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

Quantum deformed Richardson-Gaudin model

Henrik Johannesson, Alexander Stolin, P. P. Kulish

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

The Richardson-Gaudin model describes strong pairing correlations of fermions confined to a finite chain. The integrability of the Hamiltonian allows for its eigenstates to be constructed algebraically. In this work, we show that quantum group theory provides a possibility to deform the Hamiltonian preserving integrability. More precisely, we use the so-called Jordanian r-matrix to deform the Hamiltonian of the Richardson-Gaudin model. In order to preserve its integrability, we need to insert a special nilpotent term into the auxiliary L-operator which generates integrals of motion of the system. Moreover, the quantum inverse scattering method enables us to construct the exact eigenstates of the deformed Hamiltonian. These states have a highly complex entanglement structure which require further investigation.

Related papers

Deformed Richardson-Gaudin model

P Kulish, Alexander Stolin, L H Johannesson

The Richardson-Gaudin model describes strong pairing correlations of fermions confined to a finite chain. The integrability of the Hamiltonian allows the algebraic construction of its eigenstates. In this work we show that the quantum group theory provides a possibility to deform the Hamiltonian preserving integrability. More precisely, we use the so-called Jordanian r-matrix to deform the Hamiltonian of the Richardson-Gaudin model. In order to preserve its integrability, we need to insert a special nilpotent term into the auxiliary L -operator which generates integrals of motion of the system. Moreover, the quantum inverse scattering method enables us to construct the exact eigenstates of the deformed Hamiltonian. These states have a highly complex entanglement structure which require further investigation.

arXiv:1402.4838•Journal of Physics Conference Series•Sep 10, 2014•0 citations

4 Deformed Richardson-Gaudin model

P. Kulish, Alexander Stolin, H. Johannesson

The Richardson-Gaudin model describes strong pairing correlations of fermions confined to a\nfinite chain. The integrability of the Hamiltonian allows the algebraic construction of its eigenstates. In this\nwork we show that the quantum group theory provides a possibility to deform the Hamiltonian preserving\nintegrability. More precisely, we use the so-called Jordanian r-matrix to deform the Hamiltonian of the\nRichardson-Gaudin model. In order to preserve its integrability, we need to insert a special nilpotent term\ninto the auxiliary L-operator which generates integrals of motion of the system. Moreover, the quantum\ninverse scattering method enables us to construct the exact eigenstates of the deformed Hamiltonian. These\nstates have a highly complex entanglement structure which require further investigation.

•Jan 1, 2014•0 citations

Richardson-Gaudin models from the Boundary Quantum Inverse Scattering Method

Inna Lukyanenko

Richardson–Gaudin models are a class of quantum integrable models connected to many physical systems, including pairing Hamiltonians from the theory of superconductivity. They can be obtained in the quasi-classical limit of the Quantum Inverse Scattering Method, which is based on an R-matrix and a Lax operator satisfying the Yang-Baxter equation. They can also be obtained from the Boundary Quantum Inverse Scattering Method, which relies on solutions of the reflection equations known as K-matrices. In this thesis we study these latter models systematically, explore the connections betweenthem and investigate the interpretation of the “boundary”.First of all, we consider Richardson–Gaudin models obtained from the spin-1/2 su(2) Boundary Quantum Inverse Scattering Method with diagonal K-matrices. We prove that the trigonometric boundary construction is equivalent to its rational limit, through a change of variables, rescaling, and a basis transformation. Moreover, we prove that the twisted-periodic and boundary constructions are equivalent in the trigonometric case, but not in the rational limit. Thus, including the “boundary” does not lead to a new model in this case.Next, we investigate Richardson–Gaudin models obtained from the spin-1/2 su(2) Boundary Quantum Inverse Scattering Method with non-diagonal K-matrices. Here the situation is different. The conserved operators in the boundary construction are no longer equivalent to the ones in the twisted-periodic construction. Also, the rational and the trigonometric boundary constructions are not equivalent. In the rational case this allows us to construct a generalisation of the p+ip pairing Hamiltonian with external interaction terms. In the trigonometric case the expressions for the conserved operators involve several free parameters, which can be adjusted to construct a variety of Hamiltonians. This result offers opportunities for future investigations.Finally, we study the case of the q-deformed bosonic Lax operator. This case is much more challenging than the case of the spin Lax operator. It is not straightforward to define the rational and quasi-classical limits of the bosonic Lax operator. Even after making modifications to the Lax operator for these limits to be well defined, it turns out that the limits do not commute. We state some open questions for future work.

•Aug 29, 2016•0 citations

s l 2 Gaudin model with jordanian twist

N. Cirilo António, N. Manojlović

s l 2 Gaudin model with jordanian twist is studied. This system can be obtained as the semiclassical limit of the XXX spin chain deformed by the jordanian twist. The appropriate creation operators that yield the Bethe states of the Gaudin model and consequently its spectrum are defined. Their commutation relations with the generators of the corresponding loop algebra as well as with the generating function of integrals of motion are given. The inner products and norms of Bethe states and the relation to the solutions of the Knizhnik-Zamolodchikov equations are discussed.

arXiv:nlin/0506029•Journal of Mathematical Physics•Oct 1, 2005•20 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

Quantum Group Symmetries and the Deformed Spectral Transform of Solitons

Revista, Zen, MATH, 10

This paper rigorously investigates the profound interplay between quantum group symmetries and the deformed spectral transform, particularly in the context of soliton theory. We delve into how the introduction of a deformation parameter, characteristic of quantum groups, modifies the fundamental algebraic structures underlying integrable systems and, consequently, the spectral properties of their soliton solutions. By employing the framework of the quantum inverse scattering method (QISM), we construct deformed Lax pairs and analyze their impact on the scattering data. Our analysis reveals that quantum group symmetries induce a non-trivial deformation of the scattering matrix and its associated Jost functions, leading to quantized spectral parameters and modified conservation laws. We demonstrate how this deformation manifests in the properties of solitons, suggesting a departure from classical integrability and opening avenues for understanding quantum integrable models. The implications extend to a deeper understanding of quantum field theories and statistical mechanics models where these symmetries are paramount.

•Dec 6, 2025•0 citations
The Richardson-Gaudin model describes strong pairing correlations of fermions confined to a finite chain. The integrability of the Hamiltonian allows for its eigenstates to be constructed algebraically. In this work, we show that quantum group theory provides a possibility to deform the Hamiltonian preserving integrability. More precisely, we use the so-called Jordanian r-matrix to deform the Hamiltonian of the Richardson-Gaudin model. In order to preserve its integrability, we need to insert a special nilpotent term into the auxiliary L-operator which generates integrals of motion of the system. Moreover, the quantum inverse scattering method enables us to construct the exact eigenstates of the deformed Hamiltonian. These states have a highly complex entanglement structure which require further investigation.
The Richardson-Gaudin model describes strong pairing correlations of fermions confined to a finite chain. The integrability of the Hamiltonian allows the algebraic construction of its eigenstates. In this work we show that the quantum group theory provides a possibility to deform the Hamiltonian preserving integrability. More precisely, we use the so-called Jordanian r-matrix to deform the Hamiltonian of the Richardson-Gaudin model. In order to preserve its integrability, we need to insert a special nilpotent term into the auxiliary L -operator which generates integrals of motion of the system. Moreover, the quantum inverse scattering method enables us to construct the exact eigenstates of the deformed Hamiltonian. These states have a highly complex entanglement structure which require further investigation.
The Richardson-Gaudin model describes strong pairing correlations of fermions confined to a\nfinite chain. The integrability of the Hamiltonian allows the algebraic construction of its eigenstates. In this\nwork we show that the quantum group theory provides a possibility to deform the Hamiltonian preserving\nintegrability. More precisely, we use the so-called Jordanian r-matrix to deform the Hamiltonian of the\nRichardson-Gaudin model. In order to preserve its integrability, we need to insert a special nilpotent term\ninto the auxiliary L-operator which generates integrals of motion of the system. Moreover, the quantum\ninverse scattering method enables us to construct the exact eigenstates of the deformed Hamiltonian. These\nstates have a highly complex entanglement structure which require further investigation.
Richardson–Gaudin models are a class of quantum integrable models connected to many physical systems, including pairing Hamiltonians from the theory of superconductivity. They can be obtained in the quasi-classical limit of the Quantum Inverse Scattering Method, which is based on an R-matrix and a Lax operator satisfying the Yang-Baxter equation. They can also be obtained from the Boundary Quantum Inverse Scattering Method, which relies on solutions of the reflection equations known as K-matrices. In this thesis we study these latter models systematically, explore the connections betweenthem and investigate the interpretation of the “boundary”.First of all, we consider Richardson–Gaudin models obtained from the spin-1/2 su(2) Boundary Quantum Inverse Scattering Method with diagonal K-matrices. We prove that the trigonometric boundary construction is equivalent to its rational limit, through a change of variables, rescaling, and a basis transformation. Moreover, we prove that the twisted-periodic and boundary constructions are equivalent in the trigonometric case, but not in the rational limit. Thus, including the “boundary” does not lead to a new model in this case.Next, we investigate Richardson–Gaudin models obtained from the spin-1/2 su(2) Boundary Quantum Inverse Scattering Method with non-diagonal K-matrices. Here the situation is different. The conserved operators in the boundary construction are no longer equivalent to the ones in the twisted-periodic construction. Also, the rational and the trigonometric boundary constructions are not equivalent. In the rational case this allows us to construct a generalisation of the p+ip pairing Hamiltonian with external interaction terms. In the trigonometric case the expressions for the conserved operators involve several free parameters, which can be adjusted to construct a variety of Hamiltonians. This result offers opportunities for future investigations.Finally, we study the case of the q-deformed bosonic Lax operator. This case is much more challenging than the case of the spin Lax operator. It is not straightforward to define the rational and quasi-classical limits of the bosonic Lax operator. Even after making modifications to the Lax operator for these limits to be well defined, it turns out that the limits do not commute. We state some open questions for future work.
s l 2 Gaudin model with jordanian twist is studied. This system can be obtained as the semiclassical limit of the XXX spin chain deformed by the jordanian twist. The appropriate creation operators that yield the Bethe states of the Gaudin model and consequently its spectrum are defined. Their commutation relations with the generators of the corresponding loop algebra as well as with the generating function of integrals of motion are given. The inner products and norms of Bethe states and the relation to the solutions of the Knizhnik-Zamolodchikov equations are discussed.
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.
This paper rigorously investigates the profound interplay between quantum group symmetries and the deformed spectral transform, particularly in the context of soliton theory. We delve into how the introduction of a deformation parameter, characteristic of quantum groups, modifies the fundamental algebraic structures underlying integrable systems and, consequently, the spectral properties of their soliton solutions. By employing the framework of the quantum inverse scattering method (QISM), we construct deformed Lax pairs and analyze their impact on the scattering data. Our analysis reveals that quantum group symmetries induce a non-trivial deformation of the scattering matrix and its associated Jost functions, leading to quantized spectral parameters and modified conservation laws. We demonstrate how this deformation manifests in the properties of solitons, suggesting a departure from classical integrability and opening avenues for understanding quantum integrable models. The implications extend to a deeper understanding of quantum field theories and statistical mechanics models where these symmetries are paramount.