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arXiv:1912.08658 · Journal of Physics A Mathematical and Theoretical

Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion*

L Brightmore, G P Gehér, Alexander Rudol'fovich Its, V E Korepin, F Mezzadri, M Y Mo, J A Virtanen

Published Jun 15, 2020• 8 citations
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Abstract

Abstract We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv: 1104.1004 , for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m} ∪ {2m + 1, 2m + 2, …, 3m}, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.

DOI
10.1088/1751-8121/ab9cf2

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Abstract We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv: 1104.1004 , for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m} ∪ {2m + 1, 2m + 2, …, 3m}, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.
Fisher-Hartwig formula has been successful applied to describe the von Neumann and Rényi entropies of a block of spins in the ground state of XX spin chain. It was based on a determinant representation. In this paper, we generalize the free fermion method to obtain an exact formulation for the entropy of any finite subsystem in XX spin chain. Based on this, we derive a determinant representation of the entropy of multiple disjoint intervals in the ground state of XXXXXX model.
We study the Rényi entanglement entropies of two disjoint intervals in XY chains. We exploit the exact solution of the model in terms of free Majorana fermions and we show how to construct the reduced density matrix in the spin variables by taking the Jordan–Wigner string between the two blocks properly into account. From this we can evaluate any Rényi entropy of finite integer order. We study in detail critical XX and Ising chains and we show that the asymptotic results for large blocks agree with recent conformal field theory predictions if corrections to the scaling are included in the analysis correctly. We also report results for the gapped phase and after a quantum quench.
In this paper we discuss the asymptotic behavior of the entanglement entropy for a fragment of a fermionic chain in an excited state. We first study the case in which the subsystem consists of a single interval. We review the analytical tools that make the computation feasible, derived from the asymptotics of the determinant of Toeplitz matrices, we also compare with the numerical results and give a physical interpretation of them. Next we move to the case in which the fragment is made of several disjoint intervals. We compute numerically the entropy and, based on these results, we establish a conjecture on the determinant of a submatrix of a Toeplitz matrix that leads to a general expression for the entropy in the multi-interval case. We check that our results agree with those derived from conformal field theory when the latter are available.
Abstract The statistical mechanics characterization of finite subsystems embedded in an infinite system is a fundamental question of quantum physics. Nevertheless, a full closed form for all required entropic measures does not exist in the general case, even for free systems, when the finite system in question is composed of several disjoint intervals. Here we develop a mathematical framework based on the Riemann–Hilbert approach to treat this problem in the one-dimensional case where the finite system is composed of two disjoint intervals and in the thermodynamic limit (both intervals and the space between them contains an infinite number of lattice sites and the result is given as a thermodynamic expansion). To demonstrate the usefulness of our method, we compute the change in the entanglement and negativity spectra namely the spectrum of eigenvalues of the reduced density matrix with or without time reversal of one of the intervals. We do this in the case that the distance between the intervals is much larger than their size. The method we use can be easily applied to compute any power in an expansion in the ratio of the distance between the intervals to their size. We expect these results to provide the necessary mathematical apparatus to address relevant questions in concrete physical scenarios, namely the structure and extent of quantum correlations in fermionic systems subject to local environments.
The analysis of the entanglement entropy of a subsystem of a one-dimensional quantum system is a powerful tool for unravelling its critical nature. For instance, the scaling behaviour of the entanglement entropy determines the central charge of the associated Virasoro algebra. For a free fermion system, the entanglement entropy depends essentially on two sets, namely the set A of sites of the subsystem considered and the set K of excited momentum modes. In this work we make use of a general duality principle establishing the invariance of the entanglement entropy under exchange of the sets A and K to tackle complex problems by studying their dual counterparts. The duality principle is also a key ingredient in the formulation of a novel conjecture for the asymptotic behavior of the entanglement entropy of a free fermion system in the general case in which both sets A and K consist of an arbitrary number of blocks. We have verified that this conjecture reproduces the numerical results with excellent precision for all the configurations analyzed. We have also applied the conjecture to deduce several asymptotic formulas for the mutual and r-partite information generalizing the known ones for the single block case.