Extreme points of the set of density matrices with positive partial transpose

We present a necessary and sufficient condition for a finite dimensional density matrix to be an extreme point of the convex set of density matrices with positive partial transpose with respect to a subsystem. We also give an algorithm for finding such extreme points and illustrate this by some examples.Comment: 4 pages, 2 figure

Low rank positive partial transpose states and their relation to product vectors

It is known that entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. In a previous paper we presented a classification of rank four entangled PPT states which we believe to be complete. In the present paper we continue our investigations of the low rank entangled PPT states. We use perturbation theory in order to construct rank five entangled PPT states close to the known rank four states, and in order to compute dimensions and study the geometry of surfaces of low rank PPT states. We exploit the close connection between low rank PPT states and product vectors. In particular, we show how to reconstruct a PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.Comment: 29 pages, 4 figure

Correlators and fractional statistics in the quantum Hall bulk

We derive single-particle and two-particle correlators of anyons in the presence of a magnetic field in the lowest Landau level. We show that the two-particle correlator exhibits signatures of fractional statistics which can distinguish anyons from their fermionic and bosonic counterparts. These signatures include the zeroes of the two-particle correlator and its exclusion behavior. We find that the single-particle correlator in finite geometries carries valuable information relevant to experiments in which quasiparticles on the edge of a quantum Hall system tunnel through its bulk.Comment: 4 pages, 3 figures, RevTe

Gravity and Matter in Causal Set Theory

The goal of this paper is to propose an approach to the formulation of dynamics for causal sets and coupled matter fields. We start from the continuum version of the action for a Klein-Gordon field coupled to gravity, and rewrite it first using quantities that have a direct correspondent in the case of a causal set, namely volumes, causal relations, and timelike lengths, as variables to describe the geometry. In this step, the local Lagrangian density $L(f;x)$ for a set of fields $f$ is recast into a quasilocal expression $L_0(f;p,q)$ that depends on pairs of causally related points $p \prec q$ and is a function of the values of $f$ in the Alexandrov set defined by those points, and whose limit as $p$ and $q$ approach a common point $x$ is $L(f;x)$. We then describe how to discretize $L_0(f;p,q)$, and use it to define a discrete action.Comment: 13 pages, no figures; In version 2, friendlier results than in version 1 are obtained following much shorter derivation

The Random Discrete Action for 2-Dimensional Spacetime

A one-parameter family of random variables, called the Discrete Action, is defined for a 2-dimensional Lorentzian spacetime of finite volume. The single parameter is a discreteness scale. The expectation value of this Discrete Action is calculated for various regions of 2D Minkowski spacetime. When a causally convex region of 2D Minkowski spacetime is divided into subregions using null lines the mean of the Discrete Action is equal to the alternating sum of the numbers of vertices, edges and faces of the null tiling, up to corrections that tend to zero as the discreteness scale is taken to zero. This result is used to predict that the mean of the Discrete Action of the flat Lorentzian cylinder is zero up to corrections, which is verified. The ``topological'' character of the Discrete Action breaks down for causally convex regions of the flat trousers spacetime that contain the singularity and for non-causally convex rectangles.Comment: 20 pages, 10 figures, Typos correcte

A numerical study of the correspondence between paths in a causal set and geodesics in the continuum

This paper presents the results of a computational study related to the path-geodesic correspondence in causal sets. For intervals in flat spacetimes, and in selected curved spacetimes, we present evidence that the longest maximal chains (the longest paths) in the corresponding causal set intervals statistically approach the geodesic for that interval in the appropriate continuum limit.Comment: To the celebration of the 60th birthday of Rafael D. Sorki

Bethe Ansatz solution of a new class of Hubbard-type models

We define one-dimensional particles with generalized exchange statistics. The exact solution of a Hubbard-type Hamiltonian constructed with such particles is achieved using the Coordinate Bethe Ansatz. The chosen deformation of the statistics is equivalent to the presence of a magnetic field produced by the particles themselves, which is present also in a ``free gas'' of these particles.Comment: 4 pages, revtex. Essentially modified versio

Haldane exclusion statistics and second virial coefficient

We show that Haldanes new definition of statistics, when generalised to infinite dimensional Hilbert spaces, is equal to the high temperature limit of the second virial coefficient. We thus show that this exclusion statistics parameter, g , of anyons is non-trivial and is completely determined by its exchange statistics parameter $\alpha$. We also compute g for quasiparticles in the Luttinger model and show that it is equal to $\alpha$.Comment: 11 pages, REVTEX 3.