Quantum Quenches in Extended Systems

TL;DR

This work develops a unified framework for the non-equilibrium dynamics of extended quantum systems after a sudden quench, by mapping real-time evolution to a d+1 dimensional boundary problem and employing boundary conformal field theory in 1D. It reveals a robust horizon (light-cone) structure and an effective finite-temperature behavior at long times, encoded by an extrapolation length τ0 and a generalized Gibbs ensemble for integrable models. The authors validate the approach with exact real-time solutions in solvable chains (harmonic oscillators and Ising-XY), and extend the analysis to higher dimensions with mean-field results, examining boundary conditions (Dirichlet and fixed) and lattice effects. The findings provide a coherent physical picture based on quasi-particle propagation, with clear implications for equilibration, entanglement growth, and the role of integrals of motion in long-time steady states.

Abstract

We study in general the time-evolution of correlation functions in a extended quantum system after the quench of a parameter in the hamiltonian. We show that correlation functions in d dimensions can be extracted using methods of boundary critical phenomena in d+1 dimensions. For d=1 this allows to use the powerful tools of conformal field theory in the case of critical evolution. Several results are obtained in generic dimension in the gaussian (mean-field) approximation. These predictions are checked against the real-time evolution of some solvable models that allows also to understand which features are valid beyond the critical evolution. All our findings may be explained in terms of a picture generally valid, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate with a finite speed through the system. Furthermore we show that the long-time results can be interpreted in terms of a generalized Gibbs ensemble. We discuss some open questions and possible future developments.

Figures (4)

• Figure 1: Left: Space-imaginary time region in (\ref{['Oexp']}). ${\rm Im} w_i= \tau$, that will be analytically continued to $\tau\to \tau_0+it$. Right: Conformal mapping of the left geometry to the upper half-plane (c.f. Eq. (\ref{['logmap']})). Note that ${\rm arg} z_i= \theta=\pi \tau/2\tau_0$.
• Figure 2: Left: Space-time region for the correlation functions at different times. Right: Conformal mapping to the upper half-plane.
• Figure 3: Left: Space-time region for the one-point function in a boundary (at $r=0$) geometry. Note $w=\tau+ir$. Right: Conformal mapping to the upper-half plane, c.f. Eq. (\ref{['mapsin']}).
• Figure 4: Left: $G(r,t)/m_0$ given by Eq. (\ref{['Gxt']}) as function of $t$, at fixed $r=1$. Three different values of $m_0=10,3,1$ (from the bottom to the top) are shown. Inset: Lattice effects showing the $\cos 4t$ oscillations on top of the continuum result. Right: $G(r,t)/m_0$ given by the numerical integral of Eq. (\ref{['Gmfin']}) as function of $t$, at fixed $m,r=1$. It is compared with the asymptotic behavior for $0<2t-r\ll m^{-1}$ and for $t\gg r$.