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Journal of High Energy Physics

, 2019:181 | Cite as

Subsystem trace distance in low-lying states of (1 + 1)-dimensional conformal field theories

  • Jiaju Zhang
  • Paola RuggieroEmail author
  • Pasquale Calabrese
Open Access
Regular Article - Theoretical Physics

Abstract

We report on a systematic replica approach to calculate the subsystem trace distance for a quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero and P. Calabrese, Phys. Rev. Lett.122 (2019) 141602], of which this work is a completion. The trace distance between two reduced density matrices ρA and σA is obtained from the moments tr(ρA− σA)n and taking the limit n → 1 of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of length embedded in the low lying eigenstates of a one-dimensional critical system of length L, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be a scale invariant universal function of ℓ/L. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schatten n-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers of ℓ/L. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results in two-dimensional CFT are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.

Keywords

Conformal Field Theory Field Theories in Lower Dimensions 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

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© The Author(s) 2019

Authors and Affiliations

  1. 1.SISSA and INFNTriesteItaly
  2. 2.International Centre for Theoretical Physics (ICTP)TriesteItaly

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