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

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Path Integration in Statistical Field Theory:

from QM to Interacting Fermion Systems

Andreas Wipf

Theoretisch-Physikalisches Institut Friedrich-Schiller-University Jena

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

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

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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 ind=1+1

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why do we discretize quantum (field) theories? 2 / 79

weakly coupled subsystems:perturbation theory 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

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theoretical approaches 3 / 79

exactly soluble models(large symmetry, QFT, TFT) approximations

mean field, strong coupling expansion, . . . restiction toeffective degrees of freedom

Born-Oppenheimer approximation, Landau-theory, . . . functional methods

Schwinger-Dyson equations

functional renormalization group equation numerical simulations

lattice field theories = particular classical spin systems

⇒powerful methods of statistical physics and stochastics

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Quantum mechanical system in thermal equilibrium

HamiltonianHˆ :H 7→ H

system inthermal equilibriumwith heat bath canonical %ˆβ= 1

Zβ

Kˆ(β), Kˆ(β) = e^−β^H^ˆ, β= 1 kT normalizingpartition function

Zβ =trKˆ(β) expectation value of observableOˆin ensemble

hOiˆ _β=tr %ˆ_βOˆ

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thermodynamic potentials 5 / 79

inner and free energy

U=hHiˆ β=− ∂

∂β logZβ , Fβ=−kTlogZβ

⇒all thermodynamic potentials, entropyS=−∂_TF, . . . specific heat

C_V =hHˆ²iβ− hHiˆ ²_β=−∂U

∂β >0 system of particles: specify Hilbert space andHˆ identical bosons: symmetric states

identical fermions: antisymmetric states traces on different Hilbert spaces

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quantum statistics: canonical approach 6 / 79

Path integral for partition function in quantum mechanics

euclidean evolution operatorKˆ satisfiesdiffusiontype equation Kˆ(β) = e^−β^H^ˆ=⇒ d

dβ

Kˆ(β) =−HˆKˆ(β)

compare with time-evolution operator and Schrödinger equation U(tˆ ) = e⁻ⁱ^H/^ˆ ^~=⇒ i~d

dtU(t) = ˆˆ HU(tˆ ) formally:U(tˆ =−i~β) = ˆK(β), imaginary time

~quantum fluctuations,kT thermal fluctuations

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heat kernel 7 / 79

evaluate trace in position space

hq|e^−β^H^ˆ|q⁰i=K(β,q,q⁰) =⇒Zβ= Z

dq K(β,q,q)

“initial condition” for kernel: limβ→0K(β,q,q⁰) =δ(q,q⁰)

free particle in d dimensions (Brownian motion)

Hˆ₀=−~²

2m∆=⇒K₀(β,q,q⁰) = m 2π~²β

d/2

e⁻^2~^m²^β^(q⁰^−q)²

HamiltonianHˆ = ˆH0+ ˆV bounded from below⇒ e^{−β( ˆ}^H⁰^{+ ˆ}^V) =s− lim

n→∞

e^−β^H^ˆ⁰^/ne^−β^V/n^ˆ n

, Vˆ =V( ˆq)

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derivation of path integral 8 / 79

insert for every identity1in e⁻^βⁿ^H^ˆ⁰e⁻^βⁿ^V^ˆ

1 e⁻^βⁿ^H^ˆ⁰e⁻^βⁿ^V^ˆ

1· · ·1 e⁻^βⁿ^H^ˆ⁰e⁻^βⁿ^V^ˆ the resolution1=R

dq|qihq| ⇒ K β,q⁰,q

= lim

n→∞

q⁰

e⁻^βⁿ^H^ˆ⁰e⁻^βⁿ^V^ˆn

q

= lim

n→∞

Z

dq1· · ·dqn−1 j=n−1

Y

j=0

q_j+1

e⁻^βⁿ^H^ˆ⁰e⁻^βⁿ^V^ˆ q_j

,

initial and final positionsq0=q andqn=q⁰ define smallε=~β/nand finally use

e⁻^βⁿ^V^{( ˆ}^q) q_j

= q_j

e⁻^βⁿ^V(q^j⁾ q_j+1

e⁻^βⁿ^H^ˆ⁰ q_j

= m 2π~ε

d/2

e⁻^2~ε^m ^(q^j+1^−q^j⁾²

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derivation of path integral 9 / 79

discretized “path integral”

K(β,q⁰,q) = lim

n→∞

Z

dq1· · ·dqn−1

m 2π~ε

^n/2

·exp







−ε

~

j=n−1

X

j=0

m 2

qj+1−qj

2

+V(qj)





 divide interval[0,~β]intonsub-intervals of lengthε=~β/n consider pathq(τ)with sampling pointsq(τ=kε) =q_k

s q

0 ε 2ε nε=~β

q=q0

q1

q2 qn=q^′

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interpretation as path integral 10 / 79

Riemann sum in exponent approximatesRiemann integral

SE[q] =

β

Z

0

dτm

2q˙²(τ)+V(q(τ)) S_EisEuclidean action(∝action for imaginary time)

integration over all sampling points^n→∞−→ formal path integralDq path integral with real and positive density

K(β,q⁰,q) =C

q(~β)=q⁰

Z

q(0)=q

Dq e^−S^E^[q]/^~

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partition function as path integral 11 / 79

on diagonal = integration over all pathq→q

K(β,q,q) =hq|e^−β^H^ˆ|qi=C

q(~β)=q

Z

q(0)=q

Dq e^−S^E^[q]/^~

trace

tr e^−β^H^ˆ= Z

dqhq|e^−β^H^ˆ|qi=C I

q(0)=q(~β)

Dq e^−S^E^[q]/^~

partition functionZ(β) integral overall periodic pathswith period~β.

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

measure(continuous paths)=1

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not only classical paths contribute 12 / 79

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Mehler’s formula 13 / 79

exercise (Mehler formula)

show that the harmonic oscillator with Hamiltonian Hˆω=−~²

2m d²

dq²+mω² 2 q² has heat kernel

Kω(β,q⁰,q) =

r mω

2π~sinh(~ωβ)expn

−mω 2~

(q²+q⁰²)coth(~ωβ)− 2qq⁰ sinh(~ωβ)

o

equation of euclidean motionq¨=ω²qhas for givenq,q⁰the solution q(τ) =qcosh(ωτ) + q⁰−qcosh(ωβ)sinh(ωτ)

sinh(ωβ) action

S= m 2

Z β

0

( ˙q²+ω²q²) = mω 2 sinhωβ

(q²+q⁰²)coshωβ−2qq⁰

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second derivative

∂²S

∂q∂q⁰ =− mω sinh(ωβ) semiclassical formula exact for harmonic oscillator

K(β,q⁰,q) = s

− 1 2π

∂²S

∂q∂q⁰ e^−S

yields above results for heat kernel diagonal elements

Kω(β,q,q) =

r mω

2π~sinh(~ωβ) exp (

−2mωq²

~

sinh²(~ωβ/2) sin(~ωβ)

)

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spectrum of harmonic oscillator 15 / 79

partition function Zβ= 1

2 sinh(~ωβ/2) = e⁻^~^ωβ/2

1− e⁻^~^ωβ = e⁻^~^ωβ/2

∞

X

n=0

e^−n~^ωβ

evaluate trace withenergy eigenbasisofHˆ ⇒ Z(β) =tr e^−β^H^ˆ=hn|e^−β^H^ˆ|ni=X

n

e^−βEⁿ

comparison of two sums⇒ En=~ω

n+1

2

, n∈N

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thermal correlation functions 16 / 79

position operator

q(t) =ˆ e^it^H/^ˆ ^~qˆe^−it^H/^ˆ ^~, q(0) = ˆˆ q

imaginary timet =−iτ⇒euclidean operator

qˆ_E(τ) = e^τ^H/^ˆ ^~ˆqe^−τ^H/^ˆ ^~, qˆ_E(0) = ˆq

correlations at different0≤τ₁≤τ₂≤ · · · ≤τ_n≤β in ensemble qˆ_E(τ_n)· · ·qˆ_E(τ₁)

β≡ 1

Z(β) tr

e^−β^H^ˆqˆ_E(τ_n)· · ·qˆ_E(τ₁) considerthermal two-point function(now we set~=1)

hqˆ_E(τ₂) ˆq_E(τ₁)iβ= 1 Z(β)tr

e^−(β−τ²^{) ˆ}^Hˆq e^−(τ²^−τ¹^{) ˆ}^Hqˆ e^−τ¹^H^ˆ

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spectral decomposition:|niorthonormal eigenstates ofHˆ⇒ h. . .iβ= 1

Z(β) X

n

e^−(β−τ²^)Eⁿhn|qˆ e^−(τ²^−τ¹^{) ˆ}^Hq|niˆ e^−τ¹^Eⁿ

insert1=P

|mihm| ⇒ h. . .iβ= 1

Z(β) X

n,m

e^−(β−τ²^+τ¹^)Eⁿe^−(τ²^−τ¹^)E^mhn|ˆq|mihm|ˆq|ni

low temperatureβ→ ∞: contribution of excited states toP

n(. . .) exponentially suppressed,Z(β)→exp(−βE0)⇒

hqˆ_E(τ₂) ˆq_E(τ₁)i_β ^β→∞−→ X

m≥0

e^−(τ²^−τ¹^)(E^m^−E⁰⁾|h0|ˆq|mi|² likewise

hqˆ_E(τ)iβ−→ h0|ˆq|0i

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connected correlation functions 18 / 79

connected two-point function

hqˆ_E(τ₂) ˆq_E(τ₁)i_c,β ≡ hqˆ_E(τ₂) ˆq_E(τ₁)i_β− hˆq_E(τ₂)i_βhˆq_E(τ₁)i_β term withm=0 inP

m(. . .)cancels⇒exponential decay withτ₁−τ₂:

β→∞lim hqˆ_E(τ₂) ˆq_E(τ₁)i_c,β =X

m≥1

e^−(τ²^−τ¹^)(E^m^−E⁰⁾|h0|ˆq|mi|²

energy gapE₁−E₀and matrix element|h0|q|1i|²from

hqˆ_E(τ₂) ˆq_E(τ₁)i_c,β→∞−→e^−(E¹^−E⁰^)(τ²^−τ¹⁾|h0|q|1i|ˆ ², τ₂−τ₁→ ∞

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path integral for thermal correlation function 19 / 79

for path-integral representation consider matrix elements Dq⁰|e^−β^H^ê^τ²^H^ˆqê^−τ²^H^ê^τ¹^H^ˆqê^−τ¹^H^ˆ|qE resolution of the identity andq|uiˆ =u|ui:

h. . .i= Z

dvduhq⁰|e^−(β−τ²^{) ˆ}^H|vivhv|e^−(τ²^−τ¹^{) ˆ}^H|uiuhu|e^−τ¹^H^ˆ|qi path integral representations each propagator (β > τ₂> τ₁):

sum over paths withq(0) =qandq(τ1) =u sum over paths withq(τ1) =uandq(τ2) =v sum over paths withq(τ₂) =v andq(β) =q⁰ multiply with intermediate positionsq(τ1)andq(τ2)

Rdudv: path integral over all paths withq(0) =q andq(β) =q⁰

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insertion ofq(τ₂)q(τ₁in path integral hqˆ_E(τ₂) ˆq_E(τ₁)i_β= 1

Z(β) I

Dqe^−S^E^[q]q(τ₂)q(τ₁) similarly: thermaln−point correlation functions

hqˆ_E(τ_n)· · ·qˆ_E(τ₁)iβ= 1 Z(β)

I

Dqe^−S^E^[q]q(τn)· · ·q(τ1)

conclusion

there exist a path integral representation for all equilibrium quantities, e.g.

thermodynamic potentials, equation of state, correlation functions

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real and imaginary time 21 / 79

real time: quantum mechanics action from mechanics

S= Z

dtm

2q˙²−V(q) real time path integral

hq⁰|e^−it^H/^ˆ ^~|qi

=C Z q(t)=q⁰

q(0)=q

Dq e^iS[q]/~]

correlation functions h0|Tqˆ(t1) ˆq(t2)|0i

=C Z

Dq e^iS[q]/~]q(t1)q(t2) oscillatory integrals

imaginary time: quantum statistics euclidean action

SE = Z

dτm

2q˙²+V(q) imaginary time path integral

hq⁰|e^−β^H/^ˆ ^~|qi

=C

Z q(~β)=q⁰

q(0)=q

Dq e^−S^E^[q]/~]

correlation functions hqˆ_E(τ1) ˆq_E(τ2)iβ

=C Z

Dq e^−S^E^[q]/~]q(τ1)q(τ2) exponentially damped integrals

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stochastic methods are required 22 / 79 numerical simulations: discrete (euclidean) time

system on time lattice =classical spin system Zβ = lim

n→∞

Z

dq1· · ·dqn

m 2π~ε

^n/2

e^−S^E^(q¹^,...,qⁿ^)/^~ expectation values of observables

Z

dq1. . .dqnF(q1, . . . ,qn) high-dimensional integral (sometimesn=10⁶required)

curse of dimension: analytical and numerical approaches do not work stochastic methods, e.g. Monte-Carloimportant 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 . . .

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harmonic and anharmonic oscillators 23 / 79

m=1, µ=1, λ=0 m=1, µ=−3, λ=1

q

|ψ0(q)|²

Hˆ= pˆ²

2m +µˆq²+λqˆ⁴ Monte-Carlo simulation (Metropolis algorithm) square of the ground state wave function

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

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path integral for linear chain 24 / 79

exercise: harmonic chain

find free energy for periodic chain of coupled harmonic oscillators H= 1

2m

N

X

i=1

p²_i +mω² 2

X

i

qi+1−qi2

, qi =qi+N

periodicq(τ)⇒may integrate by parts in LE = m

2 Z

dτ q˙²+ω² qi+1−qi2 matrix notation

L_E = m 2

Z

dτq^T

− d² dτ²+A

q, A=ω² 2δ_ij−δ_i_,j+1−δ_i,j₋₁ hint: non-negative eigenvalues and orthonormal eigenvectors ofA:

ω_k =2ωsinπk

N and ek

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free energy of linear chain 25 / 79 expandq(τ) =P

c_k(τ)ek

LE =X

k

m 2

Z

dτ( ˙c²_k+ω²_kc_k²

Ndecoupled oscillatorswith frequenciesω_k ⇒ hq|e^−β^H^ˆ|qi=Y

k

Kω_k(β,qk,qk)

results for one-dimensional oscillator⇒ Zβ=Y

k

e^βω^k^/2 e^βω^k−1 =Y

k

e^−βω^k^/2

1−e^−βω^k, ω_k =2ωsinπk N

free energy contains zero-point energy

Fβ= 1 2

X

k

~ω_k+kTX

k

log 1− e⁻^~^ω^k^/kT

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field theories 26 / 79

spin 0:scalar field(Higgs particle, inflaton,. . . ) spin ¹₂: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 fieldφ(t,x)satisfiesKlein-Gordon type equation(~=c =1) 2φ+V⁰(φ) =0

Lagrangian = integral of Lagrangian density over space L[φ] =

Z

space

dxL(φ, ∂_µφ), L=1

2∂_µφ∂^µφ−V(φ)

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field theories 27 / 79

momentum field, Legendre transform⇒Hamiltonian(fixed timet) π(x) = δL

δφ(˙ x) = ∂L

∂φ(˙ x)= ˙φ(x) H=

Z

dx πφ˙−L

= Z

dxH, H=1 2π²+1

2(∇φ)²+V(φ) free particle:V ∝φ²⇒Klein-Gordan2φ+m²φ=0

infinitely many dof: one at each space point

one of many possible regularizations:discretize space field theory onspace lattice:x =εnwithn∈Z^d−1

φ(t,x)−→φ_x=εn(t) , Z

dx −→ε^d−1X

n

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from field theory to point mechanics 28 / 79

N1

N2

ε

N1

N2

ε

N1

N2

ε

N1

N2

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N1

N2

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N1

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N1

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N1

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N1

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N1

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N1

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N1

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N1

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N1

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N1

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N1

N2

ε

N1

N2

ε

N1

N2

ε

e.g. periodic bc lattice constantε

# of lattice sitesN=Q Ni

linear extendsL_i =εN_i physical volumeV =ε^d−1N

finitehypercubic lattice in space x =εn with n_i ∈ {1,2, . . . ,N_i} continuum fieldφ(x)→lattice fieldφ_x integral→Riemann sum

Z

dx −→ε^d−1X

n

derivative→difference quotient

∂φ(x)

∂xi

−→(∂_iφ)_x

example: symmetric “lattice derivative”

(∂_iφ)_x = φ_x+εei−φ_x_−ε_e_i 2ε

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from field theory to point mechanics 29 / 79

finite lattice→mechanical system with finite number of dof H=ε^d−1 X

x∈lattice

1 2π_x²+1

2(∂φ)²_x +V(φ_x) path integral quantizationknown

{φ⁰_x} e⁻ⁱ^H/^ˆ ^~

{φ_x}

=C Z Y

x

Dφ_x e^iS[{φ^x^}]/^~ (formal) path integral over paths{φ_x(t)}in configuration space

φ_x(0) =φ_x and φ_x(t) =φ⁰_x, ∀x =εn high-dimensionalquantum mechanical system with action

S[{φ_x}] = Z

dtε^d−1X

x

1 2φ˙²_x −1

2(∂φ)²_x −V(φ_x)

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quantum field theory at finite temperature 30 / 79

canonical partition function

Zβ=C I Y

x

Dφ_x e^−S^E^[{φ^x^}]/^~, φ_x(τ) =φ_x(τ+~β) real euclidean action

SE[{φ_x}] = Z

dτ ε^d−1X

x

1 2

φ˙²_x +1

2(∂φ)²_x +V(φ_x) path-integral well-defined afterdiscretization of “time”

convenient:same lattice constantεin time and spatial directions replaceτ∈[0,~β]−→τ∈ {ε,2ε, . . . ,N0ε}withN0ε=~β lattice sites(x^µ) = (τ,x) = (εn^µ)withnµ ∈ {1,2, . . . ,Nµ}

⇒d-dimensionalhypercubic space-time lattice

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space-time lattice 31 / 79

lattice fieldφxdefined on sites ofspace-time latticeΛ

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lattice regularization 32 / 79

d-dimensional Euclid’sche space-time→latticeΛ, sitesx ∈Λ continuous fieldφ(x)→lattice fieldφ_x,x ∈Λ

finite lattice: extend in directionµ:Lµ=εNµ

finite temperature:L1=· · ·=Ld−1L0≡β=1/(kT) scalar fieldperiodic in imaginary time direction

φ_x=(x0+εN₀,x)=φ_x=(x0,x)=⇒temperature-dependence typically: also periodic in spatial directions

⇒identificationx^µ∼x^µ+Lµ(torus) space-time volumeV=ε^dN1N2· · ·Nd

some freedom inchoice of lattice derivative(use symmetries)

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space-time lattice 33 / 79

dimensionless fields and couplings (~=c=1)

natural units~=c=1⇒all units in powers of lengthL dimensionless action (unitL⁰)

SE = Z

d^dx1

2(∂φ)²+X

a

λ^ph_a φ^a

Rd^dx(∂φ)²dimensionless⇒[∂φ] =L^−d/2⇒[φ] =L^1−d/2 λ^ph_a R

d^dxφ^adimensionless⇒[λ^ph_a] =L^{−d−a+ad/2} in particularλ^ph₂ ∝m²⇒[m] =L⁻¹

4 space-time dimensions⇒λ^ph₄ dimensionless

dimensionless lattice field and lattice constants (x =εn) φ_x =ε^1−d/2φ_n, λ^ph_a =ε^{−d−a+ad/2}λ_a

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dimensions 34 / 79

lattice action with dimensionful quantities S_L^ph=ε^dX

x

1 2

φ_x_+εe_µ−φ_x−εe_µ 2ε

2

+X

a

λ^ph_aφ^a_x

!

⇒lattice action with dimensionless quantities S_L=X

n

1

2 φ_n+e_µ−φx−e_µ2

+X

a

λ_aφ^a_n

partition function

Zβ =C

Z ^N⁰^N¹^···

Y

n=1

dφne^−S^L^[{φⁿ^}]

finite-dimensional well-defined integral(lattice regularization) lattice formulation without any dimensionful quantity

processor knows numbers, not units!

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renormalization 35 / 79

merely lettingε→0: no meaningful continuum limit λ_amust be changed asε→0

condition:dimensionful observables approach well-defined finite limits existence of suchcontinuum limitnot guaranteed

example: consider correlation length in hφ(n)φ(m)i_c ∝ e^{−|n−m|/ξ}, 1

ξ =m=dimnsionless mass ξdepends on dimensionless couplingsξ=ξ(λ_a)

relates to (given) dimensionful massm^ph =1/(εξ)⇒ε m^ph from experiment,ξ(λ_a)measured on lattice

renormalization:keepm^ph (and further observables) fixed⇒λ_a

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renormalizable field theories 36 / 79

extend of physical objectsseparation of lattice points extend of physical objectsbox size

conditions(scaling window) small discretization effectsξ1 small finite size effectsξNµ

strict continuum limit:ξ→ ∞

2’nd order phase transition required in system withN_spatial → ∞ theory renormalizable:only a small number ofλ_amust be tuned relevantrenormalizable field theories

non-Abelian gauge theories ind ≤4 scalar field theories ind<4 four-Fermi theories ind≤3 non-linear sigma-models ind≤3 Einstein-gravity ind≤4 (???)

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simulation 37 / 79

input in simulations:only a few observables (masses) simulate with stochastic algorithms in scaling window repeat simulations with same observables but decreasingε output:many (dimensionful) observables

extrapolate toε→0

if theoryrenormalizable: converge to a continuum limit asε→0 finite temperature:N0given,εfrom 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, . . .

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lattice field theory as spin model 38 / 79

path integral for finite temperature QFT= classical spin model

no non-commutative operators, instead: path or functional integration over fields scalar field:

assignφ_n∈Rto each lattice site sigma models:

φ_n∈Sphere

discretespin models:

φ_n∈ discrete group example: Potts-model:

φ_n∈Zq

figures: 3−state Potts-type model

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relativistic fermions 39 / 79

electron, muon, quarks, . . . are described by 4-component spinor fieldψα(x) metric tensor in Minkowski space-time

(η_µν) =diag(1,−1,−1,−1) 4×4 gamma-matrices

γ⁰, . . . , γ³, {γ^µ, γ^ν}=2η^µν1 covariantDirac equation for free massive fermions

i∂/−m

ψ(x) =0, ∂/=γ^µ∂µ

Euler-Lagrange equation of invariant action S=

Z

d⁴xψ(i¯ ∂/−m)ψ, ψ¯=ψ^†γ⁰=⇒πψ=−iψ^†

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quantization of Dirac field 40 / 79

quantization:ψ(x)→ψ(xˆ )

satisfiesanti-commutation relation

{ψˆ_α(t,x),ψˆ_β^†(t,y)}=δαβδ(x −y) Hamilton operator:β=γ⁰, α=γ⁰γ:

Hˆ = Z

dx ψˆ^†(x)(ˆhψ)(ˆ x), ˆh=iα· ∇+mβ derive path integral representation ofpartition function

Zβ=tr e^−β^H^ˆ leads to imaginary time path integral replacet→ −iτ and

γ⁰_E=γ⁰ and γⁱ_E=iγⁱ

(42)

path integral for partition function 41 / 79 ACR with euclidean metric

{γ^µ_E, γ_E^ν}=2δ^µν1, γ_E^µhermitean lattice regularization(drop index E)

space-timeR⁴→finite (hypercubic) latticeΛ continuum fieldψ(x)onR⁴→lattice fieldψ_x expected path integral

Zβ=trrege^−β^H^ˆ=C I Y

α,x∈Λ

dψα,x^† dψα,x e^−S^L^[ψ,ψ^†^]

integration overanti-periodic fields(ACR forψ, see below) ψ_x(τ+β) =−ψ_x(τ), also on time lattice S_Lsome lattice regularization of

S_E= Z

d^dxψ^†(i∂/+im)ψ

(43)

Grassmann variables 42 / 79

quantized scalar field obey equal-time CR φ(t,ˆ x),φ(tˆ ,y)

=0, x 6=y

⇒commuting fields in path integral

[φ(x), φ(y)] =0, ∀x,y quantized fermion field obey equal-time ACR

ψˆ_α(t,x),ψˆ^†_β(t,y) =0, x 6=y ,

⇒anti-commuting fields in path integral

ψ_α(x), ψ_β^†(y) =0, ∀x,y

variables{ψ_α,n, ψ^†_α,n}in fermion path integral:Grassmann variables

(44)

Graussian integrals 43 / 79

free theories have quadratic action

Gaussian integralswithA=A^T positive matrix;exercise⇒ Z ^N

Y

n=1

dφ_n exp

−1 2

Xφ_nA_nmφ_m

= (2π)^n/2

√detA what do we get for fermions?

simplify notation:ψ_α,n≡η_i andψ_α,n^† ≡η¯_i withi =1, . . . ,m objects{η_i,η¯_i}form complex Grassmann algebra:

{η_i, η_j}={η¯_i,η¯_j}={η_i,η¯_j}=0=⇒η²_i = ¯η_i²=0

Grassmann integration defined by (a, b ∈ C )

Z

linear, Z

dη_i(a+bη_i) =b, Z

dη¯_i(a+bη¯_i) =b

(45)

Gaussian integrals with Grassmann variables 44 / 79

Grassmann integrals with

D¯ηDη ≡

m

Y

i=1

dη¯_idη_i

free fermions⇒Gaussian Grassmann integral Z =

Z

D¯ηDη e^−¯^ηAη, ηAη¯ =X

i,j

¯ η_iA_ijη_j

expand exponential function:R

D¯ηDη ηAη¯ k

=0 fork 6=m remaining contribution (useη¯_i²=0)

1 n!

Z

DηDη¯ (¯ηAη)^m = Z

D¯ηDη X

i₁,...,im

(¯η1A1i₁ηi₁)· · ·(¯ηmAmimηim)

= Z

D¯ηDηY

i

(¯ηiηi) X

i₁,...,im

εi₁...imA1i₁· · ·Amim

= (−1)^m Z

Y

i

(d¯ηiη¯idηiηi)detA= (−1)^mdetA

(46)

generating function 45 / 79

simple formula

Z

D¯ηDη e^−¯^ηAη=detA

generalization:generating function Z( ¯α, α) =

Z

D¯ηDη e^−¯^{ηAη+ ¯}^{αη+ ¯}^ηα= e⁻^αA^¯ ⁻¹^α detA expand in powers ofα,¯ α⇒

h¯η_iη_ji ≡ 1 Z

Z

D¯ηDη e^−¯^ηAηη¯_iη_j = (A⁻¹)_ij application to Dirac fields: abovepartition function

Zβ= I

DψDψ¯ e^−S^L, DψDψ¯ =Y

α,n

dψ^†α,ndψα,n

(47)

Graussian integrals for fermions and bosons 46 / 79

dimensionless field and couplings S_L=X

n∈Λ

ψ_n^†(i∂/_nm+imδnm)ψ_n=X

n

ψ¯_nD_nmψ_m

lattice partition function

Zβ=CdetD expectation value in canonical ensemble

hAiˆ _β= 1 Zβ

I

DψDψ¯ A( ¯ψ, ψ)e^−S^L^(ψ,^ψ)^¯ formula forcomplex scalar field

Zβ= I

DφDφ¯exp

−Xφ¯_mC_mnφ_n

∝ 1

detC boson fields: periodic in imaginary time

fermion fields: anti-periodic in imaginary time

(48)

Thermodynamic potentials for Gas or relativistic particles 47 / 79

neutral scalars (+: periodic bc) S_E = 1

2 Z

φ(−∆+m²)φ=⇒Fβ=kT

2 log det+(−∆+m²) +. . . Dirac fermions (−: anti-periodic bc)

S_E = Z

ψ^†(i∂/+im)ψ=⇒Fβ=−2kTlog det−(−∆+m²) +. . .

exercise

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

zeta-function for second order operator A > 0

ζ_A(s) =X

n

λ^−s_n , eigenvaluesλ_n

(49)

ζ−function regularization 48 / 79

absolute convergent series in half-plane<(s)>d/2

meromorphic analytic continuation, analytic in neighborhood ofs=0 definesζ−function regularized determinant Dowker, Hawking

log detA=trlogA=X

logλ_n=−dζ_A(s) ds

s=0

correct for matrices Mellin transformations

Z ∞

0

dt t^s−1e^−tλ= Γ(s)λ^−s

⇒relation to heat kernel ζ_A(s) =X

n

1 Γ(s)

Z ∞

0

dt t^s−1e^−tλⁿ = 1 Γ(s)

Z ∞

0

dt t^s−1tr e^−tA

(50)

heat kernel 49 / 79

coordinate representation ζ_A(s) = 1

Γ(s) Z ∞

0

dt t^s−1Z

dx K(t;x,x), K(t) = e^−tA

heat kernel ofA=−∆+m²oncylinder[0, β]×R^d−1 K^±(t;x,x⁰) = e^−m²^t

(4πt)^d/2 X

n∈Z

(±1)ⁿe⁻{(τ−τ⁰+nβ)²+(x−x⁰)²}/4t

integrate over diagonal elements ζ_A^±(s) = βV

(4π)^d/2Γ(s) Z

dt t^s−1−d/2e^−m²^t

∞

X

n=−∞

(±)ⁿe⁻ⁿ²^β²^/4t

Jacobitheta function

(51)

integral representation ofKelvin functions Z ∞

0

dt t^ae^−bt−c/t =2c b

(a+1)/2

Ka+1

2√

bc

⇒series representation; ind=4 ζ_A^±(s) = βV

16π² m^4−2s

Γ(s) Γ(s−2) +4

∞

X

1

(±)ⁿ nmβ

2 ^s−2

K2−s(nmβ)

!

identities

Γ(s−2)

Γ(s) = 1

(s−1)(s−2) and 1

Γ(s) =s+O(s²)

(52)

derivative ats=0⇒ F_β^±=−m⁴VC_±

128π²



3−2 logm²

µ² +64 X

n=1,2...

(±)ⁿK2(nmβ) (nmβ)²





real scalarsC+=1, complex fermionsC₋=−4 well-known results for massless particlesK₂(x)∼2/x²

m→0lim f⁺(β) =−π²

90T⁴ , lim

m→0f⁻(β) =− 2 45π²T⁴

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 ind dimensions?

(53)

Interacting relativistic fermions 52 / 79

condensed matter systems ind =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 ind =1+1

conducting polymers (Trans- and

Cis-polyacetylen) Su, Schrieffer, Heeger

quasi-one-dimensional inhomogeneous superconductor Mertsching, Fischbeck

relativistic dispersion-relationsfor electronic excitations on

honeycomb lattice

from Castro Neto et al.

(54)

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, . . . , ψ_N_f)

relativistic fermions

L_GN= ¯Ψi∂Ψ +/ imΨΨ +¯ L_Int(Ψ,Ψ),¯ e.g.ΨΨ =¯ Xψ¯_iψ_i

parity invariant models L_Int=g_GN²

2N_f( ¯ΨΨ)( ¯ΨΨ) scalar-scalar,Gross-Neveu L_Int=−g²_Th

2Nf( ¯Ψγ^µΨ)( ¯ΨγµΨ) vector-vector,Thirring L_Int= g_PS²

2N_f( ¯Ψγ_∗Ψ)( ¯Ψγ_∗Ψ) pseudoscalar-pseudoscalar in even dimensionsγ∗∝Q

γµ

Hubbard-Stratonovich trickwith scalar, vector and pseudscalar field

(55)

Relativistic four-Fermi Theories 54 / 79

combinations thereof ind =4

non-renormalizableFermi theoryof weak interaction

effective models for chiral phase transition in QCD (Jona Lasino) 2 spacetime dimensions:[g] =L⁰

massless ThM:soluble Thirring

massless ThM incurved space withµ:soluble Sachs+AW, . . .

GNM:asymptotically free, integrable Gross-Neveu, Coleman, . . .

3 spacetime dimensions:[g] =Ls not renormalizable in PT

renormalizable in large-Nexpansion Gawedzki, Kupiainen; Park, Rosenstein, Warr

interacting UV fixed point→asymptotically safe de Veiga; da Calen; Gies, Janssen

can exhibit parity breaking at lowT lattice theories:

generically: sign problem even forµ=0

partial solution of sign problem Schmidt, Wellegehausen, Lenz, AW

(56)

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 parameteriΣ =hΨΨi¯

ψ_a→iγ_∗ψ_a, ψ¯_a→iψ¯_aγ_∗=⇒iΣ =hΨΨi → −h¯ ΨΨi¯ equivalent formulation with auxiliary scalar fieldHubbard-Stratonovich transformation

L_GN=Lσ= ¯Ψ iD⊗1Nf

Ψ +N_f 2g( ¯ΨΨ)² Lσ= ¯Ψ iD⊗1Nf

Ψ +λN_fσ², D=∂/−σ6=D^†

(57)

chemical potential for fermion number charge 56 / 79

conservedfermion charge Q= Z

spacedxj⁰= Z

spacedxψ^†ψ partition function ofgrand canonical ensemble

Zβ,µ=tr e^{−β( ˆ}^H−µ^Q)^ˆ, functional integral with aboveLσwherein

D=∂/+σ+µγ⁰

expectation values

hOi= 1 Zβ,µ

Z

DψDψDσ¯ e^−S^σO

(58)

chemical potential for fermion charge 57 / 79

fermion integral in Zβ,µ=

Z

DψDψDσ¯ e^−S^σ^[σ,ψ,^ψ]^¯ = Z

Dσe^−N^f^S^eff^[σ]

Nf fermion species couple identically to auxiliary field⇒ det iD⊗1

= (detiD)^N^f ψanti-periodic in imaginary time,σperiodic effective action after fermion integral

S_eff=λ Z

d²xσ²−log(detiD)

Ward identity(lattice regularization) 1

Zβ,µ

Z

DψDψ¯Dσ d dσ(x)

e^{−S[σ,ψ,ψ}^†^]

=−D dS dσ(x)

E

=0

(59)

homogeneous phases 58 / 79

exact relation

Σ≡ −ihψ(x¯ )ψ(x)i= Nf

g²hσ(x)i forNf→ ∞saddle point (steepest descend) approximation

Zβ,µ= Z

Dσe⁻^N^f^S^eff]^[σ]^N−→^f^→∞ e⁻^N^f^min^S^eff^[σ]

translation invariance⇒minimizingσconstant:S_eff= (N_fβL)U_eff U_eff= σ²

4π

logσ² σ₀ −1

−1 π

Z ∞

0

dp p² ε_p

1

1+ e^β(ε^p^+µ) + 1 1+e^β(ε^p^−µ)

one-particle energies_p=p p²+σ² IR-scaleσ₀=hσi_T_=µ=0

(60)

condensate in the(µ,T)-plane 59 / 79

'data.dat' u 1:2:3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

µ/σ0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

T/σ0

0 0.2 0.4 0.6 0.8 1

symmetric phase for largeT, µ

homogeneously broken phase for smallT, µ Wolff, Barducci

special points:(Tc, µ) = (e^γ/π,0),(T,µc) = (0,1/√ 2) Lifschitz-Punkt bei(T, µ0)≈(0.608,0.318)

(61)

is homogeneity assumption really justified? 60 / 79

possible QCD-phase diagram

crystalline LOFF phase (color superconductive phase)?

problem:µ6=0⇒complex fermion determinant/ largeµbeyond reach in simulations

are there inhomogeneouscrystallic phasesin model systems?

(62)

space-dependent condensate in GN model 61 / 79

discreteε_nenergies of Dirac Hamiltonian on[0,L]

hσ=γ⁰γ¹∂_x +γ⁰σ(x)

hidden supersymmetry h²_σ=−d²

dx² +σ²(x)−γ¹σ⁰(x) = AA^† 0 0 A^†A

!

, A=−d dx +σ renormalization: fix (constant) condensateσ₀atµ=T =0

introduce constantcompanion field

¯ σ²= 1

L Z

dxσ²(x) constantσ⇒¯σ=σ