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 2ln L instead of 2L. 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.
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Souza, A.M.C., Rapčan, P. & Tsallis, C. Area-law-like systems with entangled states can preserve ergodicity. Eur. Phys. J. Spec. Top. 229, 759–772 (2020). https://doi.org/10.1140/epjst/e2020-900003-3
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DOI: https://doi.org/10.1140/epjst/e2020-900003-3