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arXiv:1502.03263

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems

Fernando G. S. L. Brandão, M. Cramer

Published Feb 11, 2015• 35 citations
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Abstract

We consider the problem of whether the canonical and microcanonical ensembles are locally equivalent for short-ranged quantum Hamiltonians of $N$ spins arranged on a $d$-dimensional lattices. For any temperature for which the system has a finite correlation length, we prove that the canonical and microcanonical state are approximately equal on regions containing up to $O(N^{1/(d+1)})$ spins. The proof rests on a variant of the Berry--Esseen theorem for quantum lattice systems and ideas from quantum information theory.

DOI
10.48550/arxiv.1502.03263

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We consider the problem of whether the canonical and microcanonical ensembles are locally equivalent for short-ranged quantum Hamiltonians of NNN spins arranged on a ddd-dimensional lattices. For any temperature for which the system has a finite correlation length, we prove that the canonical and microcanonical state are approximately equal on regions containing up to O(N1/(d+1))O(N^{1/(d+1)})O(N1/(d+1)) spins. The proof rests on a variant of the Berry--Esseen theorem for quantum lattice systems and ideas from quantum information theory.
Ensemble inequivalence, where a system's thermodynamic properties depend on the statistical ensemble used to describe it, is a well-established phenomenon in classical systems with long-range interactions. Here, we present a detailed analysis of a quantum ferromagnet spin model that exhibits this same behavior. We find that the microcanonical and canonical phase diagrams are identical at zero temperature ( T = 0 ), but differ significantly at finite temperatures. Our results highlight a breakdown of ensemble equivalence in this quantum long-range model, contrasting with what is observed for short range. These findings have important implications for the study and manipulation of long-range interacting quantum platforms, such as those used in atomic, molecular, and optical physics.
It is expected that the statistical fluctuations of local observables in large quantum systems obey the central limit theorem, and approximate a normal distribution as their size grows. Here, we prove a version of the Berry-Esseen theorem for quantum lattice systems, which strengthens that central limit theorem by providing a rigorous convergence estimate towards the normal distribution for large but finite system size. Given a local quantum Hamiltonian on NNN particles and a quantum state with a finite correlation length, the result states that the measurement of local observables such as the energy follows a normal distribution, up to an error scaling as O(N−12polylog(N))\mathcal{O}\left(N^{-\frac{1}{2}} \text{polylog}(N)\right)O(N−21​polylog(N)), which is optimal up to logarithmic factors.
It is typically assumed, without justification, that a weak coupling between a system and a bath is a necessary condition for the equivalence of a canonical ensemble and a microcanonical ensemble. For instance, in a canonical ensemble, temperature emerges if the system and the bath are uncoupled or weakly coupled. We investigate the validity region of this weak-coupling approximation, using a coupled composite-spin system. Our results show that the spin coupling strength can be as large as the level spacing of the system, indicating that the weak-coupling approximation has a much wider region of validity than usually expected.
We review several properties of thermal states of spin Hamiltonians with short range interactions. In particular, we focus on those aspects in which the application of tools coming from quantum information theory has been specially successful in the recent years. This comprises the study of the correlations at finite and zero temperature, the stability against distant and/or weak perturbations, the locality of temperature and their classical simulatability. For the case of states with a finite correlation length, we review the results on their energy distribution and the equivalence of the canonical and microcanonical ensemble.
We study the problem of ensemble equivalence in spin systems with short-range interactions under the existence of a first-order phase transition. The spherical model with nonlinear nearest-neighbour interactions is solved exactly for both canonical and microcanonical ensembles. The result reveals apparent ensemble inequivalence at the first-order transition point in the sense that the microcanonical entropy is non-concave as a function of the energy and consequently the specific heat is negative. In order to resolve the paradox, we show that an unconventional saddle point should be chosen in the microcanonical calculation that represents a phase separation. The XY model with nonlinear interactions is also studied by microcanonical Monte Carlo simulations in two dimensions to see how this model behaves in comparison with the spherical model.
Due to the equivalence of the statistical ensembles, thermostatic properties of physical systems with short-range interactions can be calculated in different ensembles leading to the same physics. In particular, the ensemble equivalence holds for systems that undergo a continuous phase transition in the infinite volume limit with the result that the properties of the transition can also be investigated in the microcanonical approach. Considering as an example the spherical model, the ensemble equivalence is explicitly demonstrated by calculating the critical properties in the microcanonical ensemble and comparing them to the well known canonical results.