Entropic Fluctuations in XY Chains and Reflectionless Jacobi Matrices

AbstractWe study entropic functionals/fluctuations of the XY chain with Hamiltonian $$\begin{array}{ll} \frac{1}{2} \sum\limits_{x \in \mathbb{Z}}J_x( \sigma_x^{(1)} \sigma_{x+1}^{(1)} +\sigma_x^{(2)} \sigma_{x+1}^{(2)}) + \lambda_x \sigma_x^{(3)}\end{array}$$where initially the left (x ≤ 0)/right (x > 0) part of the chain is in thermal equilibrium at inverse temperature βl/βr. The temperature differential results in a non-trivial energy/entropy flux across the chain. The Evans–Searles (ES) entropic functional describes fluctuations of the flux observable with respect to the initial state while the Gallavotti–Cohen (GC) functional describes these fluctuations with respect to the steady state (NESS) the chain reaches in the large time limit. We also consider the full counting statistics (FCS) of the energy/entropy flux associated with a repeated measurement protocol, the variational entropic functional (VAR) that arises as the quantization of the variational characterization of the classical Evans–Searles functional and a natural class of entropic functionals that interpolate between FCS and VAR. We compute these functionals in closed form in terms of the scattering data of the Jacobi matrix hux = Jxux+1 + λxux + Jx−1ux−1 canonically associated with the XY chain. We show that all these functionals are identical if and only if h is reflectionless (we call this phenomenon entropic identity). If h is not reflectionless, then the ES and GC functionals remain equal but differ from the FCS, VAR and interpolating functionals. Furthermore, in the non-reflectionless case, the ES/GC functional does not vanish at α = 1 (i.e., the Kawasaki identity fails) and does not have the celebrated α ↔ 1 − α symmetry. The FCS, VAR and interpolating functionals always have this symmetry. In the Schrödinger case, where Jx = J for all x, the entropic identity leads to some unexpected open problems in the spectral theory of one-dimensional discrete Schrödinger operators.