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arXiv:2507.10856 · Physical review. B./Physical review. B

Higher-spin Richardson-Gaudin model with time-dependent coupling: Exact dynamics

Suvendu Barik, Lieuwe Bakker, Vladimir Gritsev, Jiří Minář, Emil A. Yuzbashyan

Published May 4, 2026• 0 citations
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Abstract

We determine the exact asymptotic many-body wavefunction of a spin-$s$ Richardson-Gaudin model with a coupling inversely proportional to time, for time evolution starting from the ground state at $t = 0^+$ and for arbitrary $s$. Contrary to common belief, the resulting wavefunction cannot be derived from the spin-$1/2$ case by merging spins, but instead requires independent treatment for each spin size. The steady state is non-thermal and, in contrast to the spin-$1/2$ case, does not conform to a natural Generalized Gibbs Ensemble. We show that mean-field theory is exact for any product of a finite number of spin operators on different sites. We discuss how these findings can be probed in cavity QED and trapped ion experiments.

DOI
10.1103/c8vm-hz26

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Abstract We establish the most general class of spin- integrable Richardson–Gaudin models including an arbitrary magnetic field, returning a fully anisotropic (XYZ) model. The restriction to spin- 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.
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Given a time-independent Hamiltonian H~\widetilde HH , one can construct a time-dependent Hamiltonian HtH_tHt​ by means of the gauge transformation Ht=UtH~Ut†−iUt∂tUt†H_t=U_t\kern1pt \widetilde H \kern1pt U^\dagger_t-i\kern1pt U_t\kern1pt\partial_t U_t^\daggerHt​=Ut​HUt†​−iUt​∂t​Ut†​ . Here UtU_tUt​ is the unitary transformation that relates the solutions of the corresponding Schrödinger equations. In the many-body case one is usually interested in Hamiltonians with few-body (often, at most two-body) interactions. We refer to such Hamiltonians as physical. We formulate sufficient conditions on UtU_tUt​ ensuring that HtH_tHt​ is physical as long as H~\widetilde HH is physical (and vice versa). This way we obtain a general method for finding pairs of physical Hamiltonians HtH_tHt​ and H~\widetilde HH such that the driven many-body dynamics governed by HtH_tHt​ can be reduced to the quench dynamics due to the time-independent H~\widetilde HH . We apply this method to a number of many-body systems. First we review the mapping of a spin system with isotropic Heisenberg interaction and arbitrary time-dependent magnetic field to a time-independent system without a magnetic field [F. Yan, L. Yang, and B. Li, Phys. Lett. A 251, 289–293; 259, 207–211 (1999)]. Then we demonstrate that essentially the same gauge transformation eliminates an arbitrary time-dependent magnetic field from a system of interacting fermions. Further, we apply the method to the quantum Ising spin system and a spin coupled to a bosonic environment. We also discuss a more general situation where H~=H~t\widetilde H = \widetilde H_tH=Ht​ is time-dependent but dynamically integrable.