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Analysis on entanglement entropy for two-dimensional lattice gauge theories with matter fields

毅 横矢

Published Mar 25, 2019• 0 citations
Publisher

Abstract

We study entanglement entropy for 2-dimensional lattice gauge theories with matter fields.For gauge theories, even in lattice space we fail to define entanglement entropy as usual way due to non-local excitations characteristic of theories with constraints.To solve this problem, we apply extended Hilbert space formalism and define entanglement entropy in the generalized way.At first we review the formulation and outcomes by using it.As a result, in addition to ordinary quantum correlation (Bell pair part), two new types of contribution we call Shannon part and color part, will appear.After outlining this effect explicitly, we analyze entanglement entropy of the ground state for 2-dimensional SU (N ) gauge theory with fundamental scalar matter field.As tools to perform that, we use transfer matrix method and hopping parameter expansion(HPE), roughly the hopping parameter corresponds to inverse square of the mass for the matter.The evaluation is carried out in the perturbation from infinity mass limit.In the analysis, we observe that all of the three types of contribution emerge, while Bell pair part does in higher order than Shannon and color part.We discuss the implication by the result.

DOI
10.18910/72644

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We study entanglement entropy for 2-dimensional lattice gauge theories with matter fields.For gauge theories, even in lattice space we fail to define entanglement entropy as usual way due to non-local excitations characteristic of theories with constraints.To solve this problem, we apply extended Hilbert space formalism and define entanglement entropy in the generalized way.At first we review the formulation and outcomes by using it.As a result, in addition to ordinary quantum correlation (Bell pair part), two new types of contribution we call Shannon part and color part, will appear.After outlining this effect explicitly, we analyze entanglement entropy of the ground state for 2-dimensional SU (N ) gauge theory with fundamental scalar matter field.As tools to perform that, we use transfer matrix method and hopping parameter expansion(HPE), roughly the hopping parameter corresponds to inverse square of the mass for the matter.The evaluation is carried out in the perturbation from infinity mass limit.In the analysis, we observe that all of the three types of contribution emerge, while Bell pair part does in higher order than Shannon and color part.We discuss the implication by the result.
We investigate the entanglement entropy in 1+11+11+1-dimensional SU(N)SU(N)SU(N) gauge theories with various matter fields using the lattice regularization. Here we use extended Hilbert space definition for entanglement entropy, which contains three contributions; (1) classical Shannon entropy associated with superselection sector distribution, where sectors are labeled by irreducible representations of boundary penetrating fluxes, (2) logarithm of the dimensions of their representations, which is associated with ``color entanglement,'' and (3) EPR Bell pairs, which give ``genuine'' entanglement. We explicitly show that entanglement entropies (1) and (2) above indeed appear for various multiple ``meson'' states in gauge theories with matter fields. Furthermore, we employ transfer matrix formalism for gauge theory with fundamental matter field and analyze its ground state using hopping parameter expansion (HPE), where the hopping parameter KKK is roughly the inverse square of the mass for the matter. We evaluate the entanglement entropy for the ground state and show that all (1), (2), (3) above appear in the HPE, though the Bell pair part (3) appears in higher order than (1) and (2) do. With these results, we discuss how the ground state entanglement entropy in the continuum limit can be understood from the lattice ground state obtained in the HPE.
Entanglement entropy in SU(2)SU(2)SU(2) lattice gauge theory
The entropy of entanglement between a three-dimensional slab of thickness l and its complement is studied numerically for four-dimensional SU (2) lattice gauge theory. We find a signature of a nonanalytic behavior of the entanglement entropy, which was predicted recently for large Nc confining gauge theories in the framework of AdS/CFT correspondence. The derivative of the entanglement entropy over l is likely to have a discontinuity at some l = lc ≈ 0.5 fm. The application of the concepts of quantum information theory to quantum field theories in continuous space-times and on the lattices has led recently to many important advances in our understanding of their quantum behavior [1–3]. One such concept is the quantum entanglement of states of systems with many degrees of freedom, which turned out to be a very useful modelindependent characteristic of the structure of the ground state of quantum fields. In particular, for quantum fields on the lattice in the vicinity of a quantum phase transition (i.e. a phase transition which occurs at zero temperature when some parameters of the system, such as the coupling constants, are varied) the ground state is a strongly entangled superposition of the states of all elementary lattice degrees of freedom (such as spins in Heisenberg model, or link variables in lattice gauge theory), and different phases of lattice theories can be characterized by different patterns of entanglement [1–3]. Quantum entanglement thus appears to be an adequate concept for the description of the emergence of collective degrees of freedom in quantum field theories [1–3]. A commonly used measure of quantum entanglement of the ground state of quantum fields in (D − 1) + 1-dimensional space-time is the entropy of entanglement S [A] between some (D − 1)-dimensional region A and its (D − 1)-dimensional complement B, which characterizes the amount of information shared between A and B [1, 2, 4]. Entanglement entropy is defined as the usual von Neumann entropy for the reduced density matrix ρA associated with the region A: S [A] = −Tr A (ρA ln ρA) (1) The reduced density matrix is obtained from the density matrix of the ground state of the theory, ρAB = |0〉〈0| , by tracing over all degrees of freedom which are localized outside of A, i.e. within B [2, 4]: ρA = Tr B ρAB = Tr B |0〉〈0| (2) This density matrix describes the state of quantum fields as seen by an observer who can only perform measurements within A. ∗ e-mail: buividovich@tut.by † e-mail: polykarp@itep.ru
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.
We report on the recent progress in theoretical and numerical studies of entanglement entropy in lattice gauge theories.It is shown that the concept of quantum entanglement between gauge fields in two complementary regions of space can only be introduced if the Hilbert space of physical states is extended in a certain way.In the extended Hilbert space, the entanglement entropy can be partially interpreted as the classical Shannon entropy of the flux of the gauge fields through the boundary between the two regions.Such an extension leads to a reduction procedure which can be easily implemented in lattice simulations by constructing lattices with special topology.This enables us to measure the entanglement entropy in lattice Monte-Carlo simulations.On the simplest example of Z 2 lattice gauge theory in (2 + 1) dimensions we demonstrate the relation between entanglement entropy and the classical entropy of the field flux.For SU (2) lattice gauge theory in four dimensions, we find a signature of non-analytic dependence of the entanglement entropy on the size of the region.We also comment on the holographic interpretation of the entanglement entropy.
Entanglement entropy has proven to be an extremely useful concept in quantum field theory. Gauge theories are of particular interest, but for these systems the entanglement entropy is not clearly defined because the physical Hilbert space does not factor as a tensor product according to regions of space. Here we review a definition of entanglement entropy that applies to abelian and nonabelian lattice gauge theories. This entanglement entropy is obtained by embedding the physical Hilbert space into a product of Hilbert spaces associated to regions with boundary. The latter Hilbert spaces include degrees of freedom on the entangling surface that transform like surface charges under the gauge symmetry. These degrees of freedom are shown to contribute to the entanglement entropy, and the form of this contribution is determined by the gauge symmetry. We test our definition using the example of two-dimensional Yang-Mills theory, and find that it agrees with the thermal entropy in de Sitter space, and with the results of the Euclidean replica trick. We discuss the possible implications of this result for more complicated gauge theories, including quantum gravity.
Entanglement entropy is a valuable tool for characterizing the correlation structure of quantum field theories. When applied to gauge theories, subtleties arise which prevent the factorization of the Hilbert space underlying the notion of entanglement entropy. Borrowing techniques from extended topological field theories, we introduce a new definition of entanglement entropy for both Abelian and non-Abelian gauge theories. Being based on the notion of excitations, it provides a completely relational way of defining regions. Therefore, it naturally applies to background independent theories, e.g. gravity, by circumventing the difficulty of specifying the position of the entangling surface. We relate our construction to earlier proposals and argue that it brings these closer to each other. In particular, it yields the non-Abelian analogue of the ‘magnetic centre choice’, as obtained through an extended-Hilbert-space method, but applied to the recently introduced fusion basis for 3D lattice gauge theories. We point out that the different definitions of entanglement entropy can be related to a choice of (squeezed) vacuum state.