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.
Annales Henri Poincare
The authors are very grateful to Yan Fyodorov for discussions on the physical background and for providing references, and to the anonymous referee for numerous valuable remarks.
Erdös L, Krüger TH, Nemish Y. Scattering in quantum dots via noncommutative rational functions. Annales Henri Poincare. 2021. doi:10.1007/s00023-021-01085-6
Erdös, L., Krüger, T. H., & Nemish, Y. (2021). Scattering in quantum dots via noncommutative rational functions. Annales Henri Poincare. Springer Nature. https://doi.org/10.1007/s00023-021-01085-6
Erdös, László, Torben H Krüger, and Yuriy Nemish. “Scattering in Quantum Dots via Noncommutative Rational Functions.” Annales Henri Poincare. Springer Nature, 2021. https://doi.org/10.1007/s00023-021-01085-6.
L. Erdös, T. H. Krüger, and Y. Nemish, “Scattering in quantum dots via noncommutative rational functions,” Annales Henri Poincare. Springer Nature, 2021.
Erdös L, Krüger TH, Nemish Y. 2021. Scattering in quantum dots via noncommutative rational functions. Annales Henri Poincare.
Erdös, László, et al. “Scattering in Quantum Dots via Noncommutative Rational Functions.” Annales Henri Poincare, Springer Nature, 2021, doi:10.1007/s00023-021-01085-6.