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 H^ : H 7! H system in thermal equilibrium with heat bath 1 ^ ^ −βH^ 1 canonical %^β = K (β); K (β) = e ; β = Zβ kT normalizing partition function ^ Zβ = tr K (β) expectation value of observable O^ in ensemble ^ ^ hOiβ = tr %^βO Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory thermodynamic potentials 5 / 79 inner and free energy @ U = hH^i = − log Z ; F = −kT log Z β @β β β β ) all thermodynamic potentials, entropy S = −@T F,... specific heat @U C = hH^ 2i − hH^i2 = − > 0 V β β @β system of particles: specify Hilbert space and H^ 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 K^ satisfies diffusion type equation ^ d K^(β) = e−βH =) K^(β) = −H^K^(β) dβ compare with time-evolution operator and Schrödinger equation − H^= d U^(t) = e i ~ =) i~ U^(t) = H^U^(t) dt formally: U^(t = −i~β) = K^(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 −βH^ 0 0 hqj e jq 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 ^ ~ 0 2 2β H0 = − ∆ =) K0(β; q; q ) = e ~ 2m 2π~2β ^ ^ ^ Hamiltonian H = H0 + V bounded from below ) ^ ^ ^ ^ n e−β(H0+V ) = s − lim e−βH0=n e−βV =n ; V^ = V (q^) n!1 Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory derivation of path integral 8 / 79 insert for every identity 1 in − β H^ − β V^ − β H^ − β V^ − β H^ − β V^ e n 0 e n 1 e n 0 e n 1 ··· 1 e n 0 e n the resolution 1 = R dq jqihqj) n 0 0 − β H^ − β V^ K β; q ; q = lim q e n 0 e n q n!1 j=n−1 Z β ^ β ^ Y − H0 − V = lim dq1 ··· dqn−1 qj+1 e n e n qj ; n!1 j=0 0 initial and final positions q0 = q and qn = q define small " = ~β=n and finally use β β − V (q^) − V (qj ) e n qj = qj e n β ^ m d=2 m 2 − H0 − (qj+1−qj ) qj+1 e n qj = e 2~" 2π~" 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!1 2π~" 8 j=n−1 9 2 < " X m qj+1 − qj = · exp − + V (q ) 2 j : ~ j=0 ; divide interval [0; ~β] into n sub-intervals of length " = ~β=n consider path q(τ) with sampling points q(τ = k") = 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−!!1 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 ^ Z K (β; q; q) = hqj e−βH jqi = C Dq e−SE [q]=~ q(0)=q trace ^ Z ^ I tr e−βH = dq hqj e−βH jqi = 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 ^ ~ d m! 2 H! = − + q 2m dq2 2 has heat kernel r 0 0 m! n m! 2 02 2qq o K!(β; q ; q) = exp − (q +q ) coth(~!β)− 2π~ sinh(~!β) 2~ sinh(~!β) equation of euclidean motion q¨ = !2q has for given q; q0 the solution sinh(!τ) q(τ) = q cosh(!τ) + q0 − q cosh(!β) sinh(!β) action m Z β m! S = (q_ 2 + !2q2) = (q2 + q02) cosh !β − 2qq0 2 0 2 sinh !β Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory second derivative @2S m! = − @q@q0 sinh(!β) 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 ) m! 2m!q sinh (~!β=2) K!(β; q; q) = exp − 2π~ sinh(~!β) ~ sin(~!β) Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory spectrum of harmonic oscillator 15 / 79 partition function 1 −~!β=2 1 e − !β=2 X −n !β Zβ = = = e ~ e ~ 2 sinh( !β=2) 1 − e−~!β ~ n=0 evaluate trace with energy eigenbasis of H^ ) ^ ^ X Z(β) = tr e−βH = hnj e−βH jni = e−βEn n comparison of two sums ) 1 E = ! n + ; n 2 N n ~ 2 Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory thermal correlation functions 16 / 79 position operator ^ ^ q^(t) = eitH=~q^ e−itH=~; q^(0) = q^ imaginary time t = −iτ ) euclidean operator ^ ^ ^ τH=~ ^ −τH=~ ^ ^ qE(τ) = e q e ; qE(0) = q correlations at different0 ≤ τ1 ≤ τ2 ≤ · · · ≤ τn ≤ β in ensemble 1 ^ q^ (τ ) ··· q^ (τ ) ≡ tr e−βH q^ (τ ) ··· q^ (τ ) E n E 1 β Z(β) E n E 1 consider thermal two-point function (now we set ~ = 1) 1 ^ ^ ^ hq^ (τ )q^ (τ )i = tr e−(β−τ2)H q^ e−(τ2−τ1)H q^ 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: jni orthonormal eigenstates of H^ ) 1 X ^ h::: i = e−(β−τ2)En hnjq^ e−(τ2−τ1)H q^jni e−τ1En β Z(β) n insert 1 = P jmihmj ) 1 X h::: i = e−(β−τ2+τ1)En e−(τ2−τ1)Em hnjq^jmihmjq^jni β Z(β) n;m P low temperature β ! 1: contribution of excited states to n(::: ) exponentially suppressed, Z(β) ! exp(−βE0) ) β!1 X ^ ^ −(τ2−τ1)(Em−E0) ^ 2 hqE(τ2)qE(τ1)iβ −! e jh0jqjmij m≥0 likewise ^ ^ hqE(τ)iβ −! h0jqj0i Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory connected correlation functions 18 / 79 connected two-point function ^ ^ ^ ^ ^ ^ 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 hq^ (τ2)q^ (τ1)i = e jh0jq^jmij β!1 E E c,β m≥1 2 energy gap E1 − E0 and matrix element jh0jqj1ij from 2 ^ ^ −(E1−E0)(τ2−τ1) ^ hqE(τ2)qE(τ1)ic,β!1 −! e jh0jqj1ij ; τ2 − τ1 ! 1 Andreas Wipf (TPI Jena) Path Integration in Statistical Field Theory path integral for thermal correlation function 19 / 79 for path-integral representation consider matrix elements D ^ ^ ^ ^ ^ E q0j e−βH eτ2H q^ e−τ2H eτ1H q^ e−τ1H jq resolution of the identity and q^jui = ujui: Z ^ ^ ^ h::: i = dvdu hq0j e−(β−τ2)H jvivhvj e−(τ2−τ1)H juiuhuj e−τ1H jqi 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 hq^ (τ )q^ (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 hq^ (τ ) ··· q^ (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.