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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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Authors (as registered SciPost users): Marcello Dalmonte · Mohammad Ali Rajabpour · Jiaju Zhang
Submission information
Preprint Link:  (pdf)
Date submitted: 2020-03-12 01:00
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.

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Reports on this Submission

Anonymous Report 2 on 2020-4-12 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2003.00315v1, delivered 2020-04-12, doi: 10.21468/SciPost.Report.1621


Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

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

The present paper concerns the reduced-density-matrix description of a large subsystem of two critical quantum chains.
More precisely,
the authors consider the spin-1/2 $XX$ chain in zero field and the spin-1/2 Ising chain in the critical transverse field;
for both exactly diagonalizable systems the reduced density matrix $\rho_A$ and the modular Hamiltonian $H_A$ can be constructed exactly.
Then, for these critical chains,
the authors construct the (approximate) Bisognano-Wichmann modular Hamiltonian $H_A^{\rm BW}$ and the corresponding reduced density matrix $\rho_A^{\rm BW}$.
The goal of their study is
to examine various distances between the reduced density matrices $\rho_A^{\rm BW}$ and $\rho_A$
(and some related quantities including formation probabilities)
when the size of the subsystem $A$ of length $l$ becomes large, i.e., $l\to\infty$.
This analysis illustrates the precision of the Bisognano-Wichmann modular Hamiltonian approach
for calculation of various measures of quantum entanglement in the critical quantum spin chains at hand.
The differences for most quantities
(calculated exactly and with the help of the approximate Bisognano-Wichmann modular Hamiltonian)
decay algebraically as a function of $l$.
The only exception is the emptiness formation probability of the $XX$ chain:
It cannot be captured by the the Bisognano-Wichmann modular Hamiltonian approach.

In my opinion,
the authors present a set of useful results which satisfy acceptance criteria and deserve publication;
the paper is clearly written;
it should be useful for the community studying entanglement questions for quantum many-body systems.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2020-3-27 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2003.00315v1, delivered 2020-03-27, doi: 10.21468/SciPost.Report.1594


1: very well written, clear, easy to follow.
2: very pedagogical, step-by-step approach to the raised issues
3: clarifying in a quantitative way on a number of explicit examples the quantitative relevance of the BW construction for investigating entanglement effects in lattice field theories at large subsystem size.
4: challenging results: the algebraic scaling (exponent-2) of the convergence; the remarkable convergence of the NFP. This clearly open new and exciting perspectives.


An introduction to BW theorem and its uses in this paper is really needed.


The authors address the question of comparing a number of observables relevant for entanglement in 1-d spin chains , using the exact modular Hamiltonian (in several of the few cases where it is available) versus an approximate Hamiltonian, obtained by applying an extension to lattice field theories of the Bisognano-Wichmann theorem of constructive field theory, developed by several of the authors in previous papers. Such a construction is always available, and it is indeed an important issue to check its validity .

The authors propose a characterization of a relevant "distance" between density matrices for lattice field theories resp. exact; and obtained by the BW theorem . They then compare the behaviour of this distance (and others quantities such as Renyi entropies) when the size of the subsystem goes to infinity. They confront results for physical observables, correction functions and formation probabilities. A surprising scaling of the distance itself, and some of the observables , is identified, and an unexpected convergence of some FP is also identified.

For specific comments see Strengths/Weaknesses.

Requested changes

1: explain more precisely what the BW formalism is, e.g. when introducing 4.10. A brief summary of the derivation in the previous paper arXiv 1807.01322 would in this respect be useful.
2: an explanation of the qualitative difference between the computation of fidelity and even-Schatten distances on one hand; and trace and generic odd-Schatten distance on the other hand, around formulae 4.16-4.18; would be appreciated.
3: I am slightly confused by the conclusion of section 4.1.2 "close to 2 but likely different from 2" ? Is there a possibility to estimate more precisely the decay exponent from numerical results ?
4: The log-difference between correlation functions in Fig. 3 exhibits strong dips with sometimes "oscillatory-like" behavior at intermediate scales for l. Do the authors have some qualitative or even quantitative understanding of this particular behaviour ?

  • validity: high
  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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