The entanglement entropy of solvable lattice models

Weston, Robert

doi:10.1088/1742-5468/2006/03/l03002

Cited by 29 publications

(54 citation statements)

References 18 publications

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“…First, the logarithmic growth of entanglement as the block size increases in conformal field theory (CFT), which is controlled by the central charge [15,16]. This behaviour has been verified for many critical quantum chains [8][9][10][11]14]. Second, the approach to saturation in massive QFT, which is controlled by the mass spectrum [1,2].…”

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confidence: 91%

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Universal corrections to the entanglement entropy in gapped quantum spin chains: A numerical study

Levi

Castro-Alvaredo

Doyon³

2013

Phys. Rev. B

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We carry out a numerical study of the bi-partite entanglement entropy in the gapped regime of two paradigmatic quantum spin chain models: the Ising chain in an external magnetic field and the anti-ferromagnetic XXZ model. The universal scaling limit of these models is described by the massive Ising field theory and the SU (2)-Thirring (sine-Gordon) model, respectively. We may therefore exploit quantum field theoretical results to predict the behaviour of the entropy. We numerically confirm that, in the scaling limit, corrections to the saturation of the entropy at large region size are proportional to K0(2mr) where m is a mass scale (the inverse correlation length) and r the length of the region under consideration. The proportionality constant is simply related to the number of particle types in the universal spectrum. This was originally predicted in [1,2] for two-dimensional quantum field theories. Away from the universal region our numerics suggest an entropic behaviour following quite closely the quantum field theory prediction, except for extra dependencies on the correlation length. Introduction and discussion. Entanglement is a fundamental property of the state of a quantum system. In the context of quantum computation, it is a crucial resource [3]. Conceptually, it characterizes the structure of quantum fluctuations in a more universal way than other widely studied objects such as correlation functions. Developing theoretical measures of entanglement is therefore important to further understand the structure of quantum states, a problem of particular interest and difficulty for quantum many-body systems. A popular measure of entanglement is the bi-partite entanglement entropy. It measures the entanglement between two complementary sets of observables in a quantum system [4]. Interestingly, the entanglement entropy exhibits universal behaviour near quantum critical points: it has features which do not depend on the details of the model but rather on its universality class. In the last decade, this property has made the study of the entanglement entropy a very active field of research.

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confidence: 91%

“…Quantum spin chains are extensively studied in the context of quantum information science [7] and their bipartite entanglement for blocks of consecutive spins has been investigated in many works (see e.g. [8][9][10][11][12][13][14]) both numerically and analytically. Most of these works concentrate on exact critical points.…”

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confidence: 99%

Universal corrections to the entanglement entropy in gapped quantum spin chains: A numerical study

Levi

Castro-Alvaredo

Doyon³

2013

Phys. Rev. B

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“…For our present purposes, prominent examples of extended one-dimensional quantum systems are quantum spin chains, which model physical systems consisting of infinitely long one-dimensional arrays of equidistant atoms characterized by their spin. Their entanglement has been extensively studied in the literature [11,12,13,14,15,16,17,18,19]. In order to provide a formal definition of the entanglement entropy, let us consider the Hilbert space of a quantum model, such as the chain above, as a tensor product of local Hilbert spaces associated to its sites.…”

Section: Introductionmentioning

confidence: 99%

Bi-partite entanglement entropy in massive (1+1)-dimensional quantum field theories

Castro-Alvaredo

Doyon

2009

J. Phys. A: Math. Theor.

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This manuscript is a review of the main results obtained in a series of papers involving the present authors and their collaborator J.L. Cardy over the last two years. In our work we have developed and applied a new approach for the computation of the bi-partite entanglement entropy in massive 1+1-dimensional quantum field theories. In most of our work we have considered these theories to be also integrable. Our approach combines two main ingredients: the "replica trick," and form factors for integrable models and more generally for massive quantum field theory. Our basic idea for combining fruitfully these two ingredients is that of the branch-point twist field. By the replica trick we obtained an alternative way of expressing the entanglement entropy as a function of the correlation functions of branch-point twist fields. On the other hand, a generalisation of the form factor program has allowed us to study, and in integrable cases to obtain exact expressions for, form factors of such twist fields. By the usual decomposition of correlation functions in an infinite series involving form factors, we obtained exact results for the infrared behaviours of the bi-partite entanglement entropy, and studied both its infrared and ultraviolet behaviours for different kinds of models: with and without boundaries and backscattering, at and out of integrability.

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“…those in Refs. [56][57][58][59][60]. It would be interesting to generalise our results for the distribution of the entanglement spectrum levels (for instance (37)) to non-integrable systems.…”

Section: Discussionmentioning

confidence: 78%

“…an interval embedded in an infinite or a finite system) the degeneracies are more complicated [50]. We mention that the corner transfer matrix techniques have been applied to the study of entanglement entropy and spectrum also to other exactly solvable models [56][57][58][59][60].…”

Section: Entanglement Spectrum Of the Xxz Chain Via The Corner Transfmentioning

confidence: 99%

Negativity spectrum in 1D gapped phases of matter

Mbeng¹,

Alba²,

Calabrese³

2017

J. Phys. A: Math. Theor.

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Abstract.We investigate the spectrum of the partial transpose (negativity spectrum) of two adjacent regions in gapped one-dimensional models. We show that, in the limit of large regions, the negativity spectrum is entirely reconstructed from the entanglement spectrum of the bipartite system. We exploit this result in the XXZ spin chain, for which the entanglement spectrum is known by means of the corner transfer matrix. We find that the negativity spectrum levels are equally spaced, the spacing being half that in the entanglement spectrum. Moreover, the degeneracy of the spectrum is described by elegant combinatorial formulas, which are related to the counting of integer partitions. We also derive the asymptotic distribution of the negativity spectrum. We provide exact results for the logarithmic negativity and for the moments of the partial transpose. They exhibit unusual scaling corrections in the limit ∆ → 1 + with a corrections exponent which is the same as that for the Rényi entropies.arXiv:1612.05172v2 [cond-mat.str-el]

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The entanglement entropy of solvable lattice models

Cited by 29 publications

References 18 publications

Universal corrections to the entanglement entropy in gapped quantum spin chains: A numerical study

Universal corrections to the entanglement entropy in gapped quantum spin chains: A numerical study

Bi-partite entanglement entropy in massive (1+1)-dimensional quantum field theories

Negativity spectrum in 1D gapped phases of matter

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