BIMSA Integrable Systems Seminar

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In this talk, I will introduce the chiral coordinate Bethe ansatz for anisotropic spin chains with periodic boundaries, including the XYZ, XY, and XX models. First, we construct a set of factorized chiral vectors with a fixed number of kinks, which form an invariant subspace of the Hilbert space. Next, we propose a modified coordinate Bethe ansatz method to solve the XYZ model, based on the action of the Hamiltonian on the chiral vectors. For the XY and XX models, we demonstrate that our Bethe ansatz yields all normalized eigenstates and the complete spectrum of the Hamiltonian. The differences between our approach and other methods are also discussed.

Brickwork circuits composed of the Yang-Baxter gates are integrable. It becomes an important tool to study the quantum many-body system out of equilibrium. I will talk about the properties of Yang-Baxter gates via the quantum information theory. We find that only certain two-qubit gates can be converted to the Yang-Baxter gates via the single-qubit gate operations. I will also talk about some possible extensions of the integrable circuits. Numerical analysis suggests that there is a broad class of circuits that are integrable, which are beyond the standard algebraic Bethe ansatz method.

Reference: [1] K. Zhang, K. Hao, K. Yu, V. Korepin, and W.-L. Yang, Geometric representations of braid and Yang-Baxter gates, J. Phys. A: Math. Theor. 57 445303, arXiv:2406.08320 (2024).

[2] K. Zhang, K. Yu, K. Hao, and V. Korepin, Optimal realization of Yang-Baxter gate on quantum computers, Adv. Quantum Technol. 2024, 2300345, arXiv:2307.16781 (2024).

Quantum non-integrability, or the absence of local conserved quantity, is a necessary condition for various empirical laws observed in macroscopic systems. Examples are thermal equilibration, the Kubo formula in linear response theory, and the Fourier law in heat conduction, all of which require non-integrability. From these facts, it is widely believed that integrable systems are highly exceptional, and non-integrability is ubiquitous in generic quantum many-body systems. Many numerical simulations also support this expectation. Nevertheless, conventional approaches in mathematical physics cannot address this belief, and establishing non-integrability of vast majority of generic quantum many-body systems is left as an open problem.

 In this study, we address this problem and provide an affirmative result for a wide class of quantum many-body systems. Precisely, we rigorously classify the integrability and non-integrability of all spin-1/2 chains with symmetric nearest-neighbor interactions [1]. Our classification demonstrates that except for the known integrable models, all systems are indeed non-integrable. This result provides a rigorous proof of the ubiquitousness of non-integrability, as well as the absence of undiscovered integrable systems which remains to be discovered. Moreover, it is proved that there is no partially integrable systems with finite number of local conserved quantities.

 In addition, recent extensions of non-integrability proofs to spin-1 systems [2] and others will be presented.

 [1] M. Yamaguchi, Y. Chiba, N. Shiraishi, ``Complete Classification of Integrability and Non-integrability for Spin-1/2 Chain with Symmetric Nearest-Neighbor Interaction,'' arXiv:2411.02162

 [2] A. Hokkyo, M. Yamaguchi, Y. Chiba, ``Proof of the absence of local conserved quantities in the spin-1 bilinear-biquadratic chain and its anisotropic extensions,'' arXiv:2411.04945