Norm-dependent Random Matrix Ensembles in External Field and Supersymmetry

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.Comment: 19 page

Arbitrary Rotation Invariant Random Matrix Ensembles and Supersymmetry

We generalize the supersymmetry method in Random Matrix Theory to arbitrary rotation invariant ensembles. Our exact approach further extends a previous contribution in which we constructed a supersymmetric representation for the class of norm-dependent Random Matrix Ensembles. Here, we derive a supersymmetric formulation under very general circumstances. A projector is identified that provides the mapping of the probability density from ordinary to superspace. Furthermore, it is demonstrated that setting up the theory in Fourier superspace has considerable advantages. General and exact expressions for the correlation functions are given. We also show how the use of hyperbolic symmetry can be circumvented in the present context in which the non-linear sigma model is not used. We construct exact supersymmetric integral representations of the correlation functions for arbitrary positions of the imaginary increments in the Green functions.Comment: 36 page

The k-Point Random Matrix Kernels Obtained from One-Point Supermatrix Models

The k-point correlation functions of the Gaussian Random Matrix Ensembles are certain determinants of functions which depend on only two arguments. They are referred to as kernels, since they are the building blocks of all correlations. We show that the kernels are obtained, for arbitrary level number, directly from supermatrix models for one-point functions. More precisely, the generating functions of the one-point functions are equivalent to the kernels. This is surprising, because it implies that already the one-point generating function holds essential information about the k-point correlations. This also establishes a link to the averaged ratios of spectral determinants, i.e. of characteristic polynomials

Invariant Manifolds and Collective Coordinates

We introduce suitable coordinate systems for interacting many-body systems with invariant manifolds. These are Cartesian in coordinate and momentum space and chosen such that several components are identically zero for motion on the invariant manifold. In this sense these coordinates are collective. We make a connection to Zickendraht's collective coordinates and present certain configurations of few-body systems where rotations and vibrations decouple from single-particle motion. These configurations do not depend on details of the interaction.Comment: 15 pages, 2 EPS-figures, uses psfig.st

Stochastic field theory for a Dirac particle propagating in gauge field disorder

Recent theoretical and numerical developments show analogies between quantum chromodynamics (QCD) and disordered systems in condensed matter physics. We study the spectral fluctuations of a Dirac particle propagating in a finite four dimensional box in the presence of gauge fields. We construct a model which combines Efetov's approach to disordered systems with the principles of chiral symmetry and QCD. To this end, the gauge fields are replaced with a stochastic white noise potential, the gauge field disorder. Effective supersymmetric non-linear sigma-models are obtained. Spontaneous breaking of supersymmetry is found. We rigorously derive the equivalent of the Thouless energy in QCD. Connections to other low-energy effective theories, in particular the Nambu-Jona-Lasinio model and chiral perturbation theory, are found.Comment: 4 pages, 1 figur

Equivalent of a Thouless energy in lattice QCD Dirac spectra

Random matrix theory (RMT) is a powerful statistical tool to model spectral fluctuations. In addition, RMT provides efficient means to separate different scales in spectra. Recently RMT has found application in quantum chromodynamics (QCD). In mesoscopic physics, the Thouless energy sets the universal scale for which RMT applies. We try to identify the equivalent of a Thouless energy in complete spectra of the QCD Dirac operator with staggered fermions and $SU_c(2)$ lattice gauge fields. Comparing lattice data with RMT predictions we find deviations which allow us to give an estimate for this scale.Comment: LATTICE99 (theor. devel.), 3 pages, 4 figure

Transition from Poisson to gaussian unitary statistics: The two-point correlation function

We consider the Rosenzweig-Porter model of random matrix which interpolates between Poisson and gaussian unitary statistics and compute exactly the two-point correlation function. Asymptotic formulas for this function are given near the Poisson and gaussian limit.Comment: 19 pages, no figure