An algebraic approach to discrete time integrability

Abstract

We propose the systematic construction of classical and quantum two-dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable systems governed by sets of consistent integrable non-linear space-time difference equations. To illustrate the proposed methodology, we derive two versions of the fully discrete non-linear Schrödinger type system. The first one is based on the existence of a rational r-matrix, whereas the second one is the fully discrete Ablowitz–Ladik model and is associated to a trigonometric r-matrix. The Darboux-dressing method is also applied for the first discretization scheme, mostly as a consistency check, and solitonic as well as general solutions, in terms of solutions of the fully discrete heat equation, are also derived. The quantization of the fully discrete systems is then quite natural in this context and the two-dimensional quantum lattice is thus also examined.

Superintegrable cellular automata and dual unitary gates from Yang-Baxter maps

“…These systems display the hallmarks of integrability, such as the existence of a set of higher charges and commuting flows (the latter is also known as multi-dimensional consistency or the cube condition), the appearance of Lax matrices and the Yang-Baxter equation in various forms [12][13][14], and also a constrained algebraic growth of the iterated functions [15]. A new algebraic approach to discrete time integrable models was formulated recently in [16].…”