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arXiv:2404.05851 · Physical review. D/Physical review. D.

Entanglement entropy in lattices with non-Abelian gauge groups

Mihael Hategan

Published May 6, 2024• 1 citation
arXivHTMLPDF

Abstract

Entanglement entropy, taken here to be geometric, requires a geometrically separable Hilbert space. In lattice gauge theories, it is not immediately clear if the physical Hilbert space is geometrically separable. In a previous paper we have shown that the physical Hilbert space in pure gauge Abelian lattice theories exhibits some form of geometric scaling with the lattice volume, which suggest that the space is locally factorizable and, therefore, geometrically separable. In this paper, we provide strong evidence that indicates that this scaling is not present when the group is non-Abelian. We do so by looking at the scaling of the dimension of the physical Hilbert space of theories with certain discrete groups. The lack of an appropriate scaling implies that the physical Hilbert space of such a theory does not admit a local factorization. We then extend the reasoning, as sensibly possible, to SU(2) and SU(N) to reach the same conclusion. Lastly, we show that the addition of matter fields to non-Abelian lattice gauge theories makes the resulting physical Hilbert space locally factorizable. Published by the American Physical Society 2024

DOI
10.1103/physrevd.109.094501

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Entanglement entropy, taken here to be geometric, requires a geometrically separable Hilbert space. In lattice gauge theories, it is not immediately clear if the physical Hilbert space is geometrically separable. In a previous paper we have shown that the physical Hilbert space in pure gauge Abelian lattice theories exhibits some form of geometric scaling with the lattice volume, which suggest that the space is locally factorizable and, therefore, geometrically separable. In this paper, we provide strong evidence that indicates that this scaling is not present when the group is non-Abelian. We do so by looking at the scaling of the dimension of the physical Hilbert space of theories with certain discrete groups. The lack of an appropriate scaling implies that the physical Hilbert space of such a theory does not admit a local factorization. We then extend the reasoning, as sensibly possible, to SU(2) and SU(N) to reach the same conclusion. Lastly, we show that the addition of matter fields to non-Abelian lattice gauge theories makes the resulting physical Hilbert space locally factorizable. Published by the American Physical Society 2024
We revisit the issue of the geometrical separability of the Hilbert space of physical states on lattice Abelian theories in the context of entanglement entropy. We discuss the conditions under which vectors in the Hilbert space, as well as the gauge-invariant algebra, admit a tensor product decomposition with a geometrical interpretation. With the exception of pure gauge lattices with periodic boundary conditions which contain topological degrees of freedom, we show that the Hilbert space is geometrically separable.
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.
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 in Pure Z2Z_2Z2​ Gauge Lattices
We show that the Hilbert space of physical states on a pure Z2Z_2Z2​ gauge lattice in 1+11 + 11+1 and 2+12 + 12+1 dimensions is geometrically separable if the fundamental physical degrees of freedom are taken to be the plaquettes. This results in a physical entanglement entropy that is not affected by gauge fixing. We introduce a lattice model that is physically equivalent to the original and whose entanglement entropy, calculated using link degrees of freedom, is the same as the entanglement entropy calculated using physical states. We also show that, for non-physical gauge link states, entanglement entropy quantifies constraints between gauge choices in plaquettes adjacent to the boundary.
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.
We focus on the issue of proper definition of entanglement entropy in lattice gauge theories, and examine a naive definition where gauge invariant states are viewed as elements of an extended Hilbert space which contains gauge non-invariant states as well. Working in the extended Hilbert space, we can define entanglement entropy associated with an arbitrary subset of links, not only for abelian but also for non-abelian theories. We then derive the associated replica formula. We also discuss the issue of gauge invariance of the entanglement entropy. In the Z N gauge theories in arbitrary space dimensions, we show that all the standard properties of the entanglement entropy, e.g. the strong subadditivity, hold in our definition. We study the entanglement entropy for special states, including the topological states for the Z N gauge theories in arbitrary dimensions. We discuss relations of our definition to other proposals.