The partial transpose ρT2A of the reduced density matrix ρA 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 ρT2A, 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 ρT2A. 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.

Negativity spectrum of one-dimensional conformal field theories

Ruggiero, Paola;Alba, Vincenzo;Calabrese, Pasquale
2016

Abstract

The partial transpose ρT2A of the reduced density matrix ρA 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 ρT2A, 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 ρT2A. 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.
Ruggiero, Paola; Alba, Vincenzo; Calabrese, Pasquale
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11568/1140881
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