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 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: 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
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 renormalization group equation numerical simulations lattice field theories = particular classical spin systems ⇒ powerful methods of statistical physics 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, entropy 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 statistics: canonical approach 6 / 79
Path integral for partition function in quantum mechanics
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