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Finite temperature entanglement negativity in conformal field theory (1408.3043v1)

Published 13 Aug 2014 in cond-mat.stat-mech and hep-th

Abstract: We consider the logarithmic negativity of a finite interval embedded in an infinite one dimensional system at finite temperature. We focus on conformal invariant systems and we show that the naive approach based on the calculation of a two-point function of twist fields in a cylindrical geometry yields a wrong result. The correct result is obtained through a four-point function of twist fields in which two auxiliary fields are inserted far away from the interval, and they are sent to infinity only after having taken the replica limit. In this way, we find a universal scaling form for the finite temperature negativity which depends on the full operator content of the theory and not only on the central charge. In the limit of low and high temperatures, the expansion of this universal form can be obtained by means of the operator product expansion. We check our results against exact numerical computations for the critical harmonic chain.

Citations (178)

Summary

  • The paper derives a universal scaling form for finite temperature negativity in CFT, showing its dependence on the full operator content rather than just the central charge.
  • It employs an advanced four-point function approach with auxiliary fields to overcome the limitations of conventional two-point twist field methods at finite temperatures.
  • Numerical analysis on the critical harmonic chain confirms the theoretical predictions and highlights the nuanced impact of thermal fluctuations on quantum entanglement.

Finite Temperature Entanglement Negativity in Conformal Field Theory

The paper examines finite temperature entanglement negativity in conformal field theory (CFT), focusing on a finite interval within an infinite one-dimensional system. This paper significantly extends the theoretical understanding of quantum entanglement in CFT by addressing a gap between zero-temperature results and entanglement characterizations at finite temperatures.

The calculation of entanglement negativity, an entanglement measure suitable for mixed states, is approached via replica techniques commonly used in CFT to obtain entanglement entropies. The authors highlight the inadequacy of the seemingly intuitive approach using two-point functions of twist fields for capturing the negativity at finite temperatures. Instead, they adopt a more intricate four-point function method, incorporating additional auxiliary fields, to properly compute the negativity. Here, the evaluation of these four-point functions aligns with similar techniques used to address the crossover between quantum and classical domains in CFTs.

One notable result of this paper is the realization that entanglement negativity at finite temperatures depends on more than just the central charge of the theory, contrasting with zero-temperature entanglement entropy which does. The paper emphasizes this as an essential aspect differentiating finite temperature behaviors from the more traditional approaches focused on simpler measures like the Rènyi entropy of order 1/2.

The analytical developments in the paper include deriving the universal scaling form of finite temperature negativity, consisting of several components: a logarithmic term, an exponential term, and a non-trivial function dependent on the operator content of the theory. This scaling form remains valid across different thermal regimes, making it applicable to both low and high-temperature situations. Notably, the scaling function encapsulates the full operator content beyond just the central charge, suggesting that entanglement negativity might reveal deeper insights into the structure of conformal field theories than previously known measures.

Verification of these finite temperature results is provided through numerical computations applied to the critical harmonic chain, a model known for its exact solvability and conformal invariance in the continuum limit. These numerical results corroborate the theoretical predictions, particularly regarding the negative influence of finite temperature on entanglement, validating both the universal scaling form derived and the subtleties associated with handling boundary conditions and finite-size effects.

In conclusion, this paper advances the theoretical understanding of entanglement in quantum systems at finite temperatures, revealing new complexities within CFT that go beyond central charge dependencies. It paves the way for future research to explore the complex interplay between quantum and thermal fluctuations, potentially impacting numerous fields that rely on precise characterizations of quantum entanglement. Researchers in quantum information theory, statistical mechanics, and related disciplines will find the approaches and results conveyed in this paper to be of substantial significance, opening up new avenues for investigating entanglement phenomena in complex quantum systems.