Quantum Quenches in Extended Systems

Quantum Quenches in Extended Systems Spyros Sotiriadis 1 Pasquale Calabrese 2 John Cardy 1,3 1 Oxford University, Rudolf Peierls Centre for Theoretical Physics, Oxford, UK 2 Dipartimento di Fisica Enrico Fermi, Università di Pisa, Pisa, Italy 3 All Souls College, Oxford, UK The Capri Spring School on Transport in Nanostructures Capri, March 29 - April 5, 2009.

Outline Introduction

Outline Introduction Linearly coupled oscillators (free fields) Simple harmonic oscillator Coupled oscillators

Outline Introduction Linearly coupled oscillators (free fields) Simple harmonic oscillator Coupled oscillators Anharmonic coupled oscillators (interacting field theory) Self-consistent approximation Time evolution

What is a Quantum Quench? A definition An instantaneous change in the parameters that determine the dynamics of an isolated quantum system.

What is a Quantum Quench? A definition An instantaneous change in the parameters that determine the dynamics of an isolated quantum system. instantaneous: change done in a time interval much shorter than any characteristic time scale of the model. dynamics: a parameter of the hamiltonian of the system. isolated: no connection to the environment - the system evolves unitarily (i.e. under quantum mechanical laws).

Simple harmonic oscillator Before (t < 0): H 0 = 1 2 p2 + 1 2 ω2 0x 2 After (t > 0): H = 1 2 p2 + 1 2 ω2 x 2

Simple harmonic oscillator Before (t < 0): H 0 = 1 2 p2 + 1 2 ω2 0x 2 ω 0 ω After (t > 0): H = 1 2 p2 + 1 2 ω2 x 2 Ψ 0 Assume that initially the system lies in the ground state Ψ 0 of H 0. After the quench Ψ 0 acts as a source of excitations.

Use CCR [x, p] = i and the initial expectation values Ψ 0 x 2 (0) Ψ 0 = 1/2ω 0 Ψ 0 p 2 (0) Ψ 0 = ω 0 /2 Ψ 0 x(0)p(0) + p(0)x(0) Ψ 0 = 0

Use CCR [x, p] = i and the initial expectation values to find Ψ 0 x 2 (0) Ψ 0 = 1/2ω 0 Ψ 0 p 2 (0) Ψ 0 = ω 0 /2 T {x(t 1 )x(t 2 )} = Ψ 0 x(0)p(0) + p(0)x(0) Ψ 0 = 0 = (ω ω 0) 2 4ω 2 ω 0 cos ω(t 1 t 2 ) + ω2 ω 2 0 4ω 2 ω 0 cos ω(t 1 + t 2 ) + 1 2ω e iω t 1 t 2

Linearly coupled oscillators (free fields) H = 1 2 d d r [π 2 + ( φ) 2 + m 2 φ 2 ] = 1 2 πk 2 + ω2 k φ2 k Set of independently evolving momentum modes with ω 2 k = k2 + m 2. k

Linearly coupled oscillators (free fields) H = 1 2 d d r [π 2 + ( φ) 2 + m 2 φ 2 ] = 1 2 πk 2 + ω2 k φ2 k Set of independently evolving momentum modes with ω 2 k = k2 + m 2. Quantum quench of the mass (or energy gap) from m 0 to m. The propagator is the Fourier transform of last expression. The (t 1 + t 2 )-dependent part d d k deik rm2 m 2 0 (2π) 4ωk 2ω cos ω k (t 1 + t 2 ) 0k decays for large times as (t 1 + t 2 ) d/2 cos m(t 1 + t 2 ) if m > 0 or even exponentially if m = 0 and d = 3. k

The propagator becomes stationary for large times.

The propagator becomes stationary for large times. Any local observable tends to a stationary value in the thermodynamic limit due to quantum interference between the infinite set of independent momentum modes.

Comparison to the thermal propagator The stationary part of the quench propagator in momentum space is (ω k ω 0k ) 2 4ω 2 k ω 0k cos ω k (t 1 t 2 ) + 1 2ω k e iω k t 1 t 2

Comparison to the thermal propagator The stationary part of the quench propagator in momentum space is (ω k ω 0k ) 2 4ω 2 k ω 0k cos ω k (t 1 t 2 ) + 1 2ω k e iω k t 1 t 2 Compare to the thermal or Matsubara propagator in real time 1 ω k (e βω cos ω k k (t 1 t 2 ) + 1 e iω k t 1 t 2 1) 2ω k The two expressions are equal if β satisfies the condition { tanh βω k 4 = ω k /ω 0k if m < m 0 ω 0k /ω k if m > m 0

The propagator exhibits thermal-like behaviour (thermalization) with β 4/m 0 for m 0 m, k. The effective temperature β eff (k) is momentum dependent - expected for free fields since the different momentum modes thermalize independently. Comparison to the thermal propagator The stationary part of the quench propagator in momentum space is (ω k ω 0k ) 2 4ω 2 k ω 0k cos ω k (t 1 t 2 ) + 1 2ω k e iω k t 1 t 2 Compare to the thermal or Matsubara propagator in real time 1 ω k (e βω cos ω k k (t 1 t 2 ) + 1 e iω k t 1 t 2 1) 2ω k The two expressions are equal if β satisfies the condition { tanh βω k 4 = ω k /ω 0k if m < m 0 ω 0k /ω k if m > m 0

Anharmonic coupled oscillators (interacting field theory) How is this behaviour affected by interactions?

Anharmonic coupled oscillators (interacting field theory) How is this behaviour affected by interactions? Rapidly switch on a φ 4 -interaction term at t = 0 as well ( 1 H = d d r 2 π2 + 1 2 ( φ)2 + 1 2 m2 φ 2 + 1 ) 4 λφ4 The 2-point correlation function in the interaction picture is ( ) Ψ 0 T {φ(0, t 1 )φ(r, t 2 )exp i dt H int (t ) } Ψ 0 ATTENTION! Time integration must be done along the appropriate Schwinger-Keldysh contour C since the interaction is not switched on and off adiabatically as usual in QFT. t = 0 C C t 1 t 2

Loop correction and the Hartree-Fock approximation Let s try to apply perturbation theory. First order Feynman diagram: loop diagram (renormalized) r 1,t 1 ( d d (ω0k ω k ) 2 k 4ωk 2ω (m2 0 ) m2 ) 0k 4ωk 2ω cos 2ω k t 0k t' r 2,t 2 As before the time dependent part decreases with time. QUESTION: Can we ignore this part for large times? If yes, what is the effect of a constant loop to the correlation function?

Resummation using Dyson equation The 1st order correction to the correlation function assuming a constant loop, increases for large times.

Resummation using Dyson equation The 1st order correction to the correlation function assuming a constant loop, increases for large times. Perturbation theory not directly applicable - Resummation of Feynman diagrams required. Dyson equation G(k; t 1, t 2 ) = G 0 (k; t 1, t 2 ) + dt G 0 (k; t 1, t)σ(t)g(k; t, t 2 ) C G G t 2 t 2 G t G 2 0 G 0 t = + t 1 t 1 t 1 Σ is the self-energy insertion assumed to tend to a constant as t Σ(t) = λ d d k G(k, t)

Result: for large times the correlation function is the same as in the free problem, but with a mass shift to be determined self-consistently m 2 eff = m2 + λ d d k G(m eff ) t

Result: for large times the correlation function is the same as in the free problem, but with a mass shift to be determined self-consistently m 2 eff = m2 + λ d d k G(m eff ) t m eff 10 8 d = 2 6 4 d = 3 d = 1 2 0 20 40 60 80 100 120 λ Figure: Numerical solution to the self-consistency equation for m = 0.

Time evolution Alternative application of Hartree-Fock approximation: Substitute the interaction term as φ 4 φ 2 (t) φ 2

Time evolution Alternative application of Hartree-Fock approximation: Substitute the interaction term as φ 4 φ 2 (t) φ 2 i.e. the hamiltonian becomes quadratic but with a time dependent mass to be determined self-consistently m 2 eff (t) = m2 + λ d d k φ 2 k (t) Methods of solution: 1. Quasi-adiabatic approximation: ṁ eff /m 2 eff 1.

Conclusions

Conclusions Effective thermalization after a quantum quench.

Conclusions Effective thermalization after a quantum quench. Thermalization also occurs in many other cases: Massless 1d theories (free or interacting). Quench of the speed of sound instead of the mass. Lattice models. Other problems also studied: Spatially inhomogeneous initial state. Thermal initial state.

References S. Sotiriadis and J. Cardy. Inhomogeneous quantum quenches. J. Stat. Mech. 2008. S. Sotiriadis, J. Cardy and P. Calabrese. Quantum quench from a thermal initial state. arxiv:0903.0895. S. Sotiriadis and J. Cardy. Quantum quench in interacting non-integrable models: A self-consistent approximation. (in preparation).