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arXiv:2610.01987

On the Metastability of the Mean-Field Interchange Model for Local Dimension $\geq 3$

Sergio Escobar, Lin Lin, Michael Ragone, Kevin D. Stubbs

quant-phmath-phSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations• Revised Oct 5, 2026 (v2)
v1 · Oct 1, 2026v2 · Oct 5, 2026
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Abstract

Many quantum systems are believed to thermalize slowly at low temperatures due to the existence of metastable phases. However, there are cases where further cooling can restore polynomial-time thermalization: we demonstrate that the Davies dynamics of the mean-field interchange model with local dimension $d\geq 3$ and single-site generalized Pauli couplings exhibits this behavior. The dynamical phase diagram is beholden to two distinguished inverse temperatures, $β_{\mathrm{min}}(d)$ and $β=d$, with the static first-order phase transition lying between them. At low and high temperatures surrounding the interval $(β_{\mathrm{min}}(d),d)$, the spectral gap of the Davies generator is bounded below by an inverse polynomial in system size $n$. Meanwhile, in this interval, the gap vanishes exponentially in $n$, indicating metastability. The proof uses Schur-Weyl duality to decompose the Davies generator into a classical Markov generator on Young diagrams, and the restriction to the remaining symmetry sectors whose gap is $Ω(n^{-1})$. Thus, exponential slowdown arises entirely from the Young diagram chain.

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49 pages, 3 figures

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On the Metastability of the Mean-Field Interchange Model for Local Dimension ≥3\geq 3≥3
Many quantum systems are believed to thermalize slowly at low temperatures due to the existence of metastable phases. However, there are cases where further cooling can restore polynomial-time thermalization: we demonstrate that the Davies dynamics of the mean-field interchange model with local dimension d≥3d\geq 3d≥3 and single-site generalized Pauli couplings exhibits this behavior. The dynamical phase diagram is beholden to two distinguished inverse temperatures, βmin(d)β_{\mathrm{min}}(d)βmin​(d) and β=dβ=dβ=d, with the static first-order phase transition lying between them. At low and high temperatures surrounding the interval (βmin(d),d)(β_{\mathrm{min}}(d),d)(βmin​(d),d), the spectral gap of the Davies generator is bounded below by an inverse polynomial in system size nnn. Meanwhile, in this interval, the gap vanishes exponentially in nnn, indicating metastability. The proof uses Schur-Weyl duality to decompose the Davies generator into a classical Markov generator on Young diagrams, and the restriction to the remaining symmetry sectors whose gap is Ω(n−1)Ω(n^{-1})Ω(n−1). Thus, exponential slowdown arises entirely from the Young diagram chain.
The mean-field Heisenberg ferromagnet is a quantum spin model on the complete graph with isotropic spin-1/2 interactions. This non-commuting Hamiltonian is permutation and SU(2)\mathsf{SU}(2)SU(2) invariant, and its Gibbs states undergo an SU(2)\mathsf{SU}(2)SU(2) symmetry breaking phase transition at inverse temperature β=2β=2β=2. We consider the associated Davies generator, a canonical model of open-system thermalization, and prove tight asymptotic estimates for its spectral gap at all noncritical temperatures. For fixed β<2β<2β<2, the gap as a function of number of qubits nnn is Θ(1)Θ(1)Θ(1), while for fixed β>2β>2β>2 the gap is Θ(n−1)Θ(n^{-1})Θ(n−1). The matching upper bound of the spectral gap is witnessed by the total magnetization order parameter, suggesting that the low-temperature (β>2β>2β>2) slowdown is associated with broken continuous symmetry. Two key ingredients in our approach are a comparison argument, which introduces auxiliary generators to bound dissipation on nontrivial representations of the symmetry groups SU(2)\mathsf{SU}(2)SU(2) and Sn\mathsf{S}_nSn​, and a decomposition of the space of observables into spherical tensor operators to reveal a form of monotonicity.
Many physical phenomena, including thermalization in open quantum systems and quantum Gibbs sampling, are modeled by Lindbladians approximating a system weakly coupled to a bath. Understanding the convergence speed of these Lindbladians to their steady states is crucial for bounding algorithmic runtimes and thermalization timescales. We study two such families of processes: one characterizing a repeated-interaction Gibbs sampling algorithm, and another modeling open many-body quantum thermalization. We prove that both converge in polynomial time for several non-commuting systems, including high-temperature local lattices, weakly interacting fermions, and 1D spin chains. These results demonstrate that simple dissipative quantum algorithms can prepare complex Gibbs states and that Lindblad dynamics accurately capture thermal relaxation. Our proofs rely on a novel technical result that extrapolates spectral gap lower bounds from quasi-local Lindbladians to the non-local generators governing these dynamics.
Abstract Quantum systems typically reach thermal equilibrium rather quickly when coupled to a thermal environment. The usual way of bounding the speed of this process is by estimating the spectral gap of the dissipative generator. However the gap, by itself, does not always yield a reasonable estimate for the thermalization time in many-body systems: without further structure, a uniform lower bound on it only constrains the thermalization time to grow polynomially with system size. Here, instead, we show that for a large class of geometrically-2-local models of Davies generators with commuting Hamiltonians, the thermalization time is much shorter than one would naïvely estimate from the gap: at most logarithmic in the system size. This yields the so-called rapid mixing of dissipative dynamics. The result is particularly relevant for 1D systems, for which we prove rapid thermalization with a system size independent decay rate only from a positive gap in the generator. We also prove that systems in hypercubic lattices of any dimension, and exponential graphs, such as trees, have rapid mixing at high enough temperatures. We do this by introducing a novel notion of clustering which we call “strong local indistinguishability” based on a max-relative entropy, and then proving that it implies a lower bound on the modified logarithmic Sobolev inequality (MLSI) for nearest neighbour commuting models. This has consequences for the rate of thermalization towards Gibbs states, and also for their relevant Wasserstein distances and transportation cost inequalities. Along the way, we show that several measures of decay of correlations on Gibbs states of commuting Hamiltonians are equivalent, a result of independent interest. At the technical level, we also show a direct relation between properties of Davies and Schmidt dynamics, that allows to transfer results of thermalization between both.
We construct lower bounds to the spectral gap of a family of Lindblad generators known as Davies maps. These maps describe the thermalization of quantum systems weakly coupled to a heat bath. The steady state of these systems is given by the Gibbs distribution with respect to the system Hamiltonian. The bounds can be evaluated explicitly, when the eigenbasis and the spectrum of the Hamiltonian is known. A crucial assumption is that the spectrum of the Hamiltonian is non-degenerate. Furthermore, we provide a counterexample to the conjecture, that the convergence rate is always determined by the gap of the associated Pauli master equation. We conclude that the full dynamics of the Lindblad generator has to be considered. Finally, we present several physical example systems for which the bound to the spectral gap is evaluated.
We study the spectral gap of recently discovered families of KMS-detailed-balanced Gibbs-sampler Lindbladians for the mean-field Heisenberg model on nnn spins. We show that the spectral gaps of these natural families of Lindbladians admit the same phase diagram: the gap is constant above the critical temperature (i.e., in the disordered phase), scales as 1/n1/\sqrt{n}1/n​ at the critical temperature, and as 1/n1/n1/n below the critical temperature (i.e., in the ordered phase). Our result builds on and extends the recent work of Basso et al.~(2026) which characterized this phase diagram for the GNS-detailed-balanced Davies generator, but only partially resolved the gap scaling at the critical temperature.
We present rigorous bounds on the thermalization time of the family of quantum mechanical spin systems known as stabilizer Hamiltonians. The thermalizing dynamics are modeled by a Davies master equation that arises from a weak local coupling of the system to a large thermal bath. Two temperature regimes are considered. First we clarify how in the low temperature regime, the thermalization time is governed by a generalization of the energy barrier between orthogonal ground states. When no energy barrier is present the Hamiltonian thermalizes in a time that is at most quadratic in the system size. Secondly, we show that above a universal critical temperature, every stabilizer Hamiltonian relaxes to its unique thermal state in a time which scales at most linearly in the size of the system. We provide an explicit lower bound on the critical temperature. Finally, we discuss the implications of these result for the problem of self-correcting quantum memories with stabilizer Hamiltonians.