Abstract
We introduce a class of integrable dynamical systems of interacting classical matrix-valued fields propagating on a discrete space-time lattice, realized as many-body circuits built from elementary symplectic two-body maps. The models provide an efficient integrable Trotterization of non-relativistic \sigmaσ-models with complex Grassmannian manifolds as target spaces, including, as special cases, the higher-rank analogues of the Landau–Lifshitz field theory on complex projective spaces. As an application, we study transport of Noether charges in canonical local equilibrium states. We find a clear signature of superdiffusive behavior in the Kardar–Parisi–Zhang universality class, irrespectively of the chosen underlying global unitary symmetry group and the quotient structure of the compact phase space, providing a strong indication of superuniversal physics.
Highlights
In Appendix A we demonstrate that the dynamics (8) preserves the Lagrangian submanifold with the anti-symplectic signature, Lagrangian Grasmannians again constitute an admissible phase space M1 ∼= L(N )
We have introduced a novel family of classical integrable models of interacting matrix-valued degrees of freedom propagating on a discrete space-time lattice, and obtained an explicit dynamical system in the form of a classical Floquet circuit composed of elementary two-body symplectic maps
In the absence of external fields, both time and space dynamics can be realized in uniform way, which reveals a particular type of space-time self-duality
Summary
Explaining how macroscopic laws of matter emerge from microscopic reversible dynamics is one the central problems of modern theoretical physics which is still quite far from being settled. We describe a simple procedure to obtain a class of many-body propagators composed of two-body sympletic maps which governs a discrete space-time evolution of interacting matrix-valued degrees of freedom This is accomplished in a systematic manner, employing the methods of algebraic geometry and the notion of Lax representation [2] which ensures integrability of the model from the outset. Aside from partial analytical [37, 39,78] and numerical evidence [79], it is not very clear what is the precise role of non-abelian symmetries and, if higher-rank symmetries could potentially alter this picture and possibly unveil new types of transport laws With these questions in mind, we design a class of integrable models whose degrees of freedom are matrix fields which take values on certain compact manifolds. There are five separate appendices which include detailed derivations and additional information on various technical aspects
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