How to prevent a quantum system from thermalization is an open and fundamentally important question. One approach to stop thermalization is to break ergodicity. In addition to integrability and localization by disorder, another possibility to break ergodicity is by local constraints. The lattice gauge theory(LGT) is a particular family of constrained systems where the local constraints are due to the gauge symmetry. It has also been observed that in certain one-dimensional LGT, ergodicity can be broken and lead to non-trivial dynamics. However, it is not clear whether such physics can be observed in a 2D interacting system. In this collaboration, we investigate this problem in a 2D LGT by applying a new numerical method. In LGT, a generic wave function can be understood as a superposition of different weighting of correlated "disorder configurations." Such interpretation naturally suggests the initial state controls the "disorder configuration" and could lead to ergodic or localization phenomena. The method overcomes the system size challenge and allows us to analyze the propagation of defects, the light cone structure and establish the localized and ergodic phase.
Spin-orbit coupling reduces the symmetry of the system. It turns out we don't really understand the qualitative effects of the low symmetry. Especially, in a constrained Hilbert space, such as ice manifold, what is the effect of anisotropic quantum exchange process?
This problem is not only theoretically challenging but also numerically challenging. The low symmetry breaks a lot of nice properties and thus strong theorem such as LSM theorem is not useful for the analysis. In addition, the system with low symmetry usually does not have exact solvable points. Therefore, the analytic approach to study related problem is challenging.
From the numerical side, low symmetry also causes a problem for the tensor-network based algorithm. Without continuous symmetry such as U(1) and SU(2), the sparse format of the wave function is no longer valid and require higher bond dimension to study. Usual model with spin-orbit coupling will have the sign problem because of the nontrivial phase factor in the exchange process. It turns out we don't have much understanding from large-scale unbiased numerical methods for strong spin-orbit coupling systems.
One particular exception is the dipolar-octupolar doublets we discovered above. Analyzing the quantum Monte Carlo results with Prof. Ying-Jer Kao and his student Kai-Hsin Wu, we found the system stay in classical configuration down to very low temperature. Why the system acts so classical down to this low temperature is not clear. Using degenerate-perturbation theory, we found the effective theory at 6-th order has not only quantum tunneling term but also diagonal terms. Comparing with previous studies on pyrochlore lattice with U(1) symmetry, our system has only Z2 symmetry. In the U(1) case the comparable diagonal and quantum terms appear at the same lowest order only for spin with S>1/2. Thus, the low symmetry and constrained Hilbert space effectively increase the spin and make it more classical.