arXiv subjects

A. Zee

Publications and source records attributed to A. Zee.

82 records · Page 5Linked to original sources

arXiv1995

Correlations between eigenvalues of large random matrices with independent entries

We derive the connected correlation functions for eigenvalues of large Hermitian random matrices with independently distributed elements using both a diagrammatic and a renormalization group (RG) inspired approach. With the diagrammatic method we obtain a general form for the one, two and three-point connected Green function for this class of ensembles when matrix elements are identically distributed, and then discuss the derivation of higher order functions by the same approach. Using the RG approach we re-derive the one and two-point Green functions and show they are unchanged by choosing certain ensembles with non-identically distributed elements. Throughout, we compare the Green functions we obtain to those from the class of ensembles with unitary invariant distributions and discuss universality in both ensemble classes.

cond-mat↗

arXiv1995

Matrix generalizations of some dynamic field theories

We introduce matrix generalizations of the Navier--Stokes (NS) equation for fluid flow, and the Kardar--Parisi--Zhang (KPZ) equation for interface growth. The underlying field, velocity for the NS equation, or the height in the case of KPZ, is promoted to a matrix that transforms as the adjoint representation of $SU(N)$. Perturbative expansions simplify in the $N\to\infty$ limit, dominated by planar graphs. We provide the results of a one--loop analysis, but have not succeeded in finding the full solution of the theory in this limit.

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

Lattices of matrices (revised)

We study a new class of matrix models, formulated on a lattice. On each site are $N$ states with random energies governed by a Gaussian random matrix Hamiltonian. The states on different sites are coupled randomly. We calculate the density of and correlation between the eigenvalues of the total Hamiltonian in the large $N$ limit. We find that this correlation exhibits the same type of universal behavior we discovered recently. Several derivations of this result are given. This class of random matrices allows us to model the transition between the ''localized" and ''extended" regimes within the limited context of random matrix theory.

cond-mat↗

arXiv1995

Quantum Hall Fluids

We review the effective field theory treatment of topological quantum fluids, focussing on the Hall fluids.

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↗

arXiv1994

Universal Behavior of Correlations between Eigenvalues of Random Matrices

The universal connected correlations proposed recently between eigenvalues of unitary random matrices is examined numerically. We perform an ensemble average by the Monte Carlo sampling. Although density of eigenvalues and a bare correlation of the eigenvalues are not universal, the connected correlation shows a universal behavior after smoothing.

cond-mat↗

arXiv1994

Sideways Tunnelling and Fractional Josephson Frequency in Double Layered Quantum Hall Systems

The $(kkk)$ double layered quantum Hall state has been shown to exhibit many superfluid properties including the appearance of an alternating tunnelling current in response to an external voltage V. Here we propose the notion of "sideways tunnelling", in which a current tunnels between two such $(kkk)$ states through an incompressible fractional quantum Hall state $(mm0)$. We predict the tunnelling current will alternate with a frequency $\om={eVø\hbar m}$, as a consequence of the tunnelling of $e/m$ fractional charges. We also discuss singularities in the tunnelling current noise spectrum and the non-linear I-V curve.

cond-mat↗

arXiv1993

Universal Spectral Correlation between Hamiltonians with Disorder

We study the correlation between the energy spectra of two disordered Hamiltonians of the form $H_a=H_{0a}+s_{a}φ$ ($a=1,2$) with $H_{0a}$ and $φ$ drawn from random distributions. We calculate this correlation function explicitly and show that it has a simple universal form for a broad class of random distributions.

cond-mat↗

arXiv1993

Correlation Functions in Disordered Systems

{Recently, we found that the correlation between the eigenvalues of random hermitean matrices exhibits universal behavior. Here we study this universal behavior and develop a diagrammatic approach which enables us to extend our previous work to the case in which the random matrix evolves in time or varies as some external parameters vary. We compute the current-current correlation function, discuss various generalizations, and compare our work with the work of other authors. We study the distribution of eigenvalues of Hamiltonians consisting of a sum of a deterministic term and a random term. The correlation between the eigenvalues when the deterministic term is varied is calculated.}

cond-mat↗