Calculation Of Entanglement Entropy Research Articles

Exact and numerical results on entanglement entropy in (5 + 1)-dimensional CFT

We calculate the shape dependence of entanglement entropy in (5+1)-dimensional conformal field theory in terms of the extrinsic curvature of the entangling surface, the opening angles of possible conical singularities, and the conformal anomaly coefficients, which are required to obey a single constraint. An important special case of this result is given by the interacting (2,0) theory describing a large number of coincident M5-branes. To derive the more general result we rely crucially on the holographic prescription for calculating entanglement entropy using Lovelock gravity. We test the conjecture by relating the entanglement entropy of the free massless (1,0) hypermultiplet in (5+1)-dimensions to the entanglement entropy of the free massive chiral multiplet in (2+1)-dimensions, which we calculate numerically using lattice techniques. We also present a numerical calculation of the (2+1)-dimensional renormalized entanglement entropy for the free massive Dirac fermion, which is shown to be consistent with the F-theorem.

Thermodynamic singularities in the entanglement entropy at a two-dimensional quantum critical point

We study the bipartite entanglement entropy of the 2D transverse-field Ising model in the thermodynamic limit. Series expansions are developed for the Renyi entropy around both the small-field and large-field limits, allowing the separate calculation of the entanglement associated with lines and corners at the boundary between subsystems. Series extrapolations are used to extract power laws and logarithmic singularities as the quantum critical point is approached, giving access to new universal quantities. In 1D, we find excellent agreement with exact results as well as quantum Monte Carlo simulations. In 2D, we find compelling evidence that the entanglement at a corner is significantly different from a free boson field theory. These results demonstrate the power of the series expansion method for calculating entanglement entropy in interacting systems, a fact that will be particularly useful in searches for exotic quantum criticality in models with and without the sign problem.

Entanglement Entropy of Black Holes.

The entanglement entropy is a fundamental quantity, which characterizes the correlations between sub-systems in a larger quantum-mechanical system. For two sub-systems separated by a surface the entanglement entropy is proportional to the area of the surface and depends on the UV cutoff, which regulates the short-distance correlations. The geometrical nature of entanglement-entropy calculation is particularly intriguing when applied to black holes when the entangling surface is the black-hole horizon. I review a variety of aspects of this calculation: the useful mathematical tools such as the geometry of spaces with conical singularities and the heat kernel method, the UV divergences in the entropy and their renormalization, the logarithmic terms in the entanglement entropy in four and six dimensions and their relation to the conformal anomalies. The focus in the review is on the systematic use of the conical singularity method. The relations to other known approaches such as ’t Hooft’s brick-wall model and the Euclidean path integral in the optical metric are discussed in detail. The puzzling behavior of the entanglement entropy due to fields, which non-minimally couple to gravity, is emphasized. The holographic description of the entanglement entropy of the blackhole horizon is illustrated on the two- and four-dimensional examples. Finally, I examine the possibility to interpret the Bekenstein-Hawking entropy entirely as the entanglement entropy.

Finite entanglement entropy from the zero-point area of spacetime

The calculation of entanglement entropy S of quantum fields in spacetimes with horizon shows that, quite generically, S (a) is proportional to the area A of the horizon and (b) is divergent. I argue that this divergence, which arises even in the case of Rindler horizon in flat spacetime, is yet another indication of a deep connection between horizon thermodynamics and gravitational dynamics. In an emergent perspective of gravity, which accommodates this connection, the fluctuations around the equipartition value in the area elements will lead to a minimal quantum of area, of the order of L_P^2, which will act as a regulator for this divergence. In a particular prescription for incorporating L_P^2 as zero-point-area of spacetime, this does happen and the divergence in entanglement entropy is regularized, leading to S proportional to (A/L_P^2) in Einstein gravity. In more general models of gravity, the surface density of microscopic degrees of freedom is different which leads to a modified regularisation procedure and the possibility that the entanglement entropy - when appropriately regularised - matches the Wald entropy.

Entanglement entropy in non-relativistic field theories

We calculate entanglement entropy in a non-relativistic field theory described by the Schr\"odinger operator. We demonstrate that the entropy is characterized by i) the area law and ii) UV divergences that are identical to those in the relativistic field theory. These observations are further supported by a holographic consideration. We use the non-relativistic symmetry and completely specify entanglement entropy in large class of non-relativistic theories described by the field operators polynomial in derivatives. We suggest that the area law of the entropy can be tested in experiments with condensed matter systems such as liquid helium.

The Fisher-Hartwig Formula and Entanglement Entropy

Toeplitz matrices have applications to different problems of statistical mechanics. Recently it was used for calculation of entanglement entropy in spin chains. In the paper we review these recent developments. We use the Fisher-Hartwig formula, as well as the recent results concerning the asymptotics of the block Toeplitz determinants, to calculate entanglement entropy of large block of spins in the ground state of XY spin chain.

Holographic avatars of entanglement entropy

This is a rendering of the blackboard lectures at the 2008 Cargese summer school, discussing some elementary facts regarding the application of AdS/CFT techniques to the computation of entanglement entropy in strongly coupled systems. We emphasize the situations where extensivity of the entanglement entropy can be used as a crucial criterion to characterize either nontrivial dynamical phenomena at large length scales, or nonlocality in the short-distance realm.

Topological entanglement entropy and holography

We study the entanglement entropy in confining theories with gravity duals using the holographic prescription of Ryu and Takayanagi. The entanglement entropy between a region and its complement is proportional to the minimal area of a bulk hypersurface ending on their border. We consider a disk in 2+1 dimensions and a ball in 3+1 dimensions and find in both cases two types of bulk hypersurfaces with different topology, similar to the case of the slab geometry considered by Klebanov, Kutasov and Murugan. Depending on the value of the radius, one or the other type of hypersurfaces dominates the calculation of entanglement entropy. In 2+1 dimensions a useful measure of topological order of the ground state is the topological entanglement entropy, which is defined to be the constant term in the entanglement entropy of a disk in the limit of large radius. We compute this quantity and find that it vanishes for confining gauge theory, in accord with our expectations. In 3+1 dimensions the analogous quantity is shown to be generically nonzero and cutoff-dependent.

COVARIANT ENTANGLEMENT ENTROPY

We review our recent formulation1,2 of computing entanglement entropy in a holographic way. The basic examples can be found by applying AdS/CFT correspondence and the holographic formula has successfully been checked in many examples of conformal field theories. We also explain the covariant formulation of holographic entanglement entropy which is closely related to the covariant entropy bound (Bousso bound) in an interesting way.

Entanglement entropy, conformal invariance and extrinsic geometry

We use the conformal invariance and the holographic correspondence to fully specify the dependence of entanglement entropy on the extrinsic geometry of the 2d surface Σ that separates two subsystems of quantum strongly coupled N=4SU(N) superconformal gauge theory. We extend this result and calculate entanglement entropy of a generic 4d conformal field theory. As a byproduct, we obtain a closed-form expression for the entanglement entropy in flat space–time when Σ is sphere S2 and when Σ is two-dimensional cylinder. The contribution of the type A conformal anomaly to entanglement entropy is always determined by topology of surface Σ while the dependence of the entropy on the extrinsic geometry of Σ is due to the type B conformal anomaly.

Calculation of Entanglement Entropy for Continuous-Variable Entangled State Based on General Two-Mode Boson Exponential Quadratic Operator in Fock Space

We obtain an explicit formula to calculate the entanglement entropy of bipartite entangled state of general two-mode boson exponential quadratic operator with continuous variables in Fock space. The simplicity and generality of our formula are shown by some examples.

Black hole entropy as entanglement entropy: a holographic derivation

We study the possibility that black hole entropy be identified as entropy of entanglement across the horizon of the vacuum of a quantum field in the presence of the black hole. We argue that a recent proposal for computing entanglement entropy using AdS/CFT holography implies that black hole entropy can be exactly equated with entanglement entropy. The implementation of entanglement entropy in this context solves all the problems (such as cutoff dependence and the species problem) typically associated with this identification.