Scattering in Quantum Dots via Noncommutative Rational Functions

Abstract

In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via $$N\ll M$$ N ≪ M channels, the density $$\rho $$ ρ 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 $$\phi := N/M\le 1$$ ϕ : = N / M ≤ 1 ; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit $$\phi \rightarrow 0$$ ϕ → 0 , we recover the formula for the density $$\rho $$ ρ 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 $$\phi <1$$ ϕ < 1 but in the borderline case $$\phi =1$$ ϕ = 1 an anomalous $$\lambda ^{-2/3}$$ λ - 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.