Scattering in Quantum Dots via Noncommutative Rational Functions

Erdos L, Krueger T, Nemish Y (2021)

Publication Type: Journal article

Publication year: 2021


DOI: 10.1007/s00023-021-01085-6


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.

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Erdos, L., Krueger, T., & Nemish, Y. (2021). Scattering in Quantum Dots via Noncommutative Rational Functions. Annales Henri Poincaré.


Erdos, Laszlo, Torben Krueger, and Yuriy Nemish. "Scattering in Quantum Dots via Noncommutative Rational Functions." Annales Henri Poincaré (2021).

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