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:1201.5851 · Annalen der Physik

Bethe Ansatz approach to the pairing fluctuations in the mesoscopic regime

Luigi Amico A, Andreas Osterloh

Published Mar 8, 2012• 16 citations
arXivHTMLPDF

Abstract

Abstract We review the exact treatment of the pairing correlation functions in the canonical ensemble. The key for the calculations has been provided by relating the discrete BCS model to known integrable theories corresponding to the so called Gaudin magnets with suitable boundary terms. In the present case the correlation functions can be accessed beyond the formal level, allowing the description of the cross‐over from few electrons to the thermodynamic limit. In particular, we summarize the results on the finite size scaling behavior of the canonical pairing clarifying some puzzles emerged in the past. Some recent developments and applications are outlined.

DOI
10.1002/andp.201100326

Related papers

Exact mesoscopic correlation functions of the Richardson pairing model

Alexandre Faribault, Pasquale Calabrese, Jean-Sébastien Caux

We study the static correlation functions of the Richardson pairing model (also known as the reduced or discrete-state BCS model) in the canonical ensemble. Making use of the algebraic Bethe ansatz formalism, we obtain exact expressions which are easily evaluated numerically for any value of the pairing strength up to large numbers of particles. We provide explicit results at half-filling and extensively discuss their finite-size scaling behavior.

arXiv:0710.4865•Physical Review B•Feb 8, 2008•58 citations

Dynamical correlation functions of the mesoscopic pairing model

Alexandre Faribault, Pasquale Calabrese, Jean-Sébastien Caux

We study the dynamical correlation functions of the Richardson pairing model (also known as the reduced- or discrete-state Bardeen, Cooper, Schrieffer model) in the canonical ensemble. We use the algebraic Bethe Ansatz formalism, which gives exact expressions for the form factors of the most important observables. By summing these form factors over a relevant set of states, we obtain very precise estimates of the correlation functions, as confirmed by global sum rules (saturation above 99% in all cases considered). Unlike the case of many other Bethe Ansatz solvable theories, simple two-particle states are sufficient to achieve such saturations, even in the thermodynamic limit. We provide explicit results at half filling and discuss their finite-size scaling behavior.

arXiv:1003.0582•Physical Review B•May 7, 2010•22 citations

Exact Correlation Functions of the BCS Model in the Canonical Ensemble

Luigi Amico A, Andreas Osterloh

We evaluate correlation functions of the BCS model for a finite number of particles. The integrability of the Hamiltonian relates it with the Gaudin algebra G[sl(2)]. Therefore, a theorem that Sklyanin proved for the Gaudin model, can be applied. Several diagonal and off-diagonal correlators are calculated. The finite-size scaling behavior of the pairing correlation function is studied.

arXiv:cond-mat/0105141•Physical Review Letters•Mar 11, 2002•44 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

Mesoscopic BCS pairing in the repulsive one-dimensional Hubbard model

Luigi Amico A, Andrea Mastellone, Andreas Osterloh

We study mesoscopic pairing in the one-dimensional repulsive Hubbard model and its interplay with the BCS model in the canonical ensemble. The key tool is comparing the Bethe ansatz equations of the two models in the limit of small Coulomb repulsion. For the ordinary Hubbard interaction the BCS Bethe equations with infinite pairing coupling are recovered; a finite pairing is obtained by considering a further density-dependent phase-correlation in the hopping amplitude of the Hubbard model. We find that spin degrees of freedom in the Hubbard ground state are arranged in a state of the BCS type, where the Cooper pairs form a noncondensed liquid on a ``lattice'' of single particle energies provided by the Hubbard charge degrees of freedom; the condensation in the BCS ground state corresponds to Hubbard excitations constituted by a sea of spin singlets.

arXiv:cond-mat/0509554•Physical Review B•Jun 13, 2006•3 citations

LARGE-N AND BETHE ANSATZ

Branislav Jurčo

We describe an integrable model, related to the Gaudin magnet, and its relation to the matrix model of Brézin, Itzykson, Parisi and Zuber. Relation is based on Bethe ansatz for the integrable model and its interpretation using orthogonal polynomials and saddle point approximation. Large-N limit of the matrix model corresponds to the thermodynamic limit of the integrable system. In this limit (functional) Bethe ansatz is the same as the generating function for correlators of the matrix models.

arXiv:hep-th/0311252•Modern Physics Letters A•Jul 12, 2004•2 citations
Abstract We review the exact treatment of the pairing correlation functions in the canonical ensemble. The key for the calculations has been provided by relating the discrete BCS model to known integrable theories corresponding to the so called Gaudin magnets with suitable boundary terms. In the present case the correlation functions can be accessed beyond the formal level, allowing the description of the cross‐over from few electrons to the thermodynamic limit. In particular, we summarize the results on the finite size scaling behavior of the canonical pairing clarifying some puzzles emerged in the past. Some recent developments and applications are outlined.
We study the dynamical correlation functions of the Richardson pairing model (also known as the reduced- or discrete-state Bardeen, Cooper, Schrieffer model) in the canonical ensemble. We use the algebraic Bethe Ansatz formalism, which gives exact expressions for the form factors of the most important observables. By summing these form factors over a relevant set of states, we obtain very precise estimates of the correlation functions, as confirmed by global sum rules (saturation above 99% in all cases considered). Unlike the case of many other Bethe Ansatz solvable theories, simple two-particle states are sufficient to achieve such saturations, even in the thermodynamic limit. We provide explicit results at half filling and discuss their finite-size scaling behavior.
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.
We study mesoscopic pairing in the one-dimensional repulsive Hubbard model and its interplay with the BCS model in the canonical ensemble. The key tool is comparing the Bethe ansatz equations of the two models in the limit of small Coulomb repulsion. For the ordinary Hubbard interaction the BCS Bethe equations with infinite pairing coupling are recovered; a finite pairing is obtained by considering a further density-dependent phase-correlation in the hopping amplitude of the Hubbard model. We find that spin degrees of freedom in the Hubbard ground state are arranged in a state of the BCS type, where the Cooper pairs form a noncondensed liquid on a ``lattice'' of single particle energies provided by the Hubbard charge degrees of freedom; the condensation in the BCS ground state corresponds to Hubbard excitations constituted by a sea of spin singlets.