On the properties of level spacings for decomposable systems

In this paper we show that the quantum theory of chaos, based on the statistical theory of energy spectra, presents inconsistencies difficult to overcome. In classical mechanics a system described by an hamiltonian $H = H_1 + H_2$ (decomposable) cannot be ergodic, because there are always two dependent integrals of motion besides the constant of energy. In quantum mechanics we prove the existence of decomposable systems \linebreak $H^q = H^q_1 + H^q_2$ whose spacing distribution agrees with the Wigner law and we show that in general the spacing distribution of $H^q$ is not the Poisson law, even if it has often the same qualitative behaviour. We have found that the spacings of $H^q$ are among the solutions of a well defined class of homogeneous linear systems. We have obtained an explicit formula for the bases of the kernels of these systems, and a chain of inequalities which the coefficients of a generic linear combination of the basis vectors must satisfy so that the elements of a particular solution will be all positive, i.e. can be considered a set of spacings.Comment: LateX, 13 page

Free fermions and the classical compact groups

There is a close connection between the ground state of non-interacting fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: i) we consider the full family of admissible quantum boundary conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture by constructing a finite temperature extension of the Haar measure on the classical compact groups. The eigenvalue statistics of the resulting grand canonical matrix models (of random size) corresponds exactly to the grand canonical measure of non-interacting free fermions with classical boundary conditions.Comment: 35 pages, 5 figures. Final versio

Comb entanglement in quantum spin chains

Bipartite entanglement in the ground state of a chain of $N$ quantum spins can be quantified either by computing pairwise concurrence or by dividing the chain into two complementary subsystems. In the latter case the smaller subsystem is usually a single spin or a block of adjacent spins and the entanglement differentiates between critical and non-critical regimes. Here we extend this approach by considering a more general setting: our smaller subsystem $S_A$ consists of a {\it comb} of $L$ spins, spaced $p$ sites apart. Our results are thus not restricted to a simple `area law', but contain non-local information, parameterized by the spacing $p$. For the XX model we calculate the von-Neumann entropy analytically when $N\to \infty$ and investigate its dependence on $L$ and $p$. We find that an external magnetic field induces an unexpected length scale for entanglement in this case.Comment: 6 pages, 4 figure

On relations between one-dimensional quantum and two-dimensional classical spin systems

We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systems with critical properties equivalent to those of the class of one-dimensional quantum systems discussed in a companion paper (J. Hutchinson, J. P. Keating, and F. Mezzadri, arXiv:1503.05732). In particular, we use three approaches: the Trotter-Suzuki mapping; the method of coherent states; and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classical system. This enables us to establish universality of certain critical phenomena by extension from the results in our previous article for the classical systems identified.Comment: 36 page