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Journal of Physics C Solid State Physics

Functional integral quantisation of an anisotropic spin-field model

Katsuhiro Nakamura, Tomohei Sasada, A. R. Bishop

Published Jul 10, 1983• 10 citations
Publisher

Abstract

The authors develop a path-integral formalism to obtain the semiclassical quantum spectrum of the pulse soliton with an internal degree of freedom in the one-dimensional continuous Heisenberg ferromagnet with an easy axis. No restriction is imposed on the spin magnitude S. For S=1/2, the pulse soliton yields the same quantum spectrum as that of the S=1/2 lattice Heisenberg-Ising ferromagnet obtained by the Bethe ansatz method. This agreement supports a plausible particle picture of the multimagnon bound state. For S>1/2, the present semiclassical spectrum can also be interpreted, aside from its prefactor, in terms of an underlying spin S(>1/2) quantum lattice model if the cut-off condition for the wave-number of the soliton is taken into account.

DOI
10.1088/0022-3719/16/19/017

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The authors develop a path-integral formalism to obtain the semiclassical quantum spectrum of the pulse soliton with an internal degree of freedom in the one-dimensional continuous Heisenberg ferromagnet with an easy axis. No restriction is imposed on the spin magnitude S. For S=1/2, the pulse soliton yields the same quantum spectrum as that of the S=1/2 lattice Heisenberg-Ising ferromagnet obtained by the Bethe ansatz method. This agreement supports a plausible particle picture of the multimagnon bound state. For S>1/2, the present semiclassical spectrum can also be interpreted, aside from its prefactor, in terms of an underlying spin S(>1/2) quantum lattice model if the cut-off condition for the wave-number of the soliton is taken into account.
An anisotropic Heisenberg ferromagnetic spin chain model is studied by using Holstain-Primakoff representation. In the semiclassical limit, the exact solutions for bright and dark solitons are found by using the coherent-state method combined with the Holstein-Primakoff bosonic representation of spin operators. These results show that the solutions can be expressed in terms of the elliptic integrals in different parameter regions. Some solutions for dark solitons are the innovation points in this paper.
Using the algebro-geometric approach, we study the structure of semi-classical eigenstates in a weakly-anisotropic quantum Heisenberg spin chain. We outline how classical nonlinear spin waves governed by the anisotropic Landau--Lifshitz equation arise as coherent macroscopic low-energy fluctuations of the ferromagnetic ground state. Special emphasis is devoted to the simplest types of solutions, describing precessional motion and elliptic magnetisation waves. The internal magnon structure of classical spin waves is resolved by performing the semi-classical quantisation using the Riemann--Hilbert problem approach. We present an expression for the overlap of two semi-classical eigenstates and discuss how correlation functions at the semi-classical level arise from classical phase-space averaging.
We develop a geometric representation for the ground state of the spin-1/2 quantum XXZ ferromagnetic chain in terms of suitably weighted random walks in a two-dimensional lattice. The path integral model so obtained admits a genuine classical statistical mechanics interpretation with a translation invariant Hamiltonian. This new representation is used to study the interface ground states of the XXZ model. We prove that the probability of having a number of down spins in the up phase decays exponentially with the sum of their distances to the interface plus the square of the number of down spins. As an application of this bound, we prove that the total third component of the spin in a large interval of even length centered on the interface does not fluctuate, i.e. has zero variance. We also show how to construct a path integral representation in higher dimensions and obtain a reduction formula for the partition functions in two dimensions in terms of the partition function of the one-dimensional model.
Abstract We consider lambda and anisotropic deformations of the principal chiral model and show how they can be quantised in the Hamiltonian formalism on a lattice as a suitable spin chain. The spin chain is related to the higher spin XXZ Heisenberg chain and can be solved by using the Bethe Ansatz. This yields the spectrum and S-matrix of the excitations. In particular, we find the S-matrix in the gapped anti-ferromagnetic regime. In this regime, a continuum limit does not exist and this suggests that the field theories in this regime, precisely ones with a cyclic RG like the Yang–Baxter deformations, may only exist as effective theories. In a certain limit, we show that the XXZ type lambda model gives the symmetric space lambda model and, hence, we are able to find its spectrum and S-matrix and show that it gives the S-matrix of the sigma model in the appropriate limit. Finally, we show the full consistency of the S-matrix and the Lagrangian formulations of the lambda model, by coupling to a conserved charge and computing the way the ground state energy changes in both pictures.