Scattering in Quantum Dots via Noncommutative Rational Functions
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Scattering in Quantum Dots via Noncommutative Rational Functions. / Erdős, László; Krüger, Torben; Nemish, Yuriy.
I: Annales Henri Poincare, Bind 22, Nr. 12, 2021, s. 4205-4269.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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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 -
ID: 307081108