Path Integration in Statistical Theory: from QM to Interacting Fermion Systems

Andreas Wipf

Theoretisch-Physikalisches Institut Friedrich-Schiller-University Jena

Methhods of Path Integration in Modern 25.-31. August 2019

Andreas Wipf (TPI Jena) Path Integration in 1 Introduction

2 Path Integral Approach to Systems in Equilibrium: Finite Number of DOF Canonical approach Path integral formulation

3 Quantized Scalar Field at Finite Temperature Lattice regularization of quantized scalar field theories Äquivalenz to classical spin systems

4 Fermionic Systems at Finite Temperature and Density Path Integral for Fermionic systems Thermodynamic potentials of relativistic particles

5 Interacting Fermions Interacting fermions in condensed matter systems Massless GN-model at Finite Density in Two Dimensions Interacting fermions at finite density in d = 1 + 1 Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory why do we discretize quantum (field) theories? 2 / 79

weakly coupled subsystems: if not: strongly coupled system

properties can only be explained by strong correlations of subsystems

example of strongly coupled systems: ultra-cold atoms in optical lattices high-temperature superconductors statistical systems near phase transitions strong interaction at low energies

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory theoretical approaches 3 / 79

exactly soluble models (large symmetry, QFT, TFT) approximations mean field, strong coupling expansion, . . . restiction to effective degrees of freedom Born-Oppenheimer approximation, Landau-theory, . . . functional methods Schwinger-Dyson equations functional group equation numerical simulations lattice field theories = particular classical spin systems ⇒ powerful methods of and stochastics

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Quantum mechanical system in thermal equilibrium

Hamiltonian Hˆ : H 7→ H system in thermal equilibrium with heat bath

1 ˆ ˆ −βHˆ 1 canonical %ˆβ = K (β), K (β) = e , β = Zβ kT

normalizing partition function ˆ Zβ = tr K (β)

expectation value of observable Oˆ in ensemble ˆ ˆ hOiβ = tr %ˆβO

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory thermodynamic potentials 5 / 79

inner and free energy ∂ U = hHˆi = − log Z , F = −kT log Z β ∂β β β β

⇒ all thermodynamic potentials, S = −∂T F,... specific heat ∂U C = hHˆ 2i − hHˆi2 = − > 0 V β β ∂β system of particles: specify Hilbert space and Hˆ identical bosons: symmetric states identical fermions: antisymmetric states traces on different Hilbert spaces

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory quantum : canonical approach 6 / 79

Path integral for partition function in

euclidean evolution operator Kˆ satisfies diffusion type equation

ˆ d Kˆ(β) = e−βH =⇒ Kˆ(β) = −HˆKˆ(β) dβ compare with time-evolution operator and Schrödinger equation

− Hˆ/ d Uˆ(t) = e i ~ =⇒ i~ Uˆ(t) = HˆUˆ(t) dt

formally: Uˆ(t = −i~β) = Kˆ(β), imaginary time ~ quantum fluctuations, kT thermal fluctuations

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory heat kernel 7 / 79

evaluate trace in position space Z −βHˆ 0 0 hq| e |q i = K (β, q, q ) =⇒ Zβ = dq K (β, q, q)

0 0 “initial condition” for kernel: limβ→0 K (β, q, q ) = δ(q, q )

free particle in d dimensions (Brownian motion) 2  m d/2 − m (q0−q)2 ˆ ~ 0 2 2β H0 = − ∆ =⇒ K0(β, q, q ) = e ~ 2m 2π~2β

ˆ ˆ ˆ Hamiltonian H = H0 + V bounded from below ⇒

ˆ ˆ  ˆ ˆ n e−β(H0+V ) = s − lim e−βH0/n e−βV /n , Vˆ = V (qˆ) n→∞

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory derivation of path integral 8 / 79

insert for every identity 1 in

− β Hˆ − β Vˆ − β Hˆ − β Vˆ − β Hˆ − β Vˆ e n 0 e n 1 e n 0 e n 1 ··· 1 e n 0 e n

the resolution 1 = R dq |qihq|⇒

n 0  0  − β Hˆ − β Vˆ K β, q , q = lim q e n 0 e n q n→∞ j=n−1 Z β ˆ β ˆ Y − H0 − V = lim dq1 ··· dqn−1 qj+1 e n e n qj , n→∞ j=0

0 initial and final positions q0 = q and qn = q define small ε = ~β/n and finally use

β β − V (qˆ) − V (qj ) e n qj = qj e n

β ˆ  m d/2 m 2 − H0 − (qj+1−qj ) qj+1 e n qj = e 2~ε 2π~ε

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory derivation of path integral 9 / 79

discretized “path integral”

Z n/2 0  m  K (β, q , q)= lim dq1 ··· dqn−1 n→∞ 2π~ε  j=n−1   2   ε X m qj+1 − qj   · exp − + V (q ) 2  j  ~ j=0 

divide interval [0, ~β] into n sub-intervals of length ε = ~β/n consider path q(τ) with sampling points q(τ = kε) = qk

q q1 q2 qn =q′ q=q0

s 0 ε 2ε nε=~β

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory interpretation as path integral 10 / 79

Riemann sum in exponent approximates Riemann integral

β Z m  S [q] = dτ q˙ 2(τ)+V (q(τ)) E 2 0

SE is Euclidean action( ∝ action for imaginary time) integration over all sampling points n−→→∞ formal path integral Dq path integral with real and positive density

q( β)=q0 ~Z K (β, q0, q) = C Dq e−SE [q]/~

q(0)=q

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory partition function as path integral 11 / 79

on diagonal = integration over all path q → q

q(~β)=q ˆ Z K (β, q, q) = hq| e−βH |qi = C Dq e−SE [q]/~

q(0)=q

trace ˆ Z ˆ I tr e−βH = dq hq| e−βH |qi = C Dq e−SE [q]/~

q(0)=q(~β)

partition function Z(β) integral over all periodic paths with period ~β.

can construct well-defined Wiener-measure measure(differentable paths)= 0 measure(continuous paths)= 1

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory not only classical paths contribute 12 / 79

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Mehler’s formula 13 / 79

exercise (Mehler formula) show that the harmonic oscillator with Hamiltonian

2 2 2 ˆ ~ d mω 2 Hω = − + q 2m dq2 2 has heat kernel r 0 0 mω n mω  2 02 2qq o Kω(β, q , q) = exp − (q +q ) coth(~ωβ)− 2π~ sinh(~ωβ) 2~ sinh(~ωβ)

equation of euclidean motion q¨ = ω2q has for given q, q0 the solution

sinh(ωτ) q(τ) = q cosh(ωτ) + q0 − q cosh(ωβ) sinh(ωβ)

action

m Z β mω   S = (q˙ 2 + ω2q2) = (q2 + q02) cosh ωβ − 2qq0 2 0 2 sinh ωβ

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory second derivative ∂2S mω = − ∂q∂q0 sinh(ωβ) semiclassical formula exact for harmonic oscillator s 1 ∂2S K (β, q0, q) = − e−S 2π ∂q∂q0 yields above results for heat kernel diagonal elements

r ( 2 2 ) mω 2mωq sinh (~ωβ/2) Kω(β, q, q) = exp − 2π~ sinh(~ωβ) ~ sin(~ωβ)

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory spectrum of harmonic oscillator 15 / 79

partition function

∞ −~ωβ/2 1 e − ωβ/2 X −n ωβ Zβ = = = e ~ e ~ 2 sinh( ωβ/2) 1 − e−~ωβ ~ n=0

evaluate trace with energy eigenbasis of Hˆ ⇒

ˆ ˆ X Z(β) = tr e−βH = hn| e−βH |ni = e−βEn n comparison of two sums ⇒

 1 E = ω n + , n ∈ N n ~ 2

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory thermal correlation functions 16 / 79

position operator

ˆ ˆ qˆ(t) = eitH/~qˆ e−itH/~, qˆ(0) = qˆ

imaginary time t = −iτ ⇒ euclidean operator

ˆ ˆ ˆ τH/~ ˆ −τH/~ ˆ ˆ qE(τ) = e q e , qE(0) = q

correlations at different0 ≤ τ1 ≤ τ2 ≤ · · · ≤ τn ≤ β in ensemble

1  ˆ  qˆ (τ ) ··· qˆ (τ ) ≡ tr e−βH qˆ (τ ) ··· qˆ (τ ) E n E 1 β Z(β) E n E 1

consider thermal two-point function (now we set ~ = 1)

1  ˆ ˆ ˆ hqˆ (τ )qˆ (τ )i = tr e−(β−τ2)H qˆ e−(τ2−τ1)H qˆ e−τ1H E 2 E 1 β Z(β)

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory thermal correlation functions 17 / 79

spectral decomposition: |ni orthonormal eigenstates of Hˆ ⇒

1 X ˆ h... i = e−(β−τ2)En hn|qˆ e−(τ2−τ1)H qˆ|ni e−τ1En β Z(β) n

insert 1 = P |mihm| ⇒

1 X h... i = e−(β−τ2+τ1)En e−(τ2−τ1)Em hn|qˆ|mihm|qˆ|ni β Z(β) n,m P low temperature β → ∞: contribution of excited states to n(... ) exponentially suppressed, Z(β) → exp(−βE0) ⇒

β→∞ X ˆ ˆ −(τ2−τ1)(Em−E0) ˆ 2 hqE(τ2)qE(τ1)iβ −→ e |h0|q|mi| m≥0

likewise ˆ ˆ hqE(τ)iβ −→ h0|q|0i

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory connected correlation functions 18 / 79

connected two-point function

ˆ ˆ ˆ ˆ ˆ ˆ hqE(τ2)qE(τ1)ic,β ≡ hqE(τ2)qE(τ1)iβ − hqE(τ2)iβ hqE(τ1)iβ P term with m = 0 in m(... ) cancels ⇒ exponential decay with τ1 − τ2:

X −(τ2−τ1)(Em−E0) 2 lim hqˆ (τ2)qˆ (τ1)i = e |h0|qˆ|mi| β→∞ E E c,β m≥1

2 energy gap E1 − E0 and matrix element |h0|q|1i| from

2 ˆ ˆ −(E1−E0)(τ2−τ1) ˆ hqE(τ2)qE(τ1)ic,β→∞ −→ e |h0|q|1i| , τ2 − τ1 → ∞

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory path integral for thermal 19 / 79

for path-integral representation consider matrix elements

D ˆ ˆ ˆ ˆ ˆ E q0| e−βH eτ2H qˆ e−τ2H eτ1H qˆ e−τ1H |q

resolution of the identity and qˆ|ui = u|ui:

Z ˆ ˆ ˆ h... i = dvdu hq0| e−(β−τ2)H |vivhv| e−(τ2−τ1)H |uiuhu| e−τ1H |qi

path integral representations each propagator (β > τ2 > τ1):

sum over paths with q(0) = q and q(τ1) = u

sum over paths with q(τ1) = u and q(τ2) = v 0 sum over paths with q(τ2) = v and q(β) = q

multiply with intermediate positions q(τ1) and q(τ2) R dudv: path integral over all paths with q(0) = q and q(β) = q0

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory thermal correlation functions 20 / 79

insertion of q(τ2)q(τ1 in path integral 1 I hqˆ (τ )qˆ (τ )i = Dq e−SE [q] q(τ )q(τ ) E 2 E 1 β Z(β) 2 1

similarly: thermal n−point correlation functions

1 I hqˆ (τ ) ··· qˆ (τ )i = Dq e−SE [q] q(τ ) ··· q(τ ) E n E 1 β Z(β) n 1

conclusion there exist a path integral representation for all equilibrium quantities, e.g. thermodynamic potentials, , correlation functions

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory real and imaginary time 21 / 79

real time: quantum mechanics imaginary time: quantum statistics action from mechanics euclidean action Z Z  m 2   m 2  S = dt q˙ − V (q) SE = dτ q˙ + V (q) 2 2

real time path integral imaginary time path integral

ˆ ˆ hq0| e−itH/~|qi hq0| e−βH/~|qi 0 0 Z q(t)=q Z q(~β)=q = C Dq e iS[q]/~] = C Dq e−SE [q]/~] q(0)=q q(0)=q

correlation functions correlation functions

ˆ ˆ ˆ ˆ h0|T q(t1)q(t2)|0i hqE(τ1)qE(τ2)iβ Z Z iS[q]/~] −SE [q]/~] = C Dq e q(t1)q(t2) = C Dq e q(τ1)q(τ2)

oscillatory integrals exponentially damped integrals

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory stochastic methods are required 22 / 79

numerical simulations: discrete (euclidean) time system on time lattice = classical spin system

Z  m n/2 −SE (q1,...,qn)/~ Zβ = lim dq1 ··· dqn e n→∞ 2π~ε expectation values of observables Z dq1 ... dqn F(q1,..., qn)

high-dimensional integral (sometimes n = 106 required) curse of dimension: analytical and numerical approaches do not work stochastic methods, e.g. Monte-Carlo important sampling

what can be determined? energies, transitions amplitudes and wave functions in QM potentials, phase transitions, condensates and critical exponents bound states, masses and structure functions in particle physics . . .

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory harmonic and anharmonic oscillators 23 / 79

2 ψ0(q) | | m=1, µ= 3, λ=1 m=1, µ=1, λ=0 −

q

pˆ2 Hˆ = + µqˆ2 + λqˆ4 2m

Monte-Carlo simulation (Metropolis ) square of the ground state wave function

parameters in units of lattice constant ε A. Wipf, Lecture Notes Physics 864 (2013)

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory path integral for linear chain 24 / 79

exercise: harmonic chain find free energy for periodic chain of coupled harmonic oscillators

N 2 1 X m ω X 2 H = p2 + q − q  , q = q 2m i 2 i+1 i i i+N i=1 i

periodic q(τ) ⇒ may integrate by parts in

m Z 2 L = dτq˙ 2 + ω2q − q   E 2 i+1 i matrix notation

Z 2 m T  d  2  LE = dτq − + A q, A = ω 2δij − δi,j+1 − δi,j−1 2 dτ 2 hint: non-negative eigenvalues and orthonormal eigenvectors of A:

πk ω = 2 ω sin and e k N k Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory free energy of linear chain 25 / 79

P expand q(τ) = ck (τ)ek

X m Z L = dτ (c˙ 2 + ω2c2 E 2 k k k k

N decoupled oscillators with frequencies ωk ⇒

q −βHˆ q Y h | e | i = Kωk (β, qk , qk ) k results for one-dimensional oscillator ⇒

Y eβωk /2 Y e−βωk /2 πk Zβ = = , ωk = 2 ω sin eβωk − 1 1 − e−βωk N k k free energy contains zero-point energy

1 X X − ω /kT  F = ω + kT log 1 − e ~ k β 2 ~ k k k

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory field theories 26 / 79

spin 0: scalar field (Higgs particle, inflaton,. . . ) 1 spin 2 : spinor field (electron, neutrinos, quarks, . . . ) spin 1: vector field (photon, W-bosons, Z-boson, gluons, . . . )

a quick way from quantum mechanics to quantized scalar field theory:

scalar field φ(t, x) satisfies Klein-Gordon type equation( ~ = c = 1) 2φ + V 0(φ) = 0

Lagrangian = integral of Lagrangian density over space

Z 1 L[φ] = dx L(φ, ∂ φ), L = ∂ φ∂µφ − V (φ) µ 2 µ space

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory field theories 27 / 79

momentum field, Legendre transform ⇒ Hamiltonian (fixed time t)

δL ∂L π(x) = = = φ˙(x) δφ˙(x) ∂φ˙(x) Z Z 1 1 H = dxπφ˙ − L = dx H, H = π2 + (∇φ)2 + V (φ) 2 2

free particle: V ∝ φ2 ⇒ Klein-Gordan 2φ + m2φ = 0 infinitely many dof: one at each space point one of many possible regularizations: discretize space field theory on space lattice: x = εn with n ∈ Zd−1 Z d−1 X φ(t, x) −→ φx=εn (t) , dx −→ ε n

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory from field theory to point mechanics 28 / 79

ε finite hypercubic lattice in space

x = ε n with ni ∈ {1, 2,..., Ni }

continuum field φ(x) → lattice field φx N2 integral → Riemann sum

Z X dx −→ εd−1 n

N1 derivative → difference quotient e.g. periodic bc ∂φ(x) ε −→ (∂i φ)x lattice constant ∂x Q i # of lattice sites N = Ni example: symmetric “lattice derivative” linear extends Li = εNi d−1 physical volume V = ε N φx+εei − φx−εei (∂ φ)x = i 2ε

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory from field theory to point mechanics 29 / 79

finite lattice → mechanical system with finite number of dof

d−1 X 1 2 1 2  H = ε πx + (∂φ)x + V (φx ) 2 2 x∈lattice path integral quantization known Z 0 −iHˆ/ Y iS[{φx }]/ {φx } e ~ {φx } = C Dφx e ~ x

(formal) path integral over paths {φx (t)} in configuration space

0 φx (0) = φx and φx (t) = φx , ∀ x = εn

high-dimensional quantum mechanical system with action Z d−1 X 1 2 1 2  S[{φx }] = dt ε φ˙x − (∂φ)x − V (φx ) x 2 2

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory quantum field theory at finite temperature 30 / 79

canonical partition function I Y −SE [{φx }]/ Zβ = C Dφx e ~, φx (τ) = φx (τ + ~β) x

real euclidean action Z   d−1 X 1 ˙2 1 2 SE [{φx }] = dτ ε φx + (∂φ)x + V (φx ) x 2 2

path-integral well-defined after discretization of “time” convenient: same lattice constant ε in time and spatial directions

replace τ ∈ [0, ~β] −→ τ ∈ {ε, 2ε, . . . , N0ε} with N0ε = ~β µ µ lattice sites (x ) = (τ, x) = (εn ) with nµ ∈ {1, 2,..., Nµ} ⇒ d-dimensional hypercubic space-time lattice

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory space-time lattice 31 / 79

lattice field φx defined on sites of space-time lattice Λ

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory lattice regularization 32 / 79

d-dimensional Euclid’sche space-time → lattice Λ, sites x ∈ Λ

continuous field φ(x) → lattice field φx , x ∈ Λ

finite lattice: extend in direction µ: Lµ = εNµ finite temperature: L1 = ··· = Ld−1  L0 ≡ β = 1/(kT ) scalar field periodic in imaginary time direction

φ 0 x = φ 0 x =⇒ temperature-dependence x=(x +εN0, ) x=(x , ) typically: also periodic in spatial directions µ µ ⇒ identification x ∼ x + Lµ (torus) d space-time volume V = ε N1N2 ··· Nd some freedom in choice of lattice derivative (use symmetries)

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory space-time lattice 33 / 79

dimensionless fields and couplings (~ = c = 1)

natural units ~ = c = 1 ⇒ all units in powers of length L dimensionless action (unit L0)

Z 1 X  S = dd x (∂φ)2 + λph φa E 2 a a

R dd x (∂φ)2 dimensionless ⇒ [∂φ] = L−d/2 ⇒ [φ] = L1−d/2 ph R d a ph −d−a+ad/2 λa d x φ dimensionless ⇒ [λa ] = L ph 2 −1 in particular λ2 ∝ m ⇒ [m] = L ph 4 space-time dimensions ⇒ λ4 dimensionless dimensionless lattice field and lattice constants (x = εn)

1−d/2 ph −d−a+ad/2 φx = ε φn, λa = ε λa

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory dimensions 34 / 79

lattice action with dimensionful quantities ! 2 X 1φx+εeµ − φx−εeµ  X Sph = εd + λphφa L 2 2ε a x x a ⇒ lattice action with dimensionless quantities

X 1 2 X  S = φ − φ  + λ φa L 2 n+eµ x−eµ a n n a

partition function Z N0N1··· Y −SL[{φn}] Zβ = C dφn e n=1 finite-dimensional well-defined integral (lattice regularization) lattice formulation without any dimensionful quantity processor knows numbers, not units!

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory renormalization 35 / 79

merely letting ε → 0: no meaningful continuum limit

λa must be changed as ε → 0 condition: dimensionful observables approach well-defined finite limits existence of such continuum limit not guaranteed example: consider correlation length in

1 hφ(n)φ(m)i ∝ e−|n−m|/ξ, = m = dimnsionless mass c ξ

ξ depends on dimensionless couplings ξ = ξ(λa) relates to (given) dimensionful mass mph = 1/(εξ) ⇒ ε ph m from experiment, ξ(λa) measured on lattice ph renormalization: keep m (and further observables) fixed ⇒ λa

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory renormalizable field theories 36 / 79

extend of physical objects  separation of lattice points extend of physical objects  box size conditions (scaling window) small discretization effects ξ  1 small finite size effects ξ  Nµ strict continuum limit: ξ → ∞

2’nd order required in system with Nspatial → ∞

theory renormalizable: only a small number of λa must be tuned relevant renormalizable field theories non-Abelian gauge theories in d ≤ 4 scalar field theories in d < 4 four-Fermi theories in d ≤ 3 non-linear sigma-models in d ≤ 3 Einstein-gravity in d ≤ 4 (???)

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory simulation 37 / 79

input in simulations: only a few observables (masses) simulate with stochastic in scaling window repeat simulations with same observables but decreasing ε output: many (dimensionful) observables extrapolate to ε → 0 if theory renormalizable: converge to a continuum limit as ε → 0

finite temperature: N0 given, ε from matching to observable ⇒ β = εN0. ⇒ temperature dependence of free energy condensates pressure, densities free energy of two static charges (confinement) phase diagram screening effects correlations in heat bath, . . .

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory lattice field theory as 38 / 79

path integral for finite temperature QFT = classical spin model no non-commutative operators, instead: path or over fields scalar field: assign φn ∈ R to each lattice site sigma models: φn ∈ Sphere discrete spin models: φn ∈ discrete group example: Potts-model: φn ∈ Zq figures: 3−state Potts-type model

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory relativistic fermions 39 / 79

electron, muon, quarks, . . . are described by 4-component spinor field ψα(x)

metric in -time

(ηµν ) = diag(1, −1, −1, −1)

4 × 4 gamma-matrices

γ0, . . . , γ3, {γµ, γν } = 2ηµν 1

covariant Dirac equation for free massive fermions

 µ i∂/ − m ψ(x) = 0, ∂/ = γ ∂µ

Euler-Lagrange equation of invariant action Z 4 † 0 † S = d xψ¯(i∂/ − m)ψ, ψ¯ = ψ γ =⇒ πψ = −iψ

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory quantization of Dirac field 40 / 79

quantization: ψ(x) → ψˆ(x) satisfies anti-commutation relation

ˆ x ˆ† y x y {ψα(t, ), ψβ(t, )} = δαβδ( − )

Hamilton operator: β = γ0, α = γ0γ: Z Hˆ = dx ψˆ†(x)(hˆ ψˆ)(x), hˆ = i α · ∇ + m β

derive path integral representation of partition function

−βHˆ Zβ = tr e

leads to imaginary time path integral replace t → −iτ and 0 0 i i γE = γ and γE = iγ

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory path integral for partition function 41 / 79

ACR with euclidean metric µ ν µν 1 µ {γE , γE } = 2δ , γE hermitean lattice regularization (drop index E) space-time R4 → finite (hypercubic) lattice Λ 4 continuum field ψ(x) on R → lattice field ψx expected path integral

ˆ I † −βH Y † −SL[ψ,ψ ] Zβ = trreg e = C dψα,x dψα,x e α,x∈Λ

integration over anti-periodic fields (ACR for ψ, see below)

ψx (τ + β) = −ψx (τ), also on time lattice

SL some lattice regularization of Z d † SE = d x ψ (i∂/ + im)ψ

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Grassmann variables 42 / 79

quantized scalar field obey equal-time CR

φˆ(t, x), φˆ(t, y) = 0, x 6= y

⇒ commuting fields in path integral

[φ(x), φ(y)] = 0, ∀x, y

quantized fermion field obey equal-time ACR

 ˆ x ˆ† y x y ψα(t, ), ψβ(t, ) = 0, 6= ,

⇒ anti-commuting fields in path integral

 † ψα(x), ψβ(y) = 0, ∀x, y

† variables {ψα,n, ψα,n} in fermion path integral: Grassmann variables

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Graussian integrals 43 / 79

free theories have quadratic action Gaussian integrals with A = AT positive matrix; exercise ⇒

N Z Y  1 X  (2π)n/2 dφ exp − φ A φ = √ n 2 n nm m n=1 det A what do we get for fermions? † simplify notation: ψα,n ≡ ηi and ψα,n ≡ η¯i with i = 1,..., m

objects {ηi , η¯i } form complex Grassmann algebra:

2 2 {ηi , ηj } = {η¯i , η¯j } = {ηi , η¯j } = 0 =⇒ ηi =η ¯i = 0

Grassmann integration defined by (a, b ∈ C) Z Z Z linear , dηi (a + b ηi ) = b, dη¯i (a + b η¯i ) = b

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Gaussian integrals with Grassmann variables 44 / 79

Grassmann integrals with

m Y Dη¯Dη ≡ dη¯i dηi i=1 free fermions ⇒ Gaussian Grassmann integral Z −η¯Aη X Z = Dη¯Dη e , η¯Aη = η¯i Aij ηj i,j

k expand exponential function: R Dη¯Dη η¯Aη = 0 for k 6= m 2 remaining contribution (use η¯i = 0) 1 Z Z X Dη¯Dη (¯ηAη)m = Dη¯Dη (¯η A η ) ··· (¯η A η ) n! 1 1i1 i1 m mim im i1,...,im Z Y X = Dη¯Dη (¯ηi ηi ) εi1...im A1i1 ··· Amim i i1,...,im Z m Y m = (−1) (dη¯i η¯i dηi ηi ) det A = (−1) det A i

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory generating function 45 / 79

simple formula Z Dη¯Dη e−η¯Aη = det A

generalization: generating function

Z −1 Z(¯α, α) = Dη¯Dη e−η¯Aη+αη ¯ +¯ηα = e−α¯A α det A

expand in powers of α¯, α ⇒

1 Z hη¯ η i ≡ Dη¯Dη e−η¯Aη η¯ η = (A−1) i j Z i j ij

application to Dirac fields: above partition function I ¯ −SL ¯ Y † Zβ = DψDψ e , DψDψ = dψα,n dψα,n α,n

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Graussian integrals for fermions and bosons 46 / 79

dimensionless field and couplings

X † X ¯ SL = ψn(i∂/nm + imδnm)ψn = ψnDnmψm n∈Λ n

lattice partition function Zβ = C det D expectation value in

1 I ¯ ˆ −SL(ψ,ψ) hAiβ = Dψ¯Dψ A(ψ,¯ ψ) e Zβ

formula for complex scalar field

I  X  1 Z = DφDφ¯ exp − φ¯ C φ ∝ β m mn n det C

boson fields: periodic in imaginary time fermion fields: anti-periodic in imaginary time

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Thermodynamic potentials for Gas or relativistic particles 47 / 79

neutral scalars (+: periodic bc)

1 Z kT S = φ(−∆ + m2) φ =⇒ F = log det (−∆ + m2) + ... E 2 β 2 +

Dirac fermions (−: anti-periodic bc) Z † 2 SE = ψ (i∂/ + im) ψ =⇒ Fβ = −2kT log det−(−∆ + m ) + ...

exercise Try to prove the results for fermions (including sign and overall factor)

zeta-function for second order operator A > 0

X −s ζA(s) = λn , eigenvalues λn n

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory ζ− function regularization 48 / 79

absolute convergent series in half-plane <(s) > d/2 meromorphic analytic continuation, analytic in neighborhood of s = 0

defines ζ−function regularized determinant Dowker, Hawking

X dζA(s) log det A = tr log A = log λn = − ds s=0 correct for matrices Mellin transformations Z ∞ dt ts−1e−tλ = Γ(s) λ−s 0 ⇒ relation to heat kernel X 1 Z ∞ 1 Z ∞ ζ (s) = dt ts−1e−tλn = dt ts−1 tr e−tA A Γ(s) Γ(s) n 0 0

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory heat kernel 49 / 79

coordinate representation Z ∞ Z 1 s−1 −tA ζA(s) = dt t dx K (t; x, x), K (t) = e Γ(s) 0

heat kernel of A = −∆ + m2 on cylinder [0, β] × Rd−1

−m2t e X 0 2 x x 0 2 K ±(t; x, x 0) = (±1)n e−{(τ−τ +nβ) +( − ) }/4t (4πt)d/2 n∈Z

integrate over diagonal elements Z ∞ βV 2 X 2 2 ζ±(s) = dt ts−1−d/2 e−m t (±)n e−n β /4t A (4π)d/2Γ(s) n=−∞

Jacobi theta function

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory integral representation of Kelvin functions

Z ∞ (a+1)/2 √ a −bt−c/t  c    dt t e = 2 Ka+1 2 bc 0 b ⇒ series representation; in d = 4

∞ s−2 ! βV m4−2s X nmβ  ζ±(s) = Γ(s − 2) + 4 (±)n K (nmβ) A 16π2 Γ(s) 2 2−s 1 identities Γ(s − 2) 1 1 = and = s + O(s2) Γ(s) (s − 1)(s − 2) Γ(s)

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory derivative at s = 0 ⇒   4 2 ± m V C± m X n K2(nmβ) F = − 3 − 2 log + 64 (±)  β 128π2 µ2 (nmβ)2 n=1,2...

real scalars C+ = 1, complex fermions C− = −4 2 well-known results for massless particles K2(x) ∼ 2/x

π2 2 lim f +(β) = − T 4 , lim f −(β) = − T 4 m→0 90 m→0 45π2 questions Why is there a relative factor of 4? What is the free energies of complex scalars, Majorana fermions and photons. What is free energy of complex fermions in d dimensions?

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Interacting relativistic fermions 52 / 79

condensed matter systems in d = 2 + 1

tight binding approximation for small excitation energies honeycomb lattice for graphen (GN): 2 atoms in every cell, 2 Dirac points ⇒ 4-component spinor field interaction-driven transition metal ↔ insolator long rang order: AF, CDW, . . . interacting fermions (symmetries!) condensed matter systems in d = 1 + 1 relativistic dispersion-relations for electronic excitations on conducting (Trans- and honeycomb lattice Cis-polyacetylen) Su, Schrieffer, Heeger from Castro Neto et al. quasi-one-dimensional inhomogeneous superconductor Mertsching, Fischbeck

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory interacting fermions in 1 + 1 and 2 + 1 dimensions 53 / 79

irreducible spinor in two and three dimensions has 2 components

Nf species (flavours) of spinors, Ψ = (ψ1, . . . , ψNf ) relativistic fermions ¯ ¯ ¯ ¯ X ¯ LGN = Ψi∂/Ψ + imΨΨ + LInt(Ψ, Ψ), e.g. ΨΨ = ψi ψi

parity invariant models

2 gGN ¯ ¯ LInt = (ΨΨ)(ΨΨ) scalar-scalar, Gross-Neveu 2Nf 2 gTh ¯ µ ¯ LInt = − (Ψγ Ψ)(ΨγµΨ) vector-vector, Thirring 2Nf 2 gPS ¯ ¯ LInt = (Ψγ∗Ψ)(Ψγ∗Ψ) pseudoscalar-pseudoscalar 2Nf Q in even dimensions γ∗ ∝ γµ Hubbard-Stratonovich trick with scalar, vector and pseudscalar field

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Relativistic four-Fermi Theories 54 / 79

combinations thereof in d = 4 non-renormalizable Fermi theory of weak interaction effective models for chiral phase transition in QCD (Jona Lasino) 2 spacetime dimensions: [g] = L0

massless ThM: soluble Thirring massless ThM in curved space with µ: soluble Sachs+AW, . . . GNM: asymptotically free, integrable Gross-Neveu, Coleman, . . . 3 spacetime dimensions: [g] = Ls not renormalizable in PT renormalizable in large-N expansion Gawedzki, Kupiainen; Park, Rosenstein, Warr interacting UV fixed point → asymptotically safe de Veiga; da Calen; Gies, Janssen can exhibit parity breaking at low T lattice theories: generically: sign problem even for µ = 0 partial solution of sign problem Schmidt, Wellegehausen, Lenz, AW

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Masselss GN-model at finite density in two dimensions 55 / 79

with J. Lenz, L. Panullo, M. Wagner and B. Wellegehausen GN shows breaking of discrete chiral symmetry order parameter iΣ = hΨΨ¯ i ¯ ¯ ¯ ¯ ψa → iγ∗ψa, ψa → iψaγ∗ =⇒ iΣ = hΨΨi → −hΨΨi

equivalent formulation with auxiliary scalar field Hubbard-Stratonovich transformation N L = L = Ψ¯ iD ⊗ 1 Ψ + f (ΨΨ)¯ 2 GN σ Nf 2g ¯ 1  2 / † Lσ = Ψ iD ⊗ Nf Ψ + λNf σ , D = ∂ − σ 6= D

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory chemical potential for fermion number charge 56 / 79

conserved fermion charge Z Z Q = dx j 0 = dx ψ†ψ space space

partition function of

−β(Hˆ−µQˆ) Zβ,µ = tr e ,

functional integral with above Lσ wherein

D = ∂/ + σ + µγ0

expectation values

1 Z hOi = Dψ¯DψDσ e−Sσ O Zβ,µ

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory chemical potential for fermion charge 57 / 79

fermion integral in

Z ¯ Z −Sσ [σ,ψ,ψ] −NfSeff[σ] Zβ,µ = Dψ¯DψDσ e = Dσ e

Nf fermion species couple identically to auxiliary field ⇒

 N det iD ⊗ 1 = (det iD) f

ψ anti-periodic in imaginary time, σ periodic effective action after fermion integral Z 2 2 Seff = λ d x σ − log(det iD)

Ward identity (lattice regularization) Z 1 d  †  D dS E Dψ Dψ¯ Dσ e−S[σ,ψ,ψ ] = − = 0 Zβ,µ dσ(x) dσ(x)

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory homogeneous phases 58 / 79

exact relation N Σ ≡ −ihψ¯(x)ψ(x)i = f hσ(x)i g2

for Nf → ∞ saddle point (steepest descend) approximation

Z →∞ −NfS ][σ] Nf −Nf min Seff[σ] Zβ,µ = Dσ e eff −→ e

translation invariance ⇒ minimizing σ constant: Seff = (NfβL) Ueff

σ2  σ2  1 Z ∞ p2  1 1  = − − + Ueff log 1 dp β(ε +µ) β(ε −µ) 4π σ0 π 0 εp 1 + e p 1 + e p

p 2 2 one-particle energies p = p + σ

IR-scale σ0 = hσiT =µ=0

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory condensate in the (µ, T )-plane 59 / 79

0.7 1 'data.dat' u 1:2:3 0.6 0.8 0.5

0.4 0.6 0 T/ σ 0.3 0.4 0.2 0.2 0.1

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

µ/σ0 symmetric phase for large T , µ

homogeneously broken phase for small T , µ Wolff, Barducci √ γ special points: (Tc , µ) = (e /π, 0), (T,µc ) = (0, 1/ 2)

Lifschitz-Punkt bei (T , µ0) ≈ (0.608, 0.318)

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory is homogeneity assumption really justified? 60 / 79

possible QCD-phase diagram

crystalline LOFF phase (color superconductive phase)? problem: µ 6= 0 ⇒ complex fermion determinant large µ beyond reach in simulations / are there inhomogeneous crystallic phases in model systems?

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory space-dependent condensate in GN model 61 / 79

discrete εn energies of Dirac Hamiltonian on [0, L]

0 1 0 hσ = γ γ ∂x + γ σ(x)

hidden ! d 2 AA† 0 d h2 = − + σ2(x) − γ1σ0(x) = , A = − + σ σ dx 2 0 A†A dx

renormalization: fix (constant) condensate σ0 at µ = T = 0 introduce constant companion field

1 Z σ¯2 = dx σ2(x) L constant σ ⇒ σ¯ = σ

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory renormalized effective action for σ = σ(x)

βL  σ¯ 2   X X  S [σ] = σ¯ 2 log − 1 + β ε − ε¯ eff 4π σ2 n n 0 n:εn<0 n:¯εn<0 X   − log 1 + e−β(εn+µ) + log 1 + e−β(εn−µ)

n:εn>0 derive gap equation for inhomogeneous field

δS 1 σ¯2 X σ(x) X eff = σ(x) log + ψ†γ0ψ − ψ¯†γ0ψ¯ δσ(x) 2π σ2 n n σ¯ n n 0 n:εn<0 n:¯εn<0   X 1 1 † 0 + + ψnγ ψn = 0 1 + eβ(εn+µ) 1 + eβ(εn−µ) n:εn>0

solution in terms of elliptic functions ⇒ crystal of baryons at large µ, low T

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory phase diagram for two-dimensional GN-model Mertsching and Fischbeck, Thies et al. 63 / 79

0.7 Lifshitz point 0.6

0.5

0 0.4

T/ σ 0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

µ/σ0 inhomogeneous condensate for small T , large µ

⇒ breaking of translation invariance (Nf → ∞) wave-length of condensate ⇔ µ all phase transitions are second order cp. Peierls-Fröhlich model, ferromagnetic superconductors

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory no-go theorems and Nf → ∞ 64 / 79

inhomogeneous hψψ¯ i breaks translation invariance → massless Goldstone-excitations → should not exist in d = 1 + 1

no-go theorems not valid for Nf → ∞

phase diagram = artifact of Nf → ∞?

is there a inhomogeneous condensate for Nf < ∞? number of massless Goldstone excitations:

nk number of type k Goldstone modes type 1: ω ∼ |k|2n+1, e.g. relativistic dispersion relation type 2: ω ∼ |k|2n, e.g. non-relativistic dispersion relation

inner symmetries n1 + 2n2 =number of broken directions

spacetime symmetries n1 + 2n2 ≤ number of broken directions large µ: dispersion relation need not be relativistic

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory fermion system for finite Nf (beyond steepest descend): MC-simulations 65 / 79

update with (nonlocal) determinant of huge matrix D potential sign-problem for finite µ can prove: fermion determinant is indeed real

⇒ no sign problem for even Nf hybrid MC algorithm, pseudo fermions rational approximation of inverse fermion matrix simulations with chiral fermions only

naive fermions for Nf = 8, 16 (→ doublers)

simulations with SLAC fermions for Nf = 2, 8, 16 action of pseudo-fermion field with parallized Fourier transformation

scale setting: condensate σ0 at T = µ = 0

simulations on large lattices Ns ≤ 1024

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory typical configurations 66 / 79

low temperature T = 0.038 σ0, medium density µ = 0.5σ0

typical configuration for Nf = 8 and L = 64

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory strong evidences for inhomogeneous condensate 67 / 79

T/ 0 = 0.082 , / 0 = 0.00

4 T/ 0 = 0.988 , / 0 = 0.00 2 0

/ spatial correlation function of ) x

( 3

C chiral condensate

r o t 2 a l 1 X e

r C(x) = hσ(y, t)σ(y + x, t)i r

o 1 L C y

0

0.0 2.5 5.0 7.5 10.0 12.5 15.0 Nf = 8, L = 64 Spatial Direction x 0 naive fermions 0.4 top: homogeneous phase / 0 = 0.5

0.3 / 0 = 0.7 2 0 µ = 0 / / 0 = 1.0 )

x 0.2 ( T /σ0 ∈ {0.082, 0.988} C

r

o 0.1 t

a bottom: inhomogeneous phase l e

r 0.0 r

o T = 0.082σ0 C 0.1 µ/σ0 ∈ {0.5, 0.7, 1.0} 0.2 0.0 2.5 5.0 7.5 10.0 12.5 15.0 Spatial Direction x 0

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory 1.1251.0 2 0 1.100 / )

k 0.8 ( 1.075 C 0.15 Fourier transform of the spatial m u

r 0.6

t correlation function c e

p 0.10

S X kx 0.4 ˜ i r C(k) ∝ e C(x) o t a l x

e 0.05

r 0.2 r o C 0.000.0 0.075 0.250 250.4 0 0.625 0.850 751.0 Nf = 8, L = 64 Wave Number k/ 0 naive fermions 2 0 top: homogeneous phases 0.20 / ) k ( µ = 0 C

m 0.15 T /σ0 ∈ {0.082, 0.988} u r t c e bottom: inhomogeneous phase p S

0.10 r o t T = 0.082 σ0 a l e r

r µ/σ ∈ { . , . , . } 0.05 0 0 5 0 7 1 0 o C

80 60 40 20 0 20 40 60 80 Wave Number k/ 0

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory ‘inhomogeneous’ phase: µ-dependence 69 / 79

spatial correlation function and Fourier-transform

Nf = 8, L = 64 SLAC-fermions

low temperature T = 0.038σ0 different chemical potentials

µ/σ0 ∈ {0, 0.4, 0.5, 0.7} violet: symmetric phase µ = 0, T = 0.61 σ0

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory comparison of fermion species 70 / 79

L = 64, 0 0.41, 0.7, Nt = 64

2.0 Naive SLAC

1.5

1.0 Correlator 0.5

0.0 Nf = 8

0 2 4 6 8 10 12 crystalline phase Spatial Coordinate spatial correlation function L = 64, 0 0.41, 0.7, Nt = 64 for naive and SLAC 0.12 Naive SLAC fermions 0.10 Fourier transform 0.08

0.06

0.04 Fourier Spectrum of Correlator 0.02

0 10 20 30 40 50 60 70 Wave Number

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory fermion number and condensate 71 / 79

L = 63, 0 = 0.4100, Nt = 64 1.000 8

0.875 7

0.750 6

0.625 5 L | 0.500 4 B | n 0.375 3

0.250 2

0.125 1

0.000 0

0.0 0.2 0.4 0.6 0.8

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory phase diagram: homogeneous phases from σ 72 / 79

, L = 64, 0 0.20 1.0 SLAC 0.5 Naive 0.8 0.4

0.6 0.3 T

0.4 0.2

0.1 0.2

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory phase diagram: all three phases from Cmin 73 / 79

Cmin, L = 64, 0 0.20 0.20 SLAC 0.5 Naive 0.15

0.4 0.10

0.05 0.3

T 0.00

0.2 0.05

0.10 0.1 0.15

0.0 0.20 0.0 0.2 0.4 0.6 0.8 1.0

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory correlations function for Nf  1

Witten V(φ) 1 C(x, y) ∼ |x − y|1/Nf may look like SSB for large Nf on small lattices dependence on system size smallest available = 2 Nf Re(φ) Im(φ) check algorithmic aspects (e.g. thermalization)

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Nf = 2, smaller lattice Ns = 125, chiral SLAC fermions 75 / 79

fit fall-off 0.04 interference ) x ( C

r

o 0.02 t a l e r r o C

0.00 l a i t a p S 0.02

0.04 0 10 20 30 40 50 60 Lattice Distance

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Nf = 2, large lattice, Ns = 525, chiral SLAC fermions 76 / 79

fit fall-off 0.04 interference ) x ( C

r

o 0.02 t a l e r r o C

0.00 l a i t a p S 0.02

0.04 0 50 100 150 200 250 Lattice Distance

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Nf = 2, comparison 77 / 79

0.2 65.0 255.0 125.0 375.0 185.0 525.0 )

x 0.1 ( C

r o t a l e r

r 0.0 o C

l a i t a

p 0.1 S

0.2

0 50 100 150 200 250 Lattice Distance x

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory summary of results in d = 1 + 1 dimensions 78 / 79

first simulation for GN model at finite µ, T , Nf

no sign problem for even Nf

comparable results for Nf = 8 and Nf = 16 naive and chiral SLAC fermions

phase diagrams are similar as for Nf → ∞ Thies wave length and amplitude of condensate

simulations for Nf = 2 on sizable lattices Goldstone-theorem, . . . situation in higher dimensions domain walls, vortices, . . . ???

Lenz, Pannullo, Wagner, Wellegehausen, AW

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory Some remarks concerning interacting fermions in d = 2 + 1 dimensions 79 / 79

asymptotically safe (1/Nf expansion, FRG)

GN model show 2nd order phase transition for all Nf

Nf odd: parity breaking Thirring models: even Nf: no phase transition crit odd Nf: phase transition for Nf ≤ Nf crit critical Nf determined spectrum of light (would be Goldstone) particles average spectral density of Dirac operator

full phase diagram in (λ, Nf)-plane B. Wellegehausen, D. Schmidt, AW, Phys.Rev. D96 (2017) 094504

J. Lenz, AW, B. Wellegehausen, arXiv:1905.00137

Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory