## Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

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

SciPost Phys. Core 2, 007 (2020) · published 7 May 2020

- doi: 10.21468/SciPostPhysCore.2.2.007
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### Abstract

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.

### Ontology / Topics

See full Ontology or Topics database.### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1 }^{2 }Jiaju Zhang, -
^{1 }^{2 }^{3 }Pasquale Calabrese, -
^{1 }^{3 }Marcello Dalmonte, -
^{4 }Mohammad Ali Rajabpour

^{1}Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies [SISSA]^{2}Istituto Nazionale di Fisica Nucleare / National Institute for Nuclear Physics [INFN]^{3}Centro Internazionale di Fisica Teorica Abdus Salam / Abdus Salam International Centre for Theoretical Physics [ICTP]^{4}Universidade Federal Fluminense / Fluminense Federal University [UFF]

- European Research Council [ERC]
- Fundação de Amparo à Pesquisa do Estado de São Paulo / São Paulo Research Foundation [FAPESP]
- Horizon 2020 (through Organization: European Commission [EC])