in V(H)}x_{v}$ and $x_{2}=\\min\\limits_{v\\in V(H)}x_{v}$. Note that $x_{0}=\\sum\\limits_{v\\in K}x_{v}$.\nBy (\\ref{equ::8}) we have $$x_{u}\\geq\\frac{(1-\\alpha)(d_{H}(u)x_{2} +x_{0})}{\\rho_{\\alpha}-\\alpha(s-1)}\\ \\ \\mbox{and} \\ \\ x_{v}< \\frac{(1-\\alpha)(d_{H}(v)x_{1} +x_{0})}{\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)}.$$\nSince $d_{H}(u)>d_{H}(v)$, by Lemma \\ref{lem::3.1} we have $d_{H}(u)x_{2}>d_{H}(v)x_{1}$.\nNow we see that\n$$\nx_{u}-x_{v}> (1-\\alpha)\\left( \\frac{d_{H}(u)x_{2} +x_{0}}{\\rho_{\\alpha}-\\alpha(s-1)}-\\frac{d_{H}(v)x_{1} +x_{0}}{\\rho_{\\alpha}-(\\alpha(t+s-1)+(1-\\alpha)t)}\\right) >0,\n$$\nsince $\\rho_{\\alpha}\\geq\\alpha(n-1)+(1-\\alpha)(s-2)$, $0<\\alpha<1$, $n$ is sufficiently large, and $s, t$ are constants.\n\\end{proof}\n\t\nBy Claim \\ref{clm::3.6}, each $u_{i}\\in M_{2}$ has a unique neighbor $v_{i} \\in M_{1}$. Thus, each $u_{i}\\in M_{2}$ has a unique non-neighbor $u_{j} \\in V(H) \\backslash M_{1}$, where $u_{j} \\in M_{2}$.\nIf $v_{i}=v_{j}$ for some $(u_{i}, u_{j}) \\in L$, then $u_{i}$ and $u_{j}$ have $t-1$ common neighbors in $H$, which implies that $H$ contains a copy of $K_{2, t-1}$, a contradiction.\nHence, $v_{i} \\neq v_{j}$ for each $\\left(u_{i}, u_{j}\\right) \\in L$. If $v_{i} v_{j} \\in E(H)$ for some $\\left(u_{i}, u_{j}\\right) \\in L$,\nthen we can also obtain a copy of $K_{2, t-1}$ by contracting the edge $v_{i} v_{j}$, a contradiction. So, $v_{i} v_{j} \\notin E(H)$ for each $\\left(u_{i}, u_{j}\\right) \\in L$. Now, Let\n$$\nH^{\\prime}=H-\\left\\{u_{i} v_{i}, u_{j} v_{j} \\mid\\left(u_{i}, u_{j}\\right) \\in L\\right\\}+\\left\\{u_{i} u_{j}, v_{i} v_{j} \\mid\\left(u_{i}, u_{j}\\right) \\in L\\right\\},\n$$\nand $G^{\\prime}=G^{*}-E(H)+E\\left(H^{\\prime}\\right)$.\nClearly, $H^{\\prime} \\cong K_{t} \\cup K_{3}$, and so $H^{\\prime}$ has the $(s,t)$-property. Furthermore, by Claim \\ref{clm::3.7} we have\n$$\n\\begin{aligned}\n\\rho_{\\alpha}(G^{\\prime})-\\rho_{\\alpha} \\geq& \\sum_{\\left(u_{i}, u_{j}\\right) \\in L} ((\\alpha x^{2}_{u_{i}} +2(1-\\alpha)x_{u_{i}}x_{u_{j}} +\\alpha x^{2}_{u_{j}})+(\\alpha x^{2}_{v_{i}}+2(1-\\alpha)x_{v_{i}}x_{v_{j}} +\\alpha x^{2}_{v_{j}})\\\\\n&-(\\alpha x^{2}_{u_{i}}+2(1-\\alpha)x_{u_{i}}x_{v_{i}} +\\alpha x^{2}_{v_{i}})-(\\alpha x^{2}_{u_{j}} +2(1-\\alpha)x_{u_{j}}x_{v_{j}}+ \\alpha x^{2}_{v_{j}})) \\\\\n=&\\sum_{\\left(u_{i}, u_{j}\\right) \\in L} 2(1-\\alpha)(x_{u_{i}}-x_{v_{j}})(x_{u_{j}}-x_{v_{i}})>0,\n\\end{aligned}\n$$\na contradiction.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem::3.8}\nIf $H$ is a component in $H_{t+2}$. Then $\\beta \\leq 2$, moreover, $H \\cong S^{1}\\left(\\overline{H_{s, t}}\\right)$ for $\\beta=2$ and $H \\cong \\overline{H^{\\star}}$ for $\\beta=1$.\n\\end{lem}\n\n\\begin{proof}\nThe proof is divided into several claims.\n\\begin{clm}\\label{clm::3.1}\n$\ne(H)=\\binom{t}{2}+2.\n$\n\\end{clm}\n\\begin{proof}\nSince $|H|=t+2$, by Lemma \\ref{lem::2.4} we have $e(H) \\leq\\binom{t }{2}+2$. Clearly, $K_{t} \\cup K_{2}$ has the $(s, t)$-property, then by Lemma \\ref{lem::3.2} we obtain $e(H) \\geq e\\left(K_{t} \\cup K_{2}\\right)=\\binom{t }{2}+1$. Suppose that $e(H)=\\binom{t }{2}+1$. Notice that $\\Delta(H) \\leq t-1$ and $\\delta(H) \\geq 1$. Then $\\pi(H) \\prec \\pi\\left(K_{t} \\cup K_{2}\\right)$. By Lemma \\ref{lem::3.4}, we obtain\n$\\pi(H)=\\pi\\left(K_{t} \\cup K_{2}\\right)=(t-1, t-1, \\ldots, t-1,1,1)$.\nThis implies that $H$ is obtained from $K_{t}$ by deleting an edge $u_{1} u_{2}$ and adding two pendant edges $u_{1} v_{1}$ and $u_{2} v_{2}$. Since $t \\geq 4$, we have $N_{H}\\left(u_{1}\\right) \\backslash\\left\\{v_{1}\\right\\} \\neq \\varnothing$ and\n$$\n\\left( \\rho_{\\alpha}-\\alpha(s+1)+1\\right) \\left(x_{u_{1}}-x_{v_{1}}\\right)=\\alpha(t-2)x_{u_{1}}+(1-\\alpha)\\sum_{v \\in N_{H}\\left(u_{1}\\right) \\backslash\\left\\{v_{1}\\right\\}} x_{v}>0,\n$$\nthus $x_{u_{1}}>x_{v_{1}}$. By symmetry, we have $x_{u_{1}}=x_{u_{2}}$ and $x_{v_{1}}=x_{v_{2}}$. Now, let $H^{\\prime}= H-\\left\\{u_{1} v_{1}, u_{2} v_{2}\\right\\}+\\left\\{u_{1} u_{2}, v_{1} v_{2}\\right\\}$ and $G^{\\prime}=G^{*}-E(H)+E\\left(H^{\\prime}\\right)$. Clearly, $H^{\\prime} \\cong K_{t} \\cup K_{2}$, thus $H^{\\prime}$ has the $(s, t)$-property. However, we see that\n$$\n\\rho_{\\alpha}\\left(G^{\\prime}\\right)-\\rho_{\\alpha} \\geq X^{T}\\left(A_{\\alpha}\\left(G^{\\prime}\\right)-A_{\\alpha}\\left(G^{*}\\right)\\right) X=2(1-\\alpha)\\left(x_{u_{1}}-x_{v_{2}}\\right)\\left(x_{u_{2}}-x_{v_{1}}\\right)>0,\n$$\na contradiction. Therefore, $e(H)=\\binom{t}{2}+2$.\n\\end{proof}\n\n\\begin{clm}\\label{clm::3.2}\n$\\beta \\leq 2$.\n\\end{clm}\n\\begin{proof}\nSuppose to the contrary that $\\beta \\geq 3$.\nThen we have $e\\left(\\overline{H_{s, t}}\\right)= \\binom{t}{2}+\\beta-1 \\geq \\binom{t}{2}+2$ by Lemma \\ref{lem::3.5}. From Lemma \\ref{lem::3.7}, there exists a component which is isomorphic to $K_{t}$ in $G^{*}-K$.\nLet $H^{\\prime} \\cong 2\\overline{H_{s, t}}$ and $V\\left(H^{\\prime}\\right)=V\\left(H \\cup K_{t}\\right)$. By Lemma \\ref{lem::2.16}(i), $H^{\\prime}$ has the $(s, t)$-property. Then by Claim \\ref{clm::3.1}, we have $e(H^{\\prime})\\geq 2\\left( \\binom{t}{2}+2\\right) >2 \\binom{t}{2}+2=e({H} \\cup K_{t})$, which contradicts\nLemma \\ref{lem::3.2}.\n\\end{proof}\n\n\\begin{clm}\\label{clm::3.3}\nIf $\\beta=2$, then $H\\cong S^{1}\\left(\\overline{H_{s, t}}\\right)$.\n\\end{clm}\n\\begin{proof}\nClearly, $|S^{1}\\left(\\overline{H_{s, t}}\\right)|=t+2$. Moreover, Lemma \\ref{lem::2.16}(i) gives that $S^{1}\\left(\\overline{H_{s, t}}\\right)$ has the $(s, t)$-property.\nSince $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor=2$, we have $2s+1 \\leq t \\leq 3s+1$.\nThus $\\gamma=\\min \\left\\{s,\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor\\right\\}=s$.\nSince $|H|=t+2$, then together with Claims \\ref{clm::3.1}, \\ref{clm::3.2} and Lemma \\ref{lem::2.15}, we find that $H$ is isomorphic to either $\\overline{H^{\\star}}$ or some $\\overline{H_{a, b, c}}$, where $a+b+c=t-1$.\nIf $H \\cong \\overline{H^{\\star}}$. Then $|H|=t+2=10$ and $H$ is $6$-regular. Thus, we obtain $t=8$ and $e(H)=30$. Moreover, since $S^{1}\\left(\\overline{H_{s, 8}}\\right)$ is a subdivision of $\\overline{H_{s, 8}}$, by Lemma \\ref{lem::3.5} we have $e\\left(S^{1}\\left(\\overline{H_{s, 8}}\\right)\\right)=e\\left(\\overline{H_{s, 8}}\\right)+1\n=\\binom{t}{2}+\\beta=30$. From Lemma \\ref{lem::2.16}(i), we get that\n$$\n\\pi\\left(S^{1}\\left(\\overline{H_{s, t}}\\right)\\right)=(t-1, \\ldots, t-1, t-s, s+1,2).\n$$\nSince $t=8$ and $s\\geq 2$, we have $t-s \\leq 6$. Hence, $\\pi(H) \\prec \\pi\\left(S^{1}\\left(\\overline{H_{s, 8}}\\right)\\right)$, which contradicts Lemma \\ref{lem::3.4}. Therefore, $H$ is isomorphic to some $\\overline{H_{a, b, c}}$. Now, we assert that\n\\begin{equation}\\label{equ::29}\n\\min \\{b, c\\} \\geq \\gamma.\n\\end{equation}\nIf $b \\leq \\gamma-1$, we contract the edge $u_{2} w$ in $\\overline{H_{a, b, c}}$ and call the new vertex $u$ in the resulting graph, then we get a complete bipartite subgraph with bipartite partition\n$$\\left(V(K_{b}) \\cup\\{u\\}, V(K_{a})\\cup V(K_{c}) \\cup\\{u_{1}\\}\\right).$$\nThis implies that $H$ contains a $K_{b+1, a+c+1}$-minor, contradicting the $(s, t)$-property. So, $b \\geq \\gamma$. By symmetry, we have $c \\geq \\gamma$.\n\nFurthermore, by Fig. \\ref{fig-1} we can see that\n$$\n\\pi\\left(\\overline{H_{a, b, c}}\\right)=\\left(t-1, \\ldots, t-1, a_{1}, a_{2}, a_{3}\\right),\n$$\nwhere $a_{1}, a_{2}, a_{3} \\in\\{a+2, b+1, c+1\\}$. By (\\ref{equ::29}) and $\\gamma=s$, we have $\\min \\{b, c\\} \\geq \\gamma=s$. Thus, we find that $a_{3} \\geq 2$ and $a_{2} \\geq s+1$. Hence, we see that $\\pi\\left(\\overline{H_{a, b, c}}\\right) \\prec \\pi\\left(S^{1}\\left(\\overline{H_{s, t}}\\right)\\right)$. Then we obtain $\\pi\\left(\\overline{H_{a, b, c}}\\right)=\\pi\\left(S^{1}\\left(\\overline{H_{s, t}}\\right)\\right)$ by Lemma \\ref{lem::3.4},. This implies that $a_{3}=2$ and $a_{2}=s+1$. Note that $\\min \\{b, c\\} \\geq s \\geq 2$. It follows that $a=0$ and $\\min \\{b, c\\}=s$, that is, $\\overline{H_{a, b, c}} \\cong S^{1}\\left(\\overline{H_{s, t}}\\right)$.\n\\end{proof}\n\n\\begin{clm}\\label{clm::3.4}\nIf $\\beta=1$, then $H \\cong \\overline{H^{\\star}}$.\n\\end{clm}\n\n\\begin{proof}\nSince $\\beta=\\left\\lfloor\\frac{t+1}{s+1}\\right\\rfloor=1$, we have $t \\leq 2s$. Thus, $\\gamma=\\min \\left\\{s,\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor\\right\\}=\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor$. Since $|H|=t+2$, then together with Claims \\ref{clm::3.1}, \\ref{clm::3.2} and Lemma \\ref{lem::2.15}, $H$ is isomorphic to either $\\overline{H^{\\star}}$ or some $\\overline{H_{a, b, c}}$, where $a+b+c=t-1$. If $H$ is isomorphic to some $\\overline{H_{a, b, c}}$, then we get (\\ref{equ::29}) in a similar way as above. It follows that $t-1\\geq b+c \\geq 2 \\gamma=2\\left\\lfloor\\frac{t+1}{2}\\right\\rfloor$, a contradiction. Hence, $H$ is only possibly isomorphic to $\\overline{H^{\\star}}$. Since $|\\overline{H^{\\star}}|=t+2=10$, we have $t=8$. Then by Lemma \\ref{lem::2.16}(ii), $\\overline{H^{\\star}}$ has the $(s, t)$-property. Therefore, $H \\cong \\overline{H^{\\star}}$, as desired.\n\\end{proof}\nCombining Claims \\ref{clm::3.2}-\\ref{clm::3.4}, we obtain the result of Lemma \\ref{lem::3.8}.\n\\end{proof}\n\n\\begin{lem}\\label{lem::3.13}\nIf $H_{t+1} \\neq \\varnothing$. Then $H_{t+2} \\cup H_{2$. Since $D_{1}$ is a subdivision of $D_{2}$, $\\delta\\left(D_{1}\\right)=2$ and its vertex of degree two is unique. Which implies that $\\pi\\left(D_{1} \\cup D_{2}\\right) \\prec \\pi\\left(H^{\\prime}\\right)$ and $\\pi\\left(D_{1} \\cup D_{2}\\right) \\neq \\pi\\left(H^{\\prime}\\right)$, this contradicts Lemma \\ref{lem::3.4}.\n\t\nIf $|D_{1}|\\beta-1$, then\n$$\ne\\left(D_{1} \\cup D_{2}\\right)=\\binom{\\left|D_{1}\\right| }{2}+\\binom{t}{2}+\\beta-1<\\binom{t }{2}+\\binom{\\left|D_{1}\\right| }{2}+\\left|D_{1}\\right|=e\\left(K_{t} \\cup K_{\\left|D_{1}\\right|+1}\\right),\n$$\nwhich contradicts Lemma \\ref{lem::3.2}. On the other hand,\nwe get that $G^{*}-K$ contains the disjoint union of a copy of $D_{1}$ and $|D_{1}|$ copies of $K_{t}$ by Lemma \\ref{lem::3.7}. We denote it by $H^{\\prime\\prime}$. Then $e(H^{\\prime\\prime})= \\binom{\\left|D_{1}\\right| }{2} +\\left|D_{1}\\right|\\binom{t}{2}$. Now let $H^{\\prime\\prime\\prime}$ be the disjoint union of $|D_{1}|$ copies of $D_{2}$. Clearly, $|H^{\\prime\\prime}|=|H^{\\prime\\prime\\prime}|=|D_{1}|(t+1)$ and $e\\left(H^{\\prime\\prime\\prime}\\right)= |D_{1}|\\left(\\binom{t}{2}+\\beta-1\\right)$. By Lemma \\ref{lem::3.2}, $e\\left(H^{\\prime\\prime\\prime}\\right) \\leq e(H^{\\prime\\prime})$.\nTherefore, we get that\n\\begin{equation}\\label{equ::32}\n\\begin{array}{ll}\n|D_{1}|\\geq 2\\beta-1,\n\\end{array}\n\\end{equation}\nwhich contradicts (\\ref{equ::31}).\n\\end{proof}\n\n\n\\begin{lem}\\label{lem::3.11}\nLet $t\\geq4$, $\\beta=1$, $n-s+1=pt+r$ and $1\\leq r\\leq t$. Then\n$$\nG^{*}-K \\cong\\left\\{\\begin{array}{ll}\n(p-1) K_{t} \\cup \\overline{H^{\\star}} & \\text { for } r=2 \\text { and } t=8; \\\\\np K_{t} \\cup K_{r} & \\text { otherwise. }\n\\end{array}\\right.\n$$\n\\end{lem}\n\n\\begin{proof}\nBy Lemmas \\ref{lem::3.6} and \\ref{lem::3.9}, $H_{>t+2}=\\varnothing$. Note that $\\beta=1$. Then by Lemma \\ref{lem::3.5}, we also have $H_{t+1}=\\varnothing$. Since any component of $G^{*}- K$ of order $t$ is isomorphic to $K_{t}$, then by Lemma \\ref{lem::3.10}, there exists at most one component $H$ of $G^{*}-K$ are not isomorphic to $K_{t}$. Notice that $|G^{*}-K|=pt+r$, where $1 \\leq r \\leq t$, we see that either $|H|=r$ or $|H|=t+r=t+2$. If $r \\neq 2$, then $H \\cong K_{r}$ and $G^{*}-K \\cong p K_{t} \\cup K_{r}$, as desired. Now assume that $r=2$. Then either $H \\cong K_{2}$ or $H \\cong \\overline{H^{\\star}}$ by Lemma \\ref{lem::3.8}. If $H \\cong \\overline{H^{\\star}}$, then $|H|=t+2=10$, and so $t=8$. Hence, if $t \\neq 8$, then $G^{*}-K \\cong p K_{t} \\cup K_{2}$. Now we consider the case $t=8$. Suppose that $H \\cong K_{2}$, then $e\\left(K_{8} \\cup K_{2}\\right)=29<30=e\\left(\\overline{H^{\\star}}\\right)$, contradicting Lemma \\ref{lem::3.2}. That is, $H\\cong \\overline{H^{\\star}}$.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem::3.14}\nLet $t\\geq4$, $\\beta \\geq 2$, $n-s+1=pt+r$ and $1\\leq r\\leq t$. Then\n$$\nG^{*}-K \\cong\\left\\{\\begin{array}{ll}\n(p-1) K_{t} \\cup S^{1}\\left(\\overline{H_{s, t}}\\right) & \\text { for } r=\\beta=2; \\\\\n(p-r) K_{t} \\cup r \\overline{H_{s, t}} & \\text { for } r \\leq 2(\\beta-1) \\text { except } r=\\beta=2; \\\\\np K_{t} \\cup K_{r} & \\text { for } r>2(\\beta-1).\n\\end{array}\\right.\n$$\n\\end{lem}\n\n\\begin{proof}\nNote that $|G^{*}-K|=pt+r$, where $1 \\leq r \\leq t$, and $H \\cong \\overline{H_{s, t}}$ for every $H \\in H_{t+1}$.\nNow, we assert that if $r \\leq 2(\\beta-1)$ then $H_{t}=\\varnothing$ and there exists exactly one component in $H_{2(\\beta-1)$, then by Lemma \\ref{lem::3.12}, $G^{*}-K \\cong pK_{t} \\cup K_{r}$.\n\nNext, we assume that $r=2$. Since $\\beta \\geq 2$, we have $r\\leq 2(\\beta-1)$ and thus $H_{2$, then $H_{t+2}=\\varnothing$ by Lemma \\ref{lem::3.8}. Thus, $G^{*}-K \\cong(p-r) K_{t} \\cup r\\overline{H_{s, t}}$. It remains to prove the case $r=\\beta=2$. Now if $H_{t+1} \\neq \\varnothing$, then $H_{t+2} \\cup H_{2$, we can easily see that $\\pi\\left(2\\overline{H_{s, t}}\\right) \\prec \\pi\\left(H^{\\prime}\\right)$, this contradicts Lemma \\ref{lem::3.4}. Thus, $H_{t+1}=\\varnothing$. It follows that there exists exactly one component in $H_{t+2}$. By Lemma \\ref{lem::3.8}, we have $G^{*}-K \\cong(p-1) K_{t} \\cup S^{1}\\left(\\overline{H_{s, t}}\\right)$.\n\\end{proof}\n\nCombining Lemmas \\ref{lem::3.1'}-\\ref{lem::3.1'''}, \\ref{lem::3.11} and \\ref{lem::3.14}, we obtain the result of Theorem \\ref{thm::1.1}.\n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{sec:intro}Introduction}\n\n\n\n\n\n\n\n\nA cold, bright and compact atomic beam source is\nan important asset for any experiment featuring ultra-cold atoms, such as atom interferometers~\\cite{Cronin09}, degenerate quantum gases for quantum simulation~\\cite{Georgescu14} and, in particular, optical atomic clocks~\\cite{Ludlow2015}. In the last case, two valence electron alkaline-earth (like) metals, like Ca~\\cite{Wilpers07}, Sr~\\cite{Ushijima2015}, Mg~\\cite{Kulosa15}, and Yb\\cite{McGrew2018}, are generally used as atomic frequency discriminators, and present low vapor pressures at room temperature thus needing high temperature ovens to generate enough atomic vapour, typically followed by a space-demanding Zeeman slower (ZS). Although compact and transportable versions of the ``oven + ZS'' atomic beam system have been developed~\\cite{Poli2014,AOsense}, some concerns about the systematic effects due to collisions with the atomic beam particles~\\cite{Gibble13} and the hot black-body radiation from the oven region~\\cite{Beloy14} can arise below the $10^{-18}$ relative uncertainty level.\n\nThe two-dimensional magneto optical trap (2D-MOT) atomic source~\\cite{Dieckmann1998, Schoser2002} can be transversely loaded, hence reducing the setup dimensions, avoiding direct exposure of the atomic reference to hot metals, and at the same time obtaining an optical shutter of the atomic beam just by turning-off its cooling beams. This avoids the use of in-vacuum mechanical shutters or optical beam deflectors~\\cite{Witte92} as done for ZS or collimated oven beams. The 2D-MOT system complexity can be further reduced by its permanent magnets implementation~\\cite{Tiecke2009, Lamporesi2013}.\n\n\nIn this work, we present a novel atomic source employing a 2D-MOT source of strontium (Sr) atoms for metrological application. The mechanical implementation of the atomic source is similar to other setups built to generate lithium~\\cite{Tiecke2009}, sodium \\cite{Lamporesi2013,Colzi2018} and strontium~\\cite{Nosske2017} atomic beams. Our system is further characterized by a collimated atomic beam transmitted by a bundle of capillaries directly towards the 2D-MOT region, and a two-frequency optical molasses to enhance the atomic flux toward the trapping region. The design, engineering and characterization of the sideband-enhanced 2D-MOT strontium source is the main result of this work. This is accomplished by looking at the loading performances of a three-dimensional MOT typically used as the first cooling and trapping stage for an optical lattice clock~\\cite{Xu2003}. Monte Carlo (MC) numerical simulations are used to find the optimal optical configuration which are then compared to the experimental results. \n\nThe article is organized as follows: \nSec.~\\ref{sec:Theory} introduces the physical interpretation and significance of adding a sideband frequency to the cooling beams of the 2D-MOT; Sec.~\\ref{sec:apparatus} depicts the experimental apparatus assembled for an optical lattice clock; in Sec.~\\ref{sec:numerical simulation} we describe the numerical modeling of the atomic source and the 2D-MOT cooling and trapping processes by Monte Carlo simulations; Sec.~\\ref{sec:atomic source characterization} shows the experimental characterization of our atomic source and in Sec.~\\ref{sec:sideband enhancement} we demonstrate how the sideband-enhancement method is able to magnify the number of trapped atoms by a magneto-optical trap.\n\n\n\n\n\n\\section{\\label{sec:Theory}Principles of sideband-enhanced 2D-MOT}\n\nA 2D-MOT atomic source relies on the radiation-pressure friction force to capture and cool thermal atoms effusing from either an oven, or a background gas. In this work, we focus our attention on the 2D-MOT loaded from a collimated atomic source, so that a 1D model offers a good insight on the expected 2D-MOT flux. For the 1D model, the MOT captured atoms per second $\\Phi_\\text{2D}$ is provided by the formula \n\\begin{equation}\\label{eq:cap_2d}\n\\Phi_\\text{2D} \\simeq n v_\\text{th} A (v_\\text{c} \/ v_\\text{th})^4, \\qquad(v_c \\ll v_\\text{th})\n\\end{equation}\nwhere $n$ is the spatial density of the thermal beam, $v_\\text{th} = \\sqrt{2 k_B T_\\text{ov} \/ m}$ is the most probable thermal velocity (for the atomic Sr vapour at $T_\\text{ov} = \\SI{460}{\\celsius}$, $v_\\text{th} = \\SI{379}{m \\per s}$ ), $A = 2\\pi w^2$ is the MOT capture surface related to the trapping beam width $w$, and $v_c$ is the capture velocity of the trap. It is clear from (\\ref{eq:cap_2d}) that the most influential parameter is the capture velocity $v_\\text{c}$, which is related to the magnetic gradient $b$, the frequency detuning $\\Delta$ from the cooling transition, and the total saturation parameter $s=I\/I_\\text{sat}$ of the MOT optical beams, where for an atomic transition at wavelength $\\lambda$ and spontaneous emission rate $\\Gamma$ the resonant saturation intensity is $I_\\text{sat} = \\pi hc\\Gamma\/3\\lambda^3$. In the 1D model one typically computes $v_\\text{c}$ numerically by solving the semiclassical equation of motion, as shown in Fig.\\ref{fig:capt_vel}(a). Here one can observe that there are two different dynamics inside the MOT region. In the outer region, the MOT behaves like a Zeeman slower, where the friction force exerted upon any atom will be effective only if the velocity $v$ at distance $r$ from the symmetry axis will be nearly resonant with the cooling laser, i.e., if the difference of the Zeeman shift and the laser detuning equals the Doppler shift. In the inner region the motion of the atoms can be described by an overdamped harmonic oscillator model. Hence the capture velocity is strictly related with the dynamics in the outer region of the MOT and, assuming perfect compensation of the Zeeman shift and Doppler shift, it can be roughly estimated as~\\cite{Tiecke2009,Zinner98}\n\n\\begin{equation}\\label{eq:vc}\nv_\\text{c} \\lesssim v_\\text{max} = \\sqrt{a_\\text{max}\\,r_\\text{max}} \n\\end{equation}\nwhere $a_\\text{max} = \\hbar k \\Gamma\/(2m)$ is the maximum acceleration at infinite saturation parameter, and $r_\\text{max}=\\sqrt{2}w$ is the maximum interaction distance with the MOT beams, taking into account the projection of the 2D-MOT beams at \\SI{45}{\\degree} from the atoms propagation axis. This $v_\\text{c}$ corresponds to the maximum velocity allowed in order to decelerate an atom to zero at the center of the trap. This oversimplified estimation gives us some hints on the 2D-MOT expected performance. In particular, even for infinite available power, the capture velocity would be bounded, while the capture mechanism is fundamentally limited by the natural linewidth of the cooling transition and the cooling beam radius. However, if one uses laser light which has several red-detuned sidebands, even faster atoms can be slowed down and the capture velocity increased. MOT loading enhancement was observed in alkali atomic systems by means of electro-optic modulation (EOM) of the cooling beams \\cite{Anderson1994,Lee2017}. This technique is generally not feasible at the wavelengths of alkaline-earth atoms by EOMs. Furthermore, because of the higher $\\Gamma$s excessive spectral broadening would reduce the radiation pressure force, making it no longer sufficient to keep the thermal atoms in the trap. A one-sideband 3D MOT has been previously realized to trap Ca atoms loaded directly from an effusing atomic oven~\\cite{Zinner98,Riehle:1999p6441}. In this case, with a total MOT saturation parameter $s \\sim 0.1$ and an atomic vapour temperature of 600 $^\\circ$C, an enhancement factor of 7 was observed~\\cite{Riehle:1999p6441}. However here only a very small fraction of the available atoms were trapped, hence that system would be very unfavourable in the case of a 2D-MOT source loading. \n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=0.49\\textwidth]{sideband_principle.pdf}\n \\caption{1D simulation of the atomic trajectories of strontium atoms (light blue line) for different capture processes in a 2D-MOT. The color map plotted on the background depicts the acceleration value at each point of the phase-space. \\textbf{(a)} Single-frequency 2D-MOT. The total saturation and the detuning of the MOT beams are $s=7$, $\\Delta\/\\Gamma = -1.6$. The estimated capture velocity is \\SI{72(1)}{m \\per s}. \n \\textbf{(b)} Sideband-enhanced 2D-MOT. The saturation parameter is $s=3.5$ for the 2D-MOT beam at with $\\Delta\/\\Gamma =-1.6 $, and $s_\\text{SB}=3.5$ for the sideband beam at $\\Delta_\\text{side}\/ \\Gamma = -3.2$. The estimated capture velocity is \\SI{90(1)}{m \\per s}. The magnetic field gradient and beam width used in this calculation are $b=\\SI{0.22}{T \\per m}$ and $w=\\SI{1}{cm}$.\n \\textbf{(c)} Acceleration profile at $r=-w\/2$ of the sideband trapping (red line) and standard 2D-MOT trapping (blue line).}\n \n \\label{fig:capt_vel}\n\\end{figure}\n\nIt is more interesting to investigate the sideband-enhanced 2D-MOT in the limit of high total saturation parameter $s\\geq 1$, where most of the low velocity class ($v\\leq v_\\text{max}$) is slowed and captured by the cooling beams. Fig.\\ref{fig:capt_vel}(a) shows a simulation of the phase-space trajectories for typical values of the experimental parameters ($\\Delta,s,b$) used in a strontium 2D-MOT~\\cite{Nosske2017}. The acceleration patterns of the sideband-enhanced 2D-MOT in the atomic phase-space are depicted in the in Fig.\\ref{fig:capt_vel}(b). As shown in the plot, the sideband beams interact with atoms from a higher velocity class, decelerating them toward the capture region of the standard MOT beam. This increment of the capture velocity is best displayed in Fig.\\ref{fig:capt_vel}(c): here we can see the MOT acceleration as function of the atomic approaching velocity. In the standard MOT (blue dashed) the force is peaked around a given velocity value, reaching $a_\\text{max}$ and the amount of power increases the spectral width of the force as $\\sqrt{s}$. On the other hand, the sideband-enhanced force (red dot-dashed) presents a second peak at higher velocity without degrading the peak acceleration. Optimal positioning of the sideband frequency thus allows an increase of the expected capture velocity $v_c$ and of the expected MOT loading rate too. \n\nAnother expected beneficial effect of the sideband-enhanced 2D-MOT with large $s$ is the reduction of the transverse temperature of the cold atomic sample compared to the standard 2D-MOT, which would yield a higher brightness (i.e. lower beam divergence). This can be explained considering that the optical power redistributed at a higher frequency weakly interacts with the atoms trapped once they reach the center of the MOT.\n\nIn order to correctly address the expected performances of a sideband-enhanced strontium 2D-MOT we performed a dedicated Monte Carlo (MC) simulation which takes into account the actual geometry of the system, the magnetic field gradient, the residual divergence of our atomic beam from the oven, and the expected loading rate for the final 3D-MOT. This is described in detail in Sec.\\ref{sec:numerical simulation}.\n\n\n\n\\section{\\label{sec:apparatus}Experimental apparatus}\n\n\\subsection{Vacuum system}\nThe schematic drawing of the vacuum system to produce and trap ultra-cold strontium atoms is depicted in Fig.~\\ref{fig:Figure1}. It has been previously described in~\\cite{Tarallo2017}, and its concept is adapted from previous works~\\cite{Tiecke2009, Lamporesi2013}. The vacuum system is conceived to host two physical regions with very different vacuum levels, the atomic source region and the science cell region and, at the same time, to be very compact. The atomic source region consists of a stainless-steel vacuum chamber with a multi-way cross at its end, where the intersection plane of the tubes forms the 2D-MOT plane. The ultra-high vacuum region hosts a small octagonal science cell with two large vertical optical accesses (DN63CF) and seven small lateral optical windows (DN16CF) for cooling, trapping and operate a Sr optical clock. The two vacuum regions are connected by a differential pumping channel (DPC) carved in a custom bellow with \\SI{2}{mm} diameter and \\SI{22.8}{mm} length and all-metal gate valve. The DPC sets the maximum divergence of the cold atomic beam at \\SI{87}{mrad}, while a conductance of \\SI{4.3e-2}{L \\per s} allows to maintain a differential pressure of \\num{e4} between the two regions. Vacuum is maintained by two ion-getter pumps, both regions reaching a pressure below \\SI{e-10}{mbar} when the oven is not heated.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.49\\textwidth]{fig_vacuum_system2.pdf}\n \\caption{Schematic drawing of the vacuum system. It hosts a high vacuum (HV) region for the atomic beam production, and an ultra-high vacuum (UHV) region for cooling and trapping the atomic sample. A DPC connects the two regions. The size of the entire vacuum apparatus is roughly \\SI{70}{cm} $\\times$ \\SI{70}{cm} $\\times$ \\SI{45}{cm}.}\n \n \\label{fig:Figure1}\n\\end{figure}\n\n\n\n\n\\subsection{Collimated atomic source}\n\nThe oven consists of a simple stainless-steel cylinder with an aperture of \\SI{16}{mm} and a conflat flange DN16CF to be attached to the main body of the vacuum system on one of its circular sides.\nThe oven is attached to the multi-way cross vacuum chamber \\SI{128}{mm} away from its center.\nIn order to produce a collimated atomic beam, an array of $N_\\text{cap} \\simeq \\num{150}$ capillaries made of nickel-based alloy Monel400, with an internal radius $r_\\text{cap}=\\SI{0.2}{mm}$ and a length $L_\\text{cap}=\\SI{20}{mm}$, is inserted at the oven aperture. The capillaries are tightened inside a holder which lays in the aperture of the oven. The heating is insured by a pair of heating cartridges. The Sr vapor is typically generated at the temperature $T_\\text{ov} =\\SI{460}{\\celsius}$. In order to avoid clogging of the capillaries with strontium, the oven hosts an extra pair of heating cartridges close to its aperture to maintain the capillaries at a temperature $T_\\text{cap}$ higher than $T_\\text{ov}$. For all experimental characterizations we maintained a differential temperature $T_\\text{cap} - T_\\text{ov}=\\SI{30}{\\celsius}$. At the typical operational oven temperature $T_\\text{ov}=\\SI{460}{\\celsius}$, the estimated vapour pressure inside is $p_\\text{ov}=\\SI{0.133}{Pa}$~\\cite{Alcock1984} from which we estimate the Sr atomic density by means of the ideal gas law $n_\\text{ov}= p_\\text{ov} \/ k_B T_\\text{ov} =\\SI{1.31e19}{atoms \\per m^3}$.\nIn the regime of negligible collisions inside the capillaries (mean free path $\\lambda_\\text{ov} = (\\sqrt{2} n_\\text{ov} \\sigma_\\text{Sr} )^{-1}\\sim \\SI{70}{mm} \\gg L_\\text{cap}$, with $\\sigma_\\text{Sr}$ = 8$\\cdot10^{-19}$ m$^2$ the elastic cross section), the atomic flux is proportional to the oven pressure $P_\\text{ov}$ and it is estimated as~\\cite{Wang1960}:\n\\begin{equation}\n \\Phi_\\text{ov} =a \\frac{4 \\sqrt{\\pi}}{3} \\frac{n_\\text{ov} v_\\text{th} r^3_\\text{cap}}{L_\\text{cap}}N_\\text{cap}\n \\label{eq:flux_oven}\n\\end{equation}\nwhere $a$ is the isotopic abundance.\nIn the case of $^{88}$Sr, the expected atomic flux at $T_\\text{ov}= \\SI{460}{\\celsius}$ is $\\Phi_\\text{ov}=\\SI{5.8e14}{atoms \\per s}$. The geometrical constraint imposed by the capillaries yield a theoretical divergence angle $\\theta_\\text{cap} \\simeq r_\\text{cap} \/ L_\\text{cap}=20\\,$mrad.\n\n\n\n\\subsection{2D-MOT and cold atomic source generation}\n\nAs sketched in Fig.~\\ref{fig:Figure1}, the 2D MOT is composed of a 2D quadrupole magnetic field in combination with two orthogonal pairs of retroreflected laser beams of opposite circular polarization.\n\nThe magnetic field gradient is generated by four stacks of permanent magnets~\\cite{Tiecke2009}. Each stack is composed of \\num{9} neodymium bar magnets with size of \\SI{25}{mm} $\\times$ \\SI{10}{mm} $\\times$ \\SI{3}{mm} and magnetization \\SI{6.6(1)e5}{A \\per m}. The stacks are placed around the center of the 2D-MOT at the positions $\\mathbf{r}_\\text{m} = \\pm \\mathbf{x}_0 \\pm \\mathbf{y}_0$ where $x_0 = \\SI{110}{mm}$ and $y_0 = \\SI{90}{mm}$. \nThe magnetization of each permanent magnet has been oriented in such a way that it has the same direction with the one along the $y$-axis and opposite direction with the one faced along the $x$ axis. We estimated the generated field upon the 2D-MOT plane by finite element analysis (FEA). This shows a uniform linear gradient $\\mathbf{B}_\\text{m} (\\mathbf{r}) = b \\mathbf{x } - b \\mathbf{z}$ close to the center of the trap $|\\mathbf{r}|<\\SI{1}{cm}$ with $b=\\SI{0.224}{T \\per m}$. As compared to the above expression the maximum deviation of the actual magnetic field is negligible within the 2D-MOT trapping volume, as it ultimately amounts to $\\Delta_\\text{Z} = 2 \\pi\\times \\SI{5.6}{MHz} = \\SI{0.17}{\\Gamma}$ in frequency detuning.\n\n\nThe two pairs of counterpropagating beams allow magneto-optical cooling and trapping of slow atoms effusing from the oven along the $x$ and $z$ axes, while they are free to drift along the $y$ direction. Hence a nearly-resonant laser ``push'' beam is directed to the 2D-MOT center along the $y$-axis toward the UHV science cell in order to launch atoms collected in the 2D-MOT towards the MOT region. The center of the MOT in the science cell is located \\SI{370}{mm} far from the 2D-MOT center. Finally, the mandatory MOT quadrupolar field is generated by a pair of coils with the current flowing in the Anti-Helmholtz configuration, which generates a typical magnetic field gradient of \\SI{0.4}{T \\per m}. \n\n\n\n\n\\subsection{Laser system}\n\nA schematic drawing of the laser system is shown in Fig.~\\ref{fig:blue_optical_setup}.\nThe \\SI{461}{nm} laser is provided by a semiconductor-based commercial laser composed of an infrared master laser, a tapered amplifier and a second harmonic generation cavity. It is able to generate up to \\SI{600}{mW} of blue power. This blue laser is split in six main optical paths and frequency manipulated by acousto-optic modulators (AOMs). The laser frequency is stabilized to the Sr atomic transition $^1$S$_0$\\,--\\,$^1$P$_1$~by performing wavelength modulation saturation spectroscopy on an hot vapour of strontium generated in a heatpipe~\\cite{Poli2006}.\nTypically we are able to deliver about half of the available power to the atoms.\n\nA detailed scheme of the various beam paths is depicted in Fig.\\ref{fig:blue_optical_setup}. For typical experimental conditions, the 2D-MOT and sideband beams share \\SI{200}{mW} and have a $1\/e^2$ beam width $w_\\text{2D} =$~\\SI{9.5}{mm}, the MOT beams have a total power of \\SI{45}{mW} with a beam width of \\SI{6.2}{mm} and a detuning from the atomic resonance of -1.2 $\\Gamma$, the push beam has a power up to \\SI{5}{mW} and a beam width of \\SI{0.81}{mm}, the spectroscopy beam sent inside the heatpipe has a power of \\SI{0.5}{mW} and beam width \\SI{0.37}{mm}, and finally the probe beam has \\SI{0.5}{mW} of power and \\SI{0.83}{mm} width. The detuning from the atomic resonance of the beams used in atomic source system (2D-MOT, sideband and push beams) have been scanned for optimal atomic flux, as described in Sec.\\ref{sec:atomic source characterization}.\n\n\n\n\n\n\n \\begin{figure}[tb]\n \\centering\n \\includegraphics[width=0.4\\textwidth]{Figure2_art.pdf}\n \\caption{Optical setup of the blue laser system used for cooling, trapping and probe ultracold strontium atoms. Each acousto-optical modulator (AOM) is drawn with its driving frequency (MHz) and the sign corresponds to the diffraction order. Beam shaping lenses are not shown.}\n \\label{fig:blue_optical_setup}\n\\end{figure}\n\n \nWe generate the 2D-MOT main and sideband beams as follows: two dedicated \\SI{200}{MHz} and \\SI{350}{MHz} AOMs are employed to shift the frequencies of two beams, which are shaped with the same telescope in order to have the same beam width. They are combined in a polarizing beam splitter (PBS) cube with orthogonal linear polarizations in such a way that the 2D-MOT (sideband) beam is completely transmitted (reflected). The two beam polarization is then rotated 45$^{\\circ}$ by a half-wavelength retarding waveplate, thus the two beams are recombined into a second PBS which yields the two beams for the branches of the 2D optical molasses. The \\SI{350}{MHz} is dedicated to the sideband beam, offering a \\SI{100}{MHz} bandwidth to find the optimal frequency which maximizes the loading of the atomic source.\n\n\n\n\\section{Numerical simulation of the 2D-MOT\\label{sec:numerical simulation}}\n\n\nMonte Carlo (MC) simulation is a powerful and versatile numerical approach because it allows to study complex physical processes in a realistic environment: from the simple MOT capture process \\cite{Wohlleben2001,Kohel2003,Chaudhuri2006,Szulc2016}, to the loading process of an optical potential \\cite{Hanley2017,Mu2010}, a molecular MOT \\cite{Comparat2014} and Rydberg-dressed MOT \\cite{Bounds2018}. Knowing the atom-light interactions and the geometry of the system, we want to extract the capture efficiency of our 2D-MOT system at a given trapping configuration, defined by the 2D-MOT beams, sideband beams and push beam, as described in the previous section. The MC algorithm is implemented in Python language.\n\n We simulate $N_\\text{sim}=2 \\times 10^4$ trajectories of atoms that interact with the trap. At initial time $t=0$ the starting positions of atoms are randomly sampled in a disk region of radius $r_0 =\\SI{7.5}{mm}$ in the $(y,z)$ plane and at $x_0=\\SI{-128}{mm}$ far from 2D-MOT trap center along the direction of hot atomic flux emitted by the oven. The velocity space is sampled from the Maxwell-Boltzmann probability distribution expressed in polar coordinates. The sampling of the absolute value of the starting atomic velocity $v_0$ is limited to $v_\\text{cut} =\\SI{90}{m \\per s}$ to speed up the calculation. The polar angle $\\theta_0$ is uniformly sampled considering the geometrical constraint imposed by the capillaries $\\theta_0 \\leq \\theta_\\text{cap}$. The azimuthal angle $\\phi_0$ is randomly chosen between 0 and $2 \\pi$.\n\nThe trajectory is discretized in time with a step size $\\delta t = \\SI{50}{\\micro s}$, and computed until $t_\\text{tot}=\\SI{4}{ms}$ by using a Runge-Kutta algoritm~\\cite{Enright1989}. The time step $\\delta t$ is chosen to be greater than the internal atomic time scale $\\tau_{^1P_1} = \\Gamma^{-1}$, so that the atom-light interaction can be calculated by using the semi-classical approximation of the Optical Bloch Equations, but shorter than the capture time for an atom moving at $v_\\text{max}$ which is about $\\Delta t_\\text{max}$ = 165 $\\mu$s. At each time step $t_i=i \\delta t$, the atom-light scattering rate with a single laser beam is computed as:\n\\begin{equation}\n R(t_i) = \\frac{\\Gamma}{2} \\frac{s(\\mathbf{r}(t_i))}{1 + s(\\mathbf{r}(t_i)) + 4\\left(\\frac{ \\Delta_\\text{eff}(\\mathbf{r}(t_i),\\mathbf{v}(t_i))}{ \\Gamma}\\right)^2}\n \\label{eq:scattergin_rate}\n\\end{equation}\nwhere $s(\\mathbf{r}(t_i))$ is the position-dependent saturation parameter, and $\\Delta(\\mathbf{r}(t_i),\\mathbf{v}(t_i))$ is the frequency detuning due to the Doppler and Zeeman shift. \nThe local saturation parameter is computed as:\n\\begin{equation}\n\\qquad s(\\mathbf{r}(t_i)) = s_0 \\exp \\left( -\\frac{ 2 |\\mathbf{r}(t_i) \\times \\mathbf{\\hat{k}}|^2}{ w^2}\\right),\n\\end{equation}\nwhere $s_0$ is the saturation peak, $w$ is the width of the optical beam and where the vector product $\\mathbf{r} \\times \\mathbf{\\hat{k}}$ is the distance between the atom position and the center of the laser line propagation described by the unitary vector $\\mathbf{\\hat{k}}$. Considering the aperture of the optics elements, a spatial cut-off of $|\\mathbf{r} \\times \\mathbf{\\hat{k}}|< \\SI{1.2}{cm}$ in the local saturation parameter is also applied.\nThe frequency detuning is computed as:\n\\begin{equation}\n \\Delta_\\text{eff}(\\mathbf{r}(t_i),\\mathbf{v}(t_i)) = \\Delta + \\mathbf{k}\\cdot \\mathbf{v}(t_i) -\\frac{\\mu_B}{\\hbar}| \\mathbf{B}(\\mathbf{r(t_i}))|\n\\end{equation}\nwhere $\\Delta$ is the laser frequency detuning from the atomic transition, $\\mathbf{k}\\cdot \\mathbf{v}(t_i)$ is the Doppler shift and the last term is the Zeeman shift induced by the atomic position in the magnetic field $\\mathbf{B}(\\mathbf{r}(t_i))$ described in Sec.\\ref{sec:apparatus}.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.49\\textwidth]{mc_sim.pdf}\n \\caption{Monte Carlo simulation of the velocity and position transverse coordinates of the 2D-MOT generated atomic beam at the end of the simulation time $t_\\text{tot}$. Density maps of the captured trajectories for (a) the single frequency 2D-MOT ($s_\\text{2D}$ = $s_{tot}$ = 6.56, $\\Delta_\\text{2D}$ = -1.6 $\\Gamma$), and (b) adding a sideband beam ($s_{SB}$ = 3.45, $s_{2D} = s_{tot} -s_{SB}$, $\\Delta_{SB}$ = -3.1 $\\Gamma$).}\n \\label{fig:mc_sim_hist}\n\\end{figure}\n\n\nThe heating induced by the spontaneous emission process is also taken into consideration in the simulated dynamics by adding a random recoil momentum $\\hbar |k| \\sqrt{R \\delta t} \\ \\hat{\\mathbf{e}}$, where ${R\\delta t}$ is the average number of scattering events in a time interval $\\delta t$, while $\\mathbf{\\hat{e}}$ is a unitary vector randomly chosen from an isotropic distribution~\\cite{Kohel2003}. The resulting atom's acceleration induced by the 2D-MOT (and sideband) beams is described according to\n\\begin{equation}\n \\mathbf{a}_\\text{2D,SB}= \\frac{\\hbar |k| }{m} \\sum_{n=0}^{4} \\frac{R_n}{4} \\left[ \\hat{\\mathbf{k}}_n + \\frac{\\hat{\\mathbf{e}}_n}{\\sqrt{R_n \\delta t \/ 4}}\\right], \n\\end{equation}\nwhere the saturation peak $s_\\text{2D,SB}$ is redistributed equally among the 4 beams of the 2D-MOT and sideband and the beam directions $\\mathbf{\\hat{k}}_n$ are described by the 4 combinations of the unitary vectors $(\\pm \\mathbf{\\hat{x}} \\pm \\mathbf{\\hat{z}}) \/ \\sqrt{2}$.\nThe acceleration induced by the push beam is computed as:\n\\begin{equation}\n \\mathbf{a}_\\text{push}= \\frac{\\hbar |k| }{m} R_\\text{push} \\left[ \\hat{\\mathbf{y}} + \\frac{\\hat{\\mathbf{e}}}{\\sqrt{R_\\text{push} \\delta t }}\\right] .\n\\end{equation}\n\nThe total acceleration $\\mathbf{a}(t_i)$ exerted on the atom at position $\\mathbf{r}(t_i)$ with velocity $\\mathbf{v}(t_i)$ is quantified as the sum of the above processes:\n\\begin{equation}\n \\mathbf{a} = \\mathbf{a}_\\text{2D} + \n \\mathbf{a}_\\text{SB} +\n \\mathbf{a}_\\text{push} \n\\end{equation}\n\n\n\nOnce $t=t_\\text{tot}$, each simulated atom is considered captured in the MOT if the divergence of the atomic trajectory computed along the push direction is lower that the geometrical constraint imposed by the MOT capture angle $\\theta_\\text{MOT} = \\SI{16}{\\milli rad}$ and if the final longitudinal velocity is below the MOT capture velocity $v^\\text{MC}_\\text{capt} =\\SI{60}{m \\per s}$. In the selection of the captured trajectories, we also considered the losses due to collisions with hot atoms from the thermal beam, whose time scale is calculated to be $\\tau_\\text{coll}=\\SI{50}{ms}$. Hence for each trajectory the collision probability is estimated as $p_\\text{coll} = 1-e^{- \\tau_\\text{2D} \/ \\tau_\\text{coll}}$ and a unitary random number $\\varepsilon$ is generated in order to accept ($\\varepsilon>p_\\text{coll}$) or reject ($\\varepsilon6$, the number of atoms in the MOT starts to saturate, however much later than the unity value. The same result is predicted by the MC simulation.\n\nWe observed an optimal push intensity around the $s_\\text{push} \\simeq 0.34$, beyond this value the MOT number of atoms decreases, as previously verified in a similar setup~\\cite{Nosske2017}. The reduced efficiency in the transfer from the 2D-MOT to the blue MOT is explained considering that atoms accelerate beyond $v_\\text{capt}$ cannot be captured in the MOT. This behaviour at higher $s_\\text{push}$ is also observed in the MC simulation considering only the atom captured in the MOT at longitudinal $v_L$ velocity below $v^\\text{MC}_\\text{capt} \\sim \\SI{60}{ \\per s}$. Fig.\\ref{fig:2DMOT_char}(c) shows the MOT number of atoms as a function of the push beam detuning. From this plot we observe that the best transfer efficiency is obtained near the atomic resonance $\\Delta_\\text{push} = 0$, but it is not a critical parameter.\n\nAt the best trapping configuration the total atomic flux generated by the 2D-MOT source is measured by detecting the fluorescence generated by a probe beam sent along the $z$-direction,\nnearly at the center of the MOT in the science chamber. The resulting atomic flux $\\Phi_\\text{2D}$ reaches a maximum value of \\SI{6(1)e8}{atoms \\per s}, as shown in Fig.~\\ref{fig:2Dflux}. This can be compared with the MOT loading rate $L_\\text{MOT}$ and with the expected flow resulting from the capture efficiency ratio resulting from MC simulations. The MOT loading rate is simply given by $L_\\text{MOT}=N_0\/\\tau =\\SI{3.1(4)e8}{atoms \\per s}$, where $\\tau$ is the MOT relaxation time, which in our system without repumping is \\SI{17(2)}{ms} and $N_0=\\SI{5.3(2)e6}{atoms}$ is the maximum number of atoms trapped in the final MOT. It corresponds to roughly \\SI{51}{\\percent} of the total flux. The expected atomic flux generated by the 2D-MOT can be estimated as \n$$\n\\Phi^{(th)}_\\text{2D} = r f_\\text{cut} p_\\text{rad} \\Phi_\\text{ov} = \\SI{1.5e9}{atoms \\per s}.\n$$ \nIn this estimate we used $r=\\num{6.9e-2}$ from MC results, $f_\\text{cut}=\\num{1.53e-3}$ is the fraction of simulated velocities from the Maxwell-Boltzmann distribution considering a cut-off at $v_\\text{cut}=\\SI{90}{m \\per s}$, $p_\\text{rad}$ is the survival probability from optical pumping to the metastable $^3$P$_2$ state~\\cite{Xu2003} that we calculated considering a typical time spent in the 2D-MOT region $\\expval{ \\tau_\\text{2D} }= \\SI{2.9(6)}{ms}$ and a pumping rate $R=\\SI{223}{Hz}$, which give us $p_\\text{rad}\\simeq 1 - R \\expval{ \\tau_\\text{2D}} = 0.35$. The estimated theoretical flux $\\Phi^{(th)}_\\text{2D}$ provides a discrepancy from the measured $\\Phi_\\text{2D}$ of only a factor 2.4, which is remarkably close. In fact our simulations do not consider effects due to experimental imperfections, such as the misalignment of the zero magnetic field of the permanent magnets and the optimal push beam direction for optical transfer to the MOT in the science chamber\n\n \n\n\n\\begin{figure}[!b]\n \\includegraphics[width=0.45\\textwidth]{fig_atomic_flux.pdf}\n \\caption{Atomic flux generated from the atomic source as a function of the 2D-MOT saturation parameter, without (blue circles) and with (red diamonds) the use of the sideband ($P_\\text{SB}$ = 200 mW - $P_\\text{2D}$). These data are taken at $s_\\text{push} $ = 0.34, $\\Delta_\\text{2D}\/\\Gamma$ = -1.6 and compared with MC estimates (shaded lines).}\n \\label{fig:2Dflux}\n\\end{figure}\n\n\n\n\n\n\\section{Sideband enhancement \\label{sec:sideband enhancement}}\n\n\\subsection{Loading a MOT with sideband-enhancement}\n\nWe demonstrated sideband-enhanced loading of a 2D-MOT atomic source by overlapping a second laser beam with higher frequency detuning to the 2D-MOT cooling lasers, as described in Sec.\\ref{sec:apparatus}. Fig.\\ref{fig:sideband} shows how the power distribution between the two frequencies affects the number of atoms collected in the MOT trap at sideband detuning $\\Delta_\\text{SB}=\\SI{ -3.13}{\\Gamma}$, while the 2D-MOT beam is tuned at its previously shown maximum $\\Delta_\\text{2D}=\\SI{-1.6}{\\Gamma}$. From Fig.\\ref{fig:sideband} we see that it exists an optimal power distribution around $s_\\text{SB} \\simeq 3.5$ that maximizes the number of atoms in trapped in the MOT, reaching up to \\SI{1.2e7}{atoms} atoms, i.e., about \\num{2.3} times higher than with the total available power sent to the 2D-MOT AOM and about \\num{4} times higher than the corresponding value with the sideband beam blocked.\n\n\n\\begin{figure}[b]\n \\centering\n \\includegraphics[width=0.49\\textwidth]{fig_enhancing.pdf}\n \\caption{Number of atoms captured in the MOT region for different power distribution of the $s_\\text{tot}=6.56$ between the sideband beam $s_\\text{SB}$ and the 2D-MOT beam $s_\\text{2D}$. All the data are measured with a 2D-MOT detuning $\\Delta_\\text{2D} = -1.6\\,\\Gamma$ and sideband detuning $\\Delta_\\text{SB}= \\SI{-3.13}{\\Gamma}$. The (red) diamonds are the number of atoms trapped in the MOT at increasing sideband beam saturation parameter $s_\\text{SB}$. The (pink) shaded area represents the MC simulation in this experimental configuration. The (blue) circles describe corresponding number of atoms trapped in the MOT with the sideband beam AOM turned off.}\n \\label{fig:sideband}\n\\end{figure}\n\n\nIn order to find the optimal working point of the sideband-enhanced 2D-MOT, we scanned over the sideband AOM frequency from \\SI{230}{MHz} to \\SI{335}{MHz}, which corresponds to a detuning range between $-5\\Gamma$ and $-2.2\\Gamma$, and measured the MOT trapped atoms $N$ at different sideband power. We performed this scan with a total power $P_\\text{tot}=\\SI{200}{mW}$ and \\SI{110}{mW}, i.e. a total saturation parameter $s_\\text{tot}=6.56$ and $s_\\text{tot}=3.61$, respectively. We introduce the enhancement parameter $\\eta$ as: \n\\begin{equation}\n \\eta(s_\\text{SB} , \\Delta_\\text{SB})= \\frac{N(s_\\text{SB}= s_\\text{tot} - s_\\text{2D} , \\Delta_\\text{SB})}{ N(s_\\text{2D} = s_\\text{tot} ) }\n \\label{eq:enhacement_factor}\n\\end{equation}\nThe so-defined $\\eta$ parameter compares the two different trapping configurations, both sharing the same total optical power $s_\\text{tot}$. When $\\eta > 1$ sideband-enhancement is achieved. \n\nFig.\\ref{fig:SE_scan}(a) and (c) show two sets of the sideband enhancement parameter scan, where we plot the enhancement parameter ($\\eta$) with respect to $s_\\text{SB}$ and $\\Delta_\\text{SB}$ when $P_\\text{tot} = \\SI{200}{mW}$ and \\SI{110}{mW} respectively. These results are compared to their respective MC simulations (Fig.\\ref{fig:SE_scan}(b) and (d)). The data show that optimum loading efficiency of the final MOT is reached tuning the sideband frequency to $\\Delta_\\text{SB} = \\SI{-3.13}{\\Gamma}$ for both the total power regimes. At $P_\\text{tot}=\\SI{200}{mW}$, we reached a maximum enhancement of $\\eta^\\text{exp}=2.3(1)$ when $s_\\text{SB} \\simeq 3.1$ ($P_\\text{SB}\\simeq \\SI{90}{mW}$). The MC numerical data present essentially the same main features as the experimental measurements, both having the maximum loading at the same sideband parameter point, reaching a slightly lower enhancement $\\eta^\\text{MC} = 2.13(4)$, as detailed in Fig.\\ref{fig:sideband}. At $P_\\text{tot}=\\SI{110}{mW}$, we obtained the best enhancement factor of $\\eta^\\text{exp}=1.48(7)$ when $s_\\text{SB}=1.3$ ($P_\\text{SB} \\simeq \\SI{40}{mW} $), while the MC numerical results show a slightly higher enhancement of $\\eta^\\text{MC}=1.87(3)$. Both numerical and experimental data suggest that for $s_\\text{tot}\\geq 1$, sideband-enhancement grows with the increasing available power, where the optimized power distribution can be more effective. Alternatively, the rate at which the atomic flux increases with respect to the laser power is significantly higher for the case where we add the higher-detuned frequency sideband.\n\nAt maximum $\\eta^\\text{exp}$ = 2.3(1), we measured $N=\\SI{1.25(4)e7}{atoms}$ trapped in the MOT, which corresponds to a loading rate of $L_\\text{MOT}^\\text{SB}=\\SI{7.3(9)e8}{atoms \\per s}$. A total flux measurement by fluorescence detection is also performed for the sideband-enhanced 2D-MOT and reported in Fig.~\\ref{fig:2Dflux}. In this case we measured an atomic flux of $\\Phi_\\text{SB} = \\SI{1.5(2)e9}{atoms \\per s}$. The related enhancement factor is \\num{2.4(4)} and it is in agreement with the experimental and numerical results reported for the MOT loading.\n\nCompared to other Sr atomic sources, our sideband-enhanced 2D-MOT source shows high transfer efficiency $L_\\text{MOT}^\\text{SB}\/\\Phi_\\text{SB}$ = 48(8)\\%, with a MOT loading rate slightly larger than a ZS-enhanced Sr 2D-MOT source~\\cite{Nosske2017}, and less than a factor ten lower than more complex and power-demanding high-flux source systems based, for instance, on a combination of Zeeamn slower, 2D-MOT and deflection~\\cite{Yang2015}. \n\n\n\n\n\\begin{figure*\n \\centering\n \\includegraphics[width=\\textwidth]{fig_sideband_enhancing_factor_all.pdf}\n \n \n \\caption{Enhancement factor $\\eta$ as function of the sideband parameter scan. (a) Experimental data compared to (b) MC numerical results at fixed total saturation parameter $s_\\text{tot}=6.56$ and $\\Delta_\\text{2D}=-1.6\\,\\Gamma$. (c) and (d) same as (a) and (b) but for $s_\\text{tot}$ = 3.6.}\n \\label{fig:SE_scan}\n\\end{figure*}\n\n\n\\subsection{Kinetic properties of the sideband-enhanced 2D-MOT}\n\nThe kinetic properties of a Sr cold atomic beam generated by a 2D-MOT has been recently studied~\\cite{Nosske2017}, in particular as function of the push beam and 2D-MOT beam intensities. We verified these findings in our setup and extended the study to the addition of the sideband beam. \n\nThe longitudinal velocity was measured by time-of-flight technique. A push beam pulse of \\SI{5}{ms} accelerate the 2D-MOT atoms towards the science cell. The longitudinal velocity distribution is estimated recording the fluorescence time distribution $f(t)$\nas measured at the MOT center\nWe compute the longitudinal velocity distribution as $f(v) = f(d\/t)$, where $d = \\SI{36.5(5)}{cm}$ is the 2D-MOT to MOT distance. Compared to the single-frequency 2D-MOT, we did not observe any change in peak velocity or in velocity dispersion. The peak velocity for optimal push saturation parameter $s_\\text{push} = 0.34$ is $v_\\text{L} = \\SI{22.5}{m \\per s}$.\n\n\\begin{figure}[b]\n \\centering\n \\includegraphics[width=0.49\\textwidth]{temp_art.pdf}\n \\caption{Doppler spectrum of the transverse spectroscopy on the 2D-MOT Sr atomic beam. The inset shows the corresponding Doppler temperatures compared to the MC simulation results and the corresponding Doppler limit.}\n \\label{fig:transvel}\n\\end{figure}\n\n\nThe atomic beam transverse velocity was measured by Doppler spectroscopy with and without powering the sideband beam. The transverse velocity was extracted from the Doppler profile by fixing the Lorentzian component due to the natural linewidth of the $^1$S$_0$\\,--\\,$^1$P$_1$~probe transition and the saturation broadening ($s_\\text{probe} = 0.1$). The measurement results are shown in Fig.\\ref{fig:transvel}, yielding a Doppler broadening $\\sigma_\\text{2D}(T) = \\SI{3.6(8)}{MHz}$ and $\\sigma_\\text{SB}(T) = 0.8(3.0)\\,\\text{MHz}$ respectively. This corresponds to a transverse temperature of \\SI{14(7)}{mK} for the 2D-MOT and $0.7(5.0)\\,\\text{mK}$ for the sideband-enhanced 2D-MOT, as shown in the inset in Fig.\\ref{fig:transvel}. Compared to the Doppler temperature at $s_\\text{2D} = 6.6$ and $\\Delta_\\text{2D}=-\\SI{1.6}{\\Gamma}$ which is equal to \\SI{2.1}{mK}, the 2D-MOT result is nearly seven times warmer, while the sideband-enhanced case shows an upper limit temperature almost three times higher. We also estimated the transverse temperature of the atomic beam resulting from MC numerical simulations, which present transverse temperatures of \\SI{2.9(1)}{mK} and \\SI{2.12(4)}{mK} respectively. While MC results confirms a colder beam for the sideband-enhanced case, they still miss the extra-heating effects which can be explained by transverse spatial intensity fluctuations of the optical molasses in the 2D-MOT~\\cite{Chaneliere05}.\n\nFrom the measured transverse and longitudinal velocities, we derive an atomic beam divergence $\\theta_\\text{2D} \\equiv v_t\/v_L = \\SI{75(17)}{ mrad}$ and $\\theta_\\text{SB}\\leq \\SI{58}{mrad}$. Finally, from the values of the beam divergence and the atomic flux, we estimated the atomic beam radiant intensity, or sometimes called beam ``brightness'', $\\mathcal{J}\\equiv \\Phi \/(\\pi\\theta^2)$. For the single-frequency 2D-MOT we obtained a brightness\n\n$$\n\\mathcal{J}_\\text{2D} = 3.5(8)\\times 10^{10} \\,\\mathrm{atoms}\\cdot\\mathrm{s}^{-1}\\cdot\\mathrm{sr}^{-1},\n$$\nwhile, for the sideband-enhanced beam, the brightness\n\n$$\n\\mathcal{J}_\\text{SB} \\geq 1.4\\times 10^{11}\\,\\mathrm{atoms}\\cdot\\mathrm{s}^{-1}\\cdot\\mathrm{sr}^{-1}.\n$$\n\nThis results represents a factor four improvement with respect to the single-frequency 2D-MOT, making the sideband-enhancement a promising technique for optimal transfer to 2D optical molasses working on narrow linewidth intercombination transition of strontium for continuous BEC production and continuous optical clock proposals~\\cite{Bennetts17}.\n\n\n\n\\subsection{Comparison with the Zeeman-slower enhancement}\n\nAn alternative method to increase the 2D-MOT capture rate is to direct another slowing beam towards the hot atomic beam generated by oven which, exploiting the decreasing tail of the 2D-MOT magnetic field, can efficiently scatter faster atoms along the beam direction similarly to a Zeeman Slower (ZS). This approach was previously demonstrated in similar setups~\\cite{Lamporesi2013,Nosske2018,Colzi2018}.\n\nWe employed the beam generated by the sideband AOM as Zeeman slowing beam, shaped to have a beam width $w_{ZS} = \\SI{6}{mm}$. We partially scanned over the ZS parameters, which resulted in a maximum number of atoms in the MOT of $N_\\text{ZS} = \\SI{3.8(1)e6}{atoms}$, obtained with $\\Delta_\\text{ZS}=\\SI{-8.1}{\\Gamma}$, $P_\\text{ZS} =\\SI{140}{mW}$, while we kept $P_\\text{2D} =\\SI{36}{mW}$ and $\\Delta_\\text{2D}=\\SI{-1.5}{\\Gamma}$ fixed. Blocking the ZS beam we observe a gain in the atomic number of the order of 4, in agreement with the experimental observation in~\\cite{Nosske2017}.\n\nBecause of the short distance (about \\SI{25}{cm}) between the oven aperture and the optical window facing it to send the ZS beam, we heated the window flange up to \\SI{350}{\\celsius} in order to prevent metalization. However we first observed a fast degradation of the atomic source flux, soon followed by the full metalization of the window. This prevented us to perform a fine optimization of the $s_\\text{ZS}$ and $\\Delta_\\text{ZS}$ as the one shown in the Fig.~\\ref{fig:SE_scan} and also to produce a stable MOT during the day.\n\nA possible way to reduce Sr metalization of the ZS window would be to increase the distance between the oven and window itself by means of a vacuum extension (at least a \\SI{1}{m} tube with shorter diameter). However this solution would compromise the compactness of the atomic source conceived by a 2D-MOT making the system more complex, power consuming and perhaps needing extra water cooling to avoid thermal stress to the vacuum system.\n\nAnother drawback of the ZS method is the fact that the quantization axis of the magnetic field along the hot atoms direction imposes that only one half of the linear polarized ZS light power has the correct circular polarization. This means that at least half of the ZS optical power is wasted in the slowing process. On the contrary, the sideband beams have a well defined polarization in the capture region so that all the employed power is effective in the cooling and trapping process.\n\n\\subsection{Application to other alkaline-earth atoms and prospects for optical clocks}\n\nIt is interesting to extend the discussion about the sideband-enhancement method to other atomic species, in particular those employed in optical clocks. We exploit the MC simulation\nin order to investigate the potential trapping performances of additional atomic species. Table~\\ref{tab:alkaline-earth} shows the main optical and atomic parameters for alkaline-earth(-like) atomic species currently in use in optical clock experiments. In particular, we consider the broad $^1$S$_0$\\,--\\,$^1$P$_1$~strong dipole transition as cooling transition, fixing the atomic vapour pressure to \\SI{0.1}{Pa} for all the species. We ran our MC simulation with these parameters, with a total available saturation parameter $s_\\text{tot}$ = 6.56, the same magnetic field gradient and the same laser beam widths as for our previously described apparatus.\n\nThe simulation workflow is the following: first we simulate the single-frequency 2D-MOT, looking for the optimal detuning $\\Delta_\\text{2D}$ at half of $s_\\text{tot}$; then we add the sideband at $\\Delta_\\text{SB} = 2\\Delta_\\text{2D}$, which is basically the result we found in Sec.\\ref{sec:sideband enhancement} for Sr, and we scan the sideband-enhanced 2D-MOT at different sideband saturation parameter $s_\\text{SB}$. Because of the extremely high saturation intensities of Cd and Hg which makes unrealistic the application of this method, they were excluded from this numerical study.\n\n\n\\begin{table*}[tb]\n\\centering\n\\begin{tabular}{l c c c c c c |c c c}\n\\hline\n\\hline\nAtom & $\\lambda$ & $\\Gamma\/2\\pi$ & $I_\\text{sat}$ & $a_\\text{max}$ & $T(p_0)$& $v_{th}(p_0)$ & $\\Delta_\\text{2D}\/\\Gamma$ & $r^\\text{MC}f_\\text{cut}$ & $\\eta$\\\\\n & (nm) & (MHz) & (mW\/cm$^2$) & ($10^6$ m\/s$^2$) & (K) & (m\/s)&&(ppm)\\\\\n\\hline\n$^{24}$Mg & 285.30 & 80.95 & 455 & 14.8& 641 & 679 & -2.28& 77 & 1.0\\\\\n$^{40}$Ca & 422.79 & 34.63 & 59.9& 2.69 & 788 & 583 & -2.5 & 103& 2.1\\\\\n$^{88}$Sr & 460.86 & 31.99 & 42.7 & 1.01 & 725 & 379 &-1.76& 260& 2.1\\\\\n$^{138}$Ba & 553.70 & 18.33 & 14.1& 0.31 & 826 & 321&-0.89& 60&2.9 \\\\\n$^{114}$Cd & 228 & 91& 1005 & 4.64 & 485 & 273\\\\\n$^{174}$Yb & 398.91 & 29 & 59.8 & 0.54 & 673 & 258&-1.42& 316& 2.2\\\\\n$^{198}$Hg & 185 & 120 & 2481 & 4.24 & 286 & 157\\\\\n\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Sideband enhancement on alkaline-earth(-like) atomic species. On the left side of the table the most relevant parameters for cooling and trapping atoms on the $^1$S$_0$\\,--\\,$^1$P$_1$~ strong dipole transition are shown, whereas we estimate the thermal and kinetic properties of every atomic species at a pressure $p_0$ = 0.1 Pa. On the right, MC optimization results of the 2D-MOT detuning $\\Delta_\\text{2D}$, the fraction of trapped atoms, and the enhancement factor $\\eta$ for each alkaline-earth species.}\n\\label{tab:alkaline-earth}\n\\end{table*}\n\nThe MC simulation results are reported in Tab.~\\ref{tab:alkaline-earth}. Here we clearly see that the sideband-enhancement is more effective for those atoms having a lower value of the maximum acceleration $a_\\text{max}$. This dependence can be understood by looking at the definition of maximum capture velocity in (\\ref{eq:vc}). In fact, it can be only achieved for a light field uniformly resonant with the atomic transition and fully saturated all along the trap diameter. This means that the broader the cooling transition linewidth $\\Gamma$ is (and thus the higher $a_{max}$), the closer the MOT is to its capture limit, implying that the expected enhancement factor is lower. Furthermore, according to Eq.~\\ref{eq:cap_2d}, we would expect that the sideband-enhancement works better for light species, in particular where $v_\\text{th}$ is higher and the quartic dependence of the loading rate on the capture velocity is a more accurate approximation. Hence we can work out a sideband-enhancement factor functional dependence\n\n$$\n\\eta(X) \\propto \\frac{v_\\text{th}(X)^2}{a_\\text{max}(X)}\n$$\nwhere $X$ is the considered atomic species. By accident, either Sr, Ca and Yb have very similar ${v_\\text{th}(X)^2}\/{a_\\text{max}(X)}$ values, and the resulting $\\eta$ is the same within the numerical error.\n\nWe also report in Tab.\\ref{tab:alkaline-earth} the absolute capture efficiency for the sideband-enhanced 2D-MOT atomic source $r(X)$. MC simulations show that the highest capture rate is predicted for Yb followed by Sr, two of the strongest candidates for a possible redefinition of the second based on optical atomic clocks~\\cite{Riehle2015}.\n\n\n\n\\section{Conclusions}\\label{sec:conclusion}\n\nIn this work we demonstrated and fully characterized a robust method to enhance the atomic flux generated by a Sr 2D-MOT by adding a second frequency to the 2D-MOT beams. The experimental implementation of the sideband-enhancement method only requires a simple optical setup and a proper alignment of the sideband beam to the main 2D-MOT beam. The resulting bright atomic source can deliver more than $1.4\\times 10^{11}\\,\\mathrm{atoms}\\cdot\\mathrm{s}^{-1}\\cdot\\mathrm{sr}^{-1}$ if the total available power for the atomic source is 200 mW. This cold atomic flux can be efficiently loaded in a 3D MOT for ultracold atoms experiments, preventing direct sight to the hot atomic oven and providing an efficient optical shutter of the atomic beam. This result represents an enhancement in MOT loading by a factor 2.3 with respect to single-frequency 2D-MOT based atomic source.\n\nA dedicated Monte Carlo simulation, which well predicts the experimental data of our Sr atomic source, shows that this technique is a valid method to increase the number of atomic sources based on the other alkaline-earth species such as Yb and Ca, paving the way for compact atomic sources suitable for transportable optical clocks or optical clock transition-based gravimeters~\\cite{Hu17,Akatsuka_2017}.\n\n\n\n\\begin{acknowledgments}\nThe authors would like to thank U. Sterr for inspirational discussions about sideband-enhanced MOT, D. Racca, E. Bertacco, M. Bertinetti and A. Barbone for laboratory assistance. We acknowledge funding of the project EMPIR-USOQS, EMPIR projects are co-funded by the European Union's Horizon2020 research and innovation programme and the EMPIR Participating States.\nWe also acknowledge QuantERA project Q-Clocks, ASI, and Provincia Autonoma di Trento (PAT) for financial support. \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nOptical Character Recognition (OCR) has long been the fundamental task in computer vision. There are many applications such as automatic content extraction for documents \\cite{burie2015icdar2015}, translation \\cite{toyama2014mixed}, text style transfer \\cite{krishnan2021textstylebrush} and assistance for robots and visually impaired users. From the perspective of research, OCR can range from relatively easy and controlled tasks such as digit recognition~\\cite{lecun1998mnist}, to difficult scenarios such as scene text with arbitrary orientations and shapes \\cite{karatzas2015icdar,ch2017total,hassner2013computation,chng2019icdar2019,singh2021textocr}, and has become an important domain for benchmarking new machine learning techniques.\n\nNevertheless, most existing OCR approaches focus on numbers and the English alphabet due to English's status as a common \\textit{lingua franca} and its subsequent wide availability in popular datasets. Thanks to the introduction of multilingual text detection and recognition datasets and benchmarks \\cite{nayef2017icdar2017,nayef2019icdar2019}, there is now a unified platform to measure the model performance on challenging scenarios containing thousands of distinctive characters. Recent methods have shown that training different languages with different recognition heads can improve end-to-end recognition accuracy compared to combining characters from all languages in the same recognition head \\cite{huang2021multiplexed}. However, it's not clear whether an individual recognition head for each language is optimal. For example, should English and Spanish be separated into two heads? The answer is probably no since they share most of the characters. Moreover, even if separating two languages into two heads does yield the best accuracy, it might not be worth it if the accuracy gain is marginal compared to the increase in number of parameters and\/or the inference time. Therefore, one of the questions our work tries to answer is how to decide whether\/how languages should be grouped together under the constraint of a limited number of models.\n\nWithout any pre-assigned grouping, we treat each of the models at initialization as a generalist agent which looks at all tasks. Then, as the scale tips, each agent is encouraged to be increasingly specialized in one or more tasks; each agent becomes a specialist, each model can still try to learn the other tasks to some lesser extent. Due to different transferability, data variation, and similarities among the tasks, each agent can have different progress in both the specialized tasks and non-specialized tasks, and the specialties will be redistributed automatically as the agents evolve. Eventually, as confirmed by our experiments, this multi-agent system will reach an equilibrium where the specialties for each agent do not change any more, and this is when the task grouping result is finalized.\n\nTo summarize, our contributions include:\n\\begin{itemize}\n\\item To our knowledge, this is {\\em the first work} exploring the grouping of languages for multilingual OCR.\n\\item We propose an automatic grouping mechanism that allows dynamic and differentiable routing of tasks to different heads during training.\n\\item We empirically show that the automatic task grouping model outperforms both the one-task-per-head and the all-tasks-in-one-head baselines.\nWe further show that when the models have different capacities, the task assignment can potentially reflect the underlying task complexity and data distribution.\n\\end{itemize}\n\nTo promote reproduction of our work, our code is publicly available at \\url{https:\/\/github.com\/facebookresearch\/MultiplexedOCR}.\n\n\n\\section{Related work}\n\n\\subsection{Multilingual text spotting}\nText spotting systems combine text detection and text recognition modules to identify the location and content of text in images. Early works approached both modules independently: text proposals for regions containing text were generated first, followed by a recognition module to identify the text given a pre-defined character dictionary. For text detection, state-of-the-art (SotA) methods are mostly based on Region Proposal Networks (RPN)~\\cite{ren2015faster}, previously proven successful for object detection. Variants of RPNs have been proposed to account for varying text orientations~\\cite{jiang2017r2cnn}, arbitrary shapes~\\cite{liao2020mask,qin2019towards}, and character masking~\\cite{baek2019character}. For text recognition, models typically use an RNN-style design to predict sequences of characters~\\cite{su2014str,he2016reading}. Representative methods include connectionist temporal classification (CTC)~\\cite{graves2006connectionist} and attention-based decoders \\cite{bahdanau2014neural,lee2016recursive,shi2016robust}. \n\nWhile earlier systems treated detection and recognition as independent modules, most of the recent works train these modules in an end-to-end manner. Given the inter-dependability of these modules, this training methodology results in performance improvements for these systems. Some representative works include Mask TextSpotter~\\cite{liao2021mask}, FOTS~\\cite{liu2018fots}, CharNet~\\cite{xing2019charnet}, etc. For deeper insights into the text spotting systems, we refer the readers to the thorough review in \\cite{long2020scene}. Similar to these works, we also employ end-to-end training for our system. For recognition, we mainly use an attention-based decoder to make fair comparisons with previous works~\\cite{liao2020mask,huang2021multiplexed}. \n\nWith the availability of reliable multilingual datasets such as MLT19~\\cite{nayef2019icdar2019}, text spotting systems have tried to address the problem of multilingual texts. In addition to detection and recognition modules, some multilingual text spotting systems also include a script identification module~\\cite{buvsta2018e2e,huang2021multiplexed} to identify the language for text recognition. While text spotting systems such as E2E-MLT~\\cite{buvsta2018e2e} and CRAFTS~\\cite{baek2020character} present results for multilingual datasets, they do not explicitly incorporate model specific components adapted for multiple languages. Instead, they combine the characters from all languages to form a larger dictionary for the recognition head. Recently, Multiplexed Multilingual Mask TextSpotter (MMMT) \\cite{huang2021multiplexed} proposed to employ different recognition heads for different scripts, routed through a script identification module at the word level. Unlike MMMT, which employs hard assignment for routing to an appropriate recognition head, we propose to group the languages by training agents to route words to different heads in a data-driven fashion. This automatic grouping mechanism allows for dynamic and differentiable shifting of tasks to optimize the language combinations for various recognition heads.\n\n\\subsection{Multitask learning and grouping}\n\nMultitask learning methods have a long history~\\cite{caruana1997multitask,evgeniou2004regularized}. As their name implies, they jointly learn solutions for multiple tasks, sharing or transferring information between tasks to improve overall performance. Recent deep learning methods assume that the parameters for early layers, which account for low-level feature extraction, are shared among different tasks, while the parameters for later layers, which account for high-level integration of visual signals, are task-specific~\\cite{zeiler2014visualizing,inkawhich2020transferable}. Hence, information relevant to all tasks is learned by a shared trunk which later splits to multiple task-specific heads. \n\nA natural question that arises when designing multitask systems is the following: How should tasks be grouped to maximise a model's accuracy? To answer this question, Kang \\emph{et al}\\onedot \\cite{kang2011learning} proposed learning shared feature representations for related tasks by formulating task grouping as a mixed integer programming problem, where binary indicator variables are used to assign tasks to groups. Unlike this hard group assignment, Kumar \\emph{et al}\\onedot \\cite{kumar2012learning} propose to allow for parameter sharing across groups through soft, latent assignment of task features as a linear combination of a finite number of underlying basis tasks. Zhong \\emph{et al}\\onedot \\cite{zhong2016flexible} extend this work by removing constraints on the size of latent basis tasks and adding regularization terms to enforce sparsity in task weights and orthogonality to prohibit commonality among unrelated tasks. Zamir \\emph{et al}\\onedot \\cite{zamir2018taskonomy} proposed a method for modeling the relationship of different visual tasks based on the {\\em transferability} between them. Instead of learning shared representation on a trunk, Strezoski \\emph{et al}\\onedot \\cite{strezoski2019routing} introduce a {\\em Task Routing layer} that masks convolutional channels based on the task, effectively creating a sub-network per task.\n\nOur work is similar in spirit to Strezoski \\emph{et al}\\onedot \\cite{strezoski2019routing} in that we allow for dynamic routing of tasks to different heads during training. In our approach, however, the routing is done using {\\em Gumbel-Softmax} \\cite{jang2016categorical} to ensure probabilistic interpretation of each task and using a novel {\\em grouping loss} for task assignment.\n\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.98\\linewidth]{figures\/system_b.png}\n \\caption{\\textbf{Our proposed task grouping framework.} Here we show a batch of $4$ inputs potentially belonging to $3$ tasks, with $2$ recognition heads. See Sec.~\\ref{sec:method} for more details.}\n \\label{fig:task_grouper}\n\\end{figure*}\n\n\\section{Methodology}\\label{sec:method}\n\nGiven a list of tasks $T = \\{T_i\\}_{1 \\leq i \\leq t}$ and a list of models $M = \\{M_j\\}_{1 \\leq j \\leq m}$, we can define the task grouping of $T$ over $M$ as a mapping $G: T \\rightarrow M$, where $G$ is a single-valued function, which means each task will be assigned to exactly one model. On the other hand, $G$ does not need to be an injection, since multiple tasks can be assigned to the same model. $G$ need not be a surjection either, in which case some models will not be assigned with any tasks. Our goal is to find out the best assignment $G$ such that the overall performance is maximized.\n\nFigure \\ref{fig:task_grouper} shows the core architecture of the proposed task grouping framework. Given an input, which could be a batch of already cropped image patches or pre-extracted features, we first pass them through a {\\em task classification network} that predicts the probabilities of each input belonging to each of the tasks.\nUnder the context of multilingual text recognition, each task $T_i$ can be described as ``recognizing the word instance $W_k$ in language set $L_i$'', where $W_k$ is the $k$-th instance in a batch of $w$ word crops. We can thus define a probability matrix of size $w \\times t$ on the likelihood of each word belonging to each task\/language set:\n\n\\begin{equation} \\label{eq:word_task_matrix}\n P_{WT} = \\{p(T_i | W_k)\\}_{1 \\leq k \\leq w, 1 \\leq i \\leq t}\n\\end{equation}\n\nAt inference time, $P_{WT}$ can be inferred from a task classification network such as a language prediction network \\cite{huang2021multiplexed}. This is a $t$-way classification problem, and the task classification loss can be computed using a cross entropy loss:\n\\begin{equation}\n L_{task}(W_k) = - \\sum_{i=1}^{t} I(T_i=T_{gt}) \\log p(T_i|W_k)\n\\end{equation}\nwhere $I(T_i=T_{gt})$ is a binary indicator of the task matching the ground truth.\n\nAt training time, $P_{WT}$ can be inferred from the ground truth, if there is an annotation of which language each word belongs to: $p(T_i|W_k)$ is $1$ if $W_k$ belongs to $T_i$ and $0$ otherwise. When the ground truth annotation for the language information is not directly available but the transcription is available, we can make an educated guess of the probability by calculating the proportion of characters in $W_k$ that are supported by language set $L_i$.\n\n\\subsection{Grouping module}\n\nSince task-model mapping $G$ is a discrete function, to be able to learn it we can define the following probability matrix, of size $t \\times m$:\n\\begin{equation} \\label{eq:task_model_matrix}\n P_{TM} = \\{p(M_j|T_i)\\}_{1 \\leq i \\leq t, 1 \\leq j \\leq m},\n\\end{equation}\nwhere $p(M_j|T_i)$ is the probability of an arbitrary word belonging to $T_i$ to be handled by model $M_j$. Then, we can compute the probability matrix of each word $W_k$ to be handled by model $M_j$ by multiplying $P_{WT}$ and $P_{TM}$:\n\n\\begin{equation} \\label{eq:word_model_matrix}\n P_{WM} = P_{WT} \\cdot P_{TM}\n\\end{equation}\nNaive task assignment to a group based on traditional SoftMax is a discrete operation and thus non-differentiable.\nBackpropagation through only the selected task-group pairing would result in high variance gradients leading to unstable learning.\nInstead, when computing $P_{WM}$ during training, we apply a soft relaxation of the assignment operation using the Gumbel-Softmax \\cite{jang2016categorical}.\nGumbel-Softmax if fully differentiable with the reparameterization trick and results in gradient backpropagating through all possible task-group pairings, not just the one with the maximum score.\nWe instantiate learnable parameters for task-model assignment as a real-valued matrix $R_{TM} \\in \\mathbb{R}^{t \\times m }$, initialized with all ones (or any equal numbers) in the beginning, and we set the temperature $\\tau = 1.0$ throughout the training. At test time, we can just pick the model corresponding to the maximum, \\emph{i.e}\\onedot the hard mode of Gumbel-Softmax.\n\n\\subsection{Integrated loss}\n\nA key difference of our approach compared to \\cite{huang2021multiplexed} is that in our framework, we do not restrict the capability of each model, or recognition head, to support any specific task, \\emph{i.e}\\onedot, a certain recognition head can only support certain characters, from the beginning. Instead, we assume each model to be omnipotent in the beginning and has the potential to handle every task. This is necessary since otherwise there is no point in doing the grouping if each model is already designed to do certain tasks. \n\nTherefore, unlike \\cite{huang2021multiplexed}, we can directly use the negative log likelihood as the recognition loss $L_{seq(j)}$ for each model $M_j$ without worrying about the unsupported characters:\n\\begin{equation}\n L_{seq} = - \\frac{1}{s} \\sum_{l=1}^{s} \\log p(S_l),\n\\end{equation}\nwhere $p(S_l)$ is the predicted probability of character at position $l$ of the sequence, and $s$ is the length of the sequence of character labels.\nWe can, however, perform the pruning at the output layer to remove any characters that do not belong to the task assigned to certain head, once the grouping is determined. This reduces the unnecessary weights in the final model.\n\nThe integrated loss across all probabilistic instance-model assignments can thus be calculated as the weighted sum of individual losses:\n\n\\begin{equation}\n L_{integrated0} = \\sum_{k=1}^{w}\\sum_{j=1}^{m} p(M_j|W_k) \\cdot L_{seq(j)}(W_k, M_j),\n \\label{eq:integrated_loss_0}\n\\end{equation}\nwhere the probability term is from $P_{WM}$ of Eq. \\eqref{eq:word_model_matrix}, which is essentially the law of total probability:\n\n\\begin{equation}\n p(M_j|W_k) = \\sum_{i=1}^{t} p(T_i|W_k) \\cdot p(M_j|T_i)\n \\label{eq:word_model_element}\n\\end{equation}\n\n\\subsection{Integrated loss with a base loss coefficient}\n\nWith the integrated loss (Eq. \\eqref{eq:integrated_loss_0}), we can see that in general, a task $T_{big}$ with a bigger probability $p(M_j|T_{big})$ to be assigned to a model $M_j$ will contribute a bigger loss than a task $T_{small}$ with a smaller probability $p(M_j|T_{small})$ to be assigned to the model, encouraging the model to optimize towards a better prediction for $T_{big}$, which then encourages $p(M_j|T_{big})$ to be bigger until it reaches $1$. A similar but opposite process applies to $p(M_j|T_{small})$, which would become smaller until it reaches $0$. As a result, the learned task-model assignment $P_{TM}$ will almost certainly be random and fully depending on the first few iterations due to the positive-feedback loop. We resolve this issue by adding a small positive base loss coefficient, $\\epsilon$:\n\n\\begin{equation}\n L_{integrated} = \\sum_{k=1}^{w}\\sum_{j=1}^{m} (p(M_j|W_k)+\\epsilon) \\cdot L_{seq(j)}(W_k, M_j).\n \\label{eq:integrated_loss}\n\\end{equation}\n\nThis ensures that the model not only tries to excel at the tasks assigned to it, but also learns the other tasks at a small but positive rate. The effect of $\\epsilon$ can be quantified from the perspective of training data ratios among different tasks. Assume the original ratio of data from any task is $1$, for any model-task pair, the maximum effective data ratio would be $1 + \\epsilon$, which is achieved when $p$ reaches $1$, and the minimum effective data ratio would be $0 + \\epsilon$, which is achieved when $p$ falls to $0$. The ratio $\\frac{1 + \\epsilon}{\\epsilon}$ can thus be used to measure how biased the model can potentially be trained towards the most vs. least important task. Based on our ablation study (Sec. \\ref{sec:ablation_study}), we set $\\epsilon=0.2$ when training from scratch, $\\epsilon=0.1$ when fine-tuning from pretrained models and $\\epsilon=0$ for the final head-wise fine-tuning.\n\n\\subsection{Grouping loss}\n\nWhile Eq. \\eqref{eq:integrated_loss} makes sure that any model has the potential to learn every task, we also would like to ensure that happens within a certain budget, i.e., given the number of different models (heads) we can support, each model is specialized in at least one task. This ensures we do not waste the modeling capacity of an idle head. Therefore, we introduce the following grouping loss\n\n\\begin{equation}\n L_{group} = \\sum_{j=1}^{m} L_{group(j)} \\\\\n = \\sum_{j=1}^{m}\\max(\\mu_j - \\sum_{i=1}^{t}p(M_j|T_i), 0),\n \\label{eq:grouping_loss}\n\\end{equation}\nwhere $\\mu_j$ is the least number of tasks model $M_j$ is expected to handle. In most experiments, we set $\\mu_j=1$, meaning that if $M_j$ completely takes over at least one task, the grouping loss for $M_j$ would reach the minimum value $0$. Note that the converse does not hold - the grouping loss can reach $0$ even when certain model do not excel in any specific task. However, in practice, as long as the number of tasks is larger than or equal to the number of models, the small penalty of the grouping loss could help us achieve the minimum task assignment goal.\n\n\n\\section{Experimentals}\n\n\\subsection{Datasets}\nOur work leverages a number of public datasets. These sets are summarized in Table~\\ref{tab:datasets}. We next offer a brief description of these sets.\n\n\\minisection{ICDAR 2013 dataset (IC13)}~\\cite{karatzas2013icdar} This is the oldest set used in this work, originally released for the ICDAR 2013 Robust Reading Competition. It offers 229 training and 233 test images of English text. Text locations are given as axis aligned, rectangular bounding boxes with text annotated at a word level. \n\n\\minisection{ICDAR 2015 dataset (IC15)}~\\cite{karatzas2015icdar}\nThis dataset was introduced in ICDAR'15 and offers more images than IC13: 1000 training and 500 test. Images in this set are of scene text in English, appearing at different orientations, where words are annotated using quadrangle bounding boxes. \n\n\n\\minisection{Total Text dataset}~\\cite{ch2017total} This collection offers 1255 training and 300 test, English scene text images. The images reflect a wide range of text orientations and shapes, including curved text examples. To accommodate different shapes, text locations are provided as polygons; recognition labels are given at word level.\n\n\\minisection{ICDAR 2017 RCTW dataset (RCTW17)}~\\cite{shi2017rctw} This set was collected to promote development of OCR methods for in the wild Chinese text. It is partitioned to 8034 and 4229 subsets of training and test images, respectively. \n\n\n\\minisection{ICDAR 2019 MLT dataset (MLT19) and SynthTextMLT}~\\cite{nayef2019icdar2019} was an extension of the ICDAR 2017 MLT dataset (MLT17) \\cite{nayef2017icdar2017} for multilingual text detection, recognition and script identification, which contains 10000 training images, 2000 validation images and 10000 test images in 7 different scripts from 10 languages. The dataset contains multi-oriented scene text annotated by quadrangle boxes. A synthetic dataset (SynthTextMLT)~\\cite{buvsta2018e2e} containing over 250k synthetic data in 7 scripts was also released along with the MLT19 benchmark. Since MLT19 training and validation sets completely covers the training and validation images in MLT17, though the split is a bit different, we only use MLT19 data for training in this paper.\n\n\n\\minisection{ICDAR 2019 ArT dataset (ArT19)}~\\cite{chng2019icdar2019} Contains 5603 training and 4563 test images in both English and Chinese, sourced from Total Text~\\cite{ch2017total} and SCUT-CTW1500~\\cite{yuliang2017detecting}. Released as part of the ICDAR 2019 Robust Reading Competition, the images in this collection depict texts in challenging shapes. Similarly to Total Text, text locations are encoded as polygons. We remove all Total Text test images from this set, ensuring that any training on this set can be applied to other sets without risk of test images influencing models trained on this set. \n\n\n\\minisection{ICDAR 2019 LSVT dataset (LSVT19)}~\\cite{sun2019lsvt}\nThis is one of the largest data sets used for developing OCR methods: 30000 training and 20000 test images. LSVT images mostly show street views with about 80\\% of them showing Chinese text and the rest examples in English. \n\n\\begin{table}[t]\n \\scriptsize\n \\centering\n \\caption{\\textbf{Datasets used in our experiments.} \\#Train: number of training images. Ratio: the relative sampling ratio when the dataset is used in training. Word \/ Phrase: Annotations given at a word or phrase level. Box type: horizontal, axis aligned (H-Box), arbitrarily rotated (R-Box), quadrangle (Quad), and Polygon. \\#Lang: Number of languages provided. Note that the Total Text dataset is fully covered in ArT19, and we removed the testing set of Total Text from ArT19.}\n \\resizebox{0.95\\columnwidth}{!}{\n \\begin{tabular}{lccccc}\n \\toprule\n Name & \\#Train & Ratio & Word \/ Phrase & Box type & \\#Lang.\\\\\n \\midrule\n ICDAR13~\\cite{karatzas2013icdar} & 229 & 20 & Word & H-Box & 1 \\\\ \n ICDAR15~\\cite{karatzas2015icdar} & 1000 & 20 & Word & Quad & 1 \\\\\n Total Text~\\cite{ch2017total} & 1255 & 50 & Word & Polygon & 1 \\\\\nRCTW17~\\cite{shi2017rctw} & 8034 & 20 & Phrase & R-Box & 2 \\\\ \nMLT19~\\cite{nayef2019icdar2019} & 10000 & 100 & Word & Quad & 10 \\\\\nSynthTextMLT~\\cite{buvsta2018e2e} & 252599 & 1 & Word & R-Box & 7 \\\\\nArT19~\\cite{chng2019icdar2019} & 5303 & 50 & Word & Polygon & 2 \\\\\nLSVT19~\\cite{sun2019lsvt} & 30000\t& 20 & Phrase & Polygon & 2 \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\label{tab:datasets}\n\\end{table}\n\n\\subsection{Model training}\nWe base our implementation on the Multiplexer codebase\\footnote{\\url{https:\/\/github.com\/facebookresearch\/MultiplexedOCR}}.\nFor fair comparison, we adopt the same segmentation-based detection and ROI mask feature extraction modules as \\cite{huang2021multiplexed}, and freeze the pretrained weights of these layers throughout training. For language classification, \\cite{huang2021multiplexed} uses 8 classes including Arabic, Bengali, Chinese, Hindi, Japanese, Korean, Latin, and Symbol, but in our experiment we only use 7 classes by discarding the Symbol class, as it does not have any dedicated dataset and the number of the samples is too small to make a difference.\n\nTo expedite the training, we first combine every dataset to train a single recognition head with hidden size 256 and embed size of 200 covering all datasets using the ratios specified in Table \\ref{tab:datasets} for 40k iterations. We then use this weight as a universal pretrained weights for the second stage of training.\n\nNext, we perform a series of experiments that jointly train the grouping module and the recognition heads, each restricting the number of recognition heads to $m$ ($2 \\leq m \\leq 7$). For each $m$, we launch three training jobs with different random seeds. Each of the training jobs runs for 20k iterations on the MLT19 training datasets only to reduce the potential data imbalance when including the other training set. We record and summarize the final grouping results in Table \\ref{tab:task_grouping_result}, which we will discuss in \\ref{sec:results:grouping}.\n\nFinally, based on the grouping result, we fine-tune each recognition head with only the datasets within the assigned group corresponding to the head. At this stage the grouping is essentially frozen and does not change any more. We can prune the output layer of the decoder so that the characters not belonging to the group are removed, to reduce the parameter number for the final model.\n\n\n\\subsection{Task grouping results}\n\\label{sec:results:grouping}\n\nTable \\ref{tab:task_grouping_result} shows the aggregation of grouping results from $18$ task grouping experiments with $2$ to $7$ recognition heads, each repeated for $3$ times. All task assignments stabilize after about 10000 iterations.\n\nThe top 14 groups are ordered first by the number of occurrences and then by the first occurrence, \\emph{i.e}\\onedot the minimum number of recognition heads when the group first occurs. All exclusive task-model assignments (one head focusing on one task) occur at least twice, showing the effectiveness of having a dedicated model for each task. Chinese ending up as an individual task occurs in 50\\% of the cases, which is expected given its high character number and the datasets, except that it's grouped together with Japanese, which shares many characters with it, only once. On the other hand, Hindi seems to be suitable to be grouped with many different languages rather than being trained by itself. \n\nSurprisingly, the most frequent task group that has more than one task is Arabic+Korean, which occurs 5 times. This suggests that there are inherent characteristics shared either by these two scripts, or by the examples in the MLT19 dataset itself, that boost the performance for each other. Another unusual cluster is the combination of 5 tasks, Bengali+Chinese+Hindi+Japanese+Latin, which is the only grouping with more than 2 tasks that occurs more than once.\nWe note however that the scattering of the grouping results shows that there can be many local optima for this specific scenario of $7$ distinctive scripts. We shall expect higher frequencies of the same grouping results if certain tasks share greater similarity, and we will leave that as one of our future work.\nWe additionally find it interesting that despite a grouping loss to encourage each head to take on at least one task, we observe that some recognition heads might not be assigned with any task in the end when there are $6$ or $7$ tasks. This means for certain combinations of tasks, training them together could outperform training them separately, even if there is spare resource for a new head.\n\n\\begin{table}[t]\n \\scriptsize\n \\centering\n \\caption{\\textbf{Task grouping result.} Task combinations that end up being grouped together. The 2nd to the 5th columns indicate task names in the final grouping, the number of tasks in the group, the number of occurrences in the 18 experiments, and the minimum number of recognition heads when the combination first occurs. }\n \\begin{tabularx}{0.98\\linewidth}{c|X|c|c|c}\n \\toprule\n Rank & Group & \\#Tasks within group & \\#Occurrences & \\#Heads at first occurrence \\\\ \n \\midrule\n 1 & Chinese (C) & 1 & 9 & 4 \\\\\n 2 & Latin (L) & 1 & 7 & 5 \\\\\n 3 & Arabic (A) & 1 & 6 & 3 \\\\\n 3 & Korean (K) & 1 & 6 & 3 \\\\\n 5 & Arabic+Korean & 2 & 5 & 2 \\\\\n 6 & Bengali (B) & 1 & 5 & 5 \\\\\n 7 & Japanese (J) & 1 & 4 & 5 \\\\\n 8 & B+C+H+J+L & 5 & 2 & 2 \\\\\n 9 & Hindi+Japanese & 2 & 3 & 3 \\\\\n 10 & Japanese+Latin & 2 & 2 & 4 \\\\\n 11 & Arabic+Hindi & 2 & 2 & 5 \\\\\n 11 & Bengali+Japanese & 2 & 2 & 5 \\\\\n 11 & Hindi+Latin & 2 & 2 & 5 \\\\\n 14 & Hindi (H) & 1 & 2 & 6 \\\\\n \\hline\n 15 & A+K+L & 3 & 1 & 2 \\\\\n 15 & H+J+K & 3 & 1 & 2 \\\\\n 15 & A+B+C+L & 4 & 1 & 2 \\\\\n 15 & B+C+H+J & 4 & 1 & 2 \\\\\n 15 & Chinese+Hindi & 2 & 1 & 3 \\\\\n 15 & Korean+Latin & 2 & 1 & 3 \\\\\n 15 & A+B+J & 3 & 1 & 3 \\\\\n 15 & B+C+L & 3 & 1 & 3 \\\\\n 15 & Arabic+Bengali & 2 & 1 & 4 \\\\\n 15 & Bengali+Hindi & 2 & 1 & 4 \\\\\n 15 & Chinese+Latin & 2 & 1 & 4 \\\\\n 15 & A+B+H & 3 & 1 & 4 \\\\\n 15 & Japanese+Korean & 2 & 1 & 5 \\\\\n 15 & B+C+K & 3 & 1 & 6 \\\\\n 15 & Hindi+Korean & 2 & 1 & 7 \\\\\n 15 & Chinese+Japanese & 2 & 1 & 7 \\\\\n \\bottomrule\n \\end{tabularx}\n \\label{tab:task_grouping_result}\n\\end{table}\n\n\n\\subsection{Ablation study}\n\\label{sec:ablation_study}\n\n\\minisection{Base integrated loss coefficient.} We train the task grouping network with different base integrated loss coefficient $\\epsilon$ defined in Eq. \\eqref{eq:integrated_loss} on MLT19 training set. The network contains 5 recognition heads that are initialized with the same pretrained weights. We record the number of task assignment changes in the first 3000 iterations. From Table \\ref{tab:base_loss_weight} we can see that, when $\\epsilon=0.0$, there's only 1 assignment change since the model does not have much chance to learn the unassigned tasks; interestingly, when $\\epsilon$ is too big ($0.3\/0.4$), there are also fewer changes happening, possibly because there is not much diversity across the models and everything moves in the same direction. The maximum number of assignment changes happen when $\\epsilon$ is $0.2$ or $0.1$. Therefore, in most of our experiments we use $0.2$ for early training and $0.1$ for fine-tuning.\n\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics[width=0.98\\linewidth]{figures\/qual_img_grping_3.png}\n \\caption{\\textbf{Qualitative results on MLT19 test set~\\cite{nayef2019icdar2019}.} The predicted transcription is rendered with green background, along with the detection confidence, language and the assigned group. The model has $5$ heads (groups): group 1 - Arabic (ar) and Hindi (hi), group 2 - Bengali (bn) and Japanese (ja), group 3 - Chinese (zh), group 4 - Latin (la), group 5 - Korean (ko). See Sec.~\\ref{sec:results:e2e} for more details.}\n \\label{fig:qualitative}\n\\end{figure*}\n\n\n\\begin{table}[t]\n \\scriptsize\n \\centering\n \\caption{\\textbf{Ablation study for base integrated loss coefficient $\\epsilon$.} }\n \\begin{tabularx}{0.98\\linewidth}{X*{5}{p{0.1\\linewidth}}}\n \\toprule\n Base integrated loss coefficient $\\epsilon$ & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 \\\\ \n \\midrule\n Assignment changes within 3k iters & 1 & 3 & 5 & 2 & 2 \\\\\n \\bottomrule\n \\end{tabularx}\n \\label{tab:base_loss_weight}\n\\end{table}\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.9\\linewidth, scale=0.8,clip,trim=0mm 2mm 0mm 2mm]{figures\/error_res_grping_scl_2.png}\n \\caption{\\textbf{Error analysis on MLT19.} Detection errors are represented in red outline and recognition errors in purple. See Sec.~\\ref{sec:results:e2e}.}\n \\label{fig:failures}\n\\end{figure} \n\n\n\\subsection{Task assignment on models with different hyper-parameters}\n\nIn this section, we perform an interesting experiment that showcases how our design can help assign different tasks to models with different hyper-parameters based on the potential difficulty and the available data. We set the number of models (recognition heads) to be equal to the number of tasks, but set the key hyper-parameters of the models, embed size and hidden size, to be different from each other. We train the overall model on the weighted combination of all datasets listed above, and Table \\ref{tab:parameter_number_comparison} shows the assigned task corresponding to each of the models. We can clearly see the correlation between the number of parameters versus the number of characters in the corresponding character set, with the exception of Latin. This illustrated that in general, when the number of characters grow, the heavier models will outperform lighter models in the long term; however, since Latin words are dominating in all the datasets including many difficult cases like curved text, the task grouping framework learns to spend the heaviest model on it to boost the overall performance.\n\n\n\\begin{table}[t]\n \\scriptsize\n \\centering\n \\caption{\\textbf{Task assignment result for models with different major hyper-parameters.} Each model supports all characters in the beginning so the total number of parameters for each head is high, but they can be pruned when the task assignment stabilizes.}\n \\begin{tabularx}{0.98\\linewidth}{c|c|c|c|c|c}\n \\toprule\n Embed Size & Hidden Size & Parameter Number & Assigned Task & Charset Size & Final Parameters\\\\\n \\hline\n 100 & 224 & 4.05M & Arabic & 80 & 1.15M \\\\\n 150 & 224 & 4.51M & Bengali & 110 & 1.18M \\\\\n 200 & 224 & 4.98M & Japanese & 2300 & 2.13M \\\\\n 100 & 256 & 4.59M & Hindi & 110 & 1.42M \\\\\n 150 & 256 & 5.06M & Korean & 1500 & 2.00M \\\\\n 200 & 256 & 5.52M & Chinese & 5200 & 3.78M \\\\\n 250 & 256 & 5.98M & Latin & 250 & 1.54M \\\\\n \\bottomrule\n \\end{tabularx}\n \\label{tab:parameter_number_comparison}\n\\end{table}\n\n\n\\subsection{E2E text recognition}\\label{sec:results:e2e}\n\nTable \\ref{tab:mlt19_task4} shows the results on MLT19 \\cite{nayef2019icdar2019} end-to-end multilingual recognition benchmark. Fig.~\\ref{fig:qualitative} additionally provides qualitative examples of these multilingual. We find that using varying numbers of grouped heads can perform similarly to (and in some cases, better than) the multiplexed approach of a separate recognition head per language~\\cite{huang2021multiplexed}.\nThis is an interesting result, as it means we can significantly cut down on the computational cost and model size with little impact or even some gains to the performance.\nNotably, we also find that increasing the number of heads from a single shared head (Mask TextSpotter V3 \\cite{liao2020mask}) to even just two grouped heads leads to a significant increase in F1-score. \n\nWe provide qualitative failure cases in Fig.~\\ref{fig:failures}. While detection errors could be attributed to arbitrary text shape, blurred text, glossy surfaces and rare fonts; recognition errors could be attributed to text ordering, text resolution and challenging scripts. \n\n\\begin{table}[t]\n \\scriptsize\n \\centering\n \\caption{\\textbf{End-to-end recognition results on MLT19.} Note that there are two versions results of CRAFTS, one from the official MLT19 website and one from paper \\cite{baek2020character}. Importantly, CRAFTS has a ResNet-based feature extraction which is much bigger than the one with $5$-Convs used in our experiments.}\n \\begin{tabularx}{0.9\\linewidth}{lX*{2}X}\n \\toprule\n Method & F & P & R \\\\ \n \\midrule\n E2E-MLT \\cite{buvsta2018e2e} & 26.5 & 37.4 & 20.5 \\\\\n RRPN+CLTDR \\cite{ma2018arbitrary} & 33.8 & 38.6 & 30.1 \\\\\n CRAFTS \\cite{baek2020character} & 51.7 & 65.7 & 42.7 \\\\\n CRAFTS (paper) \\cite{baek2020character} & \\textbf{58.2} & \\textbf{72.9} & \\textbf{48.5} \\\\\n \\hline\n Mask TextSpotter V3 (1 head) \\cite{liao2020mask} & 39.7 & \\textbf{71.8} & 27.4 \\\\ \n Multiplexed TextSpotter (8 heads) \\cite{huang2021multiplexed} & 48.2 & 68.0 & 37.3 \\\\\n \\hline\n Grouped (2 heads) & 45.5 & 67.7 & 34.3 \\\\\n Grouped (3 heads) & 47.1 & 67.0 & 36.3 \\\\\n Grouped (4 heads) & 47.9 & 66.7 & 37.4 \\\\\n Grouped (5 heads) & \\textbf{48.5} & 67.7 & \\textbf{37.8} \\\\\n Grouped (6 heads) & 48.3 & 67.8 & 37.5 \\\\\n Grouped (7 heads) & 48.2 & 68.0 & 37.3 \\\\\n \\bottomrule\n \\end{tabularx}\n \\label{tab:mlt19_task4}\n\\end{table}\n\n\\section{Conclusions}\nText is one of the most ubiquitous visual object classes in real-world scenes, making understanding it practical and critically important. Processing multiple languages, however, requires substantial resources, to accurately recognize the subtleties of appearances variations of different scripts and different intra-script characters. This ability was, therefore, previously accomplished by specializing separate network heads to specific languages. We, instead, are the first to propose automatically grouping different languages together, in the same recognition heads. Our dynamic, and differentiable task shifting approach automatically routes tasks to different heads while the network trains, optimizing for the best, bottom line accuracy across all languages. Extensive tests show our method to not only achieve SotA accuracy, but to do so with fewer recognition heads and hyperparameters, consequently making is a practical design choice for real-world OCR systems. \n\n\\minisection{Future work} Our work leaves several natural follow-up directions. One interesting question relates to the scalability of our approach: How many multitask heads, for example, would be required to effectively learn hundreds of languages? Another intriguing direction is extending our multitask learning and task grouping to include neural architecture search as part of its design. Such a solution should allow growing heads with different architectures for different languages to account for, \\emph{e.g}\\onedot, harder vs. easier languages. Finally, another potential extension could be continual language learning~\\cite{Parisi2019}: adding more languages as relevant training data becomes available, without retraining or regrouping existing languages. Alternative grouping approaches based on Bayesian nonparametric approaches like the Chinese Restaurant Process~\\cite{Aldous1985} or Indian Buffet Process~\\cite{Ghahramani2006,mehta2021continual} may be natural ways to perform groupings in such settings.\n\n\n\\clearpage\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}