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arXiv:1408.1298 · Journal of High Energy Physics

Higher spin entanglement entropy

Jiang Long

Published Dec 1, 2014• 20 citations
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Abstract

In this paper, we develop a perturbation formulation to calculate the single interval higher spin Rényi and entanglement entropy for two dimensional conformal field theory with $$ {\mathcal{W}}_{\infty}\left(\lambda \right) $$ symmetry. The system is at finite temperature and is deformed by higher spin chemical potential. We manage to compute higher spin Rényi entropy with various spin deformations up to order $$ \mathcal{O}\left({\mu}^2\right) $$ . For spin 3 deformation, we calculate exact higher spin Rényi and entanglement entropy up to $$ \mathcal{O}\left({\mu}^4\right) $$ . When λ = 3, in the large c limit, we find perfect match with tree level holographic higher spin entanglement entropy up to order μ 4 obtained by the Wilson line prescription. We also find quantum corrections to higher spin entanglement entropy which is beyond tree level holographic results. The quantum correction is universal at order μ 4 in the sense that it is independent of λ. Our computation relies on a multi-valued conformal map from n-sheeted Riemann surface $$ {\mathrm{\mathcal{R}}}_n $$ to complex plane and correlation functions of primary fields on complex plane. The method can be applied to general conformal field theories with $$ \mathcal{W} $$ symmetry.

DOI
10.1007/jhep12(2014)055

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We consider free fermion and free boson CFTs in two dimensions, deformed by a chemical potential μ for the spin-three current. For the CFT on the infinite spatial line, we calculate the finite temperature entanglement entropy of a single interval perturbatively to second order in μ in each of the theories. We find that the result in each case is given by the same non-trivial function of temperature and interval length. Remarkably, we further obtain the same formula using a recent Wilson line proposal for the holographic entanglement entropy, in holomorphically factorized form, associated to the spin-three black hole in SL(3, ℝ) × SL(3, ℝ) Chern-Simons theory. Our result suggests that the order μ 2 correction to the entanglement entropy may be universal for $ \mathcal{W} $ -algebra CFTs with spinthree chemical potential, and constitutes a check of the holographic entanglement entropy proposal for higher spin theories of gravity in AdS3.
It is generally believed that the semiclassical AdS3 higher spin gravity could be described by a two dimensional conformal field theory with W \mathcal{W} W -algebra symmetry in the large central charge limit. In this paper, we study the single interval entanglement entropy on the torus in the CFT with a W3 {\mathcal{W}}_{\mathcal{3}} W3​ deformation. More generally we develop the monodromy analysis to compute the two-point function of the light operators under a thermal density matrix with a W3 {\mathcal{W}}_{\mathcal{3}} W3​ chemical potential to the leading order. Holographically we compute the probe action of the Wilson line in the background of the spin-3 black hole with a chemical potential. We find exact agreement.
We consider free fermion and free boson CFTs in two dimensions, deformed by a chemical potential mu for the spin-three current. For the CFT on the infinite spatial line, we calculate the finite temperature entanglement entropy of a single interval perturbatively to second order in mu in each of the theories. We find that the result in each case is given by the same non-trivial function of temperature and interval length. Remarkably, we further obtain the same formula using a recent Wilson line proposal for the holographic entanglement entropy, in holomorphically factorized form, associated to the spin-three black hole in SL(3, R) x SL(3, R) Chern-Simons theory. Our result suggests that the order mu(2) correction to the entanglement entropy may be universal for W-algebra CFTs with spin-three chemical potential, and constitutes a check of the holographic entanglement entropy proposal for higher spin theories of gravity in AdS(3).
We study the time evolution of single interval Rényi and entanglement entropies following local quantum quenches in two dimensional conformal field theories at finite temperature for which the locally excited states have a finite temporal width ϵ. We show that, for local quenches produced by the action of a conformal primary field, the time dependence of Rényi and entanglement entropies at order ϵ2 is universal. It is determined by the expectation value of the stress tensor in the replica geometry and proportional to the conformal dimension of the primary field generating the local excitation. We also show that in CFTs with a gravity dual, the ϵ2 correction to the holographic entanglement entropy following a local quench precisely agrees with the CFT prediction. We then consider CFTs admitting a higher spin symmetry and turn on a higher spin chemical potential μ. We calculate the time dependence of the order ϵ2 correction to the entanglement entropy for small μ, and show that the contribution at order μ 2 is universal. We verify our arguments against exact results for minimal models and the free fermion theory.
In this paper, we study the entanglement entropy of a single interval on a cylinder in two-dimensional TT‾T\overline{T}TT-deformed conformal field theory (CFT). For such case, the (R\'enyi) entanglement entropy takes a universal form in a CFT. We compute the correction due to the deformation up to the leading order of the deformation parameter in the framework of the conformal perturbation theory. We find that the correction to the entanglement entropy is nonvanishing in the finite temperature case, while it is vanishing in the finite size case. For the deformed holographic large ccc CFT, which is proposed to be dual to a AdS3{\mathrm{AdS}}_{3}AdS3​ gravity in a finite region, we find the agreement with the holographic entanglement entropy via the Ryu-Takayanagi formula. Moreover, we compute the leading order correction to the R\'enyi entropy, and discuss its holographic picture as well.
We study the free fermion theory in 1+1 dimensions deformed by chemical potentials for holomorphic, conserved currents at finite temperature and on a spatial circle. For a spin-three chemical potential μ, the deformation is related at high temperatures to a higher spin black hole in hs[0] theory on AdS3 spacetime. We calculate the order μ2 corrections to the single interval Rényi and entanglement entropies on the torus using the bosonized formulation. A consistent result, satisfying all checks, emerges upon carefully accounting for both perturbative and winding mode contributions in the bosonized language. The order μ2 corrections involve integrals that are finite but potentially sensitive to contact term singularities. We propose and apply a prescription for defining such integrals which matches the Hamiltonian picture and passes several non-trivial checks for both thermal corrections and the Rényi entropies at this order. The thermal corrections are given by a weight six quasi-modular form, whilst the Rényi entropies are controlled by quasi-elliptic functions of the interval length with modular weight six. We also point out the well known connection between the perturbative expansion of the partition function in powers of the spin-three chemical potential and the Gross-Taylor genus expansion of large-N Yang-Mills theory on the torus. We note the absence of winding mode contributions in this connection, which suggests qualitatively different entanglement entropies for the two systems.