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Journal of Physics A Mathematical and General

The nonlinear Schrodinger model as a special continuum limit of the anisotropic Heisenberg model

B Golzer, Arno Holz

Published Aug 1, 1987• 24 citations
Publisher

Abstract

One-dimensional spin chains are investigated by constructing a mapping of the anisotropic Heisenberg model in the limit Delta to 1 onto the non-linear Schrodinger model (a gas of bosons with delta function interactions). Three applications of this mapping are given. First it is shown that the Bethe ansatz solutions of the anisotropic spin-1/2 Heisenberg model go over to the continuous Bethe ansatz solutions of the non-linear Schrodinger model. Then bound-state energies for arbitrary spin are calculated and as a third application a nearly identical mapping of the anisotropic spin-1/2 Heisenberg model in the limit Delta to 0 onto a non-linear Schrodinger model for fermion fields is given. Finally the authors construct an anisotropic SU(N) model. The continuum limit of this model is shown to be the (N-1)-component non-linear Schrodinger model, which can be solved by means of the nested Bethe ansatz.

DOI
10.1088/0305-4470/20/11/035

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•Jan 1, 2018•0 citations
One-dimensional spin chains are investigated by constructing a mapping of the anisotropic Heisenberg model in the limit Delta to 1 onto the non-linear Schrodinger model (a gas of bosons with delta function interactions). Three applications of this mapping are given. First it is shown that the Bethe ansatz solutions of the anisotropic spin-1/2 Heisenberg model go over to the continuous Bethe ansatz solutions of the non-linear Schrodinger model. Then bound-state energies for arbitrary spin are calculated and as a third application a nearly identical mapping of the anisotropic spin-1/2 Heisenberg model in the limit Delta to 0 onto a non-linear Schrodinger model for fermion fields is given. Finally the authors construct an anisotropic SU(N) model. The continuum limit of this model is shown to be the (N-1)-component non-linear Schrodinger model, which can be solved by means of the nested Bethe ansatz.
An analysis of the anisotropic Heisenberg model is carried out by solving the Bethe ansatz solution of the model numerically as a function of the anisotropy parameter for finite N. A brief introduction to the limit of the infinite chain is presented. The energy for a few special limiting cases of the anisotropy parameter in the Hamiltonian are worked out. Numerical results for finite cycles as well as for the infinite chain are given. Comparison can then be made with the case of finite increasing N. © 1997 John Wiley & Sons, Inc.
The authors derive a continuum representation of the spin-one operators which provides the mapping of the spin-one anisotropic Heisenberg chain onto the Luttinger model. They show a schematic phase diagram of the Hamiltonian, and calculate a general two-spin dynamic correlation function at finite temperatures, valid in the massless domain of the Hamiltonian. This can be applied to obtain the transverse dynamic form factor, and results are compared with neutron scattering data on CsNiF3. The behaviour of the transverse correlation function is shown to be governed by a critical exponent eta u+or- which is a measured of quantum effects in the system, eta M+or- to 0 being the classical limit. Existing data on CsNiF3 are shown to be in the classical regime, and they make suggestions for further experiments to search for quantum fluctuations. They also indicate the limits of applicability of the quantum sine-Gordon theory to CsNIF3.
A family of one-dimensional magnetic Hamiltonians is introduced, where at each site there are nnn spin-SSS operators. It is shown that, for special couplings between spins and for S=12S=\frac{1}{2}S=21​, the model contains the complete spectrum of the Heisenberg chain with spins \textonehalf{}, 1, frac32;, etc., and the ground state is that of the corresponding Heisenberg chain. By the varying of a single parameter the model allows continuous transitions between chains with different spin. We map the spin-(S+SS+SS+S) model onto the nonlinear $\ensuremath{\sigma}$ model and discuss the possibility of a finite gap in the spin-(\textonehalf{}+\textonehalf{}) model.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.