Conserved quantities enable the quantum Mpemba effect in weakly open systems

TL;DR

The paper establishes that the quantum Mpemba effect in weakly open many-body systems hinges on the number of (approximately) conserved quantities: when the Hamiltonian has only a single conserved quantity, dissipation confines dynamics to a one-dimensional Gibbs manifold and crossings are unlikely; with multiple conserved quantities, dynamics probe a multidimensional GGE space, allowing crossings to occur. The authors formulate a zeroth-order Gibbs/GGE framework and verify it with tensor-network simulations for chaotic (N_C=1) and integrable (N_C~L) regimes, as well as free-fermion analyses for the integrable case, showing Mpemba crossings only in the integrable/multiconserved scenario. Distances to the steady state are quantified via a normalized Frobenius norm on reduced states, and, for Gaussian free-fermion states, through correlation-matrix formalism leading to explicit expressions for both reduced and full-density-matrix distances in terms of mode occupations $raket{n_q}$. The results provide a unifying principle linking symmetries and integrability to non-equilibrium relaxation, with potential experimental relevance for quantum simulators and nearly integrable materials.

Abstract

Observation of the quantum Mpemba effect has spurred much interest in its enabling conditions and its relation to the classical counterpart. Here, we consider weakly open many-body quantum systems initialized in different thermal states and examine when the initially farther state relaxes to the (non-equilibrium) steady state faster. We claim that the number of conserved quantities in the unitary part plays a crucial role: the Mpemba effect is possible only when the Hamiltonian commutes with other extensive operators or is integrable. The reason lies in the dynamical evolution happening in spaces of different dimensions. When energy is the only approximately conserved quantity, dissipation pushes the dynamics within a single-parameter manifold of different thermal states. In contrast, for Hamiltonians with several conserved quantities, the dynamics drift in the multi-dimensional space of generalized Gibbs ensembles, whose distance to the steady state is less trivial. We provide numerical results for large system sizes using tensor networks and free-fermion techniques, thereby supporting our claim.

Figures (7)

• Figure 1: We consider many-body quantum systems with unitary Hamiltonian and a weak dissipative evolution, initialized in two thermal states. $N_C=1$: For Hamiltonians without additional conserved quantities, each dynamics can be approximated by a thermal ensemble with a time-dependent temperature. $N_C>1$: For Hamiltonians with more conserved quantities, dissipation can steer the evolution from thermal states in the multi-dimensional manifold of generalized Gibbs ensembles (GGEs). When projecting the dynamics onto a scalar - the distance between the time-evolved and the non-equilibrium steady state (NESS) - crossings can happen in the presence of multiple approximately conserved quantities.
• Figure 2: (a,b) Distances $d_{\ell=2}(\rho(t),\rho_\infty)$ as a function of rescaled time $\epsilon t$ for the initial thermal states with $\beta_c=0.15$ (blue) and $\beta_h=0$ (red) inverse temperature for (a) chaotic and (b) integrable transverse field Ising Hamiltonian $H$. Results for $\epsilon=0.05,0.2,0.5$ (shown in solid lines of diminishing shades for diminishing $\epsilon$) converge to the (a) Gibbs and (b) GGE results (shown with dashed lines). (c) Distances $d_{\ell=2}(\rho(t),\rho_\infty)$ for different initial temperatures $\beta\in[0, 0.2]$ and chaotic $H$ at $\epsilon=0.2$: no Mpemba effect is observed. (d) Crossing times $t_{\text{Mp}}$ as a function of subsystem size $\ell$ for the integrable $H$ extracted from $\rho_{\boldsymbol\lambda}(t)$ calculated on $L=400$. Parameters: $L=80$ (chaotic TN results) and $L=160$ (integrable TN results), with $160 \le \chi \le 240$.
• Figure 3: Distances $d_{\ell}(\rho(t),\rho_\infty)$ as a function of rescaled time $\epsilon t$ for the initial thermal states with $\beta_c=0.12$ (blue) and $\beta_h=0$ (red) inverse temperature and chemical potential $\mu(0)=0$ for chaotic Hamiltonian with two conserved quantities, Eq. \ref{['eq:H2']}, at (a) $\epsilon=0.05$ and (b) $\epsilon=0.5$. Parameters: $L=80$, $\ell=2$, $\chi=200 \text{ (a)}, 160 \text{ (b)}$.
• Figure 4: Energy density $\langle H \rangle/L$ as a function of rescaled time $\epsilon t$ for (a) the integrable Hamiltonian \ref{['eq:H']} with $N_C\sim L$ and (b) the chaotic Hamiltonian with a single additional conserved quantity \ref{['eq:H2']}, $N_C=2$. Parameters: $\epsilon=0.05,0.2,0.5$, $L=160$, $180\le \chi\le 240$ ($N_C \sim L$) and $\epsilon=0.05,0.2,0.5$$L=80$, $160\le \chi\le 200$ ($N_C=2$).
• Figure 5: Normalized overlap of different thermal states with the slowest mode and different distances (trace distance, Frobenius distance, normalized Frobenius distance) of thermal states to the steady state for (a) chaotic and (b) integrable transverse field Ising parameters used in the main text. System sizes (a) $L=8, 10, 12, 14$ and (b) $L=10, 12, 14, 16$ from lighter to darker colors.