On the determinant representations of Gaudin models' scalar products and form factors

Abstract

We propose alternative determinant representations of certain form factors and scalar products of states in rational Gaudin models realized in terms of compact spins. We use alternative pseudo-vacuums to write overlaps in terms of partition functions with domain wall boundary conditions. Contrarily to Slavnovs determinant formulas, this construction does not require that any of the involved states be solutions to the Bethe equations; a fact that could prove useful in certain non-equilibrium problems. Moreover, by using an atypical determinant representation of the partition functions, we propose expressions for the local spin raising and lowering operators form factors which only depend on the eigenvalues of the conserved charges. These eigenvalues define eigenstates via solutions of a system of quadratic equations instead of the usual Bethe equations. Consequently, the current work allows important simplifications to numerical procedures addressing decoherence in Gaudin models.

arXiv comment15 pages, 0 figures, Published version

References in corpus (31)

• Electron spin decoherence in quantum dots due to interaction with nuclei
• Spectroscopy of discrete energy levels in ultrasmall metallic grains
• The ODE/IM Correspondence
• Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations
• Electron Spin Dephasing due to Hyperfine Interactions with a Nuclear Spin Bath
• Exactly-Solvable Models Derived from a Generalized Gaudin Algebra
• Pure quantum dephasing of a solid state electron spin qubit in a large nuclear spin bath coupled by long-range hyperfine-mediated interactions
• Algebraic Bethe ansatz method for the exact calculation of energy spectra and form factors: applications to models of Bose-Einstein condensates and metallic nanograins
• Exact dynamics in the inhomogeneous central-spin model
• Gaudin models solver based on the Bethe ansatz/ordinary differential equations correspondence
• Numerical modeling of the central spin problem using the spin coherent states P-representation
• Integrability-based analysis of the hyperfine-interaction -nduced decoherence in quantum dots
• Quantum Decoherence of the Central Spin in a Sparse System of Dipolar Coupled Spins
• Superconductivity in Ultrasmall Metallic Grains
• Large N limit of the exactly solvable BCS model: analytics versus numerics
• Exponential decay in a spin bath
• Bethe Ansatz and Ordinary Differential Equation Correspondence for Degenerate Gaudin Models
• On the Bethe Ansatz for the Jaynes-Cummings-Gaudin model
• Free-induction decay and envelope modulations in a narrowed nuclear spin bath
• Solving the Richardson equations for Fermions
• Algebraic Bethe Ansatz for a discrete-state BCS pairing model
• Dynamics and decoherence in the central spin model using exact methods
• On the structure of typical states of a disordered Richardson model and many-body localization
• Nuclear spin dynamics and Zeno effect in quantum dots and defect centers
• Constraints on the anisotropy of dark energy
• Spin- and entanglement-dynamics in the central spin model with homogeneous couplings
• Solving the Richardson equations close to the critical points
• Quantum versus classical hyperfine-induced dynamics in a quantum dot
• Dynamics of the inhomogeneous Dicke model
• The classical and quantum dynamics of the inhomogeneous Dicke model and its Ehrenfest time
• Spins coupled to a Spin Bath: From Integrability to Chaos

Cited by in corpus (23)

• Integrability-based analysis of the hyperfine-interaction -nduced decoherence in quantum dots
• An eigenvalue-based method and determinant representations for general integrable XXZ Richardson-Gaudin models
• Spin decoherence due to a randomly fluctuating spin bath
• Integrable time-dependent Hamiltonians, solvable Landau-Zener models and Gaudin magnets
• Read-Green resonances in a topological superconductor coupled to a bath
• Inner products in integrable Richardson-Gaudin models
• 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
• 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
• Quadratic operator relations and Bethe equations for spin-1/2 Richardson-Gaudin models
• Exact solution of the p+ip Hamiltonian revisited: duality relations in the hole-pair picture
• Ground-state energies of the open and closed $p+ip$-pairing models from the Bethe Ansatz
• A variational method for integrability-breaking Richardson-Gaudin models
• Ground-state energy of a Richardson-Gaudin integrable BCS model
• Determinant representation of the domain-wall boundary condition partition function of a Richardson-Gaudin model containing one arbitrary spin
• Order parameter, correlation functions and fidelity susceptibility for the BCS model in the thermodynamic limit
• Steady-states of out-of-equlibrium inhomogeneous Richardson-Gaudin quantum integrable models in quantum optics
• Particle-hole duality, integrability, and Russian doll BCS model
• Quantum dynamics of Gaudin magnets
• "Bethe-Ansatz-free" eigenstates of spin-1/2 Richardson-Gaudin integrable models