Integrability of q-oscillator lattice model

Jean Michel Maillet, Frank Willem Nijhoff

Sara Vanovac, Catherine McCarthy, Federica Maria Surace +1 more

We show how to systematically construct weak integrability breaking perturbations (WIBs) for classical integrable models on the lattice. These perturbations, which allow quasi-conserved quantities, have mostly been explored in quantum systems, where they are expected to delay the onset of thermalization and diffusive transport to timescales far exceeding those predicted by Fermi's golden rule. However, accessing such long-time dynamics in quantum models is computationally challenging. Classical integrable lattice models offer a complementary setting for probing transport and long-time dynamics under WIBs. In this work, we specialize our general framework to construct several families of WIBs for the Ishimori model, the Toda chain, and the Harmonic Oscillator Chain (HOC). Such constructions can help quantify how WIBs contribute to anomalous transport and serve as a benchmark for thermalization studies in perturbed integrable models. An important example is the Fermi-Pasta-Ulam-Tsingou (FPUT) model: Starting from the HOC, we show that the cubic nonlinearity (the alpha-FPUT interaction) is a genuine WIB perturbation. Using the integrals of motion (IoMs) of the Toda lattice, we explicitly construct corrections to the entire hierarchy of the HOC IoMs, thereby obtaining an infinite tower of quasi-conserved quantities for the alpha-FPUT chain. We further identify the corresponding adiabatic gauge potential (AGP) as a nontrivial trilocal generator in real space, and show that, more generally, any cubic, translationally invariant, momentum-conserving perturbation of the HOC admits such a generator and is therefore a WIB. Together with our transport and AGP-variance studies, our results provide a unified classical framework for weak integrability breaking and for diagnosing anomalous thermalization and transport in nearly integrable Hamiltonian lattice systems.

David A. Coker

A. KUNDU, B. Basu Mallick

A systematic approach for generation of integrable quantum lattice models exploiting the underlying Uq(2) quantum group structure as well as its multiparameter generalization is presented. We find an extension of trigonometric Sklyanin algebra and also its deformation through "symmetry breaking transformation," which after consistent bosonization (or q-bosonization) construct a series of integrable lattice models. A novel quantum solvable derivative NLS, a relativistic Toda chain and a lattice model involving q-oscillators warrant special mention. As an added advantage, along with the integrable models the corresponding quantum R-matrices are also specified.

С. М. Сергеев

A lattice model of interacting q‐oscillators, proposed by V. Bazhanov and S. Sergeev in 2005 is the quantum‐mechanical integrable model in 2 + 1 dimensional space‐time. Its layer‐to‐layer transfer matrix is a polynomial of two spectral parameters, it may be regarded in terms of quantum groups both as a sum of sl(N) transfer matrices of a chain of length M and as a sum of sl(M) transfer matrices of a chain of length N for reducible representations. The aim of this paper is to derive the Bethe ansatz equations for the q‐oscillator model entirely in the framework of 2 + 1 integrability providing the evident rank‐size duality.