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S. Hikami

Publications and source records attributed to S. Hikami.

34 records · Page 2Linked to original sources
arXiv1999

Logarithmic moments of characteristic polynomials of random matrices

In a recent article we have discussed the connections between averages of powers of Riemann's $ζ$-function on the critical line, and averages of characteristic polynomials of random matrices. The result for random matrices was shown to be universal, i.e. independent of the specific probability distribution, and the results were derived for arbitrary moments. This allows one to extend the previous results to logarithmic moments, for which we derive the explicit universal expressions in random matrix theory. We then compare these results to various results and conjectures for $ζ$-functions, and the correspondence is again striking.

math-ph↗
arXiv1999

Characteristic polynomials of random matrices

Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $ζ$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices, which are the expectation values of the characteristic polynomials of the matrix. It turns out that these expectation values are quite interesting. For instance, the moments of order 2K scale, for unitary invariant ensembles, as the density of eigenvalues raised to the power $K^2$ ; the prefactor turns out to be a universal number, i.e. it is independent of the specific probability distribution. An equivalent behaviour and prefactor had been found, as a conjecture, within number theory. The moments of the characteristic determinants of random matrices are computed here as limits, at coinciding points, of multi-point correlators of determinants. These correlators are in fact universal in Dyson's scaling limit in which the difference between the points goes to zero, the size of the matrix goes to infinity, and their product remains finite.

math-ph↗
arXiv1999

Level statistics inside the vortex of a superconductor and symplectic random matrix theory in an external source

In the core of the vortex of a superconductor, energy levels appear inside the gap. We discuss here through a random matrix approach how these levels are broadened by impurities. It is first shown that the level statistics is governed by an ensemble consisting of a symplectic random potential added to a non-random matrix. A generalization of previous work on the unitary ensemble in the presence of an external source (which relied on the Itzykson-Zuber integral) is discussed for this symplectic case through the formalism introduced by Harish-Chandra and Duistermaat-Heckman. This leads to explicit formulae for the density of states and for the correlation functions, which describe the cross-over from the clean to the dirty limits.

cond-mat.mes-hall↗
arXiv1998

Universal singularity at the closure of a gap in a random matrix theory

We consider a Hamiltonian $ H = H_0+ V $, in which $ H_0$ is a given non-random Hermitian matrix,and $V$ is an $N \times N$ Hermitian random matrix with a Gaussian probability distribution.We had shown before that Dyson's universality of the short-range correlations between energy levels holds at generic points of the spectrum independently of $H_{0}$. We consider here the case in which the spectrum of $H_{0}$ is such that there is a gap in the average density of eigenvalues of $H$ which is thus split into two pieces. When the spectrum of $H_{0}$ is tuned so that the gap closes, a new class of universality appears for the energy correlations in the vicinity of this singular point.

cond-mat.stat-mech↗
arXiv1998

Level Spacing of Random Matrices in an External Source

In an earlier work we had considered a Gaussian ensemble of random matrices in the presence of a given external matrix source. The measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials, or on the Coulomb gas representation, are not available. Nevertheless the n-point correlation functions are still given in terms of the determinant of a kernel, known through an explicit integral representation. This kernel is no longer symmetric though and is not readily accessible to standard methods. In particular finding the level spacing probability is always a delicate problem in Fredholm theory, and we have to reconsider the problem within our model. We find a new class of universality for the level spacing distribution when the spectrum of the source is ajusted to produce a vanishing gap in the density of the state. The problem is solved through coupled non-linear differential equations, which turn out to form a Hamiltonian system. As a result we find that the level spacing probability $p(s)$ behaves like $\exp[ - C s^{8\over{3}}]$ for large spacing $s$; this is consistent with the asymptotic behavior $\exp[ - C s^{2 β+ 2}]$, whenever the density of state behaves near the edge as $ρ(λ)\sim λ^β$.

cond-mat.stat-mech↗
arXiv1997

An Extension of Level-spacing Universality

Dyson's short-distance universality of the correlation functions implies the universality of P(s), the level-spacing distribution. We first briefly review how this property is understood for unitary invariant ensembles and consider next a Hamiltonian H = H_0+ V , in which H_0 is a given, non-random, N by N matrix, and V is an Hermitian random matrix with a Gaussian probability distribution. n-point correlation function may still be expressed as a determinant of an n by n matrix, whose elements are given by a kernel $K(λ,μ)$ as in the H_0=0 case. From this representation we can show that Dyson's short-distance universality still holds. We then conclude that P(s) is independent of H_0.

cond-mat.mes-hall↗
arXiv1996

Fancy and facts in the (d - 2) expansion of non-linear sigma models

We review the existing results on the scaling dimensions of operators with more than two derivatives in the non-linear sigma models. We argue that the speculations on the relevance of these operators, and correspondingly on the breakdown of the $(d-2)$ expansion for the classical Heisenberg model, or for the one-parameter scaling theory of localization, are based on a dubious mathematical analysis.

cond-mat.stat-mech↗
arXiv1996

Diagrammatic analysis of the two-state quantum Hall system with chiral invariance

The quantum Hall system in the lowest Landau level with Zeeman term is studied by a two-state model, which has a chiral invariance. Using a diagrammatic analysis, we examine this two-state model with random impurity scattering, and find the exact value of the conductivity at the Zeeman energy $E = Δ$. We further study the conductivity at the another extended state $E = E_1$ ($ E_1 > Δ$). We find that the values of the conductivities at $E = 0$ and $E = E_1$ do not depend upon the value of the Zeeman energy $Δ$. We discuss also the case where the Zeeman energy $Δ$ becomes a random field.

cond-mat.mes-hall↗
arXiv1996

Spectral form factor in a random matrix theory

In the theory of disordered systems the spectral form factor $S(τ)$, the Fourier transform of the two-level correlation function with respect to the difference of energies, is linear for $τ<τ_c$ and constant for $τ>τ_c$. Near zero and near $τ_c$ its exhibits oscillations which have been discussed in several recent papers. In the problems of mesoscopic fluctuations and quantum chaos a comparison is often made with random matrix theory. It turns out that, even in the simplest Gaussian unitary ensemble, these oscilllations have not yet been studied there. For random matrices, the two-level correlation function $ρ(λ_1,λ_2)$ exhibits several well-known universal properties in the large N limit. Its Fourier transform is linear as a consequence of the short distance universality of $ρ(λ_1,λ_2)$. However the cross-over near zero and $τ_c$ requires to study these correlations for finite N. For this purpose we use an exact contour-integral representation of the two-level correlation function which allows us to characterize these cross-over oscillatory properties. The method is also extended to the time-dependent case.

cond-mat↗
arXiv1996

Correlations of Nearby Levels Induced by a Random Potential

We consider a Hamiltonian $H$ which is the sum of a deterministic part $H_0$ and of a random potential $V$. For finite $N \times N$ matrices, following a method introduced by Kazakov, we derive a representation of the correlation functions in terms of contour integrals over a finite number of variables. This allows one to analyse the level correlations, whereas the standard methods of random matrix theory, such as the method of orthogonal polynomials, are not available for such cases. At short distance we recover, for an arbitrary $H_0$, an oscillating behavior for the connected two-level correlation.

cond-mat↗
arXiv1995

Oscillating density of states near zero energy for matrices made of blocks with possible application to the random flux problem

We consider random hermitian matrices made of complex blocks. The symmetries of these matrices force them to have pairs of opposite real eigenvalues, so that the average density of eigenvalues must vanish at the origin. These densities are studied for finite $N\times N$ matrices in the Gaussian ensemble. In the large $N$ limit the density of eigenvalues is given by a semi-circle law. However, near the origin there is a region of size $1\over N$ in which this density rises from zero to the semi-circle, going through an oscillatory behavior. This cross-over is calculated explicitly by various techniques. We then show to first order in the non-Gaussian character of the probability distribution that this oscillatory behavior is universal, i.e. independent of the probability distribution. We conjecture that this universality holds to all orders. We then extend our consideration to the more complicated block matrices which arise from lattices of matrices considered in our previous work. Finally, we study the case of random real symmetric matrices made of blocks. By using a remarkable identity we are able to determine the oscillatory behavior in this case also. The universal oscillations studied here may be applicable to the problem of a particle propagating on a lattice with random magnetic flux.

cond-mat↗
arXiv1995

Complex random matrix models with possible applications to spin- impurity scattering in quantum Hall fluids

We study the one-point and two-point Green's functions in a complex random matrix model to sub-leading orders in the large N limit. We take this complex matrix models as a model for the two-state scattering problem, as applied to spin dependent scattering of impurities in quantum Hall fluids. The density of state shows a singularity at the band center due to reflection symmetry. We also compute the one-point Green's function for a generalized situation by putting random matrices on a lattice of arbitrary dimensions.

cond-mat↗
arXiv1995

Matrix model approach to the flux lattice melting in $2D$ superconductors

We investigate a gauged matrix model in the large $N$ limit which is closely related to the superconductor fluctuation and the flux lattice melting in two dimensions. With the use of saddle point method the free energy is expanded up to eighth order for the coupling constant $g$. In the case that the coefficient of quadratic term of the Ginzburg-Landau matrix model is negative, a critical point $g=g_c$ is obtained in the large $N$ limit and the relation between this phase transition and the 2D flux lattice melting transition is discussed.

cond-mat↗
arXiv1994

Universal correlations for deterministic plus random Hamiltonians

We consider the (smoothed) average correlation between the density of energy levels of a disordered system, in which the Hamiltonian is equal to the sum of a deterministic H0 and of a random potential $φ$. Remarkably, this correlation function may be explicitly determined in the limit of large matrices, for any unperturbed H0 and for a class of probability distribution P$(φ)$ of the random potential. We find a compact representation of the correlation function. From this representation one obtains readily the short distance behavior, which has been conjectured in various contexts to be universal. Indeed we find that it is totally independent of both H0 and P($φ$).

hep-th↗
arXiv1992

A naive matrix-model approach to two-dimensional quantum gravity coupled to matter of arbitrary central charge

In the usual matrix-model approach to random discretized two-dimensional manifolds, one introduces n Ising spins on each cell, i.e. a discrete version of 2D quantum gravity coupled to matter with a central charge n/2. The matrix-model consists then of an integral over $2^{n}$ matrices, which we are unable to solve for $n>1$. However for a fixed genus we can expand in the cosmological constant g for arbitrary values of n, and a simple minded analysis of the series yields for n=0,1 and 2 the expected results for the exponent $γ_{string}$ with an amazing precision given the small number of terms that we considered. We then proceed to larger values of n. Simple tests of universality are successfully applied; for instance we obtain the same exponents for n=3 or for one Ising model coupled to a one dimensional target space. The calculations are easily extended to states Potts models, through an integration over $q^{n}$ matrices. We see no sign of the tachyonic instability of the theory, but we have only considered genus zero at this stage.

hep-th↗