Phase transitions in Hubbard Model

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Phase transitions in Hubbard Model. Anti-ferromagnetic and superconducting order in the Hubbard model. A functional renormalization group study. T.Baier, E.Bick, C.Krahl, J.Mueller, S.Friederich. Phase diagram. AF. SC. Mermin-Wagner theorem ?. No spontaneous symmetry breaking - PowerPoint PPT Presentation


  • Phase transitions in Hubbard Model

  • Anti-ferromagnetic andsuperconducting order in the Hubbard modelA functional renormalization group studyT.Baier, E.Bick, C.Krahl, J.Mueller, S.Friederich

  • Phase diagramAFSC

  • Mermin-Wagner theorem ?No spontaneous symmetry breaking of continuous symmetry in d=2 !

    not valid in practice !

  • Phase diagramPseudo-criticaltemperature

  • Goldstone boson fluctuationsspin waves ( anti-ferromagnetism )electron pairs ( superconductivity )

  • Flow equation for average potential

  • Simple one loop structure nevertheless (almost) exact

  • Scaling form of evolution equationTetradis On r.h.s. :neither the scale k nor the wave function renormalization Z appear explicitly.

    Scaling solution:no dependence on t;correspondsto second order phase transition.

  • Solution of partial differential equation :yields highly nontrivial non-perturbative results despite the one loop structure !

    Example: Kosterlitz-Thouless phase transition

  • Anti-ferromagnetism in Hubbard modelSO(3) symmetric scalar model coupled to fermionsFor low enough k : fermion degrees of freedom decouple effectivelycrucial question : running of ( location of minimum of effective potential , renormalized , dimensionless )

  • Critical temperatureFor T 10-9 size of probe > 1 cm-ln(k/t)Tc=0.115T/t=0.05T/t=0.1local disorderpseudo gapSSB

  • Below the pseudocritical temperaturethe reign of the goldstone bosonseffective nonlinear O(3) - model

  • critical behaviorfor interval Tc < T < Tpc evolution as for classical Heisenberg model

    cf. Chakravarty,Halperin,Nelson

  • critical correlation length c, : slowly varying functions

    exponential growth of correlation length compatible with observation !

    at Tc : correlation length reaches sample size !

  • Mermin-Wagner theorem ?No spontaneous symmetry breaking of continuous symmetry in d=2 !

    not valid in practice !

  • Below the critical temperature :temperature in units of t antiferro-magnetic orderparameter Tc/t = 0.115U = 3Infinite-volume-correlation-length becomes larger than sample size

    finite sample finite k : order remains effectively

  • Action for Hubbard model

  • Truncation for flowing action

  • Additional bosonic fieldsanti-ferromagneticcharge density waves-wave superconductingd-wave superconducting

    initial values for flow : bosons are decoupled auxiliary fields ( microscopic action )

  • Effective potential for bosonsSYMSSBmicroscopic :only mass terms

  • Yukawa coupling between fermions and bosonsMicroscopic Yukawa couplings vanish !

  • Kinetic terms for bosonic fieldsanti-ferromagnetic bosond-wave superconductingboson

  • incommensurate anti-ferromagnetismcommensurate regime :incommensurate regime :

  • infrared cutofflinear cutoff ( Litim )

  • flowing bosonisationeffective four-fermion couplingin appropriate channel

    is translated to bosonicinteraction at every scale k H.Gies , k-dependent field redefinitionabsorbs four-fermion coupling

  • running Yukawa couplings

  • flowing boson mass termsSYM : close to phase transition

  • Pseudo-critical temperature TpcLimiting temperature at which bosonic mass term vanishes ( becomes nonvanishing )

    It corresponds to a diverging four-fermion coupling

    This is the critical temperature computed in MFT !

    Pseudo-gap behavior below this temperature

  • Pseudocritical temperatureTpcTcMFT(HF)Flow eq.

  • Critical temperatureFor T 10-9 size of probe > 1 cm-ln(k/t)Tc=0.115T/t=0.05T/t=0.1local disorderpseudo gapSSB

  • Phase diagramPseudo-criticaltemperature

  • spontaneous symmetry breaking of abelian continuous symmetry in d=2Bose Einstein condensate

    Superconductivity in Hubbard model

    Kosterlitz Thouless phase transition

  • Essential scaling : d=2,N=2Flow equation contains correctly the non-perturbative information !(essential scaling usually described by vortices)Von Gersdorff

  • Kosterlitz-Thouless phase transition (d=2,N=2)Correct description of phase with Goldstone boson ( infinite correlation length ) for T
  • Temperature dependent anomalous dimension T/Tc

  • Running renormalized d-wave superconducting order parameter in doped Hubbard (-type ) model- ln (k/)TcT>TcT
  • Renormalized order parameter and gap in electron propagator in doped Hubbard-type model100 / t T/Tcjump

  • order parameters in Hubbard model

  • Competing ordersAFSC

  • Anti-ferromagnetism suppresses superconductivity

  • coexistence of different orders ?

  • quartic couplings for bosons

  • conclusionsfunctional renormalization gives access to low temperature phases of Hubbard modelorder parameters can be computed as function of temperature and chemical potentialcompeting ordersfurther quantitative progress possible

  • changing degrees of freedom

  • flowing bosonisationadapt bosonisation to every scale k such that

    is translated to bosonic interactionH.Gies , k-dependent field redefinitionabsorbs four-fermion coupling

  • flowing bosonisationChoose k in order to absorb the four fermion coupling in corresponding channelEvolution with k-dependentfield variablesmodified flow of couplings

  • Mean Field Theory (MFT)Evaluate Gaussian fermionic integralin background of bosonic field , e.g.

  • Mean field phase diagramTcTcfor two different choices of couplings same U !

  • Mean field ambiguityTcmean field phase diagramUm= U= U/2U m= U/3 ,U = 0Artefact of approximation

    cured by inclusion ofbosonic fluctuations


  • Bosonisation and the mean field ambiguity

  • Bosonic fluctuationsfermion loopsboson loopsmean field theory

  • Bosonisation cures mean field ambiguityTcU/tMFTFlow eq.HF/SD

  • end

  • quartic couplings for bosons

  • kinetic and gradient terms for bosons

  • fermionic wave function renormalization

  • ********************************


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