Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0101029 · Physical Review Letters

Integrable Model for Interacting Electrons in Metallic Grains

Luigi Amico A, Antonio Di Lorenzo, Andreas Osterloh

Published Jun 18, 2001• 98 citations
arXivHTMLPDF

Abstract

We find an integrable generalization of the BCS model with nonuniform Coulomb and pairing interaction. The Hamiltonian is integrable by construction since it is a functional of commuting operators; these operators, which therefore are constants of motion of the model, contain the anisotropic Gaudin Hamiltonians. The exact solution is obtained diagonalizing them by means of Bethe ansatz. Uniform pairing and Coulomb interaction are obtained as the "isotropic limit" of the Gaudin Hamiltonians. We discuss possible applications of this model to a single grain and to a system of few interacting grains.

DOI
10.1103/physrevlett.86.5759

Related papers

Superconducting correlations in metallic nanoparticles: Exact solution of the BCS model by the algebraic Bethe ansatz

Huan-Qiang Zhou, Jon Links, Ross H. McKenzie +1 more

Superconducting pairing of electrons in nanoscale metallic particles with discrete energy levels and a fixed number of electrons is described by the reduced Bardeen, Cooper, and Schrieffer model Hamiltonian. We show that this model is integrable by the algebraic Bethe ansatz. The eigenstates, spectrum, conserved operators, integrals of motion, and norms of wave functions are obtained. Furthermore, the quantum inverse problem is solved, meaning that form factors and correlation functions can be explicitly evaluated. Closed form expressions are given for the form factors and correlation functions that describe superconducting pairing.

arXiv:cond-mat/0106390•Physical review. B, Condensed matter•Jan 11, 2002•98 citations

Richardson-Gaudin integrable systems (and beyond) for strongly correlated quantum many-body systems

Stijn De Baerdemacker, Paul W. Ayers, Patrick Bultinck +4 more

Integrable systems take a special niche in quantum many-body systems. Due to the underlying algebraic structure, it is possible to capture all the quantum correlations exactly within a Bethe Ansatz wavefunction at a favourably scaling computational cost. Richardson [1] and Gaudin [2] have shown that the reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian of superconductivity [3] is integrable and that the Bethe Ansatz state corresponds to a product wave function of generalized Cooper pairs [4]. In this formulation, the Bethe ansatz state is equivalent to a non-orthogonal geminal, which is widely used in quantum chemistry to describe electron pair correlations in atoms and molecules [5]. Inspired by Richardson-Gaudin integrable systems, it is possible to construct classes of non-orthogonal geminals, allowing for an accurate description of realistic and non-integrable strongly correlated atoms and molecules, while retaining the favorable computational scaling [6, 7]. In this contribution, I will discuss Richardson-Gaudin integrability and why it is useful for strongly correlated systems.

•Jan 1, 2013•0 citations

Classicalr-matrices and integrable BCS models with many types of fermions

T. Skrypnyk

Based on generalized Gaudin models and Lie algebras so (2 m ), we construct two integrable cases of the pairing BCS Hamiltonians containing m types of ‘spinless’ fermions. They are the so-called s -type and p x + i p y -type BCS Hamiltonians. In the particular case of the Lie algebra so (8), the constructed models can be viewed as N = Z proton–neutron integrable models. We find the spectrum of the constructed Hamiltonians in terms of solutions of Bethe-type equations.

Journal of Physics A Mathematical and Theoretical•Oct 3, 2012•2 citations

Electrostatic analogy for integrable pairing force Hamiltonians

Luigi Amico A, Antonio Di Lorenzo, Andrea Mastellone +2 more

For the exactly solved reduced BCS model an electrostatic analogy exists; in particular it served to obtain the exact thermodynamic limit of the model from the Richardson Bethe ansatz equations. We present an electrostatic analogy for a wider class of integrable Hamiltonians with pairing force interactions. We apply it to obtain the exact thermodynamic limit of this class of models. To verify the analytical results, we compare them with numerical solutions of the Bethe ansatz equations for finite systems at half-filling for the ground state.

•Jan 1, 2002•19 citations

Non-skew-symmetric classicalr-matrices and integrable cases of the reduced BCS model

T. Skrypnyk

We consider generalized quantum Gaudin systems in an external magnetic field associated with non-skew-symmetric sl (2)-valued classical r -matrices. We calculate spectra of the generating function of the corresponding Hamiltonians using the algebraic Bethe ansatz. We apply these results to the construction of integrable fermionic Hamiltonians of a generalized BCS type. We investigate the special cases when the corresponding integrable Hamiltonians contain only a pairing interaction term and consider an example of such a situation associated with a special non-skew-symmetric r -matrix.

Journal of Physics A Mathematical and Theoretical•Nov 6, 2009•46 citations

The elliptic Gaudin model: a numerical study

C Esebbag, J. Dukelsky

The elliptic Gaudin model describes completely anisotropic spin systems with long range interactions. The model was proven to be quantum integrable by Gaudin and latter the exact solution was found by means of the algebraic Bethe ansatz. In spite of the appealing properties of the model, it has not yet been applied to any physical problem. We here generalize the exact solution to systems with arbitrary spins, and study numerically the behavior of the Bethe roots for a system with three different spins. Then, we propose an integrable anisotropic central spin model that we study numerically for very large systems.

arXiv:1509.01451•Journal of Physics A Mathematical and Theoretical•Oct 28, 2015•5 citations
We find an integrable generalization of the BCS model with nonuniform Coulomb and pairing interaction. The Hamiltonian is integrable by construction since it is a functional of commuting operators; these operators, which therefore are constants of motion of the model, contain the anisotropic Gaudin Hamiltonians. The exact solution is obtained diagonalizing them by means of Bethe ansatz. Uniform pairing and Coulomb interaction are obtained as the "isotropic limit" of the Gaudin Hamiltonians. We discuss possible applications of this model to a single grain and to a system of few interacting grains.
Superconducting pairing of electrons in nanoscale metallic particles with discrete energy levels and a fixed number of electrons is described by the reduced Bardeen, Cooper, and Schrieffer model Hamiltonian. We show that this model is integrable by the algebraic Bethe ansatz. The eigenstates, spectrum, conserved operators, integrals of motion, and norms of wave functions are obtained. Furthermore, the quantum inverse problem is solved, meaning that form factors and correlation functions can be explicitly evaluated. Closed form expressions are given for the form factors and correlation functions that describe superconducting pairing.
Integrable systems take a special niche in quantum many-body systems. Due to the underlying algebraic structure, it is possible to capture all the quantum correlations exactly within a Bethe Ansatz wavefunction at a favourably scaling computational cost. Richardson [1] and Gaudin [2] have shown that the reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian of superconductivity [3] is integrable and that the Bethe Ansatz state corresponds to a product wave function of generalized Cooper pairs [4]. In this formulation, the Bethe ansatz state is equivalent to a non-orthogonal geminal, which is widely used in quantum chemistry to describe electron pair correlations in atoms and molecules [5]. Inspired by Richardson-Gaudin integrable systems, it is possible to construct classes of non-orthogonal geminals, allowing for an accurate description of realistic and non-integrable strongly correlated atoms and molecules, while retaining the favorable computational scaling [6, 7]. In this contribution, I will discuss Richardson-Gaudin integrability and why it is useful for strongly correlated systems.
For the exactly solved reduced BCS model an electrostatic analogy exists; in particular it served to obtain the exact thermodynamic limit of the model from the Richardson Bethe ansatz equations. We present an electrostatic analogy for a wider class of integrable Hamiltonians with pairing force interactions. We apply it to obtain the exact thermodynamic limit of this class of models. To verify the analytical results, we compare them with numerical solutions of the Bethe ansatz equations for finite systems at half-filling for the ground state.
We consider generalized quantum Gaudin systems in an external magnetic field associated with non-skew-symmetric sl (2)-valued classical r -matrices. We calculate spectra of the generating function of the corresponding Hamiltonians using the algebraic Bethe ansatz. We apply these results to the construction of integrable fermionic Hamiltonians of a generalized BCS type. We investigate the special cases when the corresponding integrable Hamiltonians contain only a pairing interaction term and consider an example of such a situation associated with a special non-skew-symmetric r -matrix.
The elliptic Gaudin model describes completely anisotropic spin systems with long range interactions. The model was proven to be quantum integrable by Gaudin and latter the exact solution was found by means of the algebraic Bethe ansatz. In spite of the appealing properties of the model, it has not yet been applied to any physical problem. We here generalize the exact solution to systems with arbitrary spins, and study numerically the behavior of the Bethe roots for a system with three different spins. Then, we propose an integrable anisotropic central spin model that we study numerically for very large systems.