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Entropia topológica de emaranhamento em teorias de Higher Gauge Abelianas

J. P. Ibieta Jimenez

Published Dec 20, 2019• 0 citations
Publisher

Abstract

We compute topological entanglement entropy for a large set of lattice models in d-dimensions.It is well known that many such quantum systems can be constructed out of lattice gauge models.For dimensionality higher than 2 there are generalizations going beyond gauge theories.They are called higher gauge theories and rely on higher-order generalizations of groups.Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories.In this work, we calculate the bipartition entanglement entropy for this class of models.Our formalism allows us to do most of the calculation for arbitrary dimension d.We show that the entanglement entropy S A in a sub-region A is proportional to log(GSD Ã), where GSD Ã is the ground state degeneracy of a particular restriction of the full model to A. When A has the topology of a d-dimensional ball, the GSD Ã counts the number of edge states.In this case, S A scales with the area of the (d -1)-dimensional boundary of A. The precise formula for the entropy we obtain is in agreement with entanglement calculations for known topological models.

DOI
10.11606/t.43.2019.tde-30102019-232405

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We compute topological entanglement entropy for a large set of lattice models in d-dimensions.It is well known that many such quantum systems can be constructed out of lattice gauge models.For dimensionality higher than 2 there are generalizations going beyond gauge theories.They are called higher gauge theories and rely on higher-order generalizations of groups.Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories.In this work, we calculate the bipartition entanglement entropy for this class of models.Our formalism allows us to do most of the calculation for arbitrary dimension d.We show that the entanglement entropy S A in a sub-region A is proportional to log(GSD Ã), where GSD Ã is the ground state degeneracy of a particular restriction of the full model to A. When A has the topology of a d-dimensional ball, the GSD Ã counts the number of edge states.In this case, S A scales with the area of the (d -1)-dimensional boundary of A. The precise formula for the entropy we obtain is in agreement with entanglement calculations for known topological models.
Abstract We compute the topological entanglement entropy for a large set of lattice models in d-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy SA in a sub-region A is proportional to log (GSDA~) \left({GSD}_{\tilde{A}}\right) (GSDA~​) GSD A ˜ , where GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ is the ground state degeneracy of a particular restriction of the full model to A. The quantity GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂A. There is also a topological contribution coming from A itself, that may be non-zero when A has a non-trivial homology. We present some examples and discuss how the topology of A affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension d. The result is in agreement with entanglement calculations for known topological models.
Nós calculamos a entropia de emaranhamento topológica para um grande conjunto de modelos em dimensão d. Sabe-se que muitos sistemas quânticos podem ser construídos a partir de teorias de gauge na rede. Em dimensões maiores a 2 existem generalizações além das teorias de gauge. Chamadas higher gauge theories, estas são baseadas em generalizações de ordem superior do conceito de grupo. O nosso objeto de estudo é um conjunto grande de modelos d-dimensionais, que são obtidos a partir de teorias Abelianas de higher gauge. Neste trabalho, calculamos a entropia de emaranhamento para dito conjunto de modelos. O nosso formalismo permite fazer a maior parte do cálculo para dimensão arbitrária d. Mostramos que a entropia de emaran- hamento S_A , em uma sub-região A do sistema, é proporcional à log(GSD_Ã ), onde GSD_Ã é a degenerescência do estado fundamental de uma restrição particular do modelo na região A. Quando A tem a topologia de uma bola de dimensão d, a quantidade GSD_Ã conta o número de estados de borda. Neste caso, S_A escala com a área da borda (d 1)-dimensional de A. O resultado exato da entropia que obtemos está em concordância com os resultados conhecidos na literatura.
We consider entanglement entropy between regions of space in lattice gauge theory. The Hilbert space corresponding to a region of space includes ``edge states'' that transform nontrivially under gauge transformations. By decomposing the edge states in irreducible representations of the gauge group, the entropy of an arbitrary state is expressed as the sum of three positive terms: a term associated with the classical Shannon entropy of the distribution of boundary representations, a term that appears only for non-Abelian gauge theories and depends on the dimension of the boundary representations, and a term representing nonlocal correlations. The first two terms are the entropy of the edge states, and depend only on observables measurable at the boundary. These results are applied to several examples of lattice gauge theory states, including the ground state in the strong coupling expansion of Kogut and Susskind. In all these examples we find that the entropy of the edge states is the dominant contribution to the entanglement entropy.
The definition of geometric entanglement entropy associated with some region in space is discussed for the case of gauge theories. It is argued that since in gauge theories elementary excitations look like loops (closed electric strings) rather than points (particles), the boundaries of the regions should also carry some nonzero entropy. This entropy counts the number of strings which cross these boundaries. Explicit calculations of such entropy are carried out in the limits of infinitely strong and weak couplings of three- and four-dimensional Z N gauge theories. In three dimensions we find that the entropy is a constant which does not depend on the region, while in four dimensions the familiar area law for the entropy is recovered.
A definition for the entanglement entropy in a gauge theory was given recently in arXiv:1501.02593. Working on a spatial lattice, it involves embedding the physical state in an extended Hilbert space obtained by taking the tensor product of the Hilbert space of states on each link of the lattice. This extended Hilbert space admits a tensor product decomposition by definition and allows a density matrix and entanglement entropy for the set of links of interest to be defined. Here, we continue the study of this extended Hilbert space definition with particular emphasis on the case of Non-Abelian gauge theories. We extend the electric centre definition of Casini, Huerta and Rosabal to the Non-Abelian case and find that it differs in an important term. We also find that the entanglement entropy does not agree with the maximum number of Bell pairs that can be extracted by the processes of entanglement distillation or dilution, and give protocols which achieve the maximum bound. Finally, we compute the topological entanglement entropy which follows from the extended Hilbert space definition and show that it correctly reproduces the total quantum dimension in a class of Toric code models based on Non-Abelian discrete groups.
It is realized recently that the entanglement entropy in gauge theories is ambiguous because the Hilbert space cannot be expressed as a simple direct product of Hilbert spaces defined on the two regions; different ways of dividing the Hilbert spaces near the boundary leads to significantly different result, to the extreme that it could annihilate the otherwise finite topological entanglement entropy between two regions altogether. In this article, we first show that the topological entanglement entropy in the Kitaev model [1] which is not a true gauge theory, is free of ambiguity. Then, we give a physical interpretation, from the perspectives of what can be measured in an experiment, to the purported ambiguity of true gauge theories, where the topological entanglement arises as redundancy in counting the degrees of freedom along the boundary separating two regions. We generalize these discussions to non-Abelian gauge theories.