A non-Hermitian critical point and the correlation

arXiv:cond-mat/0606348v1[cond-mat.str-el]14Jun2006Typesetwithjpsj2.clsver.1.2FullPaperAnon-HermitiancriticalpointandthecorrelationlengthofstronglycorrelatedquantumsystemsYuichiNakamura1∗andNaomichiHatano2†1DepartmentofPhysics,UniversityofTokyo,Komaba,Meguro,Tokyo,153-85052InstituteofIndustrialScience,UniversityofTokyo,Komaba,Meguro,Tokyo,153-8505Westudyanon-Hermitiangeneralizationofquantumsystemsinwhichanimaginaryvectorpotentialisaddedtothemomentumoperator.Inthetight-bindingapproximation,wemakethehoppingenergyasymmetricintheHermitianHamiltonian.Inapreviousarticle,weconjecturedthatthenon-HermitiancriticalpointwheretheenergygapvanishesisequaltotheinversecorrelationlengthoftheHermitiansystemandweconfirmedtheconjecturefortwoexactlysolvablesystems.Inthisarticle,wepresentmoreevidencefortheconjecture.Wealsoarguethebasisofourconjecturebynotingthedispersionrelationoftheelementaryexcitation.KEYWORDS:Non-Hermitianquantummechanics,correlationlength,dispersionrelation,isotropicXYchain,Hubbardmodel,S=1/2antiferromagneticXXZchain,Majumdar-Ghoshmodel1.IntroductionInthispaper,westudyanon-Hermitiangeneralizationofstronglycorrelatedquantumsystemsinwhichanimaginaryvectorpotentiali~g(where~gisarealvector)isaddedtothemomentumoperator.Thenon-Hermitiankineticenergyinthecontinuousspaceisgivenby1Hk=(−i~~∇+i~g)22m.(1)Itssecond-quantizedformwithinthetight-bindingapproximationisgivenbyHk=−tdXν=1X~xegν(~x)c†~x+~eνc~x+e−gν(~x)c†~xc~x+~eν.(2)Inthisarticle,wefocusontheone-dimensionalcasewithaconstantimaginaryvectorpotentialandhereafteruseHk=−tXx(egc†x+1cx+e−gc†xcx+1),(3)wheregisarealconstant;inotherwords,wemakethehoppingenergyasymmetric.Moregenerally,wemultiplytherighthoppingenergyc†x+ncxbyengandthelefthoppingenergyc†xcx+nbye−ngintheoriginalHermitianHamiltonian.∗E-mailaddress:[e-mail]†E-mailaddress:[e-mail]1/25J.Phys.Soc.Jpn.FullPaperThepurposeofthenon-HermitiangeneralizationistoobtainalengthscaleinherentinthewavefunctionoftheHermitiansystemonlyfromthenon-Hermitianenergyspectrum.Thenon-Hermitiangeneralizationwasfirstappliedtotheone-electronAndersonmodelbyHatanoandNelson.1Theirmodelis,inonedimension,Hrandom(g)=tLXx=1eg|x+1ihx|+e−g|xihx+1|
+LXx=1Vx|xihx|,(4)whereVxisarandompotentialatsitexandwerequiretheperiodicboundarycondition.Asweincreasethenon-Hermiticityg,apairofeigenvaluescollideatapointg=gcandthenbecomecomplex.Itwasrevealed1thatthenon-HermitiancriticalpointgcisequaltotheinverselocalizationlengthoftheeigenfunctionoftheoriginalHermitianHamiltonian.Wehereapplythenon-Hermitiangeneralizationtosystemswithoutrandomnessbutwithinteractions.Weconjecturedinthepreviousarticle2thatwecanobtainthecorrela-tionlengthfromthenon-Hermitiangeneralizationofstronglycorrelatedquantumsystems;thenon-Hermitiancriticalpointg=gcwheretheenergygapfromthegroundstatecollapsesisequaltotheinversecorrelationlengthofthegroundstateoftheHermitiansystem.Wecon-firmedtheconjecturefortwoone-dimensionalexactlysolvablesystems:theHubbardmodelinthehalf-filledcaseandtheS=1/2antiferromagneticXXZchainintheIsing-likeregion.Inthepresentarticle,wegivefurtherevidencefortheconjectureinvariouslevelsofcertainty.Wealsoclarifythereasonwhywecanobtaintheinversecorrelationlengthbyournon-Hermitiangeneralization,bynotingthedispersionrelationoftheelementaryexcitation.Weshowthatthenon-Hermitiangeneralizationisactuallyequivalenttoreplacingkwithk+iginthedispersionrelationoftheelementaryexcitation;wethusseekazeroofthedispersionrelationbycomputingthenon-Hermitiancriticalpointgc.In§2,weconfirmourconjectureforthefollowingexactlysolvablesystemsinonedimen-sion:theS=1/2ferromagneticisotropicXYchaininamagneticfieldin§2.1;thehalf-filledHubbardmodelin§2.2;theS=1/2antiferromagneticXXZchainintheIsing-likeregionin§2.3;theMajumdar-Ghoshmodelin§2.4.In§3,wenumericallyanalyzenon-HermitianmodelsoffinitesizeL.Wecalculatethenon-Hermitian“critical”pointgc(L)wheretheenergyoftheeigenstatecorrespondingtothegroundstateinthelimitL→∞becomescomplex;wethenobtainanextrapolatedestimategc(∞).Wenumericallyconfirmthattheestimategc(∞)andtheinversecorrelationlengthoftheHermitiansystemsareconsistentfortheHubbardmodelandfortheXXZmodel.Wealsoanalyzeanunsolvedmodel,namelyS=1/2antiferromagneticHeisenbergchainwithnearest-andnext-nearest-neighborinteractions.Inthesummary,weconcludeourdiscussionsbygivingaremarkonapplyingournon-Hermitiangeneralization.2/25J.Phys.Soc.Jpn.FullPaper2.Non-HermitiananalysisofsolvablemodelsForfourexactlysolvableone-dimensionalsystems,wehereconfirmourconjecturethatthenon-HermitiancriticalpointisequaltotheinversecorrelationlengthoftheHermitiansystem.Wealsorevealinthissectionthatthenon-Hermitiangeneralizationresultsinreplacingtherealmomentumkwithacomplexonek+iginthedispersionrelationoftheelementaryexcitation.Thatis,thenon-HermitiangeneralizationmakestheHermitianHamiltonianoftheformH=X−πkπǫ(k)η†kηk(5)transformedtoH(g)=X−πkπǫ(k+ig)η†kηk.(6)Findingthenon-Hermitiancriticalpointwheretheenergygapabovethegroundstatecollapsesistherebyequivalenttoseekingazeroofthedispersionrelationinthecomplexmoment