Capacity of entanglement and distribution of density matrix eigenvalues in gapless systems

TL;DR

This paper introduces a method to analyze the capacity of entanglement in gapless systems using the eigenvalue distribution of the reduced density matrix, providing insights into low-energy excitations and non-Fermi-liquid behavior.

Contribution

The authors derive an analytical formula for the eigenvalue distribution of the RDM based on COE scaling and validate it through numerical tests on relativistic bosons and Landau level states.

Findings

Distribution function detects COE scaling more efficiently than raw data.

In the Landau level system, the distribution suggests non-Fermi-liquid behavior.

The method offers a new way to probe gapless excitations in quantum systems.

Abstract

We propose that the properties of the capacity of entanglement (COE) in gapless systems can efficiently be investigated through the use of the distribution of eigenvalues of the reduced density matrix (RDM). The COE is defined as the fictitious heat capacity calculated from the entanglement spectrum. Its dependence on the fictitious temperature can reflect the low-temperature behavior of the physical heat capacity, and thus provide a useful probe of gapless bulk or edge excitations of the system. Assuming a power-law scaling of the COE with an exponent $\alpha$ at low fictitious temperatures, we derive an analytical formula for the distribution function of the RDM eigenvalues. We numerically test the effectiveness of the formula in relativistic free scalar boson in two spatial dimensions, and find that the distribution function can detect the expected $\alpha=3$ scaling of the COE much more efficiently than the raw data of the COE. We also calculate the distribution function in the ground state of the half-filled Landau level with short-range interactions, and find a better agreement with the $\alpha=2/3$ formula than with the $\alpha=1$ one, which indicates a non-Fermi-liquid nature of the system.