Reduced fidelities for free fermions out of equilibrium: From dynamical quantum phase transitions to Mpemba effect

TL;DR

The paper studies out-of-equilibrium dynamics of reduced fidelities in one-dimensional free-fermion systems after quantum quenches, focusing on the reduced Loschmidt echo (RLE) and the final-state fidelity (FSF) within Gaussian states on a subregion A. It develops a quasiparticle picture (QPP) for the hydrodynamic limit, revealing a novel nested lightcone structure in the RLE for certain quenches and identifying DQPT-like cusps in the sub-hydrodynamic regime. The FSF, analyzed via the QPP, provides a simpler diagnostic and a natural framework to detect quantum Mpemba effects, including cases with and without symmetry restoration. Together, these results elucidate how local subsystem dynamics encode global nonequilibrium phenomena and offer quantitative predictions for experimental observation in XY-chain quenches.

Abstract

We investigate the out-of-equilibrium dynamics after a quantum quench of the reduced fidelities between the states of a subregion $A$ at different times. Precisely, we consider the fidelity between the time-dependent state of $A$ and its initial value, as well as with the state at infinite time. We denote these fidelities as the reduced Loschmidt echo (RLE) and the final-state fidelity (FSF), respectively. If region $A$ is the full system, the RLE coincides with the standard Loschmidt echo. We focus on quenches from Gaussian states in several instances of the XY spin chain. In the hydrodynamic limit of long times and large sizes of $A$, with their ratio fixed, the reduced fidelities admit a quasiparticle picture interpretation. Interestingly, for some quenches in the hydrodynamic regime the RLE features a complicated structure with an infinite sequence of nested lightcones, corresponding to quasiparticles with arbitrary large group velocities. This leads to a ''staircase'' of cusp-like singularities in the time-derivative of the fidelity. At the sub-hydrodynamic regime for some quenches the RLE exhibits cusp-like singularities, similar to the so-called dynamical quantum phase transitions (DQPT). We conjecture a criterion for the occurrence of the DQPT and for the ''critical'' times at which the singularities occur. Finally, we discuss the hydrodynamic limit of the FSF. In particular, we show that it provides a valuable tool to detect the so-called quantum Mpemba effect.

Figures (11)

• Figure 1: Logarithmic RLE in the XX chain from the Néel state, in both panels the symbols are obtained by exact numerical diagonalization with $\ell=200$. Left: Logarithmic RLE as a function of $t/t^*$ with $t^* = \pi$. The solid line is the theoretical prediction of Eq. \ref{['eq:LRLENeel']} in the sub-hydrodynamic regime. We observe that DQPTs occur at $t=(m+1/2)t^*$ where $m$ is an integer. Right: Logarithmic RLE as a function of $t/\ell$, in the hydrodynamic regime. The solid lines are theoretical predictions. The thin orange line is Eq \ref{['eq:LRLENeel']}, whereas the thick red one is Eq. \ref{['eq:LRLENeelHydro']}. Finally, the dotted horizontal line is the stationary value $1/2 \log(2)$. We observe that the oscillations are perfectly captured by Eq. \ref{['eq:LRLENeel']}, and that they are still strong at $\ell=200$, but decrease in amplitude with $t$.
• Figure 2: Logarithmic RLE in the XX chain from the dimer state with $\ell=200$ as a function of $t$. The symbols are obtained by exact numerical diagonalization.
• Figure 3: Logarithmic RLE in the Ising model, i.e., $\gamma_0=\gamma=1$ (left) and in the XY model with $h_0=0$ and $h=2.6$ (right) as a function of $t/t^*$ with $t^*$ obtained from the main text, for various values of the quench parameters and $\ell=200$. The symbols are obtained by exact numerical diagonalization. We observe that DQPTs occur at $t=(m+1/2)t^*$.
• Figure 4: Derivative $M'_n = \mathrm{d} M_n/\mathrm{d} t$ of the moments as a function of $t/\ell$ for the quench from the dimer state. The solid lines are the analytical results of Eq. \ref{['eq:MnHydroDimer']}, and the dotted vertical lines are located at $t/\ell = 1/(2j)$ with $j=1,2,3$.
• Figure 5: Moments $\mathcal{M}_n$ for the quench from the dimer state (left) and the XY chain for $\ell=750$ (right). The initial parameters are $h_0=0.5, \ \gamma_0=0.5$ and the quenched parameters are $h=2, \ \gamma=1$.