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arXiv:2303.17837

Supersymmetry and integrability for a class of XY central spin models

W J P van Tonder, Jon Links

Published Mar 31, 2023• 0 citations
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Abstract

Several studies have exploited the integrable structure of central spin models to deepen understanding of these fundamental systems. In recent years, an underlying supersymmetry for systems with XX interactions has been uncovered. Here we report that a class of central spin models with XY interactions is also supersymmetric and integrable. The associated Bethe Ansatz solution is presented for the case where all particles are spin-1/2.

DOI
10.48550/arxiv.2303.17837

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Several studies have exploited the integrable structure of central spin models to deepen understanding of these fundamental systems. In recent years, an underlying supersymmetry for systems with XX interactions has been uncovered. Here we report that a class of central spin models with XY interactions is also supersymmetric and integrable. The associated Bethe Ansatz solution is presented for the case where all particles are spin-1/2.
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Central spin models describe a variety of quantum systems in which a spin-1 2 qubit interacts with a bath of surrounding spins, as realized in quantum dots and defect centers in diamond.We show that the fully anisotropic central spin Hamiltonian with (XX) Heisenberg interactions is integrable.Building on the class of integrable Richardson-Gaudin models, we derive an extensive set of conserved quantities and obtain the exact eigenstates using the Bethe ansatz.These states divide into two exponentially large classes: bright states, where the qubit is entangled with the bath, and dark states, where it is not.We discuss how dark states limit qubit-assisted spin bath polarization and provide a robust long-lived quantum memory for qubit states.
It was recently shown that, for central spin-1/2 and central spin-1, the XX central spin model is integrable in the presence of a magnetic field oriented perpendicular to the XY plane in which the coupling exists. In the spin-1/2 case, it was also shown, through an appropriate limit of the non-skew symmetric XXZ Richardson-Gaudin models, that it remained integrable even when the magnetic field is tilted to contain an in-plane component. Although the model has not yet been shown to explicitly belong to a known class of Richardson-Gaudin models, we show, in this work, that the spin-1 case also remains integrable in a tilted magnetic field. We do so by writing explicitly the complete set of conserved charges, then showing that these operators obey polynomial relations. It is finally demonstrated numerically that dark states, for which the central spin is completely unentangled with the bath, can emerge at strong enough coupling just as they do in the central spin-1/2 model in an arbitrarily oriented magnetic field.