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Quantum Quenches to a Critical Point in One Dimension: some further results

John Cardy

TL;DR

This work analyzes global quantum quenches to a 1+1D CFT from short-range correlated states, clarifying how a finite-interval reduced density matrix becomes thermally equilibrated with an exponential approach rate $e^{-2\pi Δ (t-ℓ/2v)/β}$. It extends the CC framework by showing that a generalized Gibbs ensemble, potentially with parafermionic charges, governs stationary states when the initial state is deformed by boundary operators, and demonstrates that irrelevant CFT perturbations renormalize propagation and cause diffusive horizon broadening. The results illuminate how integrability, boundary conditions, and weak perturbations shape non-equilibrium dynamics in low-dimensional quantum systems and supply a concrete, testable framework for exploring observable-dependent thermalization and GGEs in 1+1D CFTs. The combination of boundary-state techniques, twist operators, and a random-metric interpretation of perturbations provides a coherent picture linking horizon physics, GGEs, and perturbative corrections to critical dynamics.

Abstract

We describe several results concerning global quantum quenches from states with short-range correlations to quantum critical points whose low-energy properties are described by a 1+1-dimensional conformal field theory (CFT), extending the work of Calabrese and Cardy (2006): (a) for the special class of initial states discussed in that paper we show that, once a finite region falls inside the horizon, its reduced density matrix is exponentially close in $L_2$ norm to that of a thermal Gibbs state; (b) small deformations of this initial state in general lead to a (non-Abelian) generalized Gibbs distribution (GGE) with, however, the possibility of parafermionic conserved charges; (c) small deformations of the CFT, corresponding to curvature of the dispersion relation and (non-integrable) left-right scattering, lead to a dependence of the speed of propagation on the initial state, as well as diffusive broadening of the horizon.

Quantum Quenches to a Critical Point in One Dimension: some further results

TL;DR

This work analyzes global quantum quenches to a 1+1D CFT from short-range correlated states, clarifying how a finite-interval reduced density matrix becomes thermally equilibrated with an exponential approach rate . It extends the CC framework by showing that a generalized Gibbs ensemble, potentially with parafermionic charges, governs stationary states when the initial state is deformed by boundary operators, and demonstrates that irrelevant CFT perturbations renormalize propagation and cause diffusive horizon broadening. The results illuminate how integrability, boundary conditions, and weak perturbations shape non-equilibrium dynamics in low-dimensional quantum systems and supply a concrete, testable framework for exploring observable-dependent thermalization and GGEs in 1+1D CFTs. The combination of boundary-state techniques, twist operators, and a random-metric interpretation of perturbations provides a coherent picture linking horizon physics, GGEs, and perturbative corrections to critical dynamics.

Abstract

We describe several results concerning global quantum quenches from states with short-range correlations to quantum critical points whose low-energy properties are described by a 1+1-dimensional conformal field theory (CFT), extending the work of Calabrese and Cardy (2006): (a) for the special class of initial states discussed in that paper we show that, once a finite region falls inside the horizon, its reduced density matrix is exponentially close in norm to that of a thermal Gibbs state; (b) small deformations of this initial state in general lead to a (non-Abelian) generalized Gibbs distribution (GGE) with, however, the possibility of parafermionic conserved charges; (c) small deformations of the CFT, corresponding to curvature of the dispersion relation and (non-integrable) left-right scattering, lead to a dependence of the speed of propagation on the initial state, as well as diffusive broadening of the horizon.

Paper Structure

This paper contains 15 sections, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Convergence to a thermal ensemble.
  3. Correlation functions.
  4. Reduced density matrix.
  5. Quench from a more general boundary state and the GGE
  6. Physical consequences of the GGE.
  7. Perturbative derivation.
  8. Non-integer exponents and semi-local conservation laws
  9. Example: the transverse Ising model in a longitudinal field.
  10. Perturbations of the CFT
  11. Renormalization of propagation speed.
  12. Diffusive broadening of the horizon
  13. Correlation functions.
  14. Higher order effects.
  15. Discussion.

Figures (2)

  • Figure 1: The surface $S\oplus C$ in the numerator of (\ref{['ratio']}). The cylinder $C$ and the strip $S$ are sewn together along $A$ as shown in the inset.
  • Figure 2: The contour manipulations leading to (\ref{['SGGE']}). (a) initially the perturbing operators are integrated along the dashed lines adjacent to the upper and lower boundaries of the strip; (b) using (anti-)holomorphicity they are moved into the bulk; (c) the strip is mapped to the upper half-plane: the correlator of local operators (indicated by black circles) depends on those points and their images; (d) after continuation to real time and evolving until both operators fall within the horizon, the image points move off to infinity: at this point the boundary (now indicated by a dotted line) effectively disappears and the correlator is the same as in the full plane; (e) the two parts of the dashed contour may now be rotated together and their contributions (which differ by a phase) combined; (f) on reversing the conformal mapping, the full plane becomes a cylinder with the insertion of a defect line along which the conserved currents are integrated.