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Register a new userGeneralized random matrix model with additional interactions
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We introduce a log-gas model that is a generalization of a random matrix ensemble with an additional interaction, whose strength depends on a parameter $gamma$. The equilibrium density is computed by numerically solving the Riemann-Hilbert problem associated with the ensemble. The effect of the additional parameter $gamma$ associated with the two-body interaction can be understood in terms of an effective $gamma$-dependent single-particle confining potential.
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We consider several limiting cases of the joint probability distribution for a random matrix ensemble with an additional interaction term controlled by an exponent $gamma$ (called the $gamma$-ensembles). The effective potential, which is essentially the single-particle confining potential for an equivalent ensemble with $gamma=1$ (called the Muttalib-Borodin ensemble), is a crucial quantity defined in solution to the Riemann-Hilbert problem associated with the $gamma$-ensembles. It enables us to numerically compute the eigenvalue density of $gamma$-ensembles for all $gamma > 0$. We show that one important effect of the two-particle interaction parameter $gamma$ is to generate or enhance the non-monotonicity in the effective single-particle potential. For suitable choices of the initial single-particle potentials, reducing $gamma$ can lead to a large non-monotonicity in the effective potential, which in turn leads to significant changes in the density of eigenvalues. For a disordered conductor, this corresponds to a systematic decrease in the conductance with increasing disorder. This suggests that appropriate models of $gamma$-ensembles can be used as a possible framework to study the effects of disorder on the distribution of conductances.
The geometry of multi-parameter families of quantum states is important in numerous contexts, including adiabatic or nonadiabatic quantum dynamics, quantum quenches, and the characterization of quantum critical points. Here, we discuss the Hilbert-space geometry of eigenstates of parameter-dependent random-matrix ensembles, deriving the full probability distribution of the quantum geometric tensor for the Gaussian Unitary Ensemble. Our analytical results give the exact joint distribution function of the Fubini-Study metric and the Berry curvature. We discuss relations to Levy stable distributions and compare our results to numerical simulations of random-matrix ensembles as well as electrons in a random magnetic field.
We study analytically and numerically the dynamics of the generalized Rosenzweig-Porter model, which is known to possess three distinct phases: ergodic, multifractal and localized phases. Our focus is on the survival probability $R(t)$, the probability of finding the initial state after time $t$. In particular, if the system is initially prepared in a highly-excited non-stationary state (wave packet) confined in space and containing a fixed fraction of all eigenstates, we show that $R(t)$ can be used as a dynamical indicator to distinguish these three phases. Three main aspects are identified in different phases. The ergodic phase is characterized by the standard power-law decay of $R(t)$ with periodic oscillations in time, surviving in the thermodynamic limit, with frequency equals to the energy bandwidth of the wave packet. In multifractal extended phase the survival probability shows an exponential decay but the decay rate vanishes in the thermodynamic limit in a non-trivial manner determined by the fractal dimension of wave functions. Localized phase is characterized by the saturation value of $R(ttoinfty)=k$, finite in the thermodynamic limit $Nrightarrowinfty$, which approaches $k=R(tto 0)$ in this limit.
Theory of Random Matrix Ensembles have proven to be a useful tool in the study of the statistical distribution of energy or transmission levels of a wide variety of physical systems. We give an overview of certain q-generalizations of the Random Matrix Ensembles, which were first introduced in connection with the statistical description of disordered quantum conductors.
We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large $N$ results are obtained by using Szego theorem with a Fisher-Hartwig singularity. In the large $N$ (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space.
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