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arXiv:1401.5922 · Journal of Physics A Mathematical and Theoretical

Excited state entanglement in homogeneous fermionic chains

Filiberto Ares, J G Esteve, Fernando Falceto, Eduardo Sánchez-Burillo

Published Jun 2, 2014• 67 citations
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Abstract

We study the Rényi entanglement entropy of an interval in a periodic fermionic chain for a general eigenstate of a free, translational invariant Hamiltonian. In order to analytically compute the entropy we use two technical tools. The first is used to logarithmically reduce the complexity of the problem and the second to compute the Rényi entropy of the chosen subsystem. We introduce new strategies to perform the computations, derive new expressions for the entropy of these general states and show the perfect agreement of the analytical computations and the numerical outcome. Finally we discuss the physical interpretation of our results and generalize them to compute the entanglement entropy for a fragment of a fermionic ladder.

DOI
10.1088/1751-8113/47/24/245301

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We study the Rényi entanglement entropy of an interval in a periodic fermionic chain for a general eigenstate of a free, translational invariant Hamiltonian. In order to analytically compute the entropy we use two technical tools. The first is used to logarithmically reduce the complexity of the problem and the second to compute the Rényi entropy of the chosen subsystem. We introduce new strategies to perform the computations, derive new expressions for the entropy of these general states and show the perfect agreement of the analytical computations and the numerical outcome. Finally we discuss the physical interpretation of our results and generalize them to compute the entanglement entropy for a fragment of a fermionic ladder.
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.
The Rényi entanglement entropies of quasiparticle excitations in the many-body gapped systems show a remarkable universal picture which is independent of the model, the quasiparticle momenta, and the connectedness of the subsystem. In this letter we calculate exactly the single-interval and double-interval Rényi entanglement entropies of quasiparticle excitations in the many-body gapped fermions, bosons, and XY chains and find additional contributions to the universal Rényi entanglement entropy in the excited states with quasiparticles of different momenta. The additional terms are different in different models, depend on the momentum differences of the quasiparticles, and are different for the single interval and the double interval. We derive the analytical Rényi entanglement entropy in the extremely gapped limit, matching perfectly the numerical results as long as either the intrinsic correlation length of the model or all the de Broglie wavelengths of the quasiparticles are small. When the momentum difference of any pair of distinct quasiparticles is small, the additional terms are non-negligible. We argue that the derived formulas can be applied to a wider range of models than those discussed here.
We study general entanglement properties of the excited states of the one-dimensional translational invariant free fermions and coupled harmonic oscillators. In particular, using the integrals of motion, we prove that these Hamiltonians independent of the gap (mass) have infinite excited states that can be described by conformal field theories with integer or half-integer central charges. In the case of free fermions, we also show that because of the huge degeneracy in the spectrum, even a gapless Hamiltonian can have excited states with an area law. Finally, we study the universal average entanglement entropy over eigenstates of the Hamiltonian of the free fermions introduced recently in Vidmar et al. [Phys. Rev. Lett. 119, 020601 (2017)]. In particular, using a duality relation, we map the average bipartite entanglement entropy over all the exponential numbers of eigenstates of a generic free fermion Hamiltonian to the problem of the calculation of the average multibipartite entanglement entropy over a single eigenstate of the XX\mathit{XX}XX chain. This relation can be useful for experimental measurement of the universal average entanglement entropy. Part of our conclusions can be extended to the quantum spin chains that are associated with the free fermions via Jordan-Wigner transformation.
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.
We investigate the quasiparticle excited state R\'enyi entropy of single, double and more intervals in the fermionic chain, bosonic chain and XY chain. When the gap of the theory is large or all the momenta of the quasiparticles are large, the R\'enyi Entropy takes a super universal form, which is independent of, among others, the model, the quasiparticle momenta, and the subsystem connectedness. We calculate analytically the R\'enyi Entropy in the extremely gapped limit by writing the excited states in terms of the local excitations and find different additional contributions to the super universal R\'enyi entropy in various models. The additional contributions to the super universal R\'enyi entropy cannot be neglected when the momentum differences of the excited quasiparticles are small, however, they are negligible when all the momentum differences are large. The R\'enyi entropy with additional terms derived in the extremely gapped limit is universal in the sense that it is valid in the slightly gapped and critical models as long as all the momenta of the excited quasiparticles are large. In the case of double interval in the XY chain we find new exact super universal as well as the universal results. In the case of the bosonic chain in the extremely massive limit we find analytically a novel formula, written as a permanent of a certain matrix, for the R\'enyi entropy. We support all of our analytical results with numerical calculations.