TY - JOUR

T1 - Scattering in Quantum Dots via Noncommutative Rational Functions

AU - Erdős, László

AU - Krüger, Torben

AU - Nemish, Yuriy

N1 - Publisher Copyright: © 2021, The Author(s).

PY - 2021

Y1 - 2021

N2 - In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via N≪ M channels, the density 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 ϕ: = N/ M≤ 1 ; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit ϕ→ 0 , we recover the formula for the density 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 ϕ< 1 but in the borderline case ϕ= 1 an anomalous 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.

AB - In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via N≪ M channels, the density 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 ϕ: = N/ M≤ 1 ; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit ϕ→ 0 , we recover the formula for the density 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 ϕ< 1 but in the borderline case ϕ= 1 an anomalous 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.

U2 - 10.1007/s00023-021-01085-6

DO - 10.1007/s00023-021-01085-6

M3 - Journal article

AN - SCOPUS:85111938935

VL - 22

SP - 4205

EP - 4269

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

IS - 12

ER -