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arXiv:2610.06818

The complete phase diagram of detailed-balanced Gibbs samplers for the mean-field Heisenberg model

Levente Raikovich, József Mák, Csaba Czabán, András Gilyén

quant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

We study the spectral gap of recently discovered families of KMS-detailed-balanced Gibbs-sampler Lindbladians for the mean-field Heisenberg model on $n$ 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 $1/\sqrt{n}$ at the critical temperature, and as $1/n$ 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.

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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.
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Preparing Gibbs states through dissipative dynamics requires controlling convergence for the chosen Hamiltonian HHH and coupling operators. However, efficient implementation requires convergence guarantees from the spectral gap of the generator of the dynamics, which remains a crucial challenge. Recent results address the challenge by restricting the dynamics with structural assumptions about the Hamiltonian and its interactions. For the Chen-Kastoryano-Gilyen (CKG) construction and arbitrary finite-dimensional Hamiltonians, we prove an explicit comparison that converts a spectral gap estimate for the infinite-temperature generator into a lower bound on the spectral gap of the CKG quantum Gibbs sampler at every finite positive inverse temperature βββ. It allows a single spectral gap estimate at infinite temperature to certify convergence across Hamiltonians and temperatures. Such an estimate is often known from a classical spectral gap estimate, such as the classical random walk spectral gap. For Hodge Laplacians of simplicial complexes as Hamiltonians, we construct coupling operators whose generator at infinite temperature has the same spectral gap as a classical simplicial down-up walk. A positive spectral gap of the walk and bounded spectral width W=λmax⁡(H)−λmin⁡(H)W=λ_{\max}(H)-λ_{\min}(H)W=λmax​(H)−λmin​(H) give a generator spectral gap independent of the number of simplices at every fixed temperature. We then establish conditions under which the sampler approximately prepares zero-energy (harmonic) states in polynomial evolution time. This provides a conditional convergence guarantee for the state preparation in the thermal approach to quantum topological data analysis.