Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

M. W. AlMasri

Abstract

We construct a $\mathcal{PT}$-symmetric Richardson--Gaudin models for spin-$\tfrac{1}{2}$ systems by deforming the closed integrable Hamiltonian through complex-valued transverse magnetic fields and coupling constants. By defining parity as $\mathcal{P} = \prod_i σ_i^z$ and adopting a time-reversal operator that flips only the $y$-component of spin, we establish a consistent $\mathcal{PT}$-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 $ρ= e^{-\sum_i q_i S_i^z}$ that defines the physical inner product. Numerical diagonalization reveals the characteristic $\mathcal{PT}$ 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.

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

Abstract

We construct a -symmetric Richardson--Gaudin models for spin- systems by deforming the closed integrable Hamiltonian through complex-valued transverse magnetic fields and coupling constants. By defining parity as and adopting a time-reversal operator that flips only the -component of spin, we establish a consistent -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 that defines the physical inner product. Numerical diagonalization reveals the characteristic 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.

Paper Structure

This paper contains 13 sections, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Richardson-Gaudin Models
  3. Results
  4. $\mathcal{PT}$-symmetric Richardson-Gaudin Hamiltonians
  5. Hamiltonian Spectrum
  6. Spin Dynamics
  7. Conclusion
  8. Derivation of Integrable Couplings and Magnetic Fields
  9. Commutator Structure
  10. Three-Site Terms and Anisotropy Constraints
  11. Two-Site Terms and Magnetic Field Constraints
  12. Choice of Parametrization
  13. Final Expressions

Figures (2)

  • Figure 1: Pictorial representation of the $XXZ$ Richardson-Gaudin model. Red arrows indicate pairing interactions ($S_i^+ S_j^-$), while blue dashed lines represent spin-spin interactions ($S_i^z S_j^z$) for the first four spins.
  • Figure 2: Eigenvalue spectrum of the $\mathcal{PT}$-symmetric Hamiltonian $H = Q_1$ for $N=8$ spin-$\tfrac{1}{2}$ particles. The transverse magnetic fields and couplings are purely imaginary, while longitudinal terms are real, ensuring $\mathcal{PT}$ symmetry. Eigenvalues are either strictly real (on the horizontal axis) or occur in complex conjugate pairs (symmetric about the real axis), as required by $\mathcal{PT}$ symmetry. Notably, all negative-energy states remain real, indicating that the low-energy sector resides in the unbroken $\mathcal{PT}$-symmetric phase.