arXiv subjects

Erik Tonni

Publications and source records attributed to Erik Tonni.

At least 55 records · Page 3Linked to original sources

arXiv2017

Entanglement hamiltonian and entanglement contour in inhomogeneous 1D critical systems

Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints. A contour function for the entanglement entropies corresponding to this configuration is also considered, being closely related to the entanglement hamiltonian. The analytic expressions obtained by considering the curved spacetime which characterises the rainbow model have been checked against numerical data for the rainbow chain, finding an excellent agreement.

cond-mat.stat-mech↗

arXiv2017

Corner contributions to holographic entanglement entropy in AdS4/BCFT3

We study the holographic entanglement entropy of spatial regions with corners in the AdS4/BCFT3 correspondence by considering three dimensional boundary conformal field theories whose boundary is a timelike plane. We compute analytically the corner function corresponding to an infinite wedge having one edge on the boundary. A relation between this corner function and the holographic one point function of the stress tensor is observed. An analytic expression for the corner function of an infinite wedge having only its tip on the boundary is also provided. This formula requires to find the global minimum among two extrema of the area functional. The corresponding critical configurations of corners are studied. The results have been checked against a numerical analysis performed by computing the area of the minimal surfaces anchored to some finite domains containing corners.

hep-th↗

arXiv2017

A contour for the entanglement entropies in harmonic lattices

We construct a contour function for the entanglement entropies in generic harmonic lattices. In one spatial dimension, numerical analysis are performed by considering harmonic chains with either periodic or Dirichlet boundary conditions. In the massless regime and for some configurations where the subsystem is a single interval, the numerical results for the contour function are compared to the inverse of the local weight function which multiplies the energy-momentum tensor in the corresponding entanglement hamiltonian, found through conformal field theory methods, and a good agreement is observed. A numerical analysis of the contour function for the entanglement entropy is performed also in a massless harmonic chain for a subsystem made by two disjoint intervals.

cond-mat.stat-mech↗

arXiv2017

Entanglement spectrum degeneracy and Cardy formula in 1+1 dimensional conformal field theories

We investigate the effect of a global degeneracy in the distribution of entanglement spectrum in conformal field theories in one spatial dimension. We relate the recently found universal expression for the entanglement hamiltonian to the distribution of the entanglement spectrum. The main tool to establish this connection is the Cardy formula. It turns out that the Affleck-Ludwig non-integer degeneracy, appearing because of the boundary conditions induced at the entangling surface, can be directly read from the entanglement spectrum distribution. We also clarify the effect of the non-integer degeneracy on the spectrum of the partial transpose, which is the central object for quantifying the entanglement in mixed states. We show that the exact knowledge of the entanglement spectrum in some integrable spin-chains provides strong analytical evidences corroborating our results.

hep-th↗

arXiv2017

Entanglement negativity in a two dimensional harmonic lattice: Area law and corner contributions

We study the logarithmic negativity and the moments of the partial transpose in the ground state of a two dimensional massless harmonic square lattice with nearest neighbour interactions for various configurations of adjacent domains. At leading order for large domains, the logarithmic negativity and the logarithm of the ratio between the generic moment of the partial transpose and the moment of the reduced density matrix at the same order satisfy an area law in terms of the length of the curve shared by the adjacent regions. We give numerical evidences that the coefficient of the area law term in these quantities is related to the coefficient of the area law term in the Rényi entropies. Whenever the curve shared by the adjacent domains contains vertices, a subleading logarithmic term occurs in these quantities and the numerical values of the corner function for some pairs of angles are obtained. In the special case of vertices corresponding to explementary angles, we provide numerical evidence that the corner function of the logarithmic negativity is given by the corner function of the Rényi entropy of order 1/2.

cond-mat.stat-mech↗

arXiv2016

Entanglement hamiltonians in two-dimensional conformal field theory

We enumerate the cases in 2d conformal field theory where the logarithm of the reduced density matrix (the entanglement or modular hamiltonian) may be written as an integral over the energy-momentum tensor times a local weight. These include known examples and new ones corresponding to the time-dependent scenarios of a global and local quench. In these latter cases the entanglement hamiltonian depends on the momentum density as well as the energy density. In all cases the entanglement spectrum is that of the appropriate boundary CFT. We emphasize the role of boundary conditions at the entangling surface and the appearance of boundary entropies as universal O(1) terms in the entanglement entropy.

cond-mat.stat-mech↗

arXiv2015

Spin structures and entanglement of two disjoint intervals in conformal field theories

We reconsider the moments of the reduced density matrix of two disjoint intervals and of its partial transpose with respect to one interval for critical free fermionic lattice models. It is known that these matrices are sums of either two or four Gaussian matrices and hence their moments can be reconstructed as computable sums of products of Gaussian operators. We find that, in the scaling limit, each term in these sums is in one-to-one correspondence with the partition function of the corresponding conformal field theory on the underlying Riemann surface with a given spin structure. The analytical findings have been checked against numerical results for the Ising chain and for the XX spin chain at the critical point.

cond-mat.stat-mech↗

arXiv2015

On shape dependence of holographic entanglement entropy in AdS4/CFT3

We study the finite term of the holographic entanglement entropy of finite domains with smooth shapes and for four dimensional gravitational backgrounds. Analytic expressions depending on the unit vectors normal to the minimal area surface are obtained for both stationary and time dependent spacetimes. The special cases of AdS4, asymptotically AdS4 black holes, domain wall geometries and Vaidya-AdS backgrounds have been analysed explicitly. When the bulk spacetime is AdS4, the finite term is the Willmore energy of the minimal area surface viewed as a submanifold of the three dimensional flat Euclidean space. For the static spacetimes, some numerical checks involving spatial regions delimited by ellipses and non convex domains have been performed. In the case of AdS4, the infinite wedge has been also considered, recovering the known analytic formula for the coefficient of the logarithmic divergence.

hep-th↗

arXiv2015

Towards entanglement negativity of two disjoint intervals for a one dimensional free fermion

We study the moments of the partial transpose of the reduced density matrix of two intervals for the free massless Dirac fermion. By means of a direct calculation based on coherent state path integral, we find an analytic form for these moments in terms of the Riemann theta function. We show that the moments of arbitrary order are equal to the same quantities for the compactified boson at the self-dual point. These equalities imply the non trivial result that also the negativity of the free fermion and the self-dual boson are equal.

cond-mat.stat-mech↗

arXiv2015

Partial transpose of two disjoint blocks in XY spin chains

We consider the partial transpose of the spin reduced density matrix of two disjoint blocks in spin chains admitting a representation in terms of free fermions, such as XY chains. We exploit the solution of the model in terms of Majorana fermions and show that such partial transpose in the spin variables is a linear combination of four Gaussian fermionic operators. This representation allows to explicitly construct and evaluate the integer moments of the partial transpose. We numerically study critical XX and Ising chains and we show that the asymptotic results for large blocks agree with conformal field theory predictions if corrections to the scaling are properly taken into account.

cond-mat.stat-mech↗

arXiv2015

Entanglement entropy and negativity of disjoint intervals in CFT: Some numerical extrapolations

The entanglement entropy and the logarithmic negativity can be computed in quantum field theory through a method based on the replica limit. Performing these analytic continuations in some cases is beyond our current knowledge, even for simple models. We employ a numerical method based on rational interpolations to extrapolate the entanglement entropy of two disjoint intervals for the conformal field theories given by the free compact boson and the Ising model. The case of three disjoint intervals is studied for the Ising model and the non compact free massless boson. For the latter model, the logarithmic negativity of two disjoint intervals has been also considered. Some of our findings have been checked against existing numerical results obtained from the corresponding lattice models.

cond-mat.stat-mech↗

arXiv2014

Entanglement negativity after a global quantum quench

We study the time evolution of the logarithmic negativity after a global quantum quench. In a 1+1 dimensional conformal invariant field theory, we consider the negativity between two intervals which can be either adjacent or disjoint. We show that the negativity follows the quasi-particle interpretation for the spreading of entanglement. We check and generalise our findings with a systematic analysis of the negativity after a quantum quench in the harmonic chain, highlighting two peculiar lattice effects: the late birth and the sudden death of entanglement.

cond-mat.stat-mech↗

arXiv2014

On shape dependence of holographic mutual information in AdS4

We study the holographic mutual information in AdS4 of disjoint spatial domains in the boundary which are delimited by smooth closed curves. A numerical method which approximates a local minimum of the area functional through triangulated surfaces is employed. After some checks of the method against existing analytic results for the holographic entanglement entropy, we compute the holographic mutual information of equal domains delimited by ellipses, superellipses or the boundaries of two dimensional spherocylinders, finding also the corresponding transition curves along which the holographic mutual information vanishes.

hep-th↗

arXiv2014

Finite temperature entanglement negativity in conformal field theory

We consider the logarithmic negativity of a finite interval embedded in an infinite one dimensional system at finite temperature. We focus on conformal invariant systems and we show that the naive approach based on the calculation of a two-point function of twist fields in a cylindrical geometry yields a wrong result. The correct result is obtained through a four-point function of twist fields in which two auxiliary fields are inserted far away from the interval, and they are sent to infinity only after having taken the replica limit. In this way, we find a universal scaling form for the finite temperature negativity which depends on the full operator content of the theory and not only on the central charge. In the limit of low and high temperatures, the expansion of this universal form can be obtained by means of the operator product expansion. We check our results against exact numerical computations for the critical harmonic chain.

cond-mat.stat-mech↗

arXiv2014

Holographic thermalization with Lifshitz scaling and hyperscaling violation

A Vaidya type geometry describing gravitation collapse in asymptotically Lifshitz spacetime with hyperscaling violation provides a simple holographic model for thermalization near a quantum critical point with non-trivial dynamic and hyperscaling violation exponents. The allowed parameter regions are constrained by requiring that the matter energy momentum tensor satisfies the null energy condition. We present a combination of analytic and numerical results on the time evolution of holographic entanglement entropy in such backgrounds for different shaped boundary regions and study various scaling regimes, generalizing previous work by Liu and Suh.

hep-th↗

arXiv2013

Global properties of causal wedges in asymptotically AdS spacetimes

We examine general features of causal wedges in asymptotically AdS spacetimes and show that in a wide variety of cases they have non-trivial topology. We also prove some general results regarding minimal area surfaces on the causal wedge boundary and thereby derive constraints on the causal holographic information. We go on to demonstrate that certain properties of the causal wedge impact significantly on features of extremal surfaces which are relevant for computation of holographic entanglement entropy.

hep-th↗

arXiv2013

Holographic entanglement plateaux

We consider the entanglement entropy for holographic field theories in finite volume. We show that the Araki-Lieb inequality is saturated for large enough subregions, implying that the thermal entropy can be recovered from the knowledge of the region and its complement. We observe that this actually is forced upon us in holographic settings due to non-trivial features of the causal wedges associated with a given boundary region. In the process, we present an infinite set of extremal surfaces in Schwarzschild-AdS geometry anchored on a given entangling surface. We also offer some speculations regarding the homology constraint required for computing holographic entanglement entropy.

hep-th↗

arXiv2013

On Rényi entropies of disjoint intervals in conformal field theory

We study the Rényi entropies of N disjoint intervals in the conformal field theories given by the free compactified boson and the Ising model. They are computed as the 2N point function of twist fields, by employing the partition function of the model on a particular class of Riemann surfaces. The results are written in terms of Riemann theta functions. The prediction for the free boson in the decompactification regime is checked against exact results for the harmonic chain. For the Ising model, matrix product states computations agree with the conformal field theory result once the finite size corrections have been taken into account.

hep-th↗