Quantum Mpemba effect in long-ranged U(1)-symmetric random circuits

TL;DR

The paper probes the quantum Mpemba effect (QME) in long-range, U(1)-symmetric random unitary circuits by tracking symmetry restoration via the annealed Rényi-2 entanglement asymmetry, computed with a replica tensor-network framework. The authors show that QME robustly appears for tilted ferromagnetic states across interaction ranges, is absent for tilted Néel states, and appears for domain-wall states only in effectively short-range circuits, with the Mpemba time scaling as $t_{\rm M} \sim N_A^{z}$ where $z = \min(\alpha-1, 2)$. They argue that the presence of QME depends on the interplay between initial-state charge bias and the speed of information propagation set by $\alpha$, and that long-range circuits can erase locality to suppress QME in certain configurations. The work provides a general criterion for QME in long-range quantum chaotic systems and connects the phenomenon to transport properties, suggesting feasibility of experimental verification on digital quantum simulators. Overall, the results extend the understanding of QME to nonlocal, chaotic quantum dynamics and offer a framework for controlling relaxation by tuning interaction range and initial-state structure.

Abstract

The Mpemba effect, where a state prepared farther from equilibrium relaxes faster to equilibrium than one prepared closer, has a quantum counterpart where relaxation is resolved by conserved charge. However, the fate of the quantum Mpemba effect in systems with long-range interactions remains an open question. Here, we study the quantum Mpemba effect in long-ranged, U(1)-symmetric random unitary circuits. Using annealed Rényi-2 entanglement asymmetry computed via replica tensor networks and exact diagonalization, we track the symmetry restoration from three types of tilted product states: ferromagnetic, antiferromagnetic, and ferromagnetic with a central domain wall. The quantum Mpemba effect is present for tilted ferromagnetic states at all interaction ranges, but absent for tilted antiferromagnetic states, and occurs for the domain-wall state only in effectively short-ranged circuits, where the Mpemba time $t_{\rm M}$ is found to scale with the subsystem size $N_A$ as $t_{\rm M}\!\sim\!N_{A}^{\,z}$, with the dynamical exponent $z=\min(α-1,2)$. These results reveal how the quantum Mpemba effect is governed by the interplay between interaction range and initial-state charge bias in long-ranged chaotic systems.

Figures (5)

• Figure 1: (a) Schematic of the long-range U(1)-symmetric random quantum circuit. Starting from the tilted product state $\ket{\psi_0(\theta)}$, we apply layered U(1)-preserving two-qubit gates with power-law range $P_{r_{ij}}(\alpha)\!\propto\!r^{-\alpha}$. The number of active two-qubit gates per layer equals to the system size $N$. The evolution of the circuit is given by Eq. \ref{['eq:evolution']}. (b) Summary of key dynamical features for three representative initial states, TFS, TNS, and TDWS: whether U(1) symmetry is restored at long-time limit, whether the QME occurs, and whether the presence of QME is $\alpha$-dependent.
• Figure 2: Entanglement asymmetry dynamics for TFS/TNS/TDWS: short- vs. long-range interactions. $\overline{\Delta S_{A}^{(2)}}(t)$ for a subsystem of size $N_{A}=2$ chosen at the left edge of a chain with total length $N=48$. The main panels show numerical results obtained using the replica tensor-network method, averaged over $50$ sampling realizations. Panels (a,d), (b,e), and (c,f) correspond to TFS, TNS, and TDWS initial states, respectively. The upper row (a–c) is for a more short-ranged interaction with exponent $\alpha=5.0$, while the lower row (d–f) is for a more long-ranged interaction with $\alpha=1.5$. (See Appendix \ref{['AppendixB']} for additional numerical results.)
• Figure 3: (a)-(b) Mpemba time $t_{\rm M}$ versus subsystem size $N_A$ for TFS and TDWS initial states with $\theta_1=0.3\pi$, $\theta_2=0.4\pi$, chain lengths $N=48$, and $N_A\!=\!1,2,\dots,10$. Symbols denote replica tensor-network simulation results; dashed lines are fits to $t_{\rm M}\!=\!b\,(N_A - x_0)^{z} + c$, performed only for $N_A\geq3$ and $\alpha\geq1$. (c)-(d) Fitted scaling dynamical exponent $z_{\mathrm{fit}}$ as a function of the long-ranged interaction exponent $\alpha$, compared with the theoretical prediction $z\!=\!\min(\alpha-1,2)$longRUC2Richter_2023PhysRevX.8.031057. Fits are performed only for $\alpha\geq1$ in TFS, and for $\alpha \geq3$ in TDWS.
• Figure 4: The dynamics of entanglement asymmetry are simulated using replica tensor-network methods, TFS, TNS, and TDWS, for a system size of $N=48$ and subsystem size of $N_A=2$ with $50$ ensembles averaging. Specifically: TFS is shown in panels (a), (d), (h), (k), (n), and (q) for $\alpha = 4, 3, 2.5, 2.0, 0.5, 0$, respectively. TNS is shown in panels (b), (e), (i), (l), (o), and (r) for the same $\alpha$ values. TDWS is shown in panels (c), (f), (j), (m), (p), and (s) for the same $\alpha$ values.
• Figure 5: Time evolution of the annealed Rényi-2 entanglement asymmetry $\Delta S_A^{(2)}(t)$ for the tilted ferromagnetic state (TFS) at different power-law exponents $\alpha=5.0,\,3.0,\,1.5,\,0.5$ in panels (a)-(d), respectively. Results are shown for chain length $N=12$ and subsystem size $N_A=2$. Crosses (ED) denote exact diagonalization data, while open squares (TN) denote results from replica tensor-network simulations. Two tilt angles, $\theta=0.2\pi$ and $\theta=0.3\pi$, are compared, showing consistent relaxation behavior between the two numerical methods.