Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

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 nn-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}ℓ−2 for large length of the interval \ellℓ. 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.

Entanglement Spectroscopy and probing the Li-Haldane Conjecture in Topological Quantum Matter

“…The BW prediction also holds with remarkably accuracy for ground states of discrete lattice models [25][26][27][28][29][30], described by a Hamiltonian H = j h j with quasilocal operators h j , as realized in present quantum simulation experiments. Thus, BW suggests an efficient parametrization of the corresponding reduced states as…”

Excited state Rényi entropy and subsystem distance in two-dimensional non-compact bosonic theory. Part II. Multi-particle states

“…which is an important measure of distance. Secondly it seems that the Schatten distance for even n's capture many properties that we expect for the trace distance [57]. The present paper is the following work of [65], where we have calculated the second Rényi entropy and the second subsystem Schatten distance in the excited states produced by the current and its derivatives in the 2D free massless non-compact bosonic field theory, which is a 2D CFT with central charge c = 1, and found perfect matches of the analytical CFT results with the corresponding numerical results in the gapless limit of the harmonic chain with local interactions.…”

“…In quantum information theory, it is often important to know the quantitative difference between two density matrices, especially for two subsystem RDMs. The subsystem distance has been studied in various circumstances, see for example [51][52][53][54][55][56][57][58][59][60][61]. There are many definitions of the distance between two density matrices [62][63][64].…”

Excited state Rényi entropy and subsystem distance in two-dimensional non-compact bosonic theory. Part I. Single-particle states

“…The subsystem distance of the low-lying states in the 2D free massless fermionic and compact bosonic theories were already calculated recently and compared with the results coming from the critical Ising chain and XX chain in [75,76]. In [77,78] the subsystem distance was used to quantify the precision of the approximate entanglement Hamiltonian coming from the discretization of the Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains [79,80]. The subsystem distance was also used in [81] to characterize the local operator quench in 2D CFTs and spin chains [82][83][84].…”