Phase transitions in Phase transitions in Hubbard ModelHubbard Model

Anti-ferromagnetic Anti-ferromagnetic andand

superconducting superconducting order in the order in the

Hubbard modelHubbard modelA functional renormalization A functional renormalization

group studygroup study

T.Baier, E.Bick, …C.Krahl, J.Mueller, S.Friederich

Phase diagramPhase diagram

AFSC

Mermin-Wagner theorem Mermin-Wagner theorem ??

NoNo spontaneous symmetry spontaneous symmetry breaking breaking

of continuous symmetry in of continuous symmetry in d=2 d=2 !!

not valid in practice !not valid in practice !

Phase diagramPhase diagram

Pseudo-criticaltemperature

Goldstone boson Goldstone boson fluctuationsfluctuations

spin waves ( anti-ferromagnetism )spin waves ( anti-ferromagnetism ) electron pairs ( superconductivity )electron pairs ( superconductivity )

Flow equation for average Flow equation for average potentialpotential

Simple one loop structure –Simple one loop structure –nevertheless (almost) exactnevertheless (almost) exact

Scaling form of evolution Scaling form of evolution equationequation

Tetradis …

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 Solution of partial differential equation :equation :

yields highly nontrivial non-perturbative results despite the one loop structure !

Example: Kosterlitz-Thouless phase transition

Anti-ferromagnetism Anti-ferromagnetism in Hubbard modelin Hubbard model

SO(3) – symmetric scalar model SO(3) – symmetric scalar model coupled to fermionscoupled to fermions

For low enough k : fermion degrees For low enough k : fermion degrees of freedom decouple effectivelyof freedom decouple effectively

crucial question : running of crucial question : running of κκ ( location of minimum of effective ( location of minimum of effective potential , renormalized , potential , renormalized , dimensionless )dimensionless )

Critical temperatureCritical temperatureFor T<Tc : κ remains positive for k/t > 10-9

size of probe > 1 cm

-ln(k/t)

κ

Tc=0.115

T/t=0.05

T/t=0.1

local disorderpseudo gap

SSB

Below the pseudocritical Below the pseudocritical temperaturetemperature

the reign of the goldstone bosons

effective nonlinear O(3) – σ - model

critical behaviorcritical behavior

for interval Tc < T < Tpc

evolution as for classical Heisenberg model

cf. Chakravarty,Halperin,Nelson

critical correlation critical correlation lengthlength

c,β : slowly varying functions

exponential growth of correlation length compatible with observation !

at Tc : correlation length reaches sample size !

Mermin-Wagner theorem Mermin-Wagner theorem ??

NoNo spontaneous symmetry spontaneous symmetry breaking breaking

of continuous symmetry in of continuous symmetry in d=2 d=2 !!

not valid in practice !not valid in practice !

Below the critical Below the critical temperature :temperature :

temperature in units of t

antiferro-magnetic orderparameter

Tc/t = 0.115

U = 3

Infinite-volume-correlation-length becomes larger than sample size

finite sample ≈ finite k : order remains effectively

Action for Hubbard Action for Hubbard modelmodel

Truncation for flowing Truncation for flowing actionaction

Additional bosonic fieldsAdditional bosonic fields

anti-ferromagneticanti-ferromagnetic charge density wavecharge density wave s-wave superconductings-wave superconducting d-wave superconductingd-wave superconducting

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

Effective potential for Effective potential for bosonsbosons

SYM

SSB

microscopic :only “mass terms”

Yukawa coupling Yukawa coupling between fermions and between fermions and

bosonsbosons

Microscopic Yukawa couplings vanish !

Kinetic terms for bosonic Kinetic terms for bosonic fieldsfields

anti-ferromagnetic boson

d-wave superconductingboson

incommensurate anti-incommensurate anti-ferromagnetismferromagnetism

commensurate regime :

incommensurate regime :

infrared cutoffinfrared cutoff

linear cutoff ( Litim )

flowing bosonisationflowing bosonisation

effective four-fermion effective four-fermion couplingcoupling

in appropriate channel in appropriate channel

is translated to bosonicis translated to bosonic

interaction at every interaction at every scale k scale k

H.Gies , …

k-dependent field redefinition

absorbs four-fermion coupling

running Yukawa running Yukawa couplingscouplings

flowing boson mass flowing boson mass termsterms

SYM : close to phase transition

Pseudo-critical Pseudo-critical temperature Ttemperature Tpcpc

Limiting temperature at which bosonic mass Limiting temperature at which bosonic mass term vanishes ( term vanishes ( κκ becomes nonvanishing ) becomes nonvanishing )

It corresponds to a diverging four-fermion It corresponds to a diverging four-fermion couplingcoupling

This is the “critical temperature” computed This is the “critical temperature” computed in MFT !in MFT !

Pseudo-gap behavior below this temperaturePseudo-gap behavior below this temperature

Pseudocritical Pseudocritical temperaturetemperature

Tpc

μ

Tc

MFT(HF)

Flow eq.

Critical temperatureCritical temperatureFor T<Tc : κ remains positive for k/t > 10-9

size of probe > 1 cm

-ln(k/t)

κ

Tc=0.115

T/t=0.05

T/t=0.1

local disorderpseudo gap

SSB

Phase diagramPhase diagram

Pseudo-criticaltemperature

spontaneous symmetry spontaneous symmetry breaking of abelian breaking of abelian

continuous symmetry in continuous symmetry in d=2d=2

Bose –Einstein condensate

Superconductivity in Hubbard model

Kosterlitz – Thouless phase transition

Essential scaling : d=2,N=2Essential scaling : d=2,N=2

Flow equation Flow equation contains contains correctly the correctly the non-non-perturbative perturbative information !information !

(essential (essential scaling usually scaling usually described by described by vortices)vortices)

Von Gersdorff …

Kosterlitz-Thouless phase Kosterlitz-Thouless phase transition (d=2,N=2)transition (d=2,N=2)

Correct description of phase Correct description of phase withwith

Goldstone boson Goldstone boson

( infinite correlation ( infinite correlation length ) length )

for T<Tfor T<Tcc

Temperature dependent anomalous Temperature dependent anomalous dimension dimension ηη

T/Tc

η

Running renormalized d-wave Running renormalized d-wave superconducting order parameter superconducting order parameter κκ in in

doped Hubbard (-type ) modeldoped Hubbard (-type ) model

κ

- ln (k/Λ)

Tc

T>Tc

T<Tc

C.Krahl,… macroscopic scale 1 cm

locationofminimumof u

local disorderpseudo gap

Renormalized order parameter Renormalized order parameter κκ and gap in electron and gap in electron

propagator propagator ΔΔin doped Hubbard-type modelin doped Hubbard-type model

100 Δ / t

κ

T/Tc

jump

order parameters order parameters in Hubbard modelin Hubbard model

Competing ordersCompeting orders

AFSC

Anti-ferromagnetism Anti-ferromagnetism suppresses superconductivitysuppresses superconductivity

coexistence of different coexistence of different orders ?orders ?

quartic couplings for quartic couplings for bosonsbosons

conclusionsconclusions

functional renormalization gives access functional renormalization gives access to low temperature phases of Hubbard to low temperature phases of Hubbard modelmodel

order parameters can be computed as order parameters can be computed as function of temperature and chemical function of temperature and chemical potentialpotential

competing orderscompeting orders further quantitative progress possiblefurther quantitative progress possible

changing degrees of changing degrees of freedomfreedom

flowing bosonisationflowing bosonisation

adapt bosonisation adapt bosonisation to every scale k to every scale k such thatsuch that

is translated to is translated to bosonic interactionbosonic interaction

H.Gies , …

k-dependent field redefinition

absorbs four-fermion coupling

flowing bosonisationflowing bosonisation

Choose αk in order to absorb the four fermion coupling in corresponding channel

Evolution with k-dependentfield variables

modified flow of couplings

Mean Field Theory (MFT)Mean Field Theory (MFT)

Evaluate Gaussian fermionic integralin background of bosonic field , e.g.

Mean field phase Mean field phase diagramdiagram

μμ

TcTc

for two different choices of couplings – same U !

Mean field ambiguityMean field ambiguity

Tc

μ

mean field phase diagram

Um= Uρ= U/2

U m= U/3 ,Uρ = 0

Artefact of approximation …

cured by inclusion ofbosonic fluctuations

J.Jaeckel,…

Bosonisation and the Bosonisation and the mean field ambiguitymean field ambiguity

Bosonic fluctuationsBosonic fluctuations

fermion loops boson loops

mean field theory

Bosonisation Bosonisation cures mean field ambiguitycures mean field ambiguity

Tc

Uρ/t

MFT

Flow eq.

HF/SD

end

quartic couplings for quartic couplings for bosonsbosons

kinetic and gradient terms kinetic and gradient terms for bosonsfor bosons

fermionic wave function fermionic wave function renormalizationrenormalization