phase transitions in hubbard model. anti-ferromagnetic and superconducting order in the hubbard...
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Phase transitions in Phase transitions in Hubbard ModelHubbard Model
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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
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Phase diagramPhase diagram
AFSC
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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 !
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Phase diagramPhase diagram
Pseudo-criticaltemperature
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Goldstone boson Goldstone boson fluctuationsfluctuations
spin waves ( anti-ferromagnetism )spin waves ( anti-ferromagnetism ) electron pairs ( superconductivity )electron pairs ( superconductivity )
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Flow equation for average Flow equation for average potentialpotential
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Simple one loop structure –Simple one loop structure –nevertheless (almost) exactnevertheless (almost) exact
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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.
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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
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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 )
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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
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Below the pseudocritical Below the pseudocritical temperaturetemperature
the reign of the goldstone bosons
effective nonlinear O(3) – σ - model
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critical behaviorcritical behavior
for interval Tc < T < Tpc
evolution as for classical Heisenberg model
cf. Chakravarty,Halperin,Nelson
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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 !
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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 !
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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
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Action for Hubbard Action for Hubbard modelmodel
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Truncation for flowing Truncation for flowing actionaction
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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 )
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Effective potential for Effective potential for bosonsbosons
SYM
SSB
microscopic :only “mass terms”
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Yukawa coupling Yukawa coupling between fermions and between fermions and
bosonsbosons
Microscopic Yukawa couplings vanish !
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Kinetic terms for bosonic Kinetic terms for bosonic fieldsfields
anti-ferromagnetic boson
d-wave superconductingboson
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incommensurate anti-incommensurate anti-ferromagnetismferromagnetism
commensurate regime :
incommensurate regime :
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infrared cutoffinfrared cutoff
linear cutoff ( Litim )
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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
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running Yukawa running Yukawa couplingscouplings
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flowing boson mass flowing boson mass termsterms
SYM : close to phase transition
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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
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Pseudocritical Pseudocritical temperaturetemperature
Tpc
μ
Tc
MFT(HF)
Flow eq.
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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
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Phase diagramPhase diagram
Pseudo-criticaltemperature
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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
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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 …
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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
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Temperature dependent anomalous Temperature dependent anomalous dimension dimension ηη
T/Tc
η
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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
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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
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order parameters order parameters in Hubbard modelin Hubbard model
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Competing ordersCompeting orders
AFSC
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Anti-ferromagnetism Anti-ferromagnetism suppresses superconductivitysuppresses superconductivity
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coexistence of different coexistence of different orders ?orders ?
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quartic couplings for quartic couplings for bosonsbosons
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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
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changing degrees of changing degrees of freedomfreedom
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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
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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
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Mean Field Theory (MFT)Mean Field Theory (MFT)
Evaluate Gaussian fermionic integralin background of bosonic field , e.g.
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Mean field phase Mean field phase diagramdiagram
μμ
TcTc
for two different choices of couplings – same U !
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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,…
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Bosonisation and the Bosonisation and the mean field ambiguitymean field ambiguity
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Bosonic fluctuationsBosonic fluctuations
fermion loops boson loops
mean field theory
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Bosonisation Bosonisation cures mean field ambiguitycures mean field ambiguity
Tc
Uρ/t
MFT
Flow eq.
HF/SD
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end
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quartic couplings for quartic couplings for bosonsbosons
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kinetic and gradient terms kinetic and gradient terms for bosonsfor bosons
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fermionic wave function fermionic wave function renormalizationrenormalization
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