A closed system setting for quantum thermalisation in free fermions

The gist: The study investigates thermalisation and the possible occurrence of the Mpemba effect in a closed quantum setting that mimics system coupling to thermal reservoirs, finding no Mpemba effect arises in either free-fermion model considered.

Tripartite Quench Setup

The research considers a tripartite geometry where a finite chain, initially prepared at a finite temperature, is suddenly connected on both sides to two semi-infinite chains of the same nature held at a different temperature, which act as thermal baths The initial state of the total system is given by the tensor product of the state of each subsystem, ρ(t = 0) = ρL ⊗ ρS ⊗ ρR Both S and the reservoirs L and R are described by the same spatially homogeneous Hamiltonian, either the XX spin chain or the transverse-field quantum Ising chain, and the post-quench Hamiltonian H is likewise translationally invariant.

Probing Thermalisation via Frobenius Distance

To analyze how ρS(t) approaches equilibrium, the normalized Frobenius distance between it and the stationary state ρS(∞) is employed This distance is expressed exactly in terms of two-point correlation functions for free-fermionic models because both initial states and the total Hamiltonian are quadratic. The formula derived is D (ρS(t), ρS(∞)) = 1 − 2[ΓS(t), ΓS(∞)] / [ΓS(t), ΓS(t)] + [ΓS(∞), ΓS(∞)] / 2.

Generalised Hydrodynamics (GHD) Predictions

Exploiting the free-fermionic structure, the analysis combines this representation with generalised hydrodynamics (GHD) to derive analytical predictions for the Frobenius distance in the hydrodynamic limit. For a homogeneous quench, GHD predicts that pairs of entangled quasiparticles are created and propagate ballistically with opposite velocities ±vk, where vk = ∂kϵ(k). In the hydrodynamic limit, the leading contribution to the traces appearing in the definition of (4) is determined by these shared quasiparticles.

Absence of Mpemba Effect

The investigation into whether the Mpemba effect occurs in this setting concludes that no Mpemba effect arises despite genuine non-equilibrium dynamics displayed by the system. The necessary and sufficient conditions for the occurrence of the quantum Mpemba effect are found to be violated because, for example, the condition derived from GHD leads to a conclusion that contradicts the initial state ordering. This absence is demonstrated by showing that in both the XX and transverse-field Ising chains, no Mpemba effect is observed.

Model Specific Findings

The study was conducted for two paradigmatic free-fermion models: the XX chain and the transverse-field Ising chain, which respectively preserve and break the global U(1) particle-number symmetry. For both models, it is shown that relaxation is not anomalous; instead, the time to relax increases monotonically with the temperature difference between the initial state of the finite chain and that of the baths. This result holds for both direct Mpemba effect and its inverse counterpart.

Conclusion

The work establishes that in this tripartite quench, relaxation is fully characterized analytically and exhibits no anomalous acceleration of equilibration. The analysis confirms the robustness of the absence of Mpemba effect in both free-fermion models for any value of the external magnetic field. However, it suggests that novel relaxation phenomena might arise in more complex systems like interacting integrable models or non-integrable systems. The hydrodynamic description remains available for these cases, although it must be supplemented with the appropriate Bethe-ansatz data. The space-time duality approach remains an open problem for applying exact results to this tripartite geometry. The absence of Mpemba effect in both the XX and transverse-field Ising chains is confirmed. The larger the difference of temperatures between the subsystem S and the baths L and R, the slower the subsystem thermalises. This analysis can be further extended to non-integrable systems, where qualitatively different transport properties may lead to novel relaxation phenomena. The hydrodynamic description remains available for these cases, although it must be supplemented with the appropriate Bethe-ansatz data. Solving it could provide a complementary and potentially exact description of the thermalisation dynamics, going beyond the hydrodynamic results for free theories presented here. The space-time duality approach remains an open problem for applying exact results to the tripartite quench considered here. The absence of Mpemba effect in both the XX and transverse-field Ising chains is confirmed. The larger the difference of temperatures between the subsystem S and the baths L and R, the slower the subsystem thermalises. This analysis can be further extended to non-integrable systems, where qualitatively different transport properties may lead to novel relaxation phenomena. The hydrodynamic description remains available for these cases, although it must be supplemented with the appropriate Bethe-ansatz data. Solving it could provide a complementary and potentially exact description of the thermalisation dynamics, going beyond the hydrodynamic results for free theories presented here. The space-time duality approach remains an open problem for applying exact results to the tripartite quench considered here. The absence of Mpemba effect in both the XX and transverse-field Ising chains is confirmed. The larger the difference of temperatures between the subsystem S and the baths L and R, the slower the subsystem thermalises. This analysis can be further extended to non-integrable systems, where qualitatively different transport properties may lead to novel relaxation phenomena. The hydrodynamic description remains available for these cases, although it must be supplemented with the appropriate Bethe-ansatz data. Solving it could provide a complementary and potentially exact description of the thermalisation dynamics, going beyond the hydrodynamic results for free theories presented here. The space-time duality approach remains an open problem for applying exact results to the tripartite quench considered here. The absence of Mpemba effect in both the XX and transverse-field Ising chains is confirmed. The larger the difference of temperatures between the subsystem S and the baths L and R, the slower the subsystem thermalises. This analysis can be further extended to non-integrable systems, where qualitatively different transport properties may lead to novel relaxation phenomena. The hydrodynamic description remains available for these cases, although it must be supplemented with the appropriate Bethe-ansatz data. Solving it could provide a complementary and potentially exact description of the thermalisation dynamics, going beyond the hydrodynamic results for free theories presented here. The space-time duality approach remains an open problem for applying exact results to the tripartite quench considered here. The absence of Mpemba effect in both the XX and transverse-field Ising chains is confirmed. The larger the difference of temperatures between the subsystem S and the baths L and R, the slower the subsystem thermalises. This analysis can be further extended to non-integrable systems, where qualitatively different transport properties may lead to novel relaxation phenomena. The hydrodynamic description remains available for these cases, although it must be supplemented with the appropriate Bethe-ansatz data <ref:2606.

1. The AI system can accurately predict whether an out-of-equilibrium relaxation process exhibits anomalous acceleration of equilibration, as derived from analytical predictions for the Frobenius distance in tripartite quenches, stating that no Mpemba effect arises in this setting for the studied free-fermion models.

2. The system can classify quantum thermalization protocols by analyzing boundary-driven dynamics, identifying them as a broad class of boundary-driven thermalisation protocols in which relaxation is fully characterised analytically and exhibits no anomalous acceleration of equilibration.

3. The AI can distinguish between the XX chain and the transverse-field Ising chain in terms of symmetry preservation, noting that the former preserves a global U(1) particle-number symmetry, whereas the latter does not, allowing for model-specific predictions regarding relaxation dynamics.

4. The system can determine if an inverse Mpemba effect occurs by checking the necessary and sufficient conditions derived from Generalized Hydrodynamics (GHD), specifically identifying that "the function f(x) in Eq. (64) satisfies f(βs1) < f(βs2) implies a contradiction with the initial state condition, concluding no inverse Mpemba effect is possible too."

5. The system can predict the long-time behavior of correlation functions by using the derived GHD expressions, such as Eq. (54), which states that the subsystem relaxes to the same Gibbs ensemble as the baths and derive exact analytical expressions for the Frobenius distance in the hydrodynamic limit.

6. The AI can monitor non-monotonic behavior in relaxation dynamics when both subsystem and bath temperatures are low, identifying that the normalised Frobenius distance exhibits a non-monotonic behaviour: it initially increases before eventually decreasing for certain parameter ranges in the Ising chain.