Breakdown of thermalization in quantum spin systems with long-range interactions

Abstract

Quantum thermalization of closed systems -- i.e. the relaxation of local properties to their thermodynamic value along the Hamiltonian evolution -- is generically expected in non-integrable systems at finite energy density. In this work we show that quantum spin systems with at least one spin component coupled through long-range ferromagnetic interactions, and a second spin component coupled to a field, cannot in general thermalize to a paramagnetic state whenever they have a finite, negative energy per spin. The long-range interactions we consider decay as a power law of the distance, $\sim r^{-α}$ with $α< d$, where $d$ is the number of spatial dimensions of the system. The lack of thermalization is generic to an infinite family of Hamiltonians, and it is not associated to any form of integrability. It can instead be understood by the anomalous features of the thermodynamics of the system in the paramagnetic phase, which cannot be reflected in the properties of the eigenstates. This leads to a breakdown of eigenstate typicality and of the eigenstate thermalization hypothesis. Interestingly, the non-thermal states emerging at long times in the dynamics display long-range multipartite entanglement, elevating the breakdown of thermalization to a resource.

The paper asks whether nonintegrable quantum spin systems with nonadditive, Kac-normalized ferromagnetic interactions $r^{-\alpha}$, $\alpha<d$, can thermalize to a paramagnetic Gibbs state at finite negative energy density. It proposes a mechanism for ETH violation: transverse interaction energy is subextensive in thermal paramagnetic states, while atypical eigenstates retain extensive transverse interaction energy and long-range correlations. The numerical study concentrates on a one-dimensional spin-$1/2$ XX model with $\alpha=0.5$ and a field along $x$. Quantum Monte Carlo finds thermal transverse correlations approximately scaling as $C_y(r)=N^{-1}g(r/N)$. Exact diagonalization up to $N=22$, within a resolved symmetry sector, finds broadly distributed eigenstate expectation values despite level repulsion compatible with Wigner–Dyson statistics. Eigenstate fluctuations show little size dependence at negative energies, unlike the short-range comparison at $\alpha=3$. Threshold-based counts suggest exponentially many atypical eigenstates, although their fraction tends to zero. Quenches from a coherent spin state opposite to the field are studied using matrix-product states up to $N=48$, time-dependent variational Monte Carlo up to $N=100$, and rotor-plus-spin-wave theory. At $\Omega/\mathcal J=0.3$, the simulated magnetization remains opposite to the field and transverse fluctuations reach $\langle(J^y)^2\rangle=O(N^2)$, implying extensive quantum Fisher information density for these pure states. The general argument is supported numerically primarily for this model. Canonical–microcanonical equivalence is inferred rather than proved. The dynamical interpretation assumes dilute, stable spin waves; the supplemental analysis finds decreasing excitation density with increasing size.