Jiaju Zhang, Pasquale Calabrese, Marcello Dalmonte, M. A. Rajabpour
SciPost Phys. Core 2, 007 (2020) ·
published 7 May 2020
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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.
