Area-law-like systems with entangled states can preserve ergodicity

Abstract

We study the ground entangled state of the one-dimensional spin-1/2 Ising ferromagnet at its transverse-field critical point. When this problem is expressed in terms of independent fermions, we show that the usual thermostatistical sums emerging within Fermi-Dirac statistics can, for an L-sized subsystem, be indistinctively taken up to L terms or up to lnL terms, providing a neat understanding of the origin of the logarithmic scaling of the entanglement entropy in the system. This is interpreted as a compact occupancy of the phase-space of the L-subsystem, hence standard Boltzmann-Gibbs thermodynamics quantities with an effective system size V ∝ lnL are appropriate and are explicitly calculated. The calculations are then to be done in a Hilbert space whose effective dimension is 2^{ln L} instead of 2^L. In this we can assume ergodicity. Our analysis suggests a scenario where the physical systems are essentially grouped into three classes, in terms of their phase-space occupancy, ergodicity and Lebesgue measure.