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Journal of Physics A Mathematical and General

A q-deformed completely integrable Bose gas model

N. M. Bogoliubov, R. Bullough

Published Jul 21, 1992• 64 citations

Abstract

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.

DOI: 10.1088/0305-4470/25/14/020

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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 lattice version of the quantum nonlinear Schrodinger (NLS) equation is considered, which has a significant simple form and fulfils most of the criteria desirable for such lattice variants of field models. Unlike most of the known lattice NLS equations, the present model belongs to a class which does not exhibit the usual symmetry properties. However, this lack of symmetry itself seems to be responsible for the remarkable simplification of the relevant objects in the theory, such as the Lax operator, the Hamiltonian and other commuting conserved quantities as well as their spectra. The model allows exact quantum solution through the algebraic Bethe ansatz and also a straightforward and natural generalization to the vector case, thus giving a new exact lattice version of the vector NLS model. A deformation representing a new quantum integrable system involving Tamm-Dancoff-like q-boson operators is constructed.

A lattice version of quantum nonlinear Schrodinger (NLS) equation is considered, which has significantly simple form and fullfils most of the criteria desirable for such lattice variants of field models. Unlike most of the known lattice NLS, the present model belongs to a class which does not exhibit the usual symmetry properties. However this lack of symmetry itself seems to be responsible for the remarkable simplification of the relevant objects in the theory, such as the Lax operator, the Hamiltonian and other commuting conserved quantities as well as their spectrum. The model allows exact quantum solution through algebraic Bethe ansatz and also a straightforward and natural generalisation to the vector case, giving thus a new exact lattice version of the vector NLS model. A deformation representing a new quantum integrable system involving Tamm-Dancoff like

q

-boson operators is constructed.

A composite quantum model on a lattice which describes the system of q-bosons interacting with U q (su(2)) spin impurities is introduced and solved exactly under periodic boundary conditions. In one limit the model is shown to become a new exactly soluble quantum system on a lattice which can be interpreted as a q-deformed version of the quantum Dicke model. In the limit of infinitesimal spacing the model is further reduced to a multilevel version of the previously introduced continuum-limit Dicke model. For spin 1/2 the previous results for this particular case are confirmed.

Degenerate spinor Bose gases with repulsive density-density interaction and anti-ferromagnetic spin-spin coupling in one spatial dimension are shown to be described by a quantum integrable matrix extension of the nonlinear Schrödinger model, whose fundamental fields are described by an

m\,\times\,n

matrix of bosonic field operators. The eigenstates of this model are constructed for arbitrarily sized matrix field operators by means of algebraic Bethe-ansatz techniques, and the corresponding Bethe equations governing the spectra of conserved quantities are derived. The approach thus generalizes previously chosen techniques to account for arbitrary spin multiplets and their spin-spin interaction. Focusing on the specific case of the

2\times2

model, which is shown to correspond to a spin-

1

Bose gas, a set of integral equations is derived, which describe its equilibrium thermodynamic properties. From these, the ground state phase diagram is computed both, numerically and analytically in the parameter plane spanned by the chemical potential and an external magnetic field. Furthermore, we establish a sufficient condition for interacting bosons in one dimension to obey the Pauli exclusion principle. In particular, we show that in the presence of paired bound states, no two quasiparticle rapidities can coincide, provided that the Lieb parameter satisfies

γ>4/3

.