Surface and bulk entanglement in free-fermion chains

We consider free-fermion chains where full and empty parts are connected by a transition region with narrow surfaces. This can be caused by a linear potential or by time evolution from a step-like initial state. Entanglement spectra, entanglement entropies and fluctuations are determined for subsystems either in the surface region or extending into the bulk. In all cases there is logarithmic behaviour in the subsystem size, but the prefactors in the surface differ from those in the bulk by 3/2. A previous fluctuation result is corrected and a general scaling formula is inferred from the data.Comment: 14 pages, 6 figures, minor changes, references adde

Casimir Terms and Shape Instabilities for Two-Dimensional Critical Systems

We calculate the universal part of the free energy of certain finite two- dimensional regions at criticality by use of conformal field theory. Two geometries are considered: a section of a circle ("pie slice") of angle \phi and a helical staircase of finite angular (and radial) extent. We derive some consequences for certain matrix elements of the transfer matrix and corner transfer matrix. We examine the total free energy, including non- universal edge free energy terms, in both cases. A new, general, Casimir instability toward sharp corners on the boundary is found; other new instability behavior is investigated. We show that at constant area and edge length, the rectangle is unstable against small curvature.Comment: 15 pages PostScript, accepted for publication in Z. Phys.

Free-fermion entanglement and spheroidal functions

We consider the entanglement properties of free fermions in one dimension and review an approach which relates the problem to the solution of a certain differential equation. The single-particle eigenfunctions of the entanglement Hamiltonian are then seen to be spheroidal functions or generalizations of them. The analytical results for the eigenvalue spectrum agree with those obtained by other methods. In the continuum case, there are close connections to random matrix theory.Comment: 17 pages, 4 figures, figures update

Solution of a One-Dimensional Reaction-Diffusion Model with Spatial Asymmetry

We study classical particles on the sites of an open chain which diffuse, coagulate and decoagulate preferentially in one direction. The master equation is expressed in terms of a spin one-half Hamiltonian $H$ and the model is shown to be completely solvable if all processes have the same asymmetry. The relaxational spectrum is obtained directly from $H$ and via the equations of motion for strings of empty sites. The structure and the solvability of these equations are investigated in the general case. Two phases are shown to exist for small and large asymmetry, respectively, which differ in their stationary properties.Comment: 18 pages, latex, 1 PostScript figure, uuencode

Corrections to scaling for block entanglement in massive spin-chains

We consider the Renyi entropies S_n in one-dimensional massive integrable models diagonalizable by means of corner transfer matrices (as Heisenberg and Ising spin chains). By means of explicit examples and using the relation of corner transfer matrix with the Virasoro algebra, we show that close to a conformal invariant critical point, when the correlation length xi is finite but large, the corrections to the scaling are of the unusual form xi^(-x/n), with x the dimension of a relevant operator in the conformal theory. This is reminiscent of the results for gapless chains and should be valid for any massive one-dimensional model close to a conformal critical point.Comment: 12 pages, no figures. v2 corrected typo

Critical entanglement of XXZ Heisenberg chains with defects

We study the entanglement properties of anisotropic open spin one-half Heisenberg chains with a modified central bond. The entanglement entropy between the two half-chains is calculated with the density-matrix renormalization method (DMRG).We find a logarithmic behaviour with an effective central charge c' varying with the length of the system. It flows to one in the ferromagnetic region and to zero in the antiferromagnetic region of the model. In the XX case it has a non-universal limit and we recover previous results.Comment: 8 pages, 15 figure

Entanglement in composite free-fermion systems

We consider fermionic chains where the two halves are either metals with different bandwidths or a metal and an insulator. Both are coupled together by a special bond. We study the ground-state entanglement entropy between the two pieces, its dependence on the parameters and its asymptotic form. We also discuss the features of the entanglement Hamiltonians in both subsystems and the evolution of the entanglement entropy after joining the two parts of the system.Comment: 20 pages, 13 figures, published version, minor corrections, references adde

Properties of the entanglement Hamiltonian for finite free-fermion chains

We study the entanglement Hamiltonian for fermionic hopping models on rings and open chains and determine single-particle spectra, eigenfunctions and the form in real space. For the chain, we find a commuting operator as for the ring and compare with its properties in both cases. In particular, a scaling relation between the eigenvalues is found for large systems. We also show how the commutation property carries over to the critical transverse Ising model

Analytical results for the entanglement Hamiltonian of a free-fermion chain

We study the ground-state entanglement Hamiltonian for an interval of $N$ sites in a free-fermion chain with arbitrary filling. By relating it to a commuting operator, we find explicit expressions for its matrix elements in the large-$N$ limit. The results agree with numerical calculations and show that deviations from the conformal prediction persist even for large systems.Comment: 21 pages, 8 figures. Dedicated to John Cardy on the occasion of his 70th birthday. v2: minor corrections, published versio