The distinction of integrability and non-integrability, which are strongly related to the notion of chaos, plays a pivotal role in quantum many-body physics. Integrability and non-integrability are roughly equivalent to the presence and the absence of local conserved quantities. The presence of local conserved quantities prevents thermalization and mixing, which is relevant to broad research fields from the application of the Kubo formula [1] to the scrambling in a black hole [2]. The distinction of integrability and non-integrability is also important for the energy level statistics, which obeys the Poisson distribution in integrable systems and obeys the Wigner-Dyson distribution in fully chaotic systems [3].

Although researchers have discovered a variety of integrable systems, integrable systems have some unphysical properties as explained above, and thus almost all many-body systems in nature are considered to be non-integrable. Therefore, it might be surprising that no concrete quantum many-body system has been proven to be non-integrable in spite of its ubiquitousness. Even worse, some researchers believe that non-integrability is out of scope of analytical investigation, and non-integrability can be only presumed with help of numerical simulations.

To overcome this pessimistic belief, in this presentation, we rigorously prove that a particular quantum many-body system, the spin-1/2 XYZ chain with a magnetic field, is indeed non-integrable in the sense that this system has no nontrivial local conserved quantity [4]. The proof of non-integrability exploits a bottom-up approach, in which we demonstrate that all the candidates of local conserved quantities cannot be conserved. Any nontrivial conserved quantity in this model turns out to be a sum of operators supported by at least half of the entire system. Our approach can apply to other S=1/2 systems including the Heisenberg model with the next nearest-neighbor interaction.

[1] A. Shimizu and K. Fujikura, J. Stat. Mech. 024004 (2017).

[2] S. H. Shenker and D. Stanford, J. High Energ. Phys. 2014:67 (2014).

[3] F. Haake, “Quantum Signatures of Chaos”, Springer (2010).

[4] N. Shiraishi, arXiv:1803.02637