Eur. Phys. J. Special Topics 229, 759–772 (2020)
https://doi.org/10.1140/epjst/e2020-900003-3

Regular Article

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

¹ Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil
² Departamento de Fisica, Universidade Federal de Sergipe, 49100-000 Sao Cristovao, Brazil
³ Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 841 04 Bratislava, Slovak Republic
⁴ Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
⁵ Complexity Science Hub Vienna, Josefstadter Strasse 39, 1080 Vienna, Austria

^a e-mail: amcsouza@ufs.br

Received: 11 January 2019
Received in final form: 17 April 2019
Published online: 12 March 2020

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.