[Paper Review] Area law for the entanglement entropy of the free Fermi gas at nonzero temperature

Hajo Leschke, Alexander V. Sobolev+1 more|arXiv (Cornell University)|Jan 14, 2015

Quantum many-body systems16 references3 citations

TL;DR

This paper establishes the first rigorous asymptotic formula for the entanglement entropy (EE) of the free Fermi gas at nonzero temperature in infinite multidimensional space. It shows that the leading large-scale term of thermal EE is twice the finite-size correction to the thermal entropy density, which corresponds to the thermal entropy on the boundary surface, and reduces to a ln(1/T) singularity in the zero-temperature limit, recovering the known logarithmic correction to area law scaling.

ABSTRACT

The leading asymptotic large-scale behavior of the spatially bipartite entanglement entropy (EE) of the free Fermi gas infinitely extended in multidimensionsal Euclidean space at zero absolute temperature, T=0, is by now well understood. Here, we announce and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at T>0. The leading large-scale term of this thermal EE turns out to be twice the leading finite-size correction to the infinite-volume thermal entropy (density). Not surprisingly, this correction is just the thermal entropy on the boundary surface of the bipartition. However, it is given by a rather complicated analytical expression derived from semiclassical functional calculus and differs, at least at high temperature, from simpler expressions previously obtained by arguments based on conformal field theory. In the zero-temperature limit, the leading large-scale term of the thermal EE considerably simplifies and displays a ln(1/T)-singularity which one may identify with the known logarithmic correction at T=0 to the so-called area-law scaling. Our results extend to the whole one-parameter family of (quantum) R\'enyi entropies.

Motivation & Objective

• To rigorously characterize the large-scale asymptotic behavior of entanglement entropy in the free Fermi gas at nonzero temperature.
• To clarify the relationship between thermal entanglement entropy and finite-size corrections to the thermal entropy density.
• To extend the area law for entanglement entropy to finite temperatures and to the full family of Rényi entropies.
• To resolve discrepancies between semiclassical results and earlier conformal field theory-based approximations at high temperatures.

Proposed method

• Application of semiclassical functional calculus to derive the leading-order asymptotic behavior of the entanglement entropy in thermal equilibrium states.
• Analysis of the trace of the reduced density matrix using spectral projections and the Fermi-Dirac distribution.
• Derivation of the leading large-scale term of the entanglement entropy as twice the finite-size correction to the thermal entropy density.
• Use of the trace-class property of the reduced density matrix to ensure convergence and validity of the asymptotic expansion.
• Extension of results to the entire one-parameter family of Rényi entropies via functional calculus on the reduced density matrix.
• Asymptotic analysis in the zero-temperature limit to recover the known logarithmic correction to the area law.

Experimental results

Research questions

• RQ1 What is the leading large-scale asymptotic behavior of the entanglement entropy for the free Fermi gas at nonzero temperature?
• RQ2 How does the thermal entanglement entropy relate to the finite-size correction of the thermal entropy density?
• RQ3 Why does the high-temperature behavior of the thermal EE differ from predictions based on conformal field theory?
• RQ4 What is the zero-temperature limit of the thermal entanglement entropy, and how does it relate to the logarithmic correction in the area law?
• RQ5 How do the results generalize to the full family of Rényi entropies?

Key findings

• The leading large-scale term of the thermal entanglement entropy is twice the finite-size correction to the infinite-volume thermal entropy density.
• This correction corresponds to the thermal entropy localized on the boundary surface of the bipartition, derived via semiclassical functional calculus.
• At high temperatures, the expression for the thermal EE differs from simpler conformal field theory-based approximations.
• In the zero-temperature limit, the thermal EE exhibits a ln(1/T) singularity, which matches the known logarithmic correction to the area law.
• The results are extended to all Rényi entropies, confirming consistency across the full family of entropic measures.
• The derivation establishes a rigorous connection between entanglement entropy and boundary thermal effects in quantum many-body systems at finite temperature.