Dyson's Coulomb gas on a circle and intermediate eigenvalue statistics

A random matrix representation is proposed for the two-dimensional (2D) Coulomb gas at inverse temperature β. For 2×2 matrices with Gaussian distribution we analytically compute the nearest-neighbor spacing distribution of complex eigenvalues in radial distance. Because it does not provide such a good approximation as the Wigner surmise in 1D, we introduce an effective β_{eff}(β) in our analytic formula that describes the spacing obtained numerically from the 2D Coulomb gas well for small values of β. It reproduces the 2D Poisson distribution at β=0 exactly, that is valid for a large particle number. The surmise is used to fit data in two examples, from open quantum spin chains and ecology. The spacing distributions of complex symmetric and complex quaternion self-dual ensembles of non-Hermitian random matrices, that are only known numerically, are very well fitted by noninteger values β=1.4 and β=2.6 from a 2D Coulomb gas, respectively. These two ensembles have been suggested as the only two symmetry classes, where the 2D bulk statistics is different from the Ginibre ensemble.

We collect explicit and user-friendly expressions for one-point densities of\nthe real eigenvalues $\\{\\lambda_i\\}$ of $N\\times N$ Wishart-Laguerre and Jacobi\nrandom matrices with orthogonal, unitary and symplectic symmetry. Using these\nformulae, we compute integer moments $\\tau_n=<\\sum_{i=1}^N\\lambda_i^n>$ for all\nsymmetry classes without any large $N$ approximation. In particular, our\nresults provide exact expressions for moments of transmission eigenvalues in\nchaotic cavities with time-reversal or spin-flip symmetry and supporting a\nfinite and arbitrary number of electronic channels in the two incoming leads.\n

In this article we consider the effects on ensembles of random Hamiltonian matrices when certain restrictions are imposed on the potentials of the corresponding Schrödinger Hamiltonians. In particular, we investigate the validity of the assumption which is usually made when ensembles of random matrices are used to predict statistical properties of energy levels of complex systems, namely that the Hamiltonian matrix elements are statistically independent.

The model under consideration is an asymmetric two-dimensional Coulomb gas of positively (q1=+1) and negatively (q2=−1/2) charged pointlike particles, interacting via a logarithmic potential. This continuous system is stable against collapse of positive-negative pairs of charges for the dimensionless coupling constant (inverse temperature) β<4. The mapping of the Coulomb gas is made onto the complex Bullough–Dodd model, and recent results about that integrable 2D field theory are used. The mapping provides the full thermodynamics (the free energy, the internal energy, the specific heat) and the large-distance asymptotics of the particle correlation functions, in the whole stability regime of the plasma. The results are checked by a small-β expansion and close to the collapse β=4 point. The comparison is made with the exactly solvable symmetric version of the model (q1=+1,q2=−1), and some fundamental changes in statistics caused by the charge asymmetry are pointed out.

Stochastic properties of many-body systems

A detailed, rigorous study of the statistical mechanics-screening- and critical properties, phase diagrams, etc., of classical Coulomb monopole and dipole gases in two or more dimensions is presented. The statistical mechanics of the two-dimensionalXY and Villain models is reconsidered and related to the one of two-dimensional lattice Coulomb gases. At low temperatures and moderate densities those gases behave like dipole gases. The Kosterlitz-Thouless transition is analyzed in that perspective and characterized by an order parameter. Techniques useful for a proof of existence of such a transition in a two-dimensional hard-core Coulomb gas are developed and applied to the study of dipole gases.

I– V characteristics for 2D superconductors from simulations of the 2D Coulomb gas: density dependence and exponents

Transmission measurements through three-port microwave graphs are performed, in analogy to three-terminal voltage drop devices with orthogonal, unitary, and symplectic symmetry. The terminal used as a probe is symmetrically located between two chaotic subgraphs, and each graph is connected to one port, the input and the output, respectively. The analysis of the experimental data clearly exhibits the weak localization and antilocalization phenomena. We find a good agreement with theoretical predictions, provided that the effects of dissipation and imperfect coupling to the ports are taken into account.

The class of norm-dependent Random Matrix Ensembles is studied in the presence of an external field. The probability density in those ensembles depends on the trace of the squared random matrices, but is otherwise arbitrary. An exact mapping to superspace is performed. A transformation formula is derived which gives the probability density in superspace as a single integral over the probability density in ordinary space. This is done for orthogonal, unitary and symplectic symmetry. In the case of unitary symmetry, some explicit results for the correlation functions are derived.

We carry out Monte Carlo simulations of the two-dimensional netural Coulomb gas of integer charges on a triangular lattice, as a function of temperature and chemical potential, and find rich critical behavior. We find transitions which do not seem to follow the expectations of a symmetry analysis, and nonuniversal jumps in the inverse dielectric constant.

We study the two-dimensional Ginzburg-Landau model of a neutral superfluid in the vicinity of the vortex unbinding transition. The model is mapped onto an effective interacting vortex gas by a systematic perturbative elimination of all fluctuating degrees of freedom (amplitudeand phase of the order parameter field) except the vortex positions. In the Coulomb gas descriptions derived previously in the literature, thermal amplitude fluctuations were neglected altogether. We argue that if one includes the latter, the vortices still form a two-dimensional Coulomb gas, but the vortex fugacity can be substantially raised. Under the assumption that Minnhagen's generic phase diagram of the two-dimensional Coulomb gas is correct, our results then point to a first-order transition rather than a Kosterlitz-Thouless transition, provided the Ginzburg-Landau correlation length is large enough in units of a microscopic cutoff lenght for fluctuations. The experimental relevance of these results is briefly discussed.

Complex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two independent eigenvalue sequences distributed according to particular matrix ensembles with chiral unitary symmetry. We give decompositions of the distribution of singular values, and the decimation of the singular values – whereby only even, or odd, labels are observed – for real symmetric random matrices with an orthogonal symmetry, and even weight. This requires further specifying the functional form of the weight to one of three types – Gauss, symmetric Jacobi or Cauchy. Inter-relations between gap probabilities with orthogonal and unitary symmetry follow as a corollary. The Gauss case has appeared in a recent work of Bornemann and La Croix. The Cauchy case, when appropriately specialised and upon stereographic projection, gives decompositions for the analogue of the singular values for the circular unitary and circular orthogonal ensembles.

Calculations of correlated remote ion scattering limited mobility of modulation delta-doped GaAs-AlGaAs heterostructures are presented. The scatterers are modelled as a two-dimensional Coulomb gas whose distribution is determined directly through Monte Carlo simulations. Trends in the variation of the mobility with correlations in scatterer distributions (Coulomb gas temperature), density of the scatterers, density of electrons and the spacer width are discussed.

We carry out Monte Carlo simulations of the two-dimensional classical neutral Coulomb gas of integer charges on a square and a triangular lattice, and find rich phase diagrams as a function of temperature and chemical potential. At high densities, the ground state becomes a periodic charge lattice, and we find evidence suggesting that the melting transitions of these charge lattices are not always in the universality class expected from symmetry analysis. We compute the inverse dielectric constant ${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{\ensuremath{-}}1}$, which vanishes at the ``metal-insulator'' transition of the Coulomb gas. When the ground state is a charge vacuum, we find that ${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{\ensuremath{-}}1}$ is well described by the Kosterlitz-Thouless (KT) theory. When the ground state is a charge lattice, a larger than the KT universal jump in ${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{\ensuremath{-}}1}$(${\mathit{T}}_{\mathit{c}}$) may be present. We also carry out simulations of the f=1/2 and 1/3 fractional Coulomb gas on a triangular lattice, and find similar results.

We consider two-dimensional Coulomb gases (standard or hard-core) with activityz, and prove that for any β>8π the two-point external charges correlation function exhibits the power-law falloff characteristic of the Kosterlitz-Thouless phase at sufficiently low activity.