Entanglement negativity and conformal field theory: a Monte Carlo study

TL;DR

This study develops a Monte Carlo numerical scheme to analyze the moments of the partially transposed reduced density matrix in critical quantum spin chains, confirming conformal field theory predictions for entanglement measures.

Contribution

It introduces an efficient Monte Carlo method for calculating moments of  ho_A^{T_2} and compares numerical results with conformal field theory predictions in different spin chain models.

Findings

Monte Carlo results agree with CFT for adjacent blocks in both models.

Finite size corrections are significant for disjoint blocks in the Ising chain.

Scaling corrections differ between the Ising and Heisenberg chains.

Abstract

We investigate the behavior of the moments of the partially transposed reduced density matrix \rho^{T_2}_A in critical quantum spin chains. Given subsystem A as union of two blocks, this is the (matrix) transposed of \rho_A with respect to the degrees of freedom of one of the two. This is also the main ingredient for constructing the logarithmic negativity. We provide a new numerical scheme for calculating efficiently all the moments of \rho_A^{T_2} using classical Monte Carlo simulations. In particular we study several combinations of the moments which are scale invariant at a critical point. Their behavior is fully characterized in both the critical Ising and the anisotropic Heisenberg XXZ chains. For two adjacent blocks we find, in both models, full agreement with recent CFT calculations. For disjoint ones, in the Ising chain finite size corrections are non negligible. We demonstrate that their exponent is the same governing the unusual scaling corrections of the mutual information between the two blocks. Monte Carlo data fully match the theoretical CFT prediction only in the asymptotic limit of infinite intervals. Oppositely, in the Heisenberg chain scaling corrections are smaller and, already at finite (moderately large) block sizes, Monte Carlo data are in excellent agreement with the asymptotic CFT result.