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arXiv:2406.10967 · Journal of High Energy Physics

A note on entanglement entropy and topological defects in symmetric orbifold CFTs

Michael Gutperle A, Yan-Yan Li, Dikshant Rathore, Konstantinos Roumpedakis

Published Sep 3, 2024• 4 citations
arXivHTMLPDF

Abstract

A bstract In this brief note we calculate the entanglement entropy in M ⊗ N / S N symmetric orbifold CFTs in the presence of topological defects, which were recently constructed in [1, 2]. We consider both universal defects which realize Rep ( S N ) non-invertible symmetry and non-universal defects. We calculate the sub-leading defect entropy/g-factor for defects at the boundary of the entangling surface as well as inside it.

DOI
10.1007/jhep09(2024)010

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A bstract In this brief note we calculate the entanglement entropy in M ⊗ N / S N symmetric orbifold CFTs in the presence of topological defects, which were recently constructed in [1, 2]. We consider both universal defects which realize Rep ( S N ) non-invertible symmetry and non-universal defects. We calculate the sub-leading defect entropy/g-factor for defects at the boundary of the entangling surface as well as inside it.
In this work, we compute the defect relative entropy between topological defects in the symmetric product orbifold CFT SymN(M)=M⊗N/SN\mathrm{Sym}^N(M) = M^{\otimes N}/S_NSymN(M)=M⊗N/SN​. Our analysis covers two distinct classes of defects: universal defects, which realize the Rep(SN)\mathrm{Rep}(S_N)Rep(SN​) non-invertible symmetry, and non-universal defects. We show that the defect relative entropy reduces to a Kullback--Leibler (KL) divergence. The resulting expression decomposes naturally into two contributions: one governed by characters of the symmetric group SNS_NSN​, and the other controlled by modular SSS-matrix elements of the seed RCFT. Remarkably, both sets of data appear as probability distributions, yielding an information-theoretic interpretation of permutation group data and modular data within the symmetric orbifold. The structure of the divergence depends sensitively on the defect class. For universal defects, only the permutation group data contributes; for maximally fractional defects, both permutation and modular data enter and together define the relevant probability distributions. This feature suggests that the maximally fractional defect can be understood as a kind of product of the RCFT defect and the symmetric orbifold defect.
Entanglement entropy (EE) contains signatures of many universal properties of conformal field theories (CFTs), especially in the presence of boundaries or defects. In particular, topological defects are interesting since they reflect internal symmetries of the CFT and have been extensively analyzed with field-theoretic techniques with striking predictions. So far, however, no lattice computation of EE has been available. Here, we present an ab initio analysis of EE for the Ising model in the presence of a topological defect. While the behavior of the EE depends, as expected, on the geometric arrangement of the subsystem with respect to the defect, we find that zero-energy modes give rise to crucial finite-size corrections. Importantly, contrary to the field-theory predictions, the universal subleading term in the EE when the defect lies at the edge of the subsystem arises entirely due to these zero-energy modes and is not directly related to the modular S matrix of the Ising CFT.
Abstract In this paper we study the properties of two-dimensional CFTs defined by cyclic and symmetric orbifolds of free Dirac fermions, especially by focusing on the partition function and entanglement entropy. Via the bosonization, we construct the twist operators which glue two complex planes to calculate the partition function of ℤ2 orbifold CFT on a torus. We also find an expression of ℤN cyclic orbifold in terms of Hecke operators, which provides an explicit relation between the partition functions of cyclic orbifolds and those of symmetric ones. We compute the entanglement entropy and Renyi entropy in cyclic orbifolds on a circle both for finite temperature states and for time-dependent states under quantum quenches. We find that the replica method calculation is highly non-trivial and new because of the contributions from replicas with different boundary conditions. We find the full expression for the ℤ2 orbifold and show that the periodicity gets doubled. Finally, we discuss extensions of our results on entanglement entropy to symmetric orbifold CFTs and make a heuristic argument towards holographic CFTs.
Abstract We study the entanglement entropy of gauged internal degrees of freedom in a two dimensional symmetric product orbifold CFT, whose configurations consist of N strands sewn together into “long” strings, with wavefunctions symmetrized under permutations. In earlier work a related notion of “entwinement” was introduced. Here we treat this system analogously to a system of N identical particles. From an algebraic point of view, we point out that the reduced density matrix on k out of N particles is not associated with a subalgebra of operators, but rather with a linear subspace, which we explain is sufficient. In the orbifold CFT, we compute the entropy of a single strand in states holographically dual in the D1/D5 system to a conical defect geometry or a massless BTZ black hole and find a result identical to entwinement. We also calculate the entropy of two strands in the state that represents the conical defect; the result differs from entwinement. In this case, matching entwinement would require finding a gauge-invariant way to impose continuity across strands.
We study the entanglement entropy of gauged internal degrees of freedom in a two dimensional symmetric product orbifold CFT, whose configurations consist of N strands sewn together into “long” strings, with wavefunctions symmetrized under permutations. In earlier work a related notion of “entwinement” was introduced. Here we treat this system analogously to a system of N identical particles. From an algebraic point of view, we point out that the reduced density matrix on k out of N particles is not associated with a subalgebra of operators, but rather with a linear subspace, which we explain is sufficient. In the orbifold CFT, we compute the entropy of a single strand in states holographically dual in the D1/D5 system to a conical defect geometry or a massless BTZ black hole and find a result identical to entwinement. We also calculate the entropy of two strands in the state that represents the conical defect; the result differs from entwinement. In this case, matching entwinement would require finding a gauge-invariant way to impose continuity across strands.
A bstract We compute the defect entanglement entropy for co-dimension two superconformal monodromy defects in well known maximally symmetric holographic theories of various dimension. In each case we explicitly relate the universal part of the defect entanglement entropy to field theory data characterising the defect conformal field theory. We provide evidence that, unlike in the bulk theories in which the defects reside, the universal part of the defect entanglement entropy does not necessarily decrease along a renormalisation group flow.