Parity-Time Symmetric Spin-1/2 Richardson-Gaudin Models

Abstract

We construct a $\mathcal{P}\mathcal{T}$-symmetric Richardson--Gaudin models for spin-$\frac{1}{2}$ systems by deforming the closed integrable Hamiltonian through complex-valued transverse magnetic fields and coupling constants. By defining parity as $\mathcal{P}={\prod }_i{\sigma }_i^z$ and adopting a time-reversal operator that flips only the $y$-component of spin, we establish a consistent $\mathcal{P}\mathcal{T}$-symmetric framework distinct from open-system approaches based on Lindblad dynamics. The resulting model remains integrable, with conserved charges satisfying generalized commutativity conditions. We explicitly construct the Hermitian counterpart via a similarity transformation and identify the metric operator $\rho =e^{-{\sum }_i{q}_i{S}_i^z}$ that defines the physical inner product. Numerical diagonalization reveals the characteristic $\mathcal{P}\mathcal{T}$ spectral structure: eigenvalues are either real or form complex conjugate pairs, with partial symmetry breaking wherein low-energy states remain in the unbroken phase. We further derive exact analytical expressions for spin dynamics, showing coherent oscillations in the unbroken phase and exponentially modulated behavior in the broken phase.

12 pages; matches accepted version

References in corpus (25)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• PT-Symmetric Quantum Mechanics
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Edge states and topological phases in non-Hermitian systems
• Exactly solvable Richardson-Gaudin models for many-body quantum systems
• Gaudin Model, Bethe Ansatz and Critical Level
• Model of a PT symmetric Bose-Einstein condensate in a delta-functions double well
• Exactly-Solvable Models Derived from a Generalized Gaudin Algebra
• Integrability of the pairing hamiltonian
• PT-symmetric quantum Liouvillian dynamics
• Theory of superconductivity with non-Hermitian and parity-time reversal symmetric cooper pairing symmetry
• A spin chain model with non-Hermitian interaction: The Ising quantum spin chain in an imaginary field
• PT-symmetric Rabi model: Perturbation theory
• PT-symmetric deformations of integrable models
• Integrability and dark states in an anisotropic central spin model
• Many-Body Open Quantum Systems
• PT Symmetry of the non-Hermitian XX Spin-Chain: Non-local Bulk Interaction from Complex Boundary Fields
• Parity-time symmetry breaking in spin chains
• Quadratic operator relations and Bethe equations for spin-1/2 Richardson-Gaudin models
• PT-symmetric quantum Rabi model
• Quantum phase transitions in non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric transverse-field Ising spin chains
• Integral Transforms and $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians
• Supersymmetry of $\mathcal{P}\mathcal{T}$- symmetric tridiagonal Hamiltonians