Negativity spectrum of one-dimensional conformal field theories

TL;DR

This paper analytically derives the universal negativity spectrum for one-dimensional conformal field theories, revealing its dependence on the central charge and the state purity, and validates findings with numerical simulations.

Contribution

It provides the first analytical form of the negativity spectrum in 1D CFTs, highlighting its universality and dependence on the central charge, and compares results with numerical models.

Findings

Negativity spectrum depends only on the central charge of the CFT.

The spectrum's edge eigenvalues show strong dependence on eigenvalue sign.

Good agreement between analytical results and numerical simulations, with some finite-size corrections.

Abstract

The partial transpose $\rho_A^{T_2}$ of the reduced density matrix $\rho_A$ is the key object to quantify the entanglement in mixed states, in particular through the presence of negative eigenvalues in its spectrum. Here we derive analytically the distribution of the eigenvalues of $\rho_A^{T_2}$, that we dub negativity spectrum, in the ground sate of gapless one-dimensional systems described by a Conformal Field Theory (CFT), focusing on the case of two adjacent intervals. We show that the negativity spectrum is universal and depends only on the central charge of the CFT, similarly to the entanglement spectrum. The precise form of the negativity spectrum depends on whether the two intervals are in a pure or mixed state, and in both cases, a dependence on the sign of the eigenvalues is found. This dependence is weak for bulk eigenvalues, whereas it is strong at the spectrum edges. We also investigate the scaling of the smallest (negative) and largest (positive) eigenvalues of $\rho_A^{T_2}$. We check our results against DMRG simulations for the critical Ising and Heisenberg chains, and against exact results for the harmonic chain, finding good agreement for the spectrum, but showing that the smallest eigenvalue is affected by very large scaling corrections.