Entanglement and alpha entropies for a massive scalar field in two dimensions

We find the analytic expression of the trace of powers of the reduced density matrix on an interval of length L, for a massive boson field in 1+1 dimensions. 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 Painlev\'e 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 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.Comment: 13 pages, 2 figure

Remarks on the entanglement entropy for disconnected regions

Few facts are known about the entanglement entropy for disconnected regions in quantum field theory. We study here the property of extensivity of the mutual information, which holds for free massless fermions in two dimensions. We uncover the structure of the entropy function in the extensive case, and find an interesting connection with the renormalization group irreversibility. The solution is a function on space-time regions which complies with all the known requirements a relativistic entropy function has to satisfy. We show that the holographic ansatz of Ryu and Takayanagi, the free scalar and Dirac fields in dimensions greater than two, and the massive free fields in two dimensions all fail to be exactly extensive, disproving recent conjectures.Comment: 14 pages, 4 figures, some addition

On reduced density matrices for disjoint subsystems

We show that spin and fermion representations for solvable quantum chains lead in general to different reduced density matrices if the subsystem is not singly connected. We study the effect for two sites in XX and XY chains as well as for sublattices in XX and transverse Ising chains.Comment: 10 pages, 4 figure

Bi-partite entanglement entropy in massive (1+1)-dimensional quantum field theories

This paper is a review of the main results obtained in a series of papers involving the present authors and their collaborator J L Cardy over the last 2 years. In our work, we have developed and applied a new approach for the computation of the bi-partite entanglement entropy in massive (1+1)-dimensional quantum field theories. In most of our work we have also considered these theories to be integrable. Our approach combines two main ingredients: the 'replica trick' and form factors for integrable models and more generally for massive quantum field theory. Our basic idea for combining fruitfully these two ingredients is that of the branch-point twist field. By the replica trick, we obtained an alternative way of expressing the entanglement entropy as a function of the correlation functions of branch-point twist fields. On the other hand, a generalization of the form factor program has allowed us to study, and in integrable cases to obtain exact expressions for, form factors of such twist fields. By the usual decomposition of correlation functions in an infinite series involving form factors, we obtained exact results for the infrared behaviours of the bi-partite entanglement entropy, and studied both its infrared and ultraviolet behaviours for different kinds of models: with and without boundaries and backscattering, at and out of integrability

Reflexive Cones

Reflexive cones in Banach spaces are cones with weakly compact intersection with the unit ball. In this paper we study the structure of this class of cones. We investigate the relations between the notion of reflexive cones and the properties of their bases. This allows us to prove a characterization of reflexive cones in term of the absence of a subcone isomorphic to the positive cone of \ell_{1}. Moreover, the properties of some specific classes of reflexive cones are investigated. Namely, we consider the reflexive cones such that the intersection with the unit ball is norm compact, those generated by a Schauder basis and the reflexive cones regarded as ordering cones in a Banach spaces. Finally, it is worth to point out that a characterization of reflexive spaces and also of the Schur spaces by the properties of reflexive cones is given.Comment: 23 page

Area law and vacuum reordering in harmonic networks

We review a number of ideas related to area law scaling of the geometric entropy from the point of view of condensed matter, quantum field theory and quantum information. An explicit computation in arbitrary dimensions of the geometric entropy of the ground state of a discretized scalar free field theory shows the expected area law result. In this case, area law scaling is a manifestation of a deeper reordering of the vacuum produced by majorization relations. Furthermore, the explicit control on all the eigenvalues of the reduced density matrix allows for a verification of entropy loss along the renormalization group trajectory driven by the mass term. A further result of our computation shows that single-copy entanglement also obeys area law scaling, majorization relations and decreases along renormalization group flows.Comment: 15 pages, 6 figures; typos correcte

Analytic results on the geometric entropy for free fields

The trace of integer powers of the local density matrix 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 the density matrix 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 Painleve V type.Comment: 7 pages, minor change

Geotechnical and historical aspects on the collapse of the Tiber embankment walls in the centre of Roma (1870–1900)

This paper deals with the issues related to the construction of the Tiber’s embankment walls between years 1870–1926. The embankment walls (muraglioni) were designed by Raffaele Canevari to mitigate the effects of the river inundation in the city centre of Roma. After the flood of December 1900, several portions of the Anguillara and Alberteschi sections collapsed. The aim of this work is to investigate whether the causes of the collapse can be traced back to design approaches of the time, lacking from a point of view of the hydro-mechanical interaction of the soil in the evaluation of the total earth pressure. In particular, designed calculations are also revised accounting for more advance soil phenomena laws, based on Terzaghi effective stress and the effects of scouring and erosion. Some assumptions have been made on the mechanical characteristics of the backfill soils and on the relying on foundation materials

Universal parity effects in the entanglement entropy of XX chains with open boundary conditions

We consider the Renyi entanglement entropies in the one-dimensional XX spin-chains with open boundary conditions in the presence of a magnetic field. In the case of a semi-infinite system and a block starting from the boundary, we derive rigorously the asymptotic behavior for large block sizes on the basis of a recent mathematical theorem for the determinant of Toeplitz plus Hankel matrices. We conjecture a generalized Fisher-Hartwig form for the corrections to the asymptotic behavior of this determinant that allows the exact characterization of the corrections to the scaling at order o(1/l) for any n. By combining these results with conformal field theory arguments, we derive exact expressions also in finite chains with open boundary conditions and in the case when the block is detached from the boundary.Comment: 24 pages, 9 figure