arXiv subjects

H. -J. Sommers

Publications and source records attributed to H. -J. Sommers.

At least 19 recordsLinked to original sources

arXiv2012

Induced Ginibre ensemble of random matrices and quantum operations

A generalisation of the Ginibre ensemble of non-Hermitian random square matrices is introduced. The corresponding probability measure is induced by the ensemble of rectangular Gaussian matrices via a quadratisation procedure. We derive the joint probability density of eigenvalues for such induced Ginibre ensemble and study various spectral correlation functions for complex and real matrices, and analyse universal behaviour in the limit of large dimensions. In this limit the eigenvalues of the induced Ginibre ensemble cover uniformly a ring in the complex plane. The real induced Ginibre ensemble is shown to be useful to describe statistical properties of evolution operators associated with random quantum operations, for which the dimensions of the input state and the output state do differ.

math-ph↗

arXiv2009

Non-Hermitian Random Matrix Ensembles

This is a concise review of the complex, real and quaternion real Ginibre random matrix ensembles and their elliptic deformations. Eigenvalue correlations are exactly reduced to two-point kernels and discussed in the strongly and weakly non-Hermitian limits of large matrix size.

math-ph↗

arXiv2009

Characteristic polynomials in real Ginibre ensembles

We calculate the average of two characteristic polynomials for the real Ginibre ensemble of asymmetric random matrices, and its chiral counterpart. Considered as quadratic forms they determine a skew-symmetric kernel from which all complex eigenvalue correlations can be derived. Our results are obtained in a very simple fashion without going to an eigenvalue representation, and are completely new in the chiral case. They hold for Gaussian ensembles which are partly symmetric, with kernels given in terms of Hermite and Laguerre polynomials respectively, depending on an asymmetry parameter. This allows us to interpolate between the maximally asymmetric real Ginibre and the Gaussian Orthogonal Ensemble, as well as their chiral counterparts.

math-ph↗

arXiv2009

The chiral Gaussian two-matrix ensemble of real asymmetric matrices

We solve a family of Gaussian two-matrix models with rectangular Nx(N+v) matrices, having real asymmetric matrix elements and depending on a non-Hermiticity parameter mu. Our model can be thought of as the chiral extension of the real Ginibre ensemble, relevant for Dirac operators in the same symmetry class. It has the property that its eigenvalues are either real, purely imaginary, or come in complex conjugate eigenvalue pairs. The eigenvalue joint probability distribution for our model is explicitly computed, leading to a non-Gaussian distribution including K-Bessel functions. All n-point density correlation functions are expressed for finite N in terms of a Pfaffian form. This contains a kernel involving Laguerre polynomials in the complex plane as a building block which was previously computed by the authors. This kernel can be expressed in terms of the kernel for complex non-Hermitian matrices, generalising the known relation among ensembles of Hermitian random matrices. Compact expressions are given for the density at finite N as an example, as well as its microscopic large-N limits at the origin for fixed v at strong and weak non-Hermiticity.

hep-th↗

arXiv2009

Random Bistochastic Matrices

Ensembles of random stochastic and bistochastic matrices are investigated. While all columns of a random stochastic matrix can be chosen independently, the rows and columns of a bistochastic matrix have to be correlated. We evaluate the probability measure induced into the Birkhoff polytope of bistochastic matrices by applying the Sinkhorn algorithm to a given ensemble of random stochastic matrices. For matrices of order N=2 we derive explicit formulae for the probability distributions induced by random stochastic matrices with columns distributed according to the Dirichlet distribution. For arbitrary $N$ we construct an initial ensemble of stochastic matrices which allows one to generate random bistochastic matrices according to a distribution locally flat at the center of the Birkhoff polytope. The value of the probability density at this point enables us to obtain an estimation of the volume of the Birkhoff polytope, consistent with recent asymptotic results.

nlin.SI↗

arXiv2009

Systematic approach to statistics of conductance and shot-noise in chaotic cavities

Applying random matrix theory to quantum transport in chaotic cavities, we develop a novel approach to computation of the moments of the conductance and shot-noise (including their joint moments) of arbitrary order and at any number of open channels. The method is based on the Selberg integral theory combined with the theory of symmetric functions and is applicable equally well for systems with and without time-reversal symmetry. We also compute higher-order cumulants and perform their detailed analysis. In particular, we establish an explicit form of the leading asymptotic of the cumulants in the limit of the large channel numbers. We derive further a general Pfaffian representation for the corresponding distribution functions. The Edgeworth expansion based on the first four cumulants is found to reproduce fairly accurately the distribution functions in the bulk even for a small number of channels. As the latter increases, the distributions become Gaussian-like in the bulk but are always characterized by a power-law dependence near their edges of support. Such asymptotics are determined exactly up to linear order in distances from the edges, including the corresponding constants.

cond-mat.mes-hall↗

arXiv2008

Superbosonization of invariant random matrix ensembles

Superbosonization is a new variant of the method of commuting and anti-commuting variables as used in studying random matrix models of disordered and chaotic quantum systems. We here give a concise mathematical exposition of the key formulas of superbosonization. Conceived by analogy with the bosonization technique for Dirac fermions, the new method differs from the traditional one in that the superbosonization field is dual to the usual Hubbard-Stratonovich field. The present paper addresses invariant random matrix ensembles with symmetry group U(n), O(n), or USp(n), giving precise definitions and conditions of validity in each case. The method is illustrated at the example of Wegner's n-orbital model. Superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non-Gaussian and/or non-invariant type.

math-ph↗

arXiv2007

Nonlinear statistics of quantum transport in chaotic cavities

In the framework of the random matrix approach, we apply the theory of Selberg's integral to problems of quantum transport in chaotic cavities. All the moments of transmission eigenvalues are calculated analytically up to the fourth order. As a result, we derive exact explicit expressions for the skewness and kurtosis of the conductance and transmitted charge as well as for the variance of the shot-noise power in chaotic cavities. The obtained results are generally valid at arbitrary numbers of propagating channels in the two attached leads. In the particular limit of large (and equal) channel numbers, the shot-noise variance attends the universal value 1/(64β) that determines a universal Gaussian statistics of shot-noise fluctuations in this case.

cond-mat.mes-hall↗

arXiv2007

Statistics of conductance and shot-noise power for chaotic cavities

We report on an analytical study of the statistics of conductance, $g$, and shot-noise power, $p$, for a chaotic cavity with arbitrary numbers $N_{1,2}$ of channels in two leads and symmetry parameter $β= 1,2,4$. With the theory of Selberg's integral the first four cumulants of $g$ and first two cumulants of $p$ are calculated explicitly. We give analytical expressions for the conductance and shot-noise distributions and determine their exact asymptotics near the edges up to linear order in distances from the edges. For $0<g<1$ a power law for the conductance distribution is exact. All results are also consistent with numerical simulations.

cond-mat.mes-hall↗

arXiv2007

Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian

We investigate thermodynamics of a single classical particle placed in a spherical box of a finite radius $R$ and subject to a superposition of a $N-$dimensional Gaussian random potential and the parabolic potential with the curvature $μ>0$. Earlier solutions of $R\to \infty$ version of this model were based on combining the replica trick with the Gaussian Variational Ansatz (GVA) for free energy, and revealed a possibility of a glassy phase at low temperatures. For a general $R$, we show how to utilize instead the underlying rotational symmetry of the replicated partition function and to arrive to a compact expression for the free energy in the limit $N\to \infty$ directly, without any need for intermediate variational approximations. This method reveals striking similarity with the much-studied spherical model of spin glasses. Depending on the value of $R$ and the three types of disorder - short-ranged, long-ranged, and logarithmic - the phase diagram of the system in the $(μ,T)$ plane undergoes considerable modifications. In the limit of infinite confinement radius our analysis confirms all previous results obtained by GVA.

cond-mat.dis-nn↗

arXiv2006

Energy correlations for a random matrix model of disordered bosons

Linearizing the Heisenberg equations of motion around the ground state of an interacting quantum many-body system, one gets a time-evolution generator in the positive cone of a real symplectic Lie algebra. The presence of disorder in the physical system determines a probability measure with support on this cone. The present paper analyzes a discrete family of such measures of exponential type, and does so in an attempt to capture, by a simple random matrix model, some generic statistical features of the characteristic frequencies of disordered bosonic quasi-particle systems. The level correlation functions of the said measures are shown to be those of a determinantal process, and the kernel of the process is expressed as a sum of bi-orthogonal polynomials. While the correlations in the bulk scaling limit are in accord with sine-kernel or GUE universality, at the low-frequency end of the spectrum an unusual type of scaling behavior is found.

cond-mat.mes-hall↗

arXiv2006

Shot noise in chaotic cavities with an arbitrary number of open channels

Using the random matrix approach, we calculate analytically the average shot-noise power in a chaotic cavity at an arbitrary number of propagating modes (channels) in each of the two attached leads. A simple relationship between this quantity, the average conductance and the conductance variance is found. The dependence of the Fano factor on the channel number is considered in detail.

cond-mat.mes-hall↗

arXiv2005

Scattering, reflection and impedance of waves in chaotic and disordered systems with absorption

We review recent progress in analysing wave scattering in systems with both intrinsic chaos and/or disorder and internal losses, when the scattering matrix is no longer unitary. By mapping the problem onto a nonlinear supersymmetric sigma-model, we are able to derive closed form analytic expressions for the distribution of reflection probability in a generic disordered system. One of the most important properties resulting from such an analysis is statistical independence between the phase and the modulus of the reflection amplitude in every perfectly open channel. The developed theory has far-reaching consequences for many quantities of interest, including local Green functions and time delays. In particular, we point out the role played by absorption as a sensitive indicator of mechanisms behind the Anderson localisation transition. We also provide a random-matrix-based analysis of S-matrix and impedance correlations for various symmetry classes as well as the distribution of transmitted power for systems with broken time-reversal invariance, completing previous works on the subject. The results can be applied, in particular, to the experimentally accessible impedance and reflection in a microwave or ultrasonic cavity attached to a system of antennas.

cond-mat.mes-hall↗

arXiv2005

Universal statistics of the local Green's function in quantum chaotic systems with absorption

We establish a general relation between the statistics of the local Green's function for systems with chaotic wave scattering and a uniform energy loss (absorption) and its two-point correlation function for the same system without absorption. Within the random matrix approach this kind of a fluctuation dissipation relation allows us to derive the explicit analytical expression for the joint distribution function of the real and imaginary parts of the local Green function for all symmetry classes as well as at an arbitrary degree of the time-reversal symmetry breaking in the system. The outstanding problem of the orthogonal symmetry is further reduced to simple quadratures. The results can be applied, in particular, to the experimentally accessible impedance and reflection in a microwave cavity attached to a single-mode antenna.

cond-mat.mes-hall↗

arXiv2005

Correlation functions of impedance and scattering matrix elements in chaotic absorbing cavities

Wave scattering in chaotic systems with a uniform energy loss (absorption) is considered. Within the random matrix approach we calculate exactly the energy correlation functions of different matrix elements of impedance or scattering matrices for systems with preserved or broken time-reversal symmetry. The obtained results are valid at any number of arbitrary open scattering channels and arbitrary absorption. Elastic enhancement factors (defined through the ratio of the corresponding variance in reflection to that in transmission) are also discussed.

nlin.CD↗

arXiv2003

Distribution of reflection eigenvalues in many-channel chaotic cavities with absorption

The reflection matrix R=S^{\dagger}S, with S being the scattering matrix, differs from the unit one, when absorption is finite. Using the random matrix approach, we calculate analytically the distribution function of its eigenvalues in the limit of a large number of propagating modes in the leads attached to a chaotic cavity. The obtained result is independent on the presence of time-reversal symmetry in the system, being valid at finite absorption and arbitrary openness of the system. The particular cases of perfectly and weakly open cavities are considered in detail. An application of our results to the problem of thermal emission from random media is briefly discussed.

cond-mat.mes-hall↗

arXiv2003

Delay times and reflection in chaotic cavities with absorption

Absorption yields an additional exponential decay in open quantum systems which can be described by shifting the (scattering) energy E along the imaginary axis, E+i\hbar/2τ_{a}. Using the random matrix approach, we calculate analytically the distribution of proper delay times (eigenvalues of the time-delay matrix) in chaotic systems with broken time-reversal symmetry that is valid for an arbitrary number of generally nonequivalent channels and an arbitrary absorption rate 1/τ_{a}. The relation between the average delay time and the ``norm-leakage'' decay function is found. Fluctuations above the average at large values of delay times are strongly suppressed by absorption. The relation of the time-delay matrix to the reflection matrix S^{\dagger}S is established at arbitrary absorption that gives us the distribution of reflection eigenvalues. The particular case of single-channel scattering is explicitly considered in detail.

cond-mat.mes-hall↗

arXiv2003

Is the concept of the non-Hermitian effective Hamiltonian relevant in the case of potential scattering?

We examine the notion and properties of the non-Hermitian effective Hamiltonian of an unstable system using as an example potential resonance scattering with a fixed angular momentum. We present a consistent self-adjoint formulation of the problem of scattering on a finite-range potential, which is based on separation of the configuration space on two, internal and external, segments. The scattering amplitude is expressed in terms of the resolvent of a non-Hermitian operator H. The explicit form of this operator depends both on the radius of separation and the boundary conditions at this place which can be chosen in many different ways. We discuss this freedom and show explicitly that the physical scattering amplitude is, nevertheless, unique though not all choices are equally adequate from the physical point of view. The energy-dependent operator H should not be confused with the non-Hermitian effective Hamiltonian H_{eff} exploited usually to describe interference of overlapping resonances. We apply the developed formalism to a chain of L delta-barriers whose solution is also found independently in a closed form. For a fixed band of L overlapping resonances, the smooth energy dependence of H can be ignored so that complex eigenvalues of the LxL submatrix H_{eff} define the energies and widths of the resonances.We construct H_{eff} for the two commonly considered types of the boundary conditions (Neumann and Dirichlet) for the internal motion. Formation in the outer well of a short-lived doorway state is explicitly demonstrated together with the appearance of L-1 long-lived states trapped in the inner part of the chain.

cond-mat.mes-hall↗