Entropic fluctuations of XY quantum spin chains

TL;DR

This paper analyzes entropy fluctuations in an XY quantum spin chain with reservoirs at different temperatures, using scattering data and Jacobi matrices, revealing conditions for the equivalence of various entropic functionals in non-equilibrium states.

Contribution

It introduces a detailed analysis of entropic fluctuations in XY chains, connecting quantum statistical mechanics with scattering theory and identifying conditions for functional equivalence.

Findings

Gallavotti-Cohen and Evans-Searles functionals coincide in the large time limit.

All entropic functionals are equal if and only if the Jacobi matrix is reflectionless.

Explicit computation of entropic functionals using scattering data.

Abstract

We consider an XY quantum spin chain that consists of a left, center and right part initially at thermal equilibrium at temperatures $T_l$, $T_c$, and $T_r$, respectively. The left and right systems are infinitely extended thermal reservoirs and the central system is a small quantum system linking these two reservoirs. If there is a temperature differential, then heat and entropy will flow from one part of the chain to the other. We consider the Evans-Searles and Gallavotti-Cohen functionals which describe the fluctuations of this flux with respect to the initial state of the system and the non-equilibrium steady state reached by the system in the large time limit. We also define the full counting statistics for the XY chain and consider the associated entropic functional, as well a natural class of functionals that interpolate between the full counting statistics functional and the direct quantization of the variational characterization of the Evans-Searles functional which appears in classical non-equilibrium statistical mechanics. The Jordan-Wigner transformation associates a free Fermi gas and Jacobi matrix to our XY chain. Using this representation we are able to compute the entropic functionals in the large time limit in terms of the scattering data of the underlying Jacobi matrix. We show that the Gallavotti-Cohen and Evans-Searles functionals are identical in this limit. Furthermore, we show that all of these entropic functionals are equal in the large time limit if and only if the underlying Jacobi matrix is reflectionless.