Entanglement can preserve the compact nature of the phase-space occupancy

TL;DR

This paper investigates how entanglement in a critical quantum spin chain leads to a compact phase-space occupancy, allowing standard thermodynamic quantities to be effectively computed with a reduced phase-space dimension.

Contribution

It demonstrates that entanglement causes a phase-space occupancy that can be effectively described by a logarithmic scale, simplifying the calculation of thermodynamic quantities in critical quantum systems.

Findings

Standard BG entropy scales as log L for large subsystems.

Effective phase-space dimension reduces to 2^{log L} instead of 2^L.

Thermodynamic sums can be approximated using a logarithmic effective length.

Abstract

We study the one-dimensional transverse-field spin-1/2 Ising ferromagnet at its critical point. We consider an $L$-sized subsystem of a $N$-sized ring, and trace over the states of $(N-L)$ spins, with $N\to\infty$. The full $N$-system is in a pure state, but the $L$-system is in a statistical mixture. As well known, for $L >>1$, the Boltzmann-Gibbs-von Neumann entropy violates thermodynamical extensivity, namely $S_{BG}(L) \propto \log L$, whereas the nonadditive entropy $S_q$ is extensive for $q=q_c=\sqrt{37}-6 $, namely $S_{q_c}(L) \propto L$. When this problem is expressed in terms of independent fermions, we show that the usual thermostatistical sums emerging within Fermi-Dirac statistics can, for $L>>1$, be indistinctively taken up to $L$ terms or up to $\log L$ terms. This is interpreted as a compact occupancy of phase-space of the $L$-system, hence standard BG quantities with an effective length $V \equiv \log L$ are appropriate and are explicitly calculated. In other words, the calculations are to be done in a phase-space whose effective dimension is $2^{\log L}$ instead of $2^L$. The whole scenario is strongly reminiscent of a usual phase transition of a spin-1/2 $d$-dimensional system, where the phase-space dimension is $2^{L^d}$ in the disordered phase, and effectively $2^{L^d/2}$ in the ordered one.