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arXiv:2310.14690

Entanglement entropy and the boundary action of edge modes

Jyotirmoy Mukherjee

Published Oct 23, 2023• 0 citations
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Abstract

We consider an antisymmetric gauge field in the Minkowski space of $d$-dimension and decompose it in terms of the antisymmetric tensor harmonics and fix the gauge. The Gauss law implies that the normal component of the field strength on the spherical entangling surface will label the superselection sectors. From the two-point function of the field strength on the sphere, we evaluate the logarithmic divergent term of the entanglement entropy of edge modes of $p$-form field. We observe that the logarithmic divergent term in entanglement entropy of edge modes coincides with the edge partition function of co-exact $p$-form on the sphere when expressed in terms of the Harish-Chandra characters. We also develop a boundary path integral of the antisymmetric $p$-form gauge field. From the boundary path integral, we show that the edge mode partition function corresponds to the co-exact $(p-1)$-forms on the boundary. This boundary path integral agrees with the direct evaluation of the entanglement entropy of edge modes extracted from the two-point function of the normal component of the field strength on the entangling surface.

DOI
10.48550/arxiv.2310.14690

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We consider an antisymmetric gauge field in the Minkowski space of ddd-dimension and decompose it in terms of the antisymmetric tensor harmonics and fix the gauge. The Gauss law implies that the normal component of the field strength on the spherical entangling surface will label the superselection sectors. From the two-point function of the field strength on the sphere, we evaluate the logarithmic divergent term of the entanglement entropy of edge modes of ppp-form field. We observe that the logarithmic divergent term in entanglement entropy of edge modes coincides with the edge partition function of co-exact ppp-form on the sphere when expressed in terms of the Harish-Chandra characters. We also develop a boundary path integral of the antisymmetric ppp-form gauge field. From the boundary path integral, we show that the edge mode partition function corresponds to the co-exact (p−1)(p-1)(p−1)-forms on the boundary. This boundary path integral agrees with the direct evaluation of the entanglement entropy of edge modes extracted from the two-point function of the normal component of the field strength on the entangling surface.
A bstract We consider an antisymmetric gauge field in the Minkowski space of d -dimension and decompose it in terms of the antisymmetric tensor harmonics and fix the gauge. The Gauss law implies that the normal component of the field strength on the spherical entangling surface will label the superselection sectors. From the two-point function of the field strength on the sphere, we evaluate the logarithmic divergent term of the entanglement entropy of edge modes of p -form field. We observe that the logarithmic divergent term in entanglement entropy of edge modes coincides with the edge partition function of co-exact p -form on the sphere when expressed in terms of the Harish-Chandra characters. We also develop a boundary path integral of the antisymmetric p -form gauge field. From the boundary path integral, we show that the edge mode partition function corresponds to the co-exact ( p − 1)-forms on the boundary. This boundary path integral agrees with the direct evaluation of the entanglement entropy of edge modes extracted from the two-point function of the normal component of the field strength on the entangling surface.
Abstract We consider the linearised graviton in 4d Minkowski space and decompose it into tensor spherical harmonics and fix the gauge. The Gauss law of gravity implies that certain radial components of the Riemann tensor of the graviton on the sphere labels the superselection sectors for the graviton. We show that among these 6 normal components of the Riemann tensor, 2 are related locally to the algebra of gauge-invariant operators in the sphere. From the two-point function of these components of the Riemann tensor on S2 we compute the logarithmic coefficient of the entanglement entropy of these superselection sectors across a spherical entangling surface. For sectors labelled by each of the two components of the Riemann tensor these coefficients are equal and their total contribution is given by −163 -\frac{16}{3} −316​ − 16 3 . We observe that this coefficient coincides with that extracted from the edge partition function of the massless spin-2 field on the 4-sphere when written in terms of its Harish-Chandra character. As a preliminary step, we also evaluate the logarithmic coefficient of the entanglement entropy from the superselection sectors labelled by the radial component of the electric field of the U(1) theory in even d dimensions. We show that this agrees with the corresponding coefficient of the edge Harish-Chandra character of the massless spin-1 field on Sd.
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 consider the entanglement entropy arising from edge-modes in Abelian ppp-form topological field theories in ddd 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)(p-1)(p−1)-form and (d−p−2)(d-p-2)(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 compute the entanglement entropy of a massless spin 2 field in a sphere in flat Minkowski space. We describe the theory with a linearized metric perturbation field ${h}_{\ensuremath{\mu}\ensuremath{\nu}}$ and decompose the field into independent modes using tensor spherical harmonics. We fix the gauge such that (a) the two dynamical modes for each angular momentum decouple and have the dynamics of scalar spherical modes, and (b) the gauge-fixed field degrees of freedom inside the sphere represent gauge invariant operators of the theory localized in the same region. In this way the entanglement entropy turns out to be equivalent to the one of a pair of free massless scalars where the contributions of the l=0l=0l=0 and l=1l=1l=1 modes have been subtracted. The result for the coefficient of the universal logarithmic term is $\ensuremath{-}61/45$ and coincides with that computed using a regularization based on the mutual information.
The vacuum entanglement entropy of Maxwell theory, when evaluated by standard methods, contains an unexpected term with no known statistical interpretation. We resolve this two-decades old puzzle by showing that this term is the entanglement entropy of edge modes: classical solutions determined by the electric field normal to the entangling surface. We explain how the heat kernel regularization applied to this term leads to the negative divergent expression found by Kabat. This calculation also resolves a recent puzzle concerning the logarithmic divergences of gauge fields in 3+1 dimensions.