Using the algebraic approach to entanglement entropy, we study several dual pairs of lattice theories and show how the entropy is completely preserved across each duality. Our main result is that a maximal algebra of observables in a region typically dualizes to a non-maximal algebra in a dual region. In particular, we show how the usual notion of tracing out external degrees of freedom dualizes to a tracing out coupled to an additional summation over superselection sectors. We briefly comment on possible extensions of our results to more intricate dualities, including holographic ones.Using the algebraic approach to entanglement entropy, we study several dual pairs of lattice theories and show how the entropy is completely preserved across each duality. Our main result is that a maximal algebra of observables in a region typically dualizes to a non-maximal algebra in a dual region. In particular, we show how the usual notion of tracing out external degrees of freedom dualizes to a tracing out coupled to an additional summation over superselection sectors. We briefly comment on possible extensions of our results to more intricate dualities, including holographic ones.

arXiv:1605.09396May 30, 20166 citations

Entanglement entropy and superselection sectors. Part I. Global symmetries

Horacio Casini, Marina Huerta, Javier M. Magán +1 more

Abstract Some quantum field theories show, in a fundamental or an effective manner, an alternative between a loss of duality for algebras of operators corresponding to complementary regions, or a loss of additivity. In this latter case, the algebra contains some operator that is not generated locally, in the former, the entropies of complementary regions do not coincide. Typically, these features are related to the incompleteness of the operator content of the theory, or, in other words, to the existence of superselection sectors. We review some aspects of the mathematical literature on superselection sectors aiming attention to the physical picture and focusing on the consequences for entanglement entropy (EE). For purposes of clarity, the whole discussion is divided into two parts according to the superselection sectors classification: the present part I is devoted to superselection sectors arising from global symmetries, and the forthcoming part II will consider those arising from local symmetries. Under this perspective, here restricted to global symmetries, we study in detail different cases such as models with finite and Lie group symmetry as well as with spontaneous symmetry breaking or excited states. We illustrate the general results with simple examples. As an important application, we argue the features of holographic entanglement entropy correspond to a picture of an sub-theory with a large number of superselection sectors and suggest some ways in which this identification could be made more precise.Abstract Some quantum field theories show, in a fundamental or an effective manner, an alternative between a loss of duality for algebras of operators corresponding to complementary regions, or a loss of additivity. In this latter case, the algebra contains some operator that is not generated locally, in the former, the entropies of complementary regions do not coincide. Typically, these features are related to the incompleteness of the operator content of the theory, or, in other words, to the existence of superselection sectors. We review some aspects of the mathematical literature on superselection sectors aiming attention to the physical picture and focusing on the consequences for entanglement entropy (EE). For purposes of clarity, the whole discussion is divided into two parts according to the superselection sectors classification: the present part I is devoted to superselection sectors arising from global symmetries, and the forthcoming part II will consider those arising from local symmetries. Under this perspective, here restricted to global symmetries, we study in detail different cases such as models with finite and Lie group symmetry as well as with spontaneous symmetry breaking or excited states. We illustrate the general results with simple examples. As an important application, we argue the features of holographic entanglement entropy correspond to a picture of an sub-theory with a large number of superselection sectors and suggest some ways in which this identification could be made more precise.

arXiv:1905.10487Journal of High Energy PhysicsFeb 1, 202087 citations

Remarks on entanglement entropy for gauge fields

Horacio Casini, Marina Huerta, J. A. Rosabal

In gauge theories the presence of constraints can obstruct expressing the global Hilbert space as a tensor product of the Hilbert spaces corresponding to degrees of freedom localized in complementary regions. In algebraic terms, this is due to the presence of a center---a set of operators which commute with all others---in the gauge invariant operator algebra corresponding to a finite region. A unique entropy can be assigned to algebras with a center, giving a place to a local entropy in lattice gauge theories. However, ambiguities arise on the correspondence between algebras and regions. In particular, it is always possible to choose (in many different ways) local algebras with a trivial center, and hence a genuine entanglement entropy, for any region. These choices are in correspondence with maximal trees of links on the boundary, which can be interpreted as partial gauge fixings. This interpretation entails a gauge fixing dependence of the entanglement entropy. In the continuum limit, however, ambiguities in the entropy are given by terms local on the boundary of the region, in such a way relative entropy and mutual information are finite, universal, and gauge independent quantities.In gauge theories the presence of constraints can obstruct expressing the global Hilbert space as a tensor product of the Hilbert spaces corresponding to degrees of freedom localized in complementary regions. In algebraic terms, this is due to the presence of a center---a set of operators which commute with all others---in the gauge invariant operator algebra corresponding to a finite region. A unique entropy can be assigned to algebras with a center, giving a place to a local entropy in lattice gauge theories. However, ambiguities arise on the correspondence between algebras and regions. In particular, it is always possible to choose (in many different ways) local algebras with a trivial center, and hence a genuine entanglement entropy, for any region. These choices are in correspondence with maximal trees of links on the boundary, which can be interpreted as partial gauge fixings. This interpretation entails a gauge fixing dependence of the entanglement entropy. In the continuum limit, however, ambiguities in the entropy are given by terms local on the boundary of the region, in such a way relative entropy and mutual information are finite, universal, and gauge independent quantities.

arXiv:1312.1183Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsApr 7, 2014330 citations

Entanglement entropy in $SU(2)$ lattice gauge theory using a dual loop formulationEntanglement entropy in $SU(2)$ lattice gauge theory using a dual loop formulation

S. Adarsh, T. P. Sreeraj

One must exercise care in calculating entanglement entropy of gauge theories, because the local Gauss' laws prevent a tensor-product decomposition of the Hilbert space into spatial regions. In this work, we study entanglement entropy in lattice $SU(2)$ gauge theory using the loop formulation together with the Casini-Huerta-Rosabal (henceforth, abbreviated as CHR) algebraic entropy approach. The loop formulations involve a set of canonical transformations that trade the Kogut-Susskind link variables and the local Gauss' laws for the physical loop variables and a single global Gauss' law. Within this framework, we reformulate the CHR entropy directly in the physical Hilbert space. We identify the regional algebras and the center, which, in the loop formulation, is not generated by boundary-localized operators, but instead reduces to a global Casimir of the regional algebra. This structure reveals that the area law of entanglement entropy cannot be determined solely from the gauge constraints. The gauge constraints decompose the reduced density matrix into superselection sectors, and the locality of correlations in the physical states gives rise to the area law. Thus, the analysis separates the algebraic structure dictated by Gauss' constraints from the geometric origin of the area law.One must exercise care in calculating entanglement entropy of gauge theories, because the local Gauss' laws prevent a tensor-product decomposition of the Hilbert space into spatial regions. In this work, we study entanglement entropy in lattice $SU(2)$ gauge theory using the loop formulation together with the Casini-Huerta-Rosabal (henceforth, abbreviated as CHR) algebraic entropy approach. The loop formulations involve a set of canonical transformations that trade the Kogut-Susskind link variables and the local Gauss' laws for the physical loop variables and a single global Gauss' law. Within this framework, we reformulate the CHR entropy directly in the physical Hilbert space. We identify the regional algebras and the center, which, in the loop formulation, is not generated by boundary-localized operators, but instead reduces to a global Casimir of the regional algebra. This structure reveals that the area law of entanglement entropy cannot be determined solely from the gauge constraints. The gauge constraints decompose the reduced density matrix into superselection sectors, and the locality of correlations in the physical states gives rise to the area law. Thus, the analysis separates the algebraic structure dictated by Gauss' constraints from the geometric origin of the area law.

International Symposium on Lattice Field Theory (LATTICE)Jul 16, 20260 citations

Entanglement entropy in Jackiw-Teitelboim gravity with matter

Jennifer Lin

In this paper, I study the entanglement entropy in Hartle-Hawking states of JT gravity set up by a Euclidean path integral with an operator inserted somewhere along the Euclidean boundary. I show that the entanglement entropy between the dual SYK models can be written to leading order in $1/G_N$ in a way that resembles a formula for entanglement entropy in a compact lattice gauge theory, but where the link that we appear to compute the entanglement across depends on where we put the operator along the Euclidean boundary. I then describe a toy model where we can write down an explicit family of density matrices whose von Neumann entropies exhibit such a phase transition as we tune an interpolating parameter. This behavior relies on a certain superselection sector structure and I comment on the possibility that the gravitational result comes from a similar emergent superselection sector structure in SYK/JT holography. The main implication is that each copy of the SYK model should then contain Hilbert space sectors labeled by multiple $SL(2,R)$ representations whose entanglement gives rise to space.In this paper, I study the entanglement entropy in Hartle-Hawking states of JT gravity set up by a Euclidean path integral with an operator inserted somewhere along the Euclidean boundary. I show that the entanglement entropy between the dual SYK models can be written to leading order in $1/G_N$ in a way that resembles a formula for entanglement entropy in a compact lattice gauge theory, but where the link that we appear to compute the entanglement across depends on where we put the operator along the Euclidean boundary. I then describe a toy model where we can write down an explicit family of density matrices whose von Neumann entropies exhibit such a phase transition as we tune an interpolating parameter. This behavior relies on a certain superselection sector structure and I comment on the possibility that the gravitational result comes from a similar emergent superselection sector structure in SYK/JT holography. The main implication is that each copy of the SYK model should then contain Hilbert space sectors labeled by multiple $SL(2,R)$ representations whose entanglement gives rise to space.

arXiv:2107.11872Jul 25, 20213 citations

Entanglement entropy in lattice gauge theories

P. V. Buividovich, M I Polikarpov

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 easily be 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 dimensions we demonstrate the relation between the entanglement entropy and the classical entropy of the field flux. For 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.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 easily be 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 dimensions we demonstrate the relation between the entanglement entropy and the classical entropy of the field flux. For 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.

arXiv:0811.3824Journal of Physics A Mathematical and TheoreticalJul 14, 200926 citations

Entanglement entropy in

SU(2)

lattice gauge theory using a dual loop formulation

SU(2)

gauge theory using the loop formulation together with the Casini-Huerta-Rosabal (henceforth, abbreviated as CHR) algebraic entropy approach. The loop formulations involve a set of canonical transformations that trade the Kogut-Susskind link variables and the local Gauss' laws for the physical loop variables and a single global Gauss' law. Within this framework, we reformulate the CHR entropy directly in the physical Hilbert space. We identify the regional algebras and the center, which, in the loop formulation, is not generated by boundary-localized operators, but instead reduces to a global Casimir of the regional algebra. This structure reveals that the area law of entanglement entropy cannot be determined solely from the gauge constraints. The gauge constraints decompose the reduced density matrix into superselection sectors, and the locality of correlations in the physical states gives rise to the area law. Thus, the analysis separates the algebraic structure dictated by Gauss' constraints from the geometric origin of the area law.

1/G_N

in a way that resembles a formula for entanglement entropy in a compact lattice gauge theory, but where the link that we appear to compute the entanglement across depends on where we put the operator along the Euclidean boundary. I then describe a toy model where we can write down an explicit family of density matrices whose von Neumann entropies exhibit such a phase transition as we tune an interpolating parameter. This behavior relies on a certain superselection sector structure and I comment on the possibility that the gravitational result comes from a similar emergent superselection sector structure in SYK/JT holography. The main implication is that each copy of the SYK model should then contain Hilbert space sectors labeled by multiple

SL(2,R)

representations whose entanglement gives rise to space.