Integrable deformations of superintegrable quantum circuits

Abstract

Superintegrable models are very special dynamical systems: they possess more conservation laws than what is necessary for complete integrability. This severely constrains their dynamical processes, and it often leads to their exact solvability, even in non-equilibrium situations. In this paper we consider special Hamiltonian deformations of superintegrable quantum circuits. The deformations break superintegrability, but they preserve integrability. We focus on a selection of concrete models and show that for each model there is an (at least) one parameter family of integrable deformations. Our most interesting example is the so-called Rule54 model. We show that the model is compatible with a one parameter family of Yang-Baxter integrable spin chains with six-site interaction. Therefore, the Rule54 model does not have a unique integrability structure, instead it lies at the intersection of a family of quantum integrable models.

On Two Non-Ergodic Reversible Cellular Automata, One Classical, the Other Quantum

“…One might expect that matter-field automaton hosts an infinite tower of local conserved quantities of increasing support size, however, these may be very difficult to find empirically. This hypothesis would suggest also that the matter-field automaton may be a completely integrable system, in a similar spirit as Rule 54 [15], as it also displays similar negative-length hard-rod dynamics for a class of initial data, but preliminary attempts to find a Yang-Baxter or Lax structure failed. The other option is that the matter-field system has only a few, perhaps finitely many (or an incomplete set) conserved local charges, and would then represent a paradigmatic theory between integrable and ergodic dynamics.…”

“…increases exponentially with the size of support, say such that all r j ≤ , is reminiscent of superintegrable quantum cellular automata [13,15,16], yet the absence of an exact Yang-Baxter property suggests that the dynamical system is not completely integrable (i.e., there should be no Bethe ansatz!?). We are thus suggesting a new class of non-ergodic many-body quantum dynamics.…”

The Floquet Baxterisation

“…[69][70][71], and cellular automatons has been also anticipated, see [72] for a review. We believe that development of these ideas for quantum circuits is an interesting direction for future research, see [19,[73][74][75] for recent activity in this direction. Let us give an example of a solution to the set-theoretical Yang-Baxter equation.…”

Wrapping corrections for long range spin chains

“…defines commuting quantities [T ( ) (u), T ( ) (v)] = 0. The detailed proof can be found in the appendix of [18]. In (8) we defined the twisted trace operator Tr J, which acts on an operator X as Tr J, (X) = Tr J+1,...,J+ (XP ,J+ P −1,J+ −1 .…”