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Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

by Jiaju Zhang, Pasquale Calabrese, Marcello Dalmonte, M. A. Rajabpour

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Submission summary

Authors (as registered SciPost users): Marcello Dalmonte · Mohammad Ali Rajabpour · Jiaju Zhang
Submission information
Preprint Link:  (pdf)
Date accepted: 2020-04-28
Date submitted: 2020-04-22 02:00
Submitted by: Zhang, Jiaju
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


We carry out a comprehensive comparison between the exact modular Hamiltonian and the lattice version of the Bisognano-Wichmann (BW) one in one-dimensional critical quantum spin chains. As a warm-up, we first illustrate how the trace distance provides a more informative mean of comparison between reduced density matrices when compared to any other Schatten $n$-distance, normalized or not. In particular, as noticed in earlier works, it provides a way to bound other correlation functions in a precise manner, i.e., providing both lower and upper bounds. Additionally, we show that two close reduced density matrices, i.e. with zero trace distance for large sizes, can have very different modular Hamiltonians. This means that, in terms of describing how two states are close to each other, it is more informative to compare their reduced density matrices rather than the corresponding modular Hamiltonians. After setting this framework, we consider the ground states for infinite and periodic XX spin chain and critical Ising chain. We provide robust numerical evidence that the trace distance between the lattice BW reduced density matrix and the exact one goes to zero as $\ell^{-2}$ for large length of the interval $\ell$. This provides strong constraints on the difference between the corresponding entanglement entropies and correlation functions. Our results indicate that discretized BW reduced density matrices reproduce exact entanglement entropies and correlation functions of local operators in the limit of large subsystem sizes. Finally, we show that the BW reduced density matrices fall short of reproducing the exact behavior of the logarithmic emptiness formation probability in the ground state of the XX spin chain.

Author comments upon resubmission

We thank both referees for positive judgment on our paper. We took into account the comments by the first referee and made a few minor changes that we summarize below.

List of changes

Answer 1)
We added a few sentences and equations on page 8 and explained the BW theorem.
Answer 2)
The computation of fidelity and even-Schatten distances do not require the diagonalization of the difference between reduced density matrices. They are just traces of sums of products of Gaussian operators that can be calculated using the standard Gaussian algebra. Conversely, to the best of our knowledge, the trace distance and odd-Schatten ones can be calculated only by explicitly diagonalizing the difference between reduced density matrices. We added equations (46) and (47) on page 9 to make this clear.
Answer 3)
We added a sentence (after equation (52)) about a fit for these exponents that should be taken very carefully.
Answer 4)
The dips are due to the change in sign of the differences between BW and real correlations. We do not believe there is any relevant physics behind this change of sign. In the new version, we added a sentence (page 13) stressing the existence of the dip.

Published as SciPost Phys. Core 2, 007 (2020)

Reports on this Submission

Anonymous Report 2 on 2020-4-23 (Invited Report)


I have checked the response and the revised version of the paper; I am completely satisfied with the present version and recommend to publish the paper.

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Anonymous Report 1 on 2020-4-23 (Invited Report)


The few questions and suggestions in the referee's report have been suitably answered. The paper is now acceptable for publication as it stands

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