Abstract
In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via Nll M channels, the density rho of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio phi := N/Mle 1; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit phi rightarrow 0, we recover the formula for the density rho that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker’s formula persists for any phi <1 but in the borderline case phi =1 an anomalous lambda ^{-2/3} singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries.
Highlights
Introduction and ResultsSince the pioneering discovery of E
Wigner on the universality of eigenvalue statistics of large random matrices [38], random matrix theory has become one of the most successful phenomenological theories to study disordered quantum systems, see [2] for a broad overview. It has been used for open quantum systems and quantum transport, in particular to
Together with the linearization trick, this allows us to handle arbitrary polynomials in i.i.d. random matrices [13] and in the current work we extend our method to a large class of rational functions
Summary
One main result of the theory in [7] is that in the M N 1 regime at energy E = 0 the density of transmission eigenvalues for a quantum dot is given by ρBee(λ) = π. The goal of this paper is to revisit and substantially generalize the problem of transmission eigenvalues with very different methods than Beenakker and collaborators used While those works used invariant matrix ensembles for H and relied on explicit computations for the circular ensemble, we consider very general distribution for the matrix elements of H. The flexibility in our result stems from the fact that our method directly aims at the density of states via an extension of the MDE theory to linearizations of rational functions of random matrices It seems unnecessarily ambitious, requiring too restrictive conditions, to attempt to find the joint distribution of all eigenvalues.
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