A closed system setting for quantum thermalisation in free fermions

Why Hot Water Might Freeze Faster Than Cold

Why do some systems reach equilibrium faster than others, even when they start further away from the finish line? Scientists have long studied the Mpemba effect—a counterintuitive phenomenon where a hotter system cools more rapidly than a colder one. While this has been observed in classical fluids and certain quantum models, the rules governing how a system settles into a stable state when connected to external heat baths remain elusive.

A new study from researchers at SISSA and INFN investigates this mystery by looking at how quantum spin chains thermalize. Instead of looking at a single system being cooled, the authors study a "tripartite" setup. A central quantum chain is suddenly connected to two semi-infinite chains acting as thermal reservoirs. By analyzing two classic models—the XX chain and the transverse-field Ising chain—the researchers sought to determine if this boundary-driven cooling could trigger the Mpemba effect.

The study finds that the Mpemba effect does not occur in these specific settings. In both models, the time required to reach equilibrium increases predictably with the temperature difference between the system and its baths.

The search for anomalous cooling

In statistical physics, the standard intuition is linear. The greater the temperature gradient, the longer the journey to equilibrium. The Mpemba effect challenges this. It suggests that certain initial conditions allow a system to "skip ahead" in the relaxation process.

Previous research has explored this in "open" quantum systems, where the environment actively drains energy from the system. Other studies focus on "spatially homogeneous quenches" (sudden changes applied uniformly to the entire system). Such quenches ignore the messy reality of boundary-driven thermalization. In boundary-driven scenarios, heat enters or leaves through specific contact points.

Understanding whether boundary-driven protocols can produce anomalous acceleration is vital. It helps anyone attempting to control or predict the stability of quantum states in a laboratory.

Mapping the tripartite quench

To bridge this gap, the authors implement a tripartite geometry, illustrated in .

They begin with a finite central subsystem ($S$) at an initial temperature. They then abruptly connect it to two semi-infinite reservoirs ($L$ and $R$) held at a different temperature.

Crucially, the entire combined system then evolves unitarily. Unitary evolution means the total energy and information are conserved within the whole. Even so, the subsystem $S$ appears to exchange heat with the baths.

The researchers utilize two mathematical tools to solve this complex evolution:

1. The Jordan-Wigner transformation: This maps complicated spin interactions into a system of "free fermions" (particles that do not bump into each other). This makes the math tractable. The state remains "Gaussian," meaning it is fully described by simple two-point correlation functions.
2. Generalised Hydrodynamics (GHD): This framework treats the quantum system like a fluid. Instead of tracking every individual particle, GHD tracks the local density of "quasiparticles" (collective excitations that move through the system).

By applying GHD, the authors can predict how energy and particle densities flow. As seen in, the GHD model matches the actual movement of energy density in the XX chain.

This confirms the "fluid" approximation accurately captures the microscopic quantum dynamics.

Measuring the distance to equilibrium

To quantify how fast the system relaxes, the authors employ the normalized Frobenius distance. Think of this as a geometric ruler. It measures the "gap" between the current state of the subsystem and its eventual stationary state. A distance of zero means the system has perfectly equilibrated with the baths.

The researchers report that in the XX chain, the Frobenius distance behaves exactly as predicted by the quasiparticle picture. While the distance drops to zero over time, the path is strictly monotonic. In, the authors show that for the XX chain, a larger initial temperature difference consistently results in a larger Frobenius distance.

There are no instances where the curves cross. Crossing curves is the mathematical requirement for the Mpemba effect to exist.

The study also explores the transverse-field Ising chain. This model differs from the XX model because it breaks particle-number symmetry. In the ferromagnetic phase of this model (where $|h| < 1$), the authors observe strange behavior at low temperatures. As shown in, the Frobenius distance actually increases slightly before it begins to fall.

The authors clarify that this is a transient boundary effect. It is caused by "bound states" at the edges. It does not constitute a true Mpemba effect.

Limits of the hydrodynamic view

While the study provides a rigorous analytical framework, the current model has limits. The findings are strictly limited to "free-fermion" models. In these systems, particles move through each other without interacting. This keeps the math manageable. Most real-world materials consist of interacting particles.

Furthermore, the GHD framework is a "large-scale" approximation. It is highly effective at predicting bulk behavior. However, it fails to capture the subtle, transient fluctuations seen near the boundaries in the Ising model . For a practitioner building a quantum simulator, this is a key takeaway. GHD is a reliable guide for the center of a long chain. Yet, it may miss critical temporary behaviors occurring at the interface where the system meets the reservoir.

The verdict on anomalous relaxation

Is the Mpemba effect possible in these boundary-driven systems? Based on the evidence, the answer is no. The authors provide a formal proof using the monotonicity of a specific mathematical function. They demonstrate that for both the XX and Ising chains, the relaxation time scales predictably with the temperature gap.

If you are designing a protocol to stabilize a quantum subsystem using boundary coupling, you can rely on predictability. You won't encounter a "shortcut" where a higher initial temperature leads to faster stabilization. However, the discovery of non-monotonicity in the Ising chain suggests that boundary effects can still create complex, temporary deviations from equilibrium. Simple hydrodynamic models might overlook these moments.

Figures from the paper