Narrow Results

This paper discusses entropy production in nonequilibrium steady states for infinite quantum spin systems. Rigorous results have been obtained recently in this area, but a physical discussion shows that some questions of principle remain to be clarified.

We recover, expand, and unify quantum (and classical) large deviation results for lattice Gibbs states. The main new ingredient in this paper is a control on the overlap of spectral projections for non-commutative observables. Our proof of large deviations is based on Ruelle–Lanford functions [20, 34] which establishes the existence of a rate function directly by subadditivity arguments, as done in the classical case in [23, 32], instead of relying on Gärtner–Ellis theorem, and cluster expansion or transfer operators as done in the quantum case in [21, 13, 27, 22, 16, 28]. We assume that the Gibbs states are asymptotically decoupled [23, 32], which controls the dependence of observables localized at different spatial locations. In the companion paper [29], we discuss the characterization of rate functions in terms of relative entropies.

Product vacua with boundary states (PVBS) are cousins of the Heisenberg XXZ spin model and feature $n$ particle species on ${\mathbb{Z}}^d$. The PVBS models were originally introduced as toy models for the classification of ground state phases. A crucial ingredient for this classification is the existence of a spectral gap above the ground state sector. In this work, we derive a spectral gap for PVBS models at arbitrary species number $n$ and in arbitrary dimension $d$ in the perturbative regime of small anisotropy parameters. Instead of using the more common martingale method, the proof verifies a finite-size criterion in the spirit of Knabe.

We review our approach to the second law of thermodynamics as a theorem asserting the growth of the mean (Gibbs–von Neumann) entropy of quantum spin systems undergoing automorphic (unitary) adiabatic transformations. Non-automorphic interactions with the environment, although known to produce on the average a strict reduction of the entropy of systems with finite number of degrees of freedom, are proved to conserve the mean entropy on the average. The results depend crucially on two properties of the mean entropy, proved by Robinson and Ruelle for classical systems and Lanford and Robinson for quantum lattice systems: upper semicontinuity and affinity.

Establishing the (non)existence of a spectral gap above the ground state in the thermodynamic limit is one of the fundamental steps for characterizing the topological phase of a quantum lattice model. This is particularly challenging when a model is expected to have low-lying edge excitations, but nevertheless a positive bulk gap. We review the bulk gap strategy introduced in [S. Warzel and A. Young, The spectral gap of a fractional quantum Hall system on a thin torus, J. Math. Phys.63 (2022) 041901; S. Warzel and A. Young, A bulk spectral gap in the presence of edge states for a truncated pseudopotential, Ann. Henri Poincaré24 (2023) 133–178], while studying truncated Haldane pseudopotentials. This approach is able to avoid low-lying edge modes by separating the ground states and edge states into different invariant subspaces before applying spectral gap bounding techniques. The approach is stated in a general context, and we reformulate specific spectral gap methods in an invariant subspace context to illustrate the necessary conditions for combining them with the bulk gap strategy. We then review its application to a truncation of the 1/3-filled Haldane pseudopotential in the cylinder geometry.