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Physical review. B, Condensed matter

Exact solution of aq-boson hopping model

N. M. Bogoliubov, R. Bullough, G. D. Pang

Published May 1, 1993• 57 citations
Publisher

Abstract

Quantum-group deformation of ordinary bosons yields the q bosons or nonlinear phonons. We solve exactly, for any real q>1, including q=\ensuremath{\infty}, a one-dimensional q-boson lattice ``hopping model,'' and calculate asymptotically its correlation functions. The model is a nonlinear model of strongly interacting ordinary bosons. Our solution solves the equivalent quantum difference differential nonlinear Schr\"odinger model. Approximation reduces this to a nonintegrable boson Hubbard-type model with q bosons as fundamental particles. One special limit q\ensuremath{\rightarrow}\ensuremath{\infty} of the hopping model is the classical $XY--- chain.

DOI
10.1103/physrevb.47.11495

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Quantum-group deformation of ordinary bosons yields the q bosons or nonlinear phonons. We solve exactly, for any real q>1, including q=\ensuremath{\infty}, a one-dimensional q-boson lattice ``hopping model,'' and calculate asymptotically its correlation functions. The model is a nonlinear model of strongly interacting ordinary bosons. Our solution solves the equivalent quantum difference differential nonlinear Schr"odinger model. Approximation reduces this to a nonintegrable boson Hubbard-type model with q bosons as fundamental particles. One special limit q\ensuremath{\rightarrow}\ensuremath{\infty} of the hopping model is the classical $XY--- chain.
Abstract Quantum models of interacting bosons have a wide range of applications, including the propagation of optical modes in nonlinear media, such as the k-photon down-conversion. Many of these models are related to nonlinear deformations of finite group algebras and, in this sense, are exactly solvable. While advanced group-theoretic methods were developed to study the eigenvalue spectrum, in quantum optics, the primary focus is not on the spectrum of the Hamiltonian but rather on the evolution of an initial state—such as the generation of optical signal modes by a strong pump mode propagating through a nonlinear medium. I propose a simple and general method to solve the state evolution problem, applicable to a broad class of quantum models of interacting bosons. For the k-photon down-conversion model and its generalizations, the solution to the state evolution problem is expressed as an infinite series expansion in powers of the propagation time, with coefficients determined by a recursion relation involving only a single polynomial function. This polynomial function is unique to each nonlinear model. As an application, I compare the exact solution of the parametric down-conversion process with the semiclassical parametric approximation.
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.
The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional self-consistent equation. In this paper we obtain this equation for the Bose-Hubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent-state path integral, leads in the large connectivity limit to the mean-field treatment of Fisher et al. [Phys. Rev. B 40, 546 (1989)] at the leading order, and to the bosonic dynamical mean field theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B 77, 235106 (2008)]. We obtain an alternative form of the equation using the occupation number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary-time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite-dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions.
The quantum correlation functions for different exactly solvable (completely integrable) models are all constructed in a similar way. In Part IV our main example will be the nonlinear Schrödinger (NS) equation. We shall also comment on the other models: the Heisenberg antiferromagnet, the Hubbard model and so on. We shall discuss different approaches to quantum correlation functions. Our main approach leads to differential equations for quantum correlation functions. These are classical nonlinear differential equations; they are completely integrable and closely related to the original classical differential equation that was quantized (the NS equation in our case). In the first stage of calculation of correlation functions we represent them as a determinant of some Predholm integral operator (here we shall use the results of Part III). The correct language for the description of quantum correlation functions is the language of τ functions, which was developed in. This helps us to relate the differential equations for quantum correlation functions with the hierarchy of the original classical differential equation that was quantized (the NS equation in our example). In this way we solve the most difficult problem; that of the evaluation of time-and temperature-dependent correlation functions. This approach can be applied not only to correlation functions for the NS equation, but also to the sine-Gordon model, to non-relativistic fermions, to the Heisenberg antiferromagnetic with special anisotropy and to other models.
The Bose-Hubbard model is used to study bosons in optical lattices. In this thesis we will use an extended Bose-Hubbard model to study a type of completely localized solutions, called compactons. The compactons are a special case of the much studied solitons. The soliton is a familiar concept in non-linear physics. It is a stable, localized wave-solution, found in a range of different systems; from DNA-molecules to optical fibers. The compacton is a soliton that is completely localized, i.e. strictly zero outside a given area. The dynamics of the (extended) Bose-Hubbard model is based on the tunneling of particles between the lattice sites. The ordinary Bose-Hubbard model only accounts for one-particle tunneling processes. We will consider a model that also takes some two-particle tunneling processes into account, basically by considering long-range effects of the particle interaction. The aim of this thesis is to find and study the quantum analog of the compactons found in an extended Discrete Non-Linear Schrödinger equation. We will study analytical solutions and try to find if and under which conditions specific compactons exist. Numerical calculations are made to study the properties of the compactons and to study how compacton solutions arise in the classical limit.
Exact solutions of many-body problems hold a very special and significant place in condensed matter physics, helping us to understand the behaviors of interacting particles. In this thesis, we focus our attention on study the exact solutions of two interesting many-body problems in stronglycorrelated \n \nsystems. One is to calculate the exact correlation functions of \n \n1D chains of spinless fermions and hardcore bosons. The other is to calculate the single-particle dispersion relation of Hubbard model of spinless fermions on 2D square lattices.