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arXiv:0707.1300 · Journal of Statistical Mechanics Theory and Experiment

Analytic results on the geometric entropy for free fields

Horacio Casini, M Huerta

Published Jan 18, 2008• 25 citations
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Abstract

The trace of integer powers of the local density matrix ρ V corresponding to the vacuum state reduced to a region V can be formally expressed in terms of a functional integral on a manifold with conical singularities. Recently, some progress has been made in explicitly evaluating this type of integral for free fields. However, finding the associated geometric entropy remained, in general, a difficult task involving an analytic continuation in the conical angle. In this paper, we obtain this analytic continuation explicitly, exploiting a relation between the functional integral formulas and the Chung–Peschel expressions for ρ V in terms of correlators. The result is that the entropy is given in terms of a functional integral in flat Euclidean space with a cut on V where a specific boundary condition is imposed. As an example, we get the exact entanglement entropies for massive scalar and Dirac free fields in 1+1 dimensions in terms of the solutions of a nonlinear differential equation of the Painlevé V type.

DOI
10.1088/1742-5468/2008/01/p01012

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The trace of integer powers of the local density matrix ρ V corresponding to the vacuum state reduced to a region V can be formally expressed in terms of a functional integral on a manifold with conical singularities. Recently, some progress has been made in explicitly evaluating this type of integral for free fields. However, finding the associated geometric entropy remained, in general, a difficult task involving an analytic continuation in the conical angle. In this paper, we obtain this analytic continuation explicitly, exploiting a relation between the functional integral formulas and the Chung–Peschel expressions for ρ V in terms of correlators. The result is that the entropy is given in terms of a functional integral in flat Euclidean space with a cut on V where a specific boundary condition is imposed. As an example, we get the exact entanglement entropies for massive scalar and Dirac free fields in 1+1 dimensions in terms of the solutions of a nonlinear differential equation of the Painlevé V type.
The trace of integer powers of the local density matrix ρV corresponding to the vacuum state reduced to a region V can be formally expressed in terms of a functional integral on a manifold with conical singularities. Recently, some progress has been made in explicitly evaluating this type of integrals for free fields. However, finding the associated geometric entropy remained in general a difficult task involving an analytic continuation in the conical angle. In this paper, we obtain this analytic continuation explicitly exploiting a relation between the functional integral formulas and the Chung-Peschel expressions for ρV in terms of correlators. The result is that the entropy is given in terms of a functional integral in flat Euclidean space with a cut on V where a specific boundary condition is imposed. As an example we get the exact entanglement entropies for massive scalar and Dirac free fields in 1 + 1 dimensions in terms of the solutions of a non linear differential equation of the Painlevé V type. 1
We find the analytic expression of tr ρ n (L) for a free massive boson field in 1+1 dimensions, where ρ(L) is the reduced density matrix corresponding to an interval of length L. This is given exactly (except for a non-universal factor) in terms of a finite sum of solutions of non-linear differential equations of the Painlevé V type. Our method is a generalization of one introduced by Myers and is based on the explicit calculation of quantities related to the Green function on a plane, where boundary conditions are imposed on a finite cut. It is shown that the associated partition function is related to correlators of exponential operators in the sine–Gordon model in agreement with a result by Delfino et al. We also compute the short and long distance leading terms of the entanglement entropy. We find that the bosonic entropic c-function interpolates between the asymptotic limits of the Dirac and Majorana fermion ones given in a previous paper. Finally, we study some universal terms for the entanglement entropy in arbitrary dimensions which, in the case of free fields, can be expressed in terms of the two-dimensional entropy functions.
We present some exact results on universal quantities derived from the local density matrix ρ, for a free massive Dirac field in two dimensions. We first find tr ρ n in a novel fashion, which involves the correlators of suitable operators in the sine–Gordon model. These, in turn, can be written exactly in terms of the solutions of non-linear differential equations of the Painlevé V type. Equipped with the previous results, we find the leading terms for the entanglement entropy, both for short and long distances, and showing that in the intermediate regime it can be expanded in a series of multiple integrals. The previous results have been checked by direct numerical calculations on the lattice, finding perfect agreement. Finally, we comment on a possible generalization of the entanglement entropy c -theorem to the alpha-entropies.
Abstract I compute the conformal weights of the twist operators of free scalar fields for charged Rényi entropy in both odd and even dimensions. Explicit expressions can be found, in odd dimensions as a function of the chemical potential in the absence of a conical singularity and thence by images for all integer coverings. This method, developed some time ago, is equivalent, in results, to the replica technique. A review is given. The same method applies for even dimensions but a general form is more immediately available. For no chemical potential, the closed form in the covering order is written in an alternative way related to old trigonometric sums. Some derivatives are obtained. An analytical proof is given of a conjecture made by Bueno, Myers and Witczak-Krempa regarding the relation between the conformal weights and a corner coefficient (a universal quantity) in the Rényi entropy.
I compute the conformal weights of the twist operators of free scalar fields for charged Rényi entropy in both odd and even dimensions. Explicit expressions can be found, in odd dimensions as a function of the chemical potential in the absence of a conical singularity and thence by images for all integer coverings. This method, developed some time ago, is equivalent, in results, to the replica technique. A review is given. The same method applies for even dimensions but a general form is more immediately available. For no chemical potential, the the closed form in the covering order is written in an alternative way related to old trigonometric sums. Some derivatives are obtained. An analytical proof is given of a conjecture made by Bueno, Myers and Witczac--Krempa regarding the relation between the conformal weights and a corner coefficient (a universal quantity) in the Rényi entropy.
We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S A = −Tr ρ A logρ A corresponding to the reduced density matrix ρ A of a subsystem A . For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c , we re-derive the result of Holzhey et al when A is a finite interval of length in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξ is large but finite, we show that , where is the number of boundary points of A . These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.