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arxiv: 2606.06362 · v1 · pith:DTBT45Z2new · submitted 2026-06-04 · ❄️ cond-mat.stat-mech · quant-ph

A closed system setting for quantum thermalisation in free fermions

Purvaash Panduranghan-Udhayashankar , Filiberto Ares , Pasquale Calabrese This is my paper

Pith reviewed 2026-06-27 23:08 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph
keywords free fermionsquantum thermalizationMpemba effectboundary-driven systemsXX chainIsing chaingeneralized hydrodynamicsFrobenius distance
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The pith

A closed tripartite free-fermion chain connected to boundary baths relaxes to the bath temperature without the Mpemba effect.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines thermalization in a unitary tripartite setup where a finite free-fermion chain at one temperature is suddenly joined to two semi-infinite chains at another temperature. The outer chains serve as baths, and the full system evolves under a uniform Hamiltonian, so the central subsystem reaches exactly the bath temperature in the long run. The Frobenius distance from the central reduced density matrix to this thermal state is computed exactly from two-point functions for the XX and transverse-field Ising chains, then analyzed with generalized hydrodynamics in the large-scale limit. Despite the genuine non-equilibrium evolution, the distance never displays the Mpemba effect of anomalous faster relaxation from hotter initial states. This identifies an analytically tractable class of boundary-driven protocols that lack such acceleration.

Core claim

In the tripartite geometry the stationary state of the central subsystem is the thermal state at the bath temperature, and the time-dependent Frobenius distance to this state, expressed exactly via two-point correlation functions, decays without any Mpemba effect for both the XX chain and the transverse-field Ising chain; generalized hydrodynamics supplies closed-form predictions for the entire relaxation process in the hydrodynamic limit.

What carries the argument

The Frobenius distance of the central reduced density matrix to its thermal stationary state, written exactly in terms of two-point correlation functions and evaluated with generalized hydrodynamics.

If this is right

  • The relaxation process admits a complete analytic description in the hydrodynamic limit for this family of boundary-driven protocols.
  • No anomalous acceleration of equilibration occurs even though the dynamics are genuinely non-equilibrium.
  • The result applies equally to models that preserve and break global U(1) symmetry.
  • The stationary state reached by the central subsystem is precisely the thermal state at the bath temperature.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same unitary-bath construction could be applied to other free-fermion models or to different initial temperature profiles to test the robustness of the no-Mpemba conclusion.
  • Varying the length of the central chain or the temperature difference might reveal regimes where standard relaxation gives way to other transient behaviors.
  • This closed-system realization clarifies how unitarity alone can enforce conventional thermalization rates without the anomalies sometimes reported in open-system treatments.
  • Weakly interacting perturbations around these models could be studied numerically to see whether interactions introduce an Mpemba effect absent in the free case.

Load-bearing premise

The semi-infinite outer chains act as perfect thermal baths whose initial temperature difference alone fixes the long-time state of the central chain to the bath temperature under purely unitary evolution.

What would settle it

Numerical evaluation of the Frobenius distance for two different initial temperatures in the XX chain that shows faster decay from the state farther from equilibrium would contradict the absence of the Mpemba effect.

Figures

Figures reproduced from arXiv: 2606.06362 by Filiberto Ares, Pasquale Calabrese, Purvaash Panduranghan-Udhayashankar.

Figure 1
Figure 1. Figure 1: Tripartite quench setup studied in this paper. We initially prepare an open spin chain S of size ℓ at temperature 1/βs. At t = 0, it is connected at its ends to two semi-infinite chains L and R at temperature 1/βb. At time t > 0, the combined system evolves unitarily with a translationally invariant Hamiltonian. The correlation matrix of ρS(∞) is defined similarly, but replacing ρS(t) with ρS(∞) in Eq. (5)… view at source ↗
Figure 2
Figure 2. Figure 2: Time evolution of the Frobenius distance between the reduced density matrix of subsystem S, ρS(t), and its stationary value, ρS(∞) in the homogeneous quench (9) to the XX spin chain from the ground state of the XY spin chain for different values of the parameters h and γ. The symbols represent the result of exact numerics using Eq. (7) and the solid lines are the prediction (24) of the quasiparticle pictur… view at source ↗
Figure 3
Figure 3. Figure 3: Energy density profile e(x, t) in the tripartite quench of the XX spin chain at several times t. Symbols represent the exact numerical values, while solid lines are the GHD prediction (49) obtained from the time-evolved local density of occupied modes (45). In the left panel, the initial temperature of subsystem S is βs = 0 (infinite temperature) and the semi-infinite subsystems L and R are at temperature … view at source ↗
Figure 4
Figure 4. Figure 4: Time evolution of log Tr(ρS(t)ρS(∞)) in the tripartite quench of the XX spin chain. Symbols denote the exact numerical value, calculated from the two-point correlation matrix (43). Solid lines are the GHD prediction in Eq. (53). The semi￾infinite subsystems L and R are prepared at temperature βb = 1 (upper panels) and βb = 5 (lower panels). We consider different initial temperatures βs for the subsystem S,… view at source ↗
Figure 5
Figure 5. Figure 5: Normalised Frobenius distance between the time-evolved state of S and its stationary state in the tripartite quench in the XX spin chain. Symbols are the exact value calculated through the two-point correlation matrices using Eq. (7). Solid curves correspond to the GHD prediction (54). The temperature of the baths L and R is βb = 1 (upper panels) and βb = 5 (lower panels). We take two different lengths for… view at source ↗
Figure 6
Figure 6. Figure 6: Particle density profile ϱ(x, t) in the tripartite quench of the tranverse-field Ising spin chain at several times t. Symbols represent the exact numerical values, while solid lines are the GHD prediction (85) obtained from the time-evolved local density of occupied modes (84). In the left panel, the initial temperature of subsystem S is βs = 0.4 and the semi-infinite subsystems L and R are at temperature … view at source ↗
Figure 7
Figure 7. Figure 7: Time evolution of log Tr(ρs(t)ρs(∞)) in the tripartite quench of the quantum Ising spin chain with h = 0.5 (ferromagnetic phase, upper panels) and h = 1.5 (param￾agnetic phase, lower panels). Symbols denote the exact numerical value, calculated from the two-point correlation matrix (79). Solid lines are the GHD prediction in Eq. (53), with n(k, x, t) now the local density of Bogoliubov modes. The semi-infi… view at source ↗
Figure 8
Figure 8. Figure 8: Normalised Frobenius distance between the time-evolved state of S and its stationary state in the tripartite quench in the Ising chain with magnetic field h = 0.5 (ferromagnetic phase). Symbols are the exact values calculated with Eq. (7). Solid curves are the GHD prediction (54), using the local density of Bogoliubov modes (84). The temperature of the baths is βn = 1 (upper panels) and βb = 5 (lower panel… view at source ↗
Figure 9
Figure 9. Figure 9: Same as [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
read the original abstract

We study thermalisation and the possible occurrence of the Mpemba effect in a closed quantum setting that mimics the interaction of a system with thermal reservoirs coupled only at its boundaries. Specifically, we consider a tripartite geometry in which 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. These outer chains act as thermal baths, while the full system evolves unitarily under a homogeneous Hamiltonian. This setup provides a simple quantum realisation of a temperature quench and closely resembles the original scenario in which the classical Mpemba effect was first observed. We focus on 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. As a probe of relaxation, we consider the Frobenius distance between the time-evolved reduced density matrix of the central subsystem and its stationary state, which is the thermal state at the bath temperature. Exploiting the free-fermionic structure of both models, the dynamics remains Gaussian and the Frobenius distance can be expressed exactly in terms of two-point correlation functions. Combining this representation with generalised hydrodynamics, we derive analytical predictions for the Frobenius distance in the hydrodynamic limit, providing a complete characterisation of the thermalisation process. Using these results, we investigate the possible occurrence of the Mpemba effect. We find that, despite the genuine non-equilibrium dynamics displayed by the system, no Mpemba effect arises in this setting. Our analysis identifies a broad class of boundary-driven thermalisation protocols in which relaxation is fully characterised analytically and exhibits no anomalous acceleration of equilibration.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper considers a closed tripartite free-fermion chain in which a finite central segment, prepared at temperature T1, is suddenly coupled at its boundaries to two semi-infinite chains held at temperature T2. The entire system evolves unitarily under a homogeneous Hamiltonian (XX or TFIM). The reduced state of the central segment is claimed to relax to the thermal Gibbs state at T2; the Frobenius distance to this state is expressed exactly via two-point correlators and analysed in the hydrodynamic limit with generalised hydrodynamics. The central result is that, despite genuine non-equilibrium dynamics, the Mpemba effect is absent for the class of boundary-driven protocols studied.

Significance. If the stationary-state identification and hydrodynamic analysis hold, the work supplies an analytically tractable, parameter-free example of quantum thermalisation in which relaxation is fully characterised by GHD applied to exact correlations and exhibits no anomalous acceleration. This identifies a broad class of integrable boundary-driven protocols free of the Mpemba effect and provides reproducible predictions that can be tested against exact numerics.

major comments (2)
  1. [Setup and stationary state] Setup section: the claim that the stationary reduced density matrix of the central subsystem is exactly the thermal Gibbs state at the bath temperature (rather than a GGE fixed by all conserved charges) is load-bearing for the Frobenius-distance analysis and the no-Mpemba conclusion. For the XX chain the U(1) charge and higher conserved quantities must match those of the target thermal state; the manuscript provides no explicit verification that the bath initial conditions enforce this matching for every charge.
  2. [Hydrodynamic limit and distance formula] Hydrodynamic analysis (around the GHD derivation of the distance): the hydrodynamic limit is invoked to obtain closed-form predictions, yet the manuscript does not state the precise scaling (e.g., t/L fixed, L o∞) nor demonstrate that sub-leading corrections to the two-point functions remain negligible for the distance observable at the times relevant to possible Mpemba crossings.
minor comments (2)
  1. [Probe of relaxation] Notation: the definition of the Frobenius distance should be written explicitly (e.g., as || ho(t)− ho_th||_F^2 = Tr[( ho(t)− ho_th)^2]) rather than left implicit when the distance is expressed via correlators.
  2. [Numerical comparisons] Figure captions: the hydrodynamic curves in the relaxation plots should indicate the system sizes used for the exact numerics against which they are compared.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify key aspects of the analysis. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Setup and stationary state] Setup section: the claim that the stationary reduced density matrix of the central subsystem is exactly the thermal Gibbs state at the bath temperature (rather than a GGE fixed by all conserved charges) is load-bearing for the Frobenius-distance analysis and the no-Mpemba conclusion. For the XX chain the U(1) charge and higher conserved quantities must match those of the target thermal state; the manuscript provides no explicit verification that the bath initial conditions enforce this matching for every charge.

    Authors: We agree that an explicit verification strengthens the presentation. In the setup, the semi-infinite baths are initialized in thermal states at temperature T2. For free-fermion models this fixes the occupation numbers of all quasiparticle modes (including the U(1) density and all higher conserved charges for the XX chain) to their thermal Fermi-Dirac values. Because the central segment is finite while the baths are infinite, the long-time reduced state is completely determined by the incoming quasiparticles whose distributions are thermal; hence the stationary reduced density matrix is the Gibbs state at T2 rather than a generic GGE. To make this matching explicit we will add, in the revised manuscript, a short calculation (or appendix) that evaluates the expectation values of the relevant conserved charges on the stationary state and confirms they coincide with those of the target thermal state. This addition does not alter the conclusions but improves clarity. revision: yes

  2. Referee: [Hydrodynamic limit and distance formula] Hydrodynamic analysis (around the GHD derivation of the distance): the hydrodynamic limit is invoked to obtain closed-form predictions, yet the manuscript does not state the precise scaling (e.g., t/L fixed, L→∞) nor demonstrate that sub-leading corrections to the two-point functions remain negligible for the distance observable at the times relevant to possible Mpemba crossings.

    Authors: We thank the referee for noting the need for a precise statement of the limit. The hydrodynamic predictions are derived in the Euler-scale limit L→∞ with t/L held fixed (standard for GHD). The Frobenius distance is an extensive observable obtained by integrating the two-point correlators over the central segment; in this scaling the leading hydrodynamic contribution dominates, while sub-leading corrections (arising from diffusive or higher-order terms) are O(1/L) or smaller and vanish uniformly on the timescales where Mpemba crossings could occur. We will revise the text to state the scaling explicitly and add a brief paragraph justifying the negligibility of corrections for the distance observable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies established GHD to free-fermion correlations

full rationale

The paper's central results follow from applying generalised hydrodynamics (an externally established framework) to the exact two-point correlation functions of the free-fermion models. The Frobenius distance is expressed directly from the Gaussian structure and its hydrodynamic evolution is computed analytically; the no-Mpemba conclusion is an output of those expressions rather than an input or fit. The stationary-state assumption (thermal Gibbs at bath temperature) is an interpretive premise of the setup, not a self-referential definition that forces the relaxation result. No load-bearing self-citations, fitted parameters renamed as predictions, or ansatz smuggling are present in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on the standard properties of free-fermion models and the applicability of generalised hydrodynamics in the hydrodynamic limit; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The dynamics of free-fermion models remain Gaussian under unitary evolution with quadratic Hamiltonians.
    Invoked when stating that the Frobenius distance can be expressed exactly in terms of two-point correlation functions.
  • domain assumption Generalised hydrodynamics accurately describes the relaxation of the Frobenius distance in the hydrodynamic limit for this geometry.
    Used to obtain analytical predictions for the thermalisation process.

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discussion (0)

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