Gaudin models solver based on the Bethe ansatz/ordinary differential equations correspondence

Abstract

We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a new set of variables, the canceling divergences which occur for certain values of the coupling strength no longer appear explicitly. The problem is thus reduced to a set of quadratic algebraic equations. The required inverse transformation can then be realized using only linear operations and a standard polynomial root finding algorithm. The method is applied to Richardson's fermionic pairing model, the central spin model and generalized Dicke model.

10 pages, 3 figures, published version

References in corpus (34)

• Quantum state transfer and entanglement distribution among distant nodes in a quantum network
• Pairing in nuclear systems: from neutron stars to finite nuclei
• Spectroscopy of discrete energy levels in ultrasmall metallic grains
• Gate-Voltage Studies of Discrete Electronic States in Al Nanoparticles
• The ODE/IM Correspondence
• Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations
• Electron spin decoherence of single Nitrogen-Vacancy defects in diamond
• Electron Spin Dephasing due to Hyperfine Interactions with a Nuclear Spin Bath
• Exactly-Solvable Models Derived from a Generalized Gaudin Algebra
• Spin dynamics in InAs-nanowire quantum-dots coupled to a transmission line
• Integrability of the pairing hamiltonian
• Pure quantum dephasing of a solid state electron spin qubit in a large nuclear spin bath coupled by long-range hyperfine-mediated interactions
• Full quantum distribution of contrast in interference experiments between interacting one dimensional Bose liquids
• Algebraic Bethe ansatz method for the exact calculation of energy spectra and form factors: applications to models of Bose-Einstein condensates and metallic nanograins
• Quantum quenches from integrability: the fermionic pairing model
• Superconducting correlations in metallic nanoparticles: exact solution of the BCS model by the algebraic Bethe ansatz
• Large N limit of the exactly solvable BCS model: analytics versus numerics
• Exponential decay in a spin bath
• Algebraic Bethe Ansatz for XYZ Gaudin model
• Bethe Ansatz approach to quench dynamics in the Richardson model
• Free-induction decay and envelope modulations in a narrowed nuclear spin bath
• Exactly-solvable models for atom-molecule hamiltonians
• Solving the Richardson equations for Fermions
• Algebraic Bethe Ansatz for a discrete-state BCS pairing model
• Exact mesoscopic correlation functions of the pairing model
• Dynamics and decoherence in the central spin model using exact methods
• The BCS model and the off shell Bethe ansatz for vertex models
• Solving the Richardson equations close to the critical points
• Electrostatic analogy for integrable pairing force Hamiltonians
• Dynamics of the inhomogeneous Dicke model
• Dynamical correlation functions of the mesoscopic pairing model
• The classical and quantum dynamics of the inhomogeneous Dicke model and its Ehrenfest time
• Poor man's derivation of the Bethe-Ansatz equations for the Dicke model
• The nuclear pairing problem: new perspectives

Cited by in corpus (58)

• Integrability-based analysis of the hyperfine-interaction -nduced decoherence in quantum dots
• The Heine-Stieltjes correspondence and the polynomial approach to the standard pairing problem
• An eigenvalue-based method and determinant representations for general integrable XXZ Richardson-Gaudin models
• Bethe Ansatz and Ordinary Differential Equation Correspondence for Degenerate Gaudin Models
• Entanglement and collective flavor oscillations in a dense neutrino gas
• Spin decoherence due to a randomly fluctuating spin bath
• On the structure of typical states of a disordered Richardson model and many-body localization
• Exceptional and regular spectra of a generalized Rabi model
• Neutrino Spectral Split in the Exact Many Body Formalism
• Exact solution of the $p_x+i{p}_y$ pairing Hamiltonian by deforming the pairing algebra
• Richardson-Gaudin integrability in the contraction limit of the quasispin
• Integrability and dark states in an anisotropic central spin model
• Eigenvalues and eigenstates of the many-body collective neutrino oscillation problem
• Master equation approach to the central spin decoherence problem: the uniform coupling model and the role of projection operators
• Exact solution of the pairing problem for spherical and deformed systems
• The Heine-Stieltjes correspondence and a new angular momentum projection for many-particle systems
• On the determinant representations of Gaudin models' scalar products and form factors
• The Lipkin-Meshkov-Glick model as a particular limit of the SU(1,1) Richardson-Gaudin integrable models
• Read-Green resonances in a topological superconductor coupled to a bath
• The extended Heine-Stieltjes polynomials associated with a special LMG model
• Inner products in integrable Richardson-Gaudin models
• Non-equilibrum dynamics in the strongly excited inhomogeneous Dicke model
• Integrable spin-1/2 Richardson-Gaudin XYZ models in an arbitrary magnetic field
• Eigenvalue-based determinants for scalar products and form factors in Richardson-Gaudin integrable models coupled to a bosonic mode
• Completeness of the Bethe states for the rational, spin-1/2 Richardson--Gaudin system
• Density Matrices of Seniority-Zero Geminal Wavefunctions
• Generalised Heine-Stieltjes and Van Vleck polynomials associated with degenerate, integrable BCS models
• Reduced Density Matrices / Static Correlation Functions of Richardson-Gaudin States Without Rapidities
• Algebraic Bethe Ansätze and eigenvalue-based determinants for Dicke-Jaynes-Cummings-Gaudin quantum integrable models
• Persisting correlations of a central spin coupled to large spin baths
• Competing interactions in semiconductor quantum dots
• Separable and entangled states in the high-spin XX central spin model
• Strong-coupling emergence of dark states in XX central spin models
• Quadratic operator relations and Bethe equations for spin-1/2 Richardson-Gaudin models
• Dissipative dynamics in open XXZ Richardson-Gaudin models
• Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields
• Exact solution of the two-axis countertwisting Hamiltonian for the half-integer $J$ case
• Integrability of an extended d+id-wave pairing Hamiltonian
• Quasi-exactly solvable relativistic soft-core Coulomb models
• A variational method for integrability-breaking Richardson-Gaudin models
• Quasi-exact Treatment of the Relativistic Generalized Isotonic Oscillator
• Ground-state Bethe root densities and quantum phase transitions
• Ground-state energies of the open and closed $p+ip$-pairing models from the Bethe Ansatz
• Exact solution for the inhomogeneous Dicke model in the canonical ensemble: thermodynamical limit and finite-size corrections
• On symmetries of Hamiltonians describing systems with arbitrary spins
• Integrability, multifractality, and two-photon dynamics in disordered Tavis-Cummings models
• Ground-state energy of a Richardson-Gaudin integrable BCS model
• Constraint polynomial approach -- an alternative to the functional Bethe Ansatz method?
• Order parameter, correlation functions and fidelity susceptibility for the BCS model in the thermodynamic limit
• Determinant representation of the domain-wall boundary condition partition function of a Richardson-Gaudin model containing one arbitrary spin
• Particle-hole duality, integrability, and Russian doll BCS model
• Steady-states of out-of-equlibrium inhomogeneous Richardson-Gaudin quantum integrable models in quantum optics
• Relative Phase and Josephson Dynamics between Weakly Coupled Richardson Models
• Few-to-many-particle crossover of pair excitations in a superfluid
• A journey into localization, integrability and thermalization
• Near-exact treatment of seniority-zero ground and excited states with a Richardson-Gaudin mean-field
• "Bethe-Ansatz-free" eigenstates of spin-1/2 Richardson-Gaudin integrable models
• Transition density matrices of Richardson-Gaudin states