Tunable quantum Mpemba effect in long-range interacting systems

TL;DR

This work investigates how the quantum Mpemba effect (QME), the accelerated restoration of symmetry after a quench, can be tuned in a 1D spin-$\tfrac{1}{2}$ system with long-range, power-law interactions. By analyzing a long-range XYZ model under a strong magnetic field, the authors show that a prethermal regime with emergent $U(1)$ symmetry enables QME, while sufficiently long-range interactions can induce symmetry breaking that halts the effect, in a manner governed by the interaction range $\alpha$ and the initial-state energy density. The study introduces entanglement asymmetry as a diagnostic, derives an effective XXZ description in the high-field limit, and maps a dynamical phase diagram highlighting a tunable boundary between QME-allowed and QME-suppressed regimes. These results connect QME to the HMW theorem and offer experimentally accessible routes to observe and control dynamical symmetry restoration in platforms such as trapped ions, polar molecules, and NV centers.

Abstract

Symmetry plays a fundamental role in many-body systems, both in and out of equilibrium. The quantum Mpemba effect (QME) - a phenomenon where systems initially farther from equilibrium can thermalize faster - can be understood in terms of how rapidly a symmetry, broken by initial conditions, is dynamically restored. In this work, we study the QME in a one-dimensional spin-1/2 XYZ model with power-law decaying interactions in the presence of a magnetic field. In the prethermal regime generated by large field strengths, the system develops a continuous U(1) symmetry, enabling the QME to emerge. However, due to the Hohenberg-Mermin-Wagner theorem, the QME can only arise when interactions are sufficiently short-ranged. This leads to an interplay between the external field, interaction range, and dynamical symmetry restoration. We systematically explore this interplay and analyze the dependence of the QME on the effective temperature set by the initial state. Our results demonstrate the tunability of the QME via long-range interactions, which can be probed in experimental platforms of trapped ions, polar molecules, and NV centers.

Figures (8)

• Figure 1: (a) Schematic of a one-dimensional long-range XYZ spin chain in an external magnetic field $h_z$. The spins $i$ and $j$ interact via power-law decaying interactions, $(J_\nu/|i-j|^\alpha) \hat{\sigma}^\nu_i\hat{\sigma}^\nu_j$, where $J_\nu$, $\nu=x,y,z$ are the interaction couplings along the three directions and $\hat{\sigma}_i^\nu$ are Pauli matrices, as in Eq. (\ref{['eq:Ham']}). (b) At weak fields, $h_z\ll J_\nu$, there is no U(1) symmetry for generic values of $J_\nu$ and the corresponding order parameter, $\sigma^+ \equiv (\sigma^x + i \sigma^y)/2$, rapidly decays to zero as a function of time $t$. By contrast, in the prethermal regime when $h_z \gg J \gg J_\mathrm{U(1)} \equiv |J_x-J_y|$, where $J$ is the largest of the couplings $J_\nu$, the system develops a U(1) symmetry in the rotating frame, accompanied by the persistent non-zero value of $\langle \sigma^+(t)\rangle$---the hallmark of a continuous time crystal Else2017. This only occurs when $\alpha$ is smaller than a critical value $\alpha_c$ (see text for details). The goal of this paper is to quantitatively explore the competition between spontaneous symmetry breaking and dynamical symmetry restoration.
• Figure 2: Quench dynamics for the initial states in Eq. (\ref{['eq:Initial_state']}) under the long-range XYZ Hamiltonian in Eq. (\ref{['eq:Ham']}). The parameters are $J_x=-0.5$, $J_y=-1.5$ and $J_z=-0.75$, at $\alpha=4$ (top row) and $\alpha=1.5$ (bottom row), while the color bar shows the field strength $h_z$. Panels (a) and (b) show the expectation value $D(t) \equiv \langle \psi(t)|\hat{D}|\psi(t)\rangle$ of the U(1)-invariant effective XXZ Hamiltonian in Eq. (\ref{['eq:D']}) for the $\theta=\pi/4$ initial state at $\alpha=4$ (a) and $\alpha=1.5$ (b). We normalize the curves by $D(t=0)$. The prethermal regime, $D(t)/D(0)\approx 1$, is seen to be robust for both short- and long-range interactions if the field strength is sufficiently large ($h_z\gtrsim 2$). (b) Panels (c) and (d) show the corresponding entanglement asymmetry $\Delta S_A$ for $N_A=4$ sites for $\alpha=4$ (c) and $\alpha=1.5$ (d). The symmetry is restored for short-range interactions (c), but not for long-range (d). Panels (e) and (f) show the ratio of the entanglement asymmetry at $\theta_1=\pi/4$ and $\theta_2=\pi/8$ at $\alpha=4$ and $\alpha=1.5$, respectively. The drop of the large-$h_z$ (red) curves below unity in panel (e) signals the QME, which is absent for longer-range interactions in panel (f). All data were obtained for a subsystem of $N_A=4$ spins in an infinite chain using TDVP with translation-invariant iMPS of bond dimension $\chi=128$ and a timestep $\delta t=0.025$.
• Figure 3: (a) Ratio of the entanglement asymmetry for the initial states in Eq. (\ref{['eq:Initial_state']}) with $\theta_1=\pi/4$ and $\theta_2=\pi/8$, evolved under the XXZ Hamiltonian (\ref{['eq:D']}). Parameters are $J_x=J_y=-1$ and $J_z=-0.75$ with varying $\alpha$ (color bar). (b) The Mpemba time $\tau_M$ as a function of $\alpha$ for these states (labeled $1$-site), as well as for two tilted Néel states in Eq. (\ref{['eq:high_energy_state']}) with $\phi_1=\pi/4$ and $\phi_2=\pi/8$ (labeled $2$-site), evolved under the same Hamiltonian. (c) Mpemba time $\tau_M$ (color bar) as a function of $(J_z,\alpha)$, for tilted product states n Eq. (\ref{['eq:Initial_state']}). The line labeled $\alpha_M$ marks where $\tau_M\rightarrow\infty$ and no QME occurs. The SSB phase boundary $\alpha_c$ is defined by central charge $c_\mathrm{eff}$ deviating from unity by more than $10\%$. $c_\mathrm{eff}$ was calculated from finite DMRG comparing the difference in entanglement entropy between system sizes $N=192$ and $N=200$, following the method in Refs. Maghrebi2017Gong2016KaleidoscopeGong2016TopologicalPhases, at a maximum bond dimension $\chi=250$. An alternative estimate of the SSB boundary, $\alpha^{\mathrm{ES}}_c$, was obtained following Ref. Schneider2022, where the entanglement spectrum of the ground state was calculated for a system of $N=192$ spins using DMRG with $\chi=250$. The shaded area between $\alpha_c$ and $\alpha_c^\mathrm{ES}$ represents the uncertainty of the SSB boundary. All dynamics data were obtained for a subsystem of $N_A=4$ spins using TDVP for iMPS with time step $\delta t=0.025$ and maximum bond dimension $\chi=128$.
• Figure 4: Dynamical phase diagram of U(1) symmetry restoration in the XYZ model (\ref{['eq:Ham']}) with generic couplings that break the U(1) symmetry, as a function of the field strength $h_z$ and interaction range $\alpha$. The QME is only possible when the field strength is large, such that the spin couplings become isotropic in $x$-$y$ plane, and the U(1) symmetry is preserved by interactions being effectively short-range. We note that this phase diagram is expected to be valid over the prethermal timescale Else2017, Eq. (\ref{['eq:XXZ']}), which depends exponentially on $h_z$ and can be made much longer than the Mpemba time $\tau_M$. Moreover, the phase diagram is in principle dependent on the type of initial states, which we assume here to be product states of spins pointing in the same direction on the Bloch sphere [Eq. (\ref{['eq:Initial_state']})] or tilted Néel states [Eq. (\ref{['eq:high_energy_state']})], or their combinations (see text for details).
• Figure 5: Ratio of trace distances for the initial states defined in Eq. (\ref{['eq:Initial_state']}) at $\theta_1=\pi/4$ and $\theta_2=\pi/8$ evolved under the XYZ Hamiltonian (\ref{['eq:Ham']}) with $J_x=-0.5$, $J_y=-1.5$, $J_z=-0.75$ and $\alpha=4$, with varying magnetic field strengths $h_z$ (color bar). Results are for a finite system of $N=100$ spins and subsystem of $N_A=4$ spins. The thermal density matrix was constructed using a smaller system of $14$ spins. The ratio crossing $1$ indicates the QME has occurred. Similar to Fig. \ref{['fig:XYZ_results']}(e), no QME is observed at low fields, but increasing $h_z$ induces the QME. The dynamics data were obtained using TDVP for finite MPS, with the finite systems results using max bond dimension $\chi=100$.