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arXiv:nlin/0701030 · Symmetry Integrability and Geometry Methods and Applications

q-Boson in Quantum Integrable Systems

Anjan Kundu

Published Mar 5, 2007• 5 citations
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Abstract

q-bosonic realization of the underlying Yang-Baxter algebra is identified for a series of quantum integrable systems, including some new models like two-mode q-bosonic model leading to a coupled two-component derivative NLS model, wide range of q-deformed matter-radiation models, q-anyon model etc. Result on a new exactly solvable interacting anyon gas, linked to q-anyons on the lattice is reported.

DOI
10.3842/sigma.2007.040

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The authors construct the Hamiltonian of a new quantum integrable 'q-boson' lattice model in 1+1 dimensions which has q-bosons as dynamical variables and solve it for its energy eigenstates and energy eigenvalues under periodic boundary conditions of finite period. This model can be regarded as the q-deformation of the integrable lattice Bose gas (the quantum lattice nonlinear Schrodinger (quantum lattice NLS)) model. In appropriate continuum limits both of these lattice models become the quantum repulsive NLS model (the Bose gas). In the lattice case the distinction between the q-bosons and ordinary bosons leads to significant new features-particularly in the energy spectrum for which the q-boson lattice model has an optical branch and bound state spectra as well as the (one) acoustic branch of the quantum lattice NLS model. It is argued that this sort of distinction is generic for discrete systems with a finite number of degrees of freedom as opposed to continuum field theories like the Bose gas where quantization by quantum group quantization or by canonical quantization leads to the same results. Various properties of the q-bosons are given and used. In particular, a q-boson Primakov-Holstein transformation is given which connects the q-boson lattice model with the XXZ models of arbitrary spin.
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.
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