ArticlePublisher preview available

Integrable spin-1/2 Richardson-Gaudin XYZ models in an arbitrary magnetic field

IOP Publishing
Journal of Physics A: Mathematical and Theoretical
Authors:
Pieter W. Claeys at Max Planck Institute for the Physics of Complex Systems
Pieter W. Claeys
Claude Dimo at University of Lorraine
Claude Dimo
Stijn De baerdemacker at University of New Brunswick
Stijn De baerdemacker
Alexandre Faribault at University of Lorraine
Alexandre Faribault
Read publisher preview
Download citation
To read the full-text of this research, you can request a copy directly from the authors.

Abstract and Figures

We establish the most general class of spin- ¹2 integrable Richardson–Gaudin models including an arbitrary magnetic field, returning a fully anisotropic (XYZ) model. The restriction to spin- ¹2 relaxes the usual integrability constraints, allowing for a general solution where the couplings between spins lack the usual antisymmetric properties of Richardson–Gaudin models. The full set of conserved charges are constructed explicitly and shown to satisfy a set of quadratic equations, allowing for the numerical treatment of a fully anisotropic central spin in an external magnetic field. While this approach does not provide expressions for the exact eigenstates, it allows their eigenvalues to be obtained, and expectation values of local observables can then be calculated from the Hellmann–Feynman theorem.
Figures - available from: Journal of Physics A: Mathematical and Theoretical
This content is subject to copyright. Terms and conditions apply.
A preview of this full-text is provided by IOP Publishing.
Learn more
Content available from Journal of Physics A: Mathematical and Theoretical 
This content is subject to copyright. Terms and conditions apply.
1
Journal of Physics A: Mathematical and Theoretical
Integrable spin-
1
2
Richardson–Gaudin XYZ
models in an arbitrary magnetic eld
PieterWClaeys1,2, ClaudeDimo3, StijnDe Baerdemacker1
and AlexandreFaribault3
1 Department of Physics and Astronomy, Ghent University, Krijgslaan 281-S9,
9000 Ghent, Belgium
2 Department of Physics, Boston University, 590 Commonwealth Ave., Boston,
MA 02215, United States of America
3 Université de Lorraine, CNRS, LPCT, F-54000 Nancy, France
E-mail: pwclaeys@bu.edu and alexandre.faribault@univ-lorraine.fr
Received 24 October 2018, revised 10 January 2019
Accepted for publication 15 January 2019
Published 28 January 2019
Abstract
We establish the most general class of spin-
1
2
integrable Richardson–Gaudin
models including an arbitrary magnetic eld, returning a fully anisotropic
(XYZ) model. The restriction to spin-
1
2
relaxes the usual integrability
constraints, allowing for a general solution where the couplings between
spins lack the usual antisymmetric properties of Richardson–Gaudin models.
The full set of conserved charges are constructed explicitly and shown to
satisfy a set of quadratic equations, allowing for the numerical treatment
of a fully anisotropic central spin in an external magnetic eld. While this
approach does not provide expressions for the exact eigenstates, it allows their
eigenvalues to be obtained, and expectation values of local observables can
then be calculated from the Hellmann–Feynman theorem.
Keywords: integrability, Richardson–Gaudin models, exactly-solvable
models
(Some guresmay appear in colour only in the online journal)
1. Introduction
Gaudin models are a specic class of quantum integrable models characterised by a large set
of mutually commuting conserved charges [1–4]. Typically, each conserved charge contains
interaction terms between a single (central) spin and the full set of other (bath) spins in the
system. Such interactions can either be fully isotropic as in XXX models or fully anisotropic
as in XYZ models, while the intermediate XXZ models maintain
U(1)
-rotation symmetry in
P W Claeys et al
Integrable spin-
1
2
Richardson–Gaudin XYZ models in an arbitrary magnetic field
Printed in the UK
08LT01
JPHAC5
© 2019 IOP Publishing Ltd
52
J. Phys. A: Math. Theor.
JPA
1751-8121
10.1088/1751-8121/aafe9b
Letter
Journal of Physics A: Mathematical and Theoretical
IOP
2019
1751-8121/19/08LT01+12$33.00 © 2019 IOP Publishing Ltd Printed in the UK
J. Phys. A: Math. Theor. 52 (2019) 08LT01 (12pp) https://doi.org/10.1088/1751-8121/aafe9b
... Cambiaggio et al. investigate the integrability of pairing Hamiltonians, which are closely related to Richardson-Gaudin models, while more recent works by Skrypnyk generalize these ideas to elliptic and quasi-trigonometric r-matrices, broadening their applicability [14][15][16]. Dimo and Faribault, along with collaborators, delve into quadratic operator relations and Bethe equations, advancing the understanding of spin-1/2 Richardson-Gaudin models under anisotropic conditions and external magnetic fields [17,18] . Villazon et al. and Claeys et al. further expand these concepts to study dark states, dissipative dynamics, and open quantum systems, highlighting the relevance of integrable models in modern physics [19,20]. ...
... Conversely, when g → ∞, it yields the conserved charges for the XXZ Gaudin magnet [11]. The integrability condition [Q i , Q j ] = 0 is satisfied if and only if the following conditions hold [17,18]: ...
... For any permutation of (α, β, γ) from the set {x, y, z}, the integrability conditions are [17,18]: ...
Parity-Time Symmetric Spin-1/2 Richardson-Gaudin Models
Preprint
Full-text available
  • Jun 2025
In this work, we systematically investigate the integrable PT\mathcal{PT}-symmetric Richardson-Gaudin model for spin-1/2 particles in an arbitrary magnetic field. First, we define the parity and time-reversal transformation rules and determine the metric operator as well as the PT\mathcal{PT}-symmetric inner product. Using the metric operator, we derive the Hermitian counterparts of the PT\mathcal{PT}-symmetric conserved charges. We then compute and plot the eigenvalues of the conserved charges obtained from the PT\mathcal{PT}-symmetric Richardson-Gaudin model. As expected for any PT\mathcal{PT}-symmetric system, the spectrum exhibits both real eigenvalues and complex-conjugate pairs. We numerically study the spin dynamics of this model and compare it with the Hermitian case in both closed and open quantum systems. Our findings reveal that, in the PT\mathcal{PT}-symmetric Richardson-Gaudin model, the system fails to reach a steady state at weak coupling, demonstrating robustness against dissipation. However, at stronger coupling strengths, the system eventually reaches a steady state after some time.
View
... Cambiaggio et al. investigate the integrability of pairing Hamiltonians, which are closely related to Richardson-Gaudin models, while more recent works by Skrypnyk generalize these ideas to elliptic and quasi-trigonometric r-matrices, broadening their applicability [14][15][16]. Dimo and Faribault, along with collaborators, delve into quadratic operator relations and Bethe equations, advancing the understanding of spin-1/2 Richardson-Gaudin models under anisotropic conditions and external magnetic fields [17,18] . Villazon et al. and Claeys et al. further expand these concepts to study dark states, dissipative dynamics, and open quantum systems, highlighting the relevance of integrable models in modern physics [19,20]. ...
... Conversely, when g → ∞, it yields the conserved charges for the XXZ Gaudin magnet [11]. The integrability condition [Q i , Q j ] = 0 is satisfied if and only if the following conditions hold [17,18]: ...
... For any permutation of (α, β, γ) from the set {x, y, z}, the integrability conditions are [17,18]: ...
Parity-Time Symmetric Spin-1/2 Richardson-Gaudin Models
Preprint
Full-text available
  • Jun 2025
In this work, we systematically investigate the integrable PT-symmetric Richardson-Gaudin model for spin-1/2 particles in an arbitrary magnetic field. First, we define the parity and time-reversal transformation rules and determine the metric operator as well as the PT-symmetric inner product. Using the metric operator, we derive the Hermitian counterparts of the PT-symmetric conserved charges. We then compute and plot the eigenvalues of the conserved charges obtained from the PT-symmetric Richardson-Gaudin model. As expected for any PT-symmetric system, the spectrum exhibits both real eigenvalues and complex-conjugate pairs. We numerically study the spin dynamics of this model and compare it with the Hermitian case in both closed and open quantum systems. Our findings reveal that, in the PT-symmetric Richardson-Gaudin model, the system fails to reach a steady state at weak coupling, demonstrating robustness against dissipation. However, at stronger coupling strengths, the system eventually reaches a steady state after some time.
View
... The operators (7), when restricted to the subspaces in which the central spin is up or down, have the same form as those in [6] up to a constant shift and change of variables. Consequently they satisfy an analogous set of quadratic identities as those given by equation (13) in [6]. ...
... The operators (7), when restricted to the subspaces in which the central spin is up or down, have the same form as those in [6] up to a constant shift and change of variables. Consequently they satisfy an analogous set of quadratic identities as those given by equation (13) in [6]. For each simultaneous eigenstate of the operators given by (7), let {q j } denote the set of corresponding eigenvalues. ...
... It remains to establish the algebraic independence of the set {Q j }. We will consider only λ to be a variable and fix all other parameters in the expression (6). The algebraic independence of {Q j } is equivalent to establishing the algebraic independence of the set {Q j = f − j (2λ) −1 Q j }, with the property that lim λ→∞Q j = S y j . ...
Supersymmetry and integrability for a class of XY central spin models
Article
Full-text available
Several studies have exploited the integrable structure of central spin models to deepen understanding of these fundamental systems. In recent years, an underlying supersymmetry for systems with XX interactions has been uncovered. Here we report that a class of central spin models with XY interactions is also supersymmetric and integrable. The associated Bethe Ansatz solution is presented for the case where all particles are spin-1/2.
View
... In classical mechanics, the integrability of a system requires that the number of degrees of freedom should be equal to the number of constant of motion in involution [103,105,106]. On the other hand, quantum integrability generally associated with the exact solution of the models [107][108][109][110][111][112][113][114][115]. This is often achieved using the Yang-Baxter equation [109,[116][117][118][119][120] and techniques like Bethe ansatz [99,101,[121][122][123]. ...
Exact Solvability Of Entanglement For Arbitrary Initial State in an Infinite-Range Floquet System
Preprint
Full-text available
  • Nov 2024
Sharma and Bhosale [Phys. Rev. B 109, 014412 (2024); Phys. Rev. B 110, 064313,(2024)] recently introduced an 𝑁-spin Floquet model with infinite-range Ising interactions. There, we have shown that the model exhibits the signatures of quantum integrability for specific parameter values 𝐽 = 1, 1/2 and 𝜏 = 𝜋/4. We have found analytically the eigensystem and the time evolution of the unitary operator for finite values of 𝑁 up to 12 qubits. We have calculated the reduced density matrix, its eigensystem, time-evolved linear entropy, and the timeevolved concurrence for the initial states |0, 0⟩ and |𝜋/2, −𝜋/2⟩. For the general case 𝑁 > 12, we have provided sufficient numerical evidences for the signatures of quantum integrability, such as the degenerate spectrum, the exact periodic nature of entanglement dynamics, and the time-evolved unitary operator. In this paper, we have extended these calculations to arbitrary initial state |𝜃0, 𝜙0⟩, such that 𝜃0 ∈ [0, 𝜋] and 𝜙0 ∈ [−𝜋, 𝜋]. Along with that, we have analytically calculated the expression for the average linear entropy for arbitrary initial states. We numerically find that the average value of time-evolved concurrence for arbitrary initial states decreases with 𝑁, implying the multipartite nature of entanglement. We numerically show that the values ⟨𝑆⟩/𝑆𝑀𝑎𝑥 → 1 for Ising strength (𝐽 ≠ 1, 1/2), while for 𝐽 = 1 and 1/2, it deviates from 1 for arbitrary initial states even though the thermodynamic limit does not exist in our model. This deviation is shown to be a signature of integrability in earlier studies where the thermodynamic limit exist. We also discuss possible experiments that could be conducted to verify our results.
View
... In [11] we have also found an integrable case of the Hamiltonian (1.2). We remark, that the integrable models corresponding to the Hamiltonian of [11] have recently obtained attention in the literature [12,22,23]. ...
Elliptic BCS-Richardson model and the modified algebraic Bethe ansatz
Article
Full-text available
We consider elliptic Gaudin-type model in an external magnetic field \cite{SkrPLA2005, SkrJGP2006, SkrJMP2006, SkrJPA2007, SkrNPB2019} associated with non-skew-symmetric elliptic r-matrix \cite{SkrPLA2005, SkrJGP2006, SkrJMP2006}. Using them we construct a new integrable fermion Hamiltonian of the Richardson type. We use the modified algebraic Bethe ansatz obtained for integrable models with the considered elliptic r-matrix in \cite{SkrNew} and find the spectrum of the obtained Richardson-type Hamiltonian in terms of solutions of the modified Bethe equations. The obtained results generalize our previous results on Richardson-type models \cite{SkrNPB2022}.
View
View
View
View
Integrability and quench dynamics in the spin-1 central spin XX model
Article
Full-text available
  • Jul 2023
Central spin models provide an idealized description of interactions between a central degree of freedom and a mesoscopic environment of surrounding spins. We show that the family of models with a spin-1 at the center and XX interactions of arbitrary strength with surrounding spins is integrable. Specifically, we derive an extensive set of conserved quantities and obtain the exact eigenstates using the Bethe ansatz. As in the homogenous limit, the states divide into two exponentially large classes: bright states, in which the spin-1 is entangled with its surroundings, and dark states, in which it is not. On resonance, the bright states further break up into two classes depending on their weight on states with central spin polarization zero. These classes are probed in quench dynamics wherein they prevent the central spin from reaching thermal equilibrium. In the single spin-flip sector we explicitly construct the bright states and show that the central spin exhibits oscillatory dynamics as a consequence of the semilocalization of these eigenstates. We relate the integrability to the closely related class of integrable Richardson-Gaudin models, and conjecture that the spin-s central spin XX model is integrable for any s.
View
Elliptic Gaudin-type model in an external magnetic field and modified algebraic Bethe ansatz
Article
  • Feb 2023
  • NUCL PHYS B
We consider the elliptic Gaudin-type model in an external magnetic field [11–13,16,15,22] associated with non-skew-symmetric elliptic r-matrix [11] defined on 4:1 unramified covering of the Weierstrass cubic y2=(u+j1)(u+j2)(u+j3). We develop a modified algebraic Bethe ansatz for the considered elliptic r-matrix and for the algebra generated by the entries of the corresponding Lax operator with the aim of obtaining the spectra of the relevant Gaudin-type Hamiltonians in terms of solutions of modified Bethe equations. The applications of the obtained result to the diagonalization of the anisotropic quantum Euler top, quantum Zhukovsky-Volterrra top, quantum Steklov and Rubanovsky tops are given.
View
Ground-state energies of the open and closed p + ip-pairing models from the Bethe Ansatz
Article
Full-text available
  • Aug 2018
  • NUCL PHYS B
Using the exact Bethe Ansatz solution, we investigate methods for calculating the ground-state energy for the p+ip-pairing Hamiltonian. We first consider the Hamiltonian isolated from its environment (closed model) through two forms of Bethe Ansatz solutions, which generally have complex-valued Bethe roots. A continuum limit approximation, leading to an integral equation, is applied to compute the ground-state energy. We discuss the evolution of the root distribution curve with respect to a range of parameters, and the limitations of this method. We then consider an alternative approach that transforms the Bethe Ansatz equations to an equivalent form, but in terms of the real-valued conserved operator eigenvalues. An integral equation is established for the transformed solution. This equation is shown to admit an exact solution associated with the ground state. Next we discuss results for a recently derived Bethe Ansatz solution of the open model. With the aforementioned alternative approach based on real-valued roots, combined with mean-field analysis, we are able to establish an integral equation with an exact solution that corresponds to the ground-state for this case.
View
Quadratic operator relations and Bethe equations for spin-1/2 Richardson-Gaudin models
Article
Full-text available
In this work we demonstrate how one can, in a generic approach, derive a set of N simple quadratic Bethe equations for integrable Richardson–Gaudin (RG) models built out of N spins-1/2. These equations depend only on the N eigenvalues of the various conserved charges so that any solution of these equations defines, indirectly through the corresponding set of eigenvalues, one particular eigenstate. The proposed construction covers the full class of integrable RG models of the XYZ (including the subclasses of XXZ and XXX models) type realised in terms of spins-1/2, coupled with one another through , , terms, including, as well, magnetic field-like terms linear in the Pauli matrices. The approach exclusively requires integrability, defined here only by the requirement that N conserved charges Ri (with ) such that exist. The result is therefore valid, and equally simple, for models with or without symmetry, with or without a properly defined pseudo-vacuum as well as for models with non-skew symmetric couplings.
View
Spin Polarization through Floquet Resonances in a Driven Central Spin Model
Article
Full-text available
  • Dec 2017
We show how adiabatically varying the driving frequency of a periodically-driven many-body quantum system can induce controlled transitions between resonant eigenstates of the time-averaged Hamiltonian. It is argued that these can be understood as adiabatic transitions in the Floquet spectrum at quasi-degeneracies. Using the central spin model as an illustration, we show how such controlled driving processes can lead to a polarization-based decoupling of the central spin from its decoherence-inducing environment. While it is generally impossible to obtain the exact Floquet Hamiltonian in driven interacting systems, we exploit the integrability of the central spin model to show how techniques from quenches in integrability can be used to explicitly construct the Floquet Hamiltonian in a restricted many-body basis.
View
Completeness of the Bethe states for the rational, spin-1/2 Richardson--Gaudin system
Article
Full-text available
  • Jul 2017
Establishing the completeness of a Bethe Ansatz solution for an exactly solved model is a perennial challenge, which is typically approached on a case by case basis. For the rational, spin-1/2 Richardson--Gaudin system it will be argued that, for generic values of the system's coupling parameters, the Bethe states are complete. This method does not depend on knowledge of the distribution of Bethe roots, such as a string hypothesis, and is generalisable to a wider class of systems.
View
Inner products in integrable Richardson-Gaudin models
Article
Full-text available
  • Jun 2017
We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula. This framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.
View
Separation of variables in anisotropic models: Anisotropic Rabi and elliptic Gaudin model in an external magnetic field
Article
Full-text available
We study the problem of separation of variables for classical integrable Hamiltonian systems governed by non-skew-symmetric non-dynamical so(3) so(3)-valued elliptic r-matrices with spectral parameters. We consider several examples of such models, and perform separation of variables for classical anisotropic one- and two-spin Gaudin-type models in an external magnetic feld, and for Jaynes-Cummings-Dicke-type models without the rotating wave approximation.
View
Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields
Article
Full-text available
  • Apr 2017
In this work we demonstrate a simple way to implement the quantum inverse scattering method to find eigenstates of spin-1/2 XXX Gaudin magnets in an arbitrarily oriented magnetic field. The procedure differs vastly from the most natural approach which would be to simply orient the spin quantisation axis in the same direction as the magnetic field through an appropriate rotation. Instead, we define a modified realisation of the rational Gaudin algebra and use the quantum inverse scattering method which allows us, within a slightly modified implementation, to build an algebraic Bethe ansatz using the same unrotated reference state (pseudovacuum) for any external field. This common framework allows us to easily write determinant expressions for certain scalar products which would be highly non-trivial in the rotated system approach.
View
Steady-states of out-of-equlibrium inhomogeneous Richardson–Gaudin quantum integrable models in quantum optics
Article
  • Aug 2018
  • J STAT MECH-THEORY E
In this work we present numerical results for physical quantities in the steady-state obtained after a variety of product-states initial conditions are evolved unitarily, driven by the dynamics of quantum integrable models of the rational (XXX) Richardson-Gaudin family, which includes notably Tavis-Cummings models. The problem of interest here is one where a completely inhomogeneous ensemble of two-level systems (spins-1/2) are coupled to a single bosonic mode. The long-time averaged magnetisation along the z-axis as well as the bosonic occupation are evaluated in the diagonal ensemble by performing the complete sum over the full Hilbert space for small system sizes. These numerically exact results are independent of any particular choice of Hamiltonian and therefore describe general results valid for any member of this class of quantum integrable models built out of the same underlying conserved quantities. The collection of numerical results obtained can be qualitatively understood by a relaxation process for which, at infinitely strong coupling, every initial state will relax to a common state where each spin is in a maximally coherent superposition of its and states, i.e. they are in-plane polarised, and consequently the bosonic mode is also in a maximally coherent superposition of different occupation number states. This bosonic coherence being a feature of a superradiant state, we shall loosely use the term superradiant steady-state to describe it. A finite value of the coupling between the spins and the bosonic mode then leads to a long-time limit steady-state whose properties are qualitatively captured by a simple 'dynamical' vision in which the coupling strength plays the role of a time at which this 'relaxation process' towards the common strong coupling superradiant steady-state is interrupted.
View
Benchmarking the Variational Reduced Density Matrix Theory in the Doubly Occupied Configuration Interaction Space with Integrable Pairing Models
Article
  • Jun 2018
The variational reduced density matrix theory has been recently applied with great success to models within the truncated doubly-occupied configuration interaction space, which corresponds to the seniority zero subspace. Conservation of the seniority quantum number restricts the Hamiltonians to be based on the SU(2) algebra. Among them there is a whole family of exactly solvable Richardson-Gaudin pairing Hamiltonians. We benchmark the variational theory against two different exactly solvable models, the Richardson-Gaudin-Kitaev and the reduced BCS Hamiltonians. We obtain exact numerical results for the so-called PQGT N-representability conditions in both cases for systems that go from 10 to 100 particles. However, when random single-particle energies as appropriate for small superconducting grains are considered, the exactness is lost but still a high accuracy is obtained.
View
“Twisted” rational r-matrices and the algebraic Bethe ansatz: Applications to generalized Gaudin models, Bose–Hubbard dimers, and Jaynes–Cummings–Dicke-type models
Article
  • Oct 2016
We construct quantum integrable systems associated with the Lie algebra gl(n) and non-skew-symmetric “shifted and twisted” rational r-matrices. The obtained models include Gaudin-type models with and without an external magnetic field, n-level (n−1)-mode Jaynes–Cummings–Dicke-type models in the Λ-configuration, a vector generalization of Bose–Hubbard dimers, etc. We diagonalize quantum Hamiltonians of the constructed integrable models using a nested Bethe ansatz.
View