[Paper Review] Boundary-law scaling of entanglement entropy in diffusive metals

Andrew C. Potter|arXiv (Cornell University)|Aug 5, 2014

Quantum many-body systems13 references3 citations

TL;DR

This paper demonstrates that diffusive metals—disordered, non-interacting fermions with extended gapless excitations—exhibit only boundary-law scaling of entanglement entropy, unlike clean Fermi-liquids that show a logarithmic violation of boundary law. The absence of the log-correction reveals that the sharp Fermi-surface structure, not just a finite density of extended states, is essential for the log-divergent correction in entanglement entropy.

ABSTRACT

Entanglement structure serves as a powerful way to characterize quantum many-body phases. This is particularly so for gapless quantum liquids, where entanglement-based tools provide one of the only means to systematically characterize these complicated phases. For example, the Fermi-surface structure of Fermi-liquids is revealed in entanglement entropy by a log-correction to the typical boundary-law scaling of simpler quantum ground-states. In this paper, I analyze the entanglement structure of a disordered, but delocalized diffusive metal. Using a combination of analytic arguments and numerical calculations, I show that, despite having the same number of extended gapless excitations as a clean Fermi-liquid, the diffusive metal exhibits only boundary-law entanglement scaling. This result pinpoints the sharp Fermi-surface structure, rather than the finite density of gapless excitations, as the origin of the log-correction in the Fermi-liquid entanglement scaling.

Motivation & Objective

• To determine whether diffusive metals, despite hosting a Fermi-surface's worth of extended gapless excitations, exhibit the same logarithmic violation of boundary-law entanglement scaling as clean Fermi-liquids.
• To clarify whether the log-correction in entanglement entropy arises from the presence of extended gapless modes or from the sharp momentum-space Fermi-surface structure.
• To establish that the log-correction is not a generic feature of gapless, delocalized systems but is specifically tied to the existence of a well-defined Fermi surface.
• To provide analytic and numerical evidence for boundary-law entanglement scaling in disordered, diffusive metals using number fluctuation arguments and hydrodynamic models.
• To extend the analysis to a gapped semimetal with quadratic dispersion to further isolate the role of Fermi-surface structure in entanglement scaling.

Proposed method

• Using number fluctuations of electron density in a subsystem A, the paper derives that Renyi entropies with index α ≥ 2 follow boundary-law scaling, implying von Neumann entanglement entropy also follows boundary law via analytic continuation.
• Applying the relation σ²_N,A = Tr[M(1−M)] for fermionic correlation matrices M, the method links number variance to entanglement scaling in free fermion systems.
• Numerical computation of entanglement entropy in a non-interacting disordered fermion system confirms the boundary-law scaling predicted by number fluctuation arguments.
• A hydrodynamic description of the Fermi-liquid is used to explain the physical origin of the difference between ballistic and diffusive metals: disorder suppresses long-wavelength Fermi-surface correlations that generate the log-correction.
• A 2D quadratically dispersing semimetal is analyzed as a dual model, decomposed into transverse sub-channels with gapped 1D modes; entanglement is shown to sum to boundary-law scaling via universal scaling functions.
• Scaling arguments in the low-energy limit (z=2 dynamical scaling) are used to show that long-wavelength contributions to entanglement entropy scale at most as area law, with no logarithmic divergence.

Experimental results

Research questions

• RQ1 Does a diffusive metal with extended gapless excitations exhibit a logarithmic violation of boundary-law entanglement scaling, like a clean Fermi-liquid?
• RQ2 Is the log-correction in entanglement entropy a universal feature of systems with a finite density of extended gapless modes, or is it specifically tied to the sharp Fermi-surface structure?
• RQ3 Can number fluctuation scaling in a subsystem be used to infer entanglement entropy scaling in free fermion systems?
• RQ4 How does the hydrodynamic description of a Fermi-liquid explain the absence of the log-correction in diffusive metals?
• RQ5 Do gapped, quasi-1D sub-channels in a 2D semimetal with quadratic dispersion contribute logarithmically to entanglement entropy, or do they respect boundary-law scaling?

Key findings

• The entanglement entropy of a diffusive metal follows a strict boundary-law scaling, with no logarithmic violation, despite having a Fermi-surface's worth of extended gapless excitations.
• All Renyi entropies with index α ≥ 2 scale with the boundary area, strongly indicating that the von Neumann entropy also follows boundary-law scaling.
• Numerical simulations of a non-interacting disordered fermion system confirm the absence of log-corrections, supporting the analytic number fluctuation argument.
• The hydrodynamic picture shows that disorder destroys the long-wavelength Fermi-surface correlations responsible for the log-correction, even while preserving extended gapless modes.
• In a 2D quadratically dispersing semimetal, the entanglement entropy scales as an area law, with no logarithmic divergence, due to the gapped nature of all transverse sub-channels.
• The absence of log-correction in both the diffusive metal and the gapped semimetal demonstrates that the log-violation is not a consequence of extended gapless modes but specifically requires a sharp Fermi surface.