Exact fermionic matrix elements from the Bethe-ansatz solution of the Richardson-Gaudin model
Abstract
We introduce an effective Hamiltonian description of mesoscopic superconducting systems based on the Bethe-ansatz solution of the Richardson--Gaudin model. Our central result is a compact determinant formula for the matrix elements of fermionic creation and annihilation operators, expressed entirely in terms of eigenvalue-based variables. This formula allows single-electron tunneling terms to be represented directly in the eigenbasis of the Richardson--Gaudin model, extending the eigenvalue-based framework beyond overlaps and previously known form factors. As a benchmark, we apply the method to a superconducting Anderson impurity model comprising a quantum dot coupled to an ultrasmall superconducting grain. The resulting low-energy description, formulated naturally in the canonical ensemble, captures even-odd effects, subgap excitations, and quantum-dot physics. The same framework gives access to observables not previously computed for such grain-dot systems in the canonical ensemble, such as the fixed-particle-number optical conductivity and flux-tunable spectra.
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BibTeX
@misc{johannsen2026exact,
title = {{Exact fermionic matrix elements from the Bethe-ansatz solution of the Richardson-Gaudin model}},
author = {Peter D. Johannsen and Valerii K. Kozin and Jelena Klinovaja and Daniel Loss},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03141},
eprint = {2611.03141},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03141},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact fermionic matrix elements from the Bethe-ansatz solution of the Richardson-Gaudin model
AU - Peter D. Johannsen
AU - Valerii K. Kozin
AU - Jelena Klinovaja
AU - Daniel Loss
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03141
UR - https://arxiv.org/abs/2611.03141
AB - We introduce an effective Hamiltonian description of mesoscopic superconducting systems based on the Bethe-ansatz solution of the Richardson--Gaudin model. Our central result is a compact determinant formula for the matrix elements of fermionic creation and annihilation operators, expressed entirely in terms of eigenvalue-based variables. This formula allows single-electron tunneling terms to be represented directly in the eigenbasis of the Richardson--Gaudin model, extending the eigenvalue-based framework beyond overlaps and previously known form factors. As a benchmark, we apply the method to a superconducting Anderson impurity model comprising a quantum dot coupled to an ultrasmall superconducting grain. The resulting low-energy description, formulated naturally in the canonical ensemble, captures even-odd effects, subgap excitations, and quantum-dot physics. The same framework gives access to observables not previously computed for such grain-dot systems in the canonical ensemble, such as the fixed-particle-number optical conductivity and flux-tunable spectra.
ER -