Quantum Dilogarithms and New Integrable Lattice Models in Three Dimensions

TL;DR

This paper introduces new integrable three-dimensional lattice models based on quantum dilogarithms, enabling exact calculations of their partition functions and expanding the understanding of 3D integrable systems.

Contribution

It presents a novel class of 3D lattice models with commuting transfer matrices constructed via quantum dilogarithms, providing explicit examples and exact solutions.

Findings

Construction of integrable 3D lattice models using quantum dilogarithms

Exact calculation of partition functions in the infinite lattice limit

Identification of models related to Faddeev's modular quantum dilogarithm

Abstract

In this paper we introduce a new class of integrable 3D lattice models, possessing continuous families of commuting layer-to-layer transfer matrices. Algebraically, this commutativity is based on a very special construction of local Boltzmann weights in terms of quantum dilogarithms satisfying the inversion and pentagon identities. We give three examples of such quantum dilogarithms, leading to integrable 3D lattice models. The partition function per site in these models can be exactly calculated in the limit of an infinite lattice by using the functional relations, symmetry and factorization properties of the transfer matrix. The results of such calculations for 3D models associated with the Faddeev modular quantum dilogarithm are briefly presented.