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 $\mathsf{SU}(2)$ invariant, and its Gibbs states undergo an $\mathsf{SU}(2)$ symmetry breaking phase transition at inverse temperature $\beta=2$. 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 $\beta<2$, the gap as a function of number of qubits $n$ is $\Theta(1)$, while for fixed $\beta>2$ the gap is $\Theta(n^{-1})$. The matching upper bound of the spectral gap is witnessed by the total magnetization order parameter, suggesting that the low-temperature ($\beta>2$) 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 $\mathsf{SU}(2)$ and $\mathsf{S}_n$, and a decomposition of the space of observables into spherical tensor operators to reveal a form of monotonicity.
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...
S. Escobar, Lin Lin, Michael Ragone et al.· 0 citations
We prove the leading low-temperature free-energy asymptotics of the nearest-neighbour quantum Heisenberg ferromagnet on $\mathbb Z^2$. For every fixed $S\in\{\frac12,1,\frac32,\ldots\}$, the free energy per site satisfies \[ \lim_{\beta\to\infty}\beta^2S f_2(\beta,S)=-\frac{\pi}{24}. \] This formula was predicted by Ta...
We study the structure of Gibbs states in weakly perturbed interacting fermionic systems. First, for a sparse Hamiltonian $H=H_0+V$ with a quadratic term $H_0$ and a non-quadratic perturbation $V$ of scale $\epsilon$, we show that the Gibbs state $\rho_{\beta}$ decomposes into a convex combination of Gaussian states wh...
We prove that for the spin-1/2 Heisenberg XXZ chain in the range $\Delta\in[-1,1/2]$, the ground state on the torus of length $L$ converges to an infinite volume ground state $\langle\cdot\rangle$ as $L\to\infty$, and that the spin-spin correlation $\langle S_0^{(1)}S_x^{(1)}\rangle$ decays polynomially fast in $x$. In...
We prove lower bounds on the relaxation time $1/\gamma_{N,\beta}$ of Langevin dynamics for mixed spherical spin glasses with even mixture $\xi(x)=\sum_{p\ge4}\gamma_p^2x^p$, under a one-step-replica-symmetry-breaking-type standing assumption of strict threshold separation $E_0(\xi)>E_1(\xi)>E_2(\xi)$; the pure spherica...
We prove that for every fixed inverse temperature $\beta<1 / 2$, with high probability over the disorder, the single-site Glauber dynamics for the $n$-spin Sherrington-Kirkpatrick model mixes from every initial configuration to within total variation distance $\varepsilon$ in $O_{\beta}\left(n \log\left(n / \varepsilon...
Si-Hang Wang· 2 citations
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