Quantum quenches in 1+1 dimensional conformal field theories

TL;DR

This review develops and applies the imaginary-time path integral framework to 1+1D conformal field theories undergoing global and local quantum quenches, establishing a boundary-state formalism with an extrapolation length $\tau_0$ that regularizes the quench and enables analytic continuation to real time. By mapping quenched geometries to the upper-half plane and employing the replica trick for entanglement, it yields explicit universal results for correlation functions, one- and two-point functions, and entanglement entropy, including light-cone spreading and finite-size revivals, while highlighting limits when comparing to realistic condensed-matter systems. It also analyzes generalizations to perturbed CFTs, different initial states leading to GGEs, and local quenches, showing how entanglement and correlations evolve in time and how these predictions connect to experiments and holographic descriptions. The work provides a cohesive, technically detailed framework that captures universal dynamical features, clarifies the role of initial-state data, and guides the interpretation of quench dynamics in near-critical systems, with practical implications for cold-atom experiments and numerical approaches based on entanglement growth.

Abstract

We review the imaginary time path integral approach to the quench dynamics of conformal field theories. We show how this technique can be applied to the determination of the time dependence of correlation functions and entanglement entropy for both global and local quenches. We also briefly review other quench protocols. We carefully discuss the limits of applicability of these results to realistic models of condensed matter and cold atoms.

Figures (5)

• Figure 1: Exact entanglement entropy in the the transverse field Ising model in the thermodynamic limit. Left: Time evolution of the entanglement entropy as function of time for a quench from $h_0=\infty$ to the critical point $h=1$ for different lengths $L$ of the subsystem. Right: The same but for fixed length of the subsystem ($L=100$) and as a function of the post-quench magnetic field $h$. Both panels reprinted with permission from cc-05.
• Figure 2: Equal time two-point correlation function following a quench in the Ising chain. Left: a ferro-to-ferro quench $h_0=0.3\rightarrow h=0.5$ at fixed distance $\ell=20$ and $\ell=40$ against the prediction (\ref{['eq:prediction']}). Right: Ferro-to-para quench showing large oscillations inside the light cone. Reprinted with permission from cef-11 and cef-12a.
• Figure 3: Left: Log of the return amplitude for the Ising CFT starting from a disordered state for $0<2t/L<2$, with $\pi\tau_0/L=0.05$. The vertical axis has been shifted so as to expose the mean plateau behaviour. This shows the initial gaussian decay and revival at $t=L$. The negative peak at $t=L/2$ is due to destructive interference between two kinds of quasiparticles. Smaller gaussian peaks are seen at rational values with small denominators. The positive peaks are mapped by the modular group to the initial peak, and the negative ones to the feature at $2t/L=1$. Right: Same as left with $\pi\tau_0/L=0.005$. Now there is structure at more rational values, and we see the predicted $1/m^2$ dependence of the heights of nearby peaks with denominators $m$. Reprinted with permission from Ref. c-14
• Figure 4: Entanglement entropy (left) and the logarithmic return amplitude (right) after a local quench in the middle of a system of size $L = 128$ for the XX spin-chain. Four oscillations of the finite-size system are clearly observed after the local quench. They are very well described by the CFT formulas (\ref{['Sfss']}) and (\ref{['Ffss']}). Reprinted with permission from ds-11.
• Figure 5: Left: Imaginary time representation of the domain-wall initial state quench. Right: Density profile in the strip geometry measured numerically for $R = 128$. Reprinted with permission from adsv-16.