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Functional renormalization group for the effective average action

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physics at different length scales microscopic theories : where the laws are formulated microscopic theories : where the laws are formulated effective theories : where observations are made effective theories : where observations are made effective theory may involve different degrees of freedom as compared to microscopic theory effective theory may involve different degrees of freedom as compared to microscopic theory example: the motion of the earth around the sun does not need an understanding of nuclear burning in the sun example: the motion of the earth around the sun does not need an understanding of nuclear burning in the sun

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QCD : Short and long distance degrees of freedom are different ! Short distances : quarks and gluons Short distances : quarks and gluons Long distances : baryons and mesons Long distances : baryons and mesons How to make the transition? How to make the transition? confinement/chiral symmetry breaking confinement/chiral symmetry breaking

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collective degrees of freedom

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Hubbard model Electrons on a cubic lattice Electrons on a cubic lattice here : on planes ( d = 2 ) here : on planes ( d = 2 ) Repulsive local interaction if two electrons are on the same site Repulsive local interaction if two electrons are on the same site Hopping interaction between two neighboring sites Hopping interaction between two neighboring sites

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Hubbard model Functional integral formulation U > 0 : repulsive local interaction next neighbor interaction External parameters T : temperature μ : chemical potential (doping )

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In solid state physics : “ model for everything “ Antiferromagnetism Antiferromagnetism High T c superconductivity High T c superconductivity Metal-insulator transition Metal-insulator transition Ferromagnetism Ferromagnetism

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Antiferromagnetism in d=2 Hubbard model temperature in units of t antiferro- magnetic order parameter T c /t = 0.115 U/t = 3 T.Baier, E.Bick,… μ = 0

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antiferromagnetic order is finite size effect here size of experimental probe 1 cm here size of experimental probe 1 cm vanishing order for infinite volume vanishing order for infinite volume consistency with Mermin-Wagner theorem consistency with Mermin-Wagner theorem dependence on probe size very weak dependence on probe size very weak

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Collective degrees of freedom are crucial ! for T < T c nonvanishing order parameter nonvanishing order parameter gap for fermions gap for fermions low energy excitations: low energy excitations: antiferromagnetic spin waves antiferromagnetic spin waves

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effective theory / microscopic theory sometimes only distinguished by different values of couplings sometimes only distinguished by different values of couplings sometimes different degrees of freedom sometimes different degrees of freedom need for methods that can cope with such situations need for methods that can cope with such situations

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Functional Renormalization Group describes flow of effective action from small to large length scales perturbative renormalization : case where only couplings change, and couplings are small

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How to come from quarks and gluons to baryons and mesons ? How to come from electrons to spin waves ? Find effective description where relevant degrees of freedom depend on momentum scale or resolution in space. Find effective description where relevant degrees of freedom depend on momentum scale or resolution in space. Microscope with variable resolution: Microscope with variable resolution: High resolution, small piece of volume: High resolution, small piece of volume: quarks and gluons quarks and gluons Low resolution, large volume : hadrons Low resolution, large volume : hadrons

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/ Wegner, Houghton

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effective average action

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Unified picture for scalar field theories with symmetry O(N) in arbitrary dimension d and arbitrary N linear or nonlinear sigma-model for chiral symmetry breaking in QCD or: scalar model for antiferromagnetic spin waves (linear O(3) – model ) fermions will be added later fermions will be added later

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Effective potential includes all fluctuations

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Scalar field theory Scalar field theory

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Flow equation for average potential

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Simple one loop structure – nevertheless (almost) exact

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Infrared cutoff

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Partial differential equation for function U(k,φ) depending on two ( or more ) variables Z k Z k = c k -η

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Regularisation For suitable R k : Momentum integral is ultraviolet and infrared finite Momentum integral is ultraviolet and infrared finite Numerical integration possible Numerical integration possible Flow equation defines a regularization scheme ( ERGE –regularization ) Flow equation defines a regularization scheme ( ERGE –regularization )

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Integration by momentum shells Momentum integral is dominated by q 2 ~ k 2. Flow only sensitive to physics at scale k

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Wave function renormalization and anomalous dimension for Z k (φ,q 2 ) : flow equation is exact !

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Scaling form of evolution equation Tetradis … On r.h.s. : neither the scale k nor the wave function renormalization Z appear explicitly. Scaling solution: no dependence on t; corresponds to second order phase transition.

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decoupling of heavy modes threshold functions vanish for large w : large mass as compared to k Flow involves effectively only modes with mass smaller or equal k unnecessary heavy modes are eliminated automatically effective theories addition of new collective modes still needs to be done

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unified approach choose N choose N choose d choose d choose initial form of potential choose initial form of potential run ! run !

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Flow of effective potential Ising model CO 2 T * =304.15 K p * =73.8.bar ρ * = 0.442 g cm-2 Experiment : S.Seide … Critical exponents

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Critical exponents, d=3 ERGE world

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derivative expansion good results already in lowest order in derivative expansion : one function u to be determined good results already in lowest order in derivative expansion : one function u to be determined second order derivative expansion - include field dependence of wave function renormalization : three functions to be determined second order derivative expansion - include field dependence of wave function renormalization : three functions to be determined

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apparent convergence of derivative expansion from talk by Bervilliers

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anomalous dimension

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Solution of partial differential equation : yields highly nontrivial non-perturbative results despite the one loop structure ! Example: Kosterlitz-Thouless phase transition

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

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Kosterlitz-Thouless phase transition (d=2,N=2) Correct description of phase with Goldstone boson Goldstone boson ( infinite correlation length ) ( infinite correlation length ) for T<T c for T<T c

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Running renormalized d-wave superconducting order parameter κ in doped Hubbard model κ - ln (k/Λ) TcTc T>T c T<T c C.Krahl,… macroscopic scale 1 cm

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Renormalized order parameter κ and gap in electron propagator Δ in doped Hubbard model 100 Δ / t κ T/T c jump

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Temperature dependent anomalous dimension η T/T c η

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convergence and errors for precise results: systematic derivative expansion in second order in derivatives for precise results: systematic derivative expansion in second order in derivatives includes field dependent wave function renormalization Z(ρ) includes field dependent wave function renormalization Z(ρ) fourth order : similar results fourth order : similar results apparent fast convergence : no series resummation apparent fast convergence : no series resummation rough error estimate by different cutoffs and truncations rough error estimate by different cutoffs and truncations

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Effective average action and exact renormalization group equation

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Generating functional Generating functional

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Loop expansion : perturbation theory with infrared cutoff in propagator Effective average action

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Quantum effective action

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Exact renormalization group equation

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Proof of exact flow equation

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Truncations Truncations Functional differential equation – cannot be solved exactly cannot be solved exactly Approximative solution by truncation of most general form of effective action most general form of effective action

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non-perturbative systematic expansions

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Exact flow equation for effective potential Evaluate exact flow equation for homogeneous field φ. Evaluate exact flow equation for homogeneous field φ. R.h.s. involves exact propagator in homogeneous background field φ. R.h.s. involves exact propagator in homogeneous background field φ.

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many models have been studied along these lines … several fields several fields complicated phase structure ( e.g. 3 He ) complicated phase structure ( e.g. 3 He ) replica trick N=0 replica trick N=0 shift in critical temperature for Bose-Einstein condensate with interaction ( needs resolution for momentum dependence of propagator ) shift in critical temperature for Bose-Einstein condensate with interaction ( needs resolution for momentum dependence of propagator ) gauge theories gauge theories

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disordered systems Canet, Delamotte, Tissier, …

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including fermions : no particular problem ! no particular problem !

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Universality in ultra-cold fermionic atom gases with S. Diehl, H.Gies, J.Pawlowski S. Diehl, H.Gies, J.Pawlowski

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BEC – BCS crossover Bound molecules of two atoms on microscopic scale: Bose-Einstein condensate (BEC ) for low T Bose-Einstein condensate (BEC ) for low T Fermions with attractive interactions (molecules play no role ) : BCS – superfluidity at low T BCS – superfluidity at low T by condensation of Cooper pairs by condensation of Cooper pairs Crossover by Feshbach resonance Crossover by Feshbach resonance as a transition in terms of external magnetic field as a transition in terms of external magnetic field

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chemical potential BEC BCS inverse scattering length

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BEC – BCS crossover qualitative and partially quantitative theoretical understanding qualitative and partially quantitative theoretical understanding mean field theory (MFT ) and first attempts beyond mean field theory (MFT ) and first attempts beyond concentration : c = a k F reduced chemical potential : σ˜ = μ/ε F Fermi momemtum : k F Fermi energy : ε F binding energy : T = 0 BCSBEC

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concentration c = a k F, a(B) : scattering length needs computation of density n=k F 3 /(3π 2 ) BCSBEC dilute dense T = 0 non- interacting Fermi gas non- interacting Bose gas

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Quantum Monte Carlo different methods

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QFT for non-relativistic fermions functional integral, action functional integral, action τ : euclidean time on torus with circumference 1/T σ : effective chemical potential perturbation theory: Feynman rules

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parameters detuning ν(B) detuning ν(B) Yukawa or Feshbach coupling h φ Yukawa or Feshbach coupling h φ

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fermionic action equivalent fermionic action, in general not local

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scattering length a broad resonance : pointlike limit broad resonance : pointlike limit large Feshbach coupling large Feshbach coupling a= M λ/4π

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collective di-atom states collective degrees of freedom can be introduced by partial bosonisation partial bosonisation ( Hubbard - Stratonovich transformation )

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units and dimensions c = 1 ; ħ =1 ; k B = 1 c = 1 ; ħ =1 ; k B = 1 momentum ~ length -1 ~ mass ~ eV momentum ~ length -1 ~ mass ~ eV energies : 2ME ~ (momentum) 2 energies : 2ME ~ (momentum) 2 ( M : atom mass ) ( M : atom mass ) typical momentum unit : Fermi momentum typical momentum unit : Fermi momentum typical energy and temperature unit : Fermi energy typical energy and temperature unit : Fermi energy time ~ (momentum) -2 time ~ (momentum) -2 canonical dimensions different from relativistic QFT ! canonical dimensions different from relativistic QFT !

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rescaled action M drops out M drops out all quantities in units of k F if all quantities in units of k F if

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effective action integrate out all quantum and thermal fluctuations integrate out all quantum and thermal fluctuations quantum effective action quantum effective action generates full propagators and vertices generates full propagators and vertices richer structure than classical action richer structure than classical action

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T = 0 gap parameter BCSBEC Δ

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limits BCS for gap condensate fraction for bosons with scattering length 0.9 a

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temperature dependence of condensate

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c -1 =1 c -1 =0 T/T c free BEC universal critical behavior condensate fraction : second order phase transition

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changing degrees of freedom

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Antiferromagnetic order in the Hubbard model A functional renormalization group study T.Baier, E.Bick, …

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Hubbard model Functional integral formulation U > 0 : repulsive local interaction next neighbor interaction External parameters T : temperature μ : chemical potential (doping )

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lattice propagator

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Fermion bilinears Introduce sources for bilinears Functional variation with respect to sources J yields expectation values and correlation functions

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Partial Bosonisation collective bosonic variables for fermion bilinears collective bosonic variables for fermion bilinears insert identity in functional integral insert identity in functional integral ( Hubbard-Stratonovich transformation ) ( Hubbard-Stratonovich transformation ) replace four fermion interaction by equivalent bosonic interaction ( e.g. mass and Yukawa terms) replace four fermion interaction by equivalent bosonic interaction ( e.g. mass and Yukawa terms) problem : decomposition of fermion interaction into bilinears not unique ( Grassmann variables) problem : decomposition of fermion interaction into bilinears not unique ( Grassmann variables)

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Partially bosonised functional integral equivalent to fermionic functional integral if Bosonic integration is Gaussian or: solve bosonic field equation as functional of fermion fields and reinsert into action

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fermion – boson action fermion kinetic term boson quadratic term (“classical propagator” ) Yukawa coupling

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source term is now linear in the bosonic fields

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

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Effective potential in mean field theory

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Mean field phase diagram μ μ TcTc TcTc for two different choices of couplings – same U !

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Mean field ambiguity TcTc μ mean field phase diagram U m = U ρ = U/2 U m = U/3,U ρ = 0 Artefact of approximation … cured by inclusion of bosonic fluctuations J.Jaeckel,…

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Rebosonization and the mean field ambiguity

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Bosonic fluctuations fermion loops boson loops mean field theory

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Rebosonization adapt bosonization to every scale k such that adapt bosonization to every scale k such that is translated to bosonic interaction is translated to bosonic interaction H.Gies, … k-dependent field redefinition absorbs four-fermion coupling

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Modification of evolution of couplings … Choose α k such that no four fermion coupling is generated Evolution with k-dependent field variables Rebosonisation

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…cures mean field ambiguity TcTc U ρ /t MFT Flow eq. HF/SD

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conclusions Flow equation for effective average action: Does it work? Does it work? Why does it work? Why does it work? When does it work? When does it work? How accurately does it work? How accurately does it work?

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end

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Flow equation for the Hubbard model T.Baier, E.Bick, …

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Truncation Concentrate on antiferromagnetism Potential U depends only on α = a 2

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scale evolution of effective potential for antiferromagnetic order parameter boson contribution fermion contribution effective masses depend on α ! gap for fermions ~α

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running couplings

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unrenormalized mass term Running mass term four-fermion interaction ~ m -2 diverges -ln(k/t)

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dimensionless quantities renormalized antiferromagnetic order parameter κ

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evolution of potential minimum -ln(k/t) U/t = 3, T/t = 0.15 κ 10 -2 λ

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Critical temperature For T 10 -9 size of probe > 1 cm -ln(k/t) κ T c =0.115 T/t=0.05 T/t=0.1

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Below the critical temperature : temperature in units of t antiferro- magnetic order parameter T c /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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Pseudocritical temperature T pc Limiting temperature at which bosonic mass term vanishes ( κ becomes nonvanishing ) It corresponds to a diverging four-fermion coupling This is the “critical temperature” computed in MFT !

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Pseudocritical temperature T pc μ TcTc MFT(HF) Flow eq.

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Below the pseudocritical temperature the reign of the goldstone bosons effective nonlinear O(3) – σ - model

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critical behavior for interval T c < T < T pc evolution as for classical Heisenberg model cf. Chakravarty,Halperin,Nelson

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critical correlation length c,β : slowly varying functions exponential growth of correlation length compatible with observation ! at T c : correlation length reaches sample size !

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critical behavior for order parameter and correlation function

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

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crossover phase diagram

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shift of BEC critical temperature

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