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arXiv:1801.01158 · Journal of High Energy Physics

On the entanglement entropy of Maxwell theory: a condensed matter perspective

Michael Pretko

Published Dec 1, 2018• 16 citations
arXivHTMLPDF

Abstract

A bstract Despite the seeming simplicity of the theory, calculating (and even defining) entanglement entropy for the Maxwell theory of a U(1) gauge field in (3+1) dimensions has been the subject of controversy. It is generally accepted that the ground state entanglement entropy for a region of linear size L behaves as an area law with a subleading logarithm, S = αL 2 − γ log L . While the logarithmic coefficient γ is believed to be universal, there has been disagreement about its precise value. After carefully accounting for subtle boundary corrections, multiple analyses in the high energy literature have converged on an answer related to the conformal trace anomaly, which is only sensitive to the local curvature of the partition. In contrast, a condensed matter treatment of the problem yielded a topological contribution which is not captured by the conformal field theory calculation. In this perspective piece, we review aspects of the various calculations and discuss the resolution of the discrepancy, emphasizing the important role played by charged states (the “extended Hilbert space”) in defining entanglement for a gauge theory. While the trace anomaly result is sufficient for a strictly pure gauge field, coupling the gauge field to dynamical charges of mass m gives a topological contribution to γ which survives even in the m → ∞ limit. For many situations, the topological contribution from dynamical charges is physically meaningful and should be taken into account. We also comment on other common issues of entanglement in gauge theories, such as entanglement distillation, algebraic definitions of entanglement, and gauge-fixing procedures.

DOI
10.1007/jhep12(2018)102

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A bstract Despite the seeming simplicity of the theory, calculating (and even defining) entanglement entropy for the Maxwell theory of a U(1) gauge field in (3+1) dimensions has been the subject of controversy. It is generally accepted that the ground state entanglement entropy for a region of linear size L behaves as an area law with a subleading logarithm, S = αL 2 − γ log L . While the logarithmic coefficient γ is believed to be universal, there has been disagreement about its precise value. After carefully accounting for subtle boundary corrections, multiple analyses in the high energy literature have converged on an answer related to the conformal trace anomaly, which is only sensitive to the local curvature of the partition. In contrast, a condensed matter treatment of the problem yielded a topological contribution which is not captured by the conformal field theory calculation. In this perspective piece, we review aspects of the various calculations and discuss the resolution of the discrepancy, emphasizing the important role played by charged states (the “extended Hilbert space”) in defining entanglement for a gauge theory. While the trace anomaly result is sufficient for a strictly pure gauge field, coupling the gauge field to dynamical charges of mass m gives a topological contribution to γ which survives even in the m → ∞ limit. For many situations, the topological contribution from dynamical charges is physically meaningful and should be taken into account. We also comment on other common issues of entanglement in gauge theories, such as entanglement distillation, algebraic definitions of entanglement, and gauge-fixing procedures.
We revisit the issue of defining the entropy of a spatial region in a broad class of quantum theories. In theories with explicit regularizations, working within an elementary but general algebraic framework applicable to matter and gauge theories alike, we give precise path integral expressions for three known types of entanglement entropy that we call full, distillable, and gauge-invariant. For a class of gauge theories that do not necessarily have a regularization in our framework, including Chern-Simons theory, we describe a related approach to defining entropies based on locally extending the Hilbert space at the entangling edge, and we discuss its connections to other calculational prescriptions. Based on results from both approaches, we conjecture that it is always the full entanglement entropy that is calculated by standard holographic techniques in strongly coupled conformal theories.
We consider the entanglement entropy for a free U(1) theory in 3+1 dimensions in the extended Hilbert space definition. By taking the continuum limit carefully we obtain a replica trick path integral which calculates this entanglement entropy. The path integral is gauge invariant, with a gauge fixing delta function accompanied by a Faddeev -Popov determinant. For a spherical region it follows that the result for the logarithmic term in the entanglement, which is universal, is given by the a anomaly coefficient. We also consider the extractable part of the entanglement, which corresponds to the number of Bell pairs which can be obtained from entanglement distillation or dilution. For a spherical region we show that the coefficient of the logarithmic term for the extractable part is different from the extended Hilbert space result. We argue that the two results will differ in general, and this difference is accounted for by a massless scalar living on the boundary of the region of interest.
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 is a powerful tool for characterizing spin liquids and other systems with long-range entanglement. For gapped spin liquids, this problem has been well-studied, finding a universal subleading constant term in the entanglement entropy, known as the topological entanglement entropy. In this paper, the idea of topological entanglement entropy is generalized to the gapless UUU(1) spin liquid. First, it is shown that the entanglement entropy separates into two pieces, one due to emergent charges of the theory (assumed to be gapped) and one due to the emergent gapless gauge mode. Both give universal subheading contributions to the entanglement entropy, which are logarithmic in the size of the region considered. The photon contribution is due to the local physics of the gapless mode. The particle contribution, on the other hand, is a topological term, generalizing the notion of topological entanglement entropy. A construction is presented which can isolate this topological piece by eliminating the photon contribution and other local terms, such as the area law. This topological term in the entanglement entropy can serve as a powerful diagnostic tool for UUU(1) spin liquids.
A bstract We consider the entanglement entropy arising from edge-modes in Abelian p -form topological field theories in d dimensions on arbitrary spatial topology and across arbitrary entangling surfaces. We find a series of descending area laws plus universal corrections proportional to the Betti numbers of the entangling surface, which can be taken as a higher- dimensional version of the “topological entanglement entropy”. Our calculation comes in two flavors: firstly, through an induced edge-mode theory appearing on the regulated entangling surface in a replica path integral and secondly through a more rigorous definition of the entanglement entropy through an extended Hilbert space. Along the way we establish several key results that are of their own merit. We explain how the edge-mode theory is a novel combination of ( p − 1)-form and ( d – p − 2)-form Maxwell theories linked by a chirality condition, in what we coin a chiral mixed Maxwell theory . We explicitly evaluate the thermal partition function of this theory. Additionally we show that the extended Hilbert space is completely organized into representations of an infinite-dimensional, centrally extended current algebra which naturally generalizes 2d Kac-Moody algebras to arbitrary dimension and topology. We construct the Verma modules and the representation characters of this algebra. Lastly, we connect the two approaches, showing that the thermal partition function of the chiral mixed Maxwell theory is precisely an extended representation character of our current algebra, establishing an exact correspondence of the edge-mode theory and the entanglement spectrum.
We study entanglement entropy (EE) for a Maxwell field in (2+1)(2+1)(2+1) dimensions. We do numerical calculations in two-dimensional lattices. This gives a concrete example of the general results of our recent work [1] on entropy for lattice gauge fields using an algebraic approach. To evaluate the entropies we extend the standard calculation methods for the entropy of Gaussian states in canonical commutation algebras to the more general case of algebras with center and arbitrary numerical commutators. We find that while the entropy depends on the details of the algebra choice, mutual information has a well defined continuum limit as predicted in [1]. We study several universal terms for the entropy of the Maxwell field and compare with the case of a massless scalar field. We find some interesting new phenomena: an ``evanescent'' logarithmically divergent term in the entropy with topological coefficient which does not have any correspondence with ultraviolet entanglement in the universal quantities, and a nonstandard way in which strong subadditivity is realized. Based on the results of our calculations we propose a generalization of strong subadditivity for the entropy on some algebras that are not in tensor product.