Abstract
This work is concerned with thermal quantum states of Hamiltonians on spin and fermionic lattice systems with short range interactions. We provide results leading to a local definition of temperature, thereby extending the notion of "intensivity of temperature" to interacting quantum models. More precisely, we derive a perturbation formula for thermal states. The influence of the perturbation is exactly given in terms of a generalized covariance. For this covariance, we prove exponential clustering of correlations above a universal critical temperature that upper bounds physical critical temperatures such as the Curie temperature. As a corollary, we obtain that above the critical temperature, thermal states are stable against distant Hamiltonian perturbations. Moreover, our results imply that above the critical temperature, local expectation values can be approximated efficiently in the error and the system size.
Highlights
The ongoing miniaturization of devices, with structures reaching the nanoscale, has lead to the development of extremely small thermometers [1,2], some of which are so small that they can only be read out with powerful electron microscopes [3]
In order to understand the working of such devices, it is necessary to formulate a theory of statistical mechanics and thermodynamics at the microscopic and mesoscopic scales
In the Appendix, we provide a detailed proof of two bounds on truncated cluster expansions, one of which is an important ingredient to the proof of clustering of correlations
Summary
The ongoing miniaturization of devices, with structures reaching the nanoscale, has lead to the development of extremely small thermometers [1,2], some of which are so small that they can only be read out with powerful electron microscopes [3]. Our rigorous results help to delineate the boundary between these two regimes They build upon and go significantly beyond previous results on the clustering of correlations in classical systems [11], for quantum gases [12], i.e., translation-invariant Hamiltonians in the continuum, and cubic lattices [13,14,15]. If a Hamiltonian has a unique ground state and is gapped, correlations in its ground state cluster exponentially and faraway regions become essentially uncorrelated This clustering of correlations is rigorously proven using information-theory-inspired methods such as Lieb-Robinson bounds and quasiadiabatic continuation [27,28,29]. These rigorous results allow for certified algorithms that efficiently approximate ground states of gapped Hamiltonians on classical computers [30]. In the Appendix, we provide a detailed proof of two bounds on truncated cluster expansions, one of which is an important ingredient to the proof of clustering of correlations
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