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Journal of Mathematical Physics

Some Simple Lattice-Spin Systems

Jong-Won Lieh

Published Jul 1, 1970• 8 citations

Abstract

A simple mathematical scheme is derived here. With this scheme, three problems of lattice-spin systems are solved exactly. The first one is the problem of solving, thermodynamically, a linear chain of the Lenz-Ising model in a zero magnetic field with nearest, as well as next-nearest, neighbor couplings. The problem turns out to be equivalent to the problem of a linear chain with only nearest-neighbor couplings but in a finite magnetic field. The second one is to solve an imperfect 1-dimensional Heisenberg-Dirac model, similar to the partially solved ``Ising-Heisenberg'' model of Lieb-Mattis-Schultz, in a zero magnetic field. The problem is solved completely in the sense that all the elementary excitations of this model are shown in terms of some pseudofermions and the spectra are given as ε1(q)=[(Aq+Bq)2+Eq2]12,ε2(q)=[(Aq−Bq)2+Eq2]12,where Aq, Bq, and Eq are three different functions of sin (q), cos (q), and coupling strengths involved. The third one, the XY model, is used to study the contribution of ``inhomogeneity'' in the coupling strengths to the system, as compared to the anisotropicity contribution to it.

DOI: 10.1063/1.1665371

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A simple mathematical scheme is derived here. With this scheme, three problems of lattice-spin systems are solved exactly. The first one is the problem of solving, thermodynamically, a linear chain of the Lenz-Ising model in a zero magnetic field with nearest, as well as next-nearest, neighbor couplings. The problem turns out to be equivalent to the problem of a linear chain with only nearest-neighbor couplings but in a finite magnetic field. The second one is to solve an imperfect 1-dimensional Heisenberg-Dirac model, similar to the partially solved

Ising-Heisenberg'' model of Lieb-Mattis-Schultz, in a zero magnetic field. The problem is solved completely in the sense that all the elementary excitations of this model are shown in terms of some pseudofermions and the spectra are given as ε1(q)=[(Aq+Bq)2+Eq2]12,ε2(q)=[(Aq−Bq)2+Eq2]12,where Aq, Bq, and Eq are three different functions of sin (q), cos (q), and coupling strengths involved. The third one, the XY model, is used to study the contribution of

inhomogeneity'' in the coupling strengths to the system, as compared to the anisotropicity contribution to it.

The authors consider several Ising models with anisotropic interactions which can be solved exactly in the thermodynamic limit, for instance using a transfer matrix method. These include a two-dimensional system where each spin interacts directly with the two nearest neighbours along one of the principal axes of the lattice and has a coherent-field coupling with the spins along the second principal direction and a multidimensional system with coherent-field couplings along any of the principal axes.

The one-dimensional spin-\textonehalf{} Ising model is solved exactly with very long-range, equivalent-neighbor ferromagnetic interactions of strength

{J}_{\mathrm{LR}}

superimposed on nearest-neighbor anti-ferromagnetic interactions of strength $\ensuremath{-}{J}_{\mathrm{sr}}$. For $\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}} >R=3.1532$... there is a critical point of classical type in zero field. For $2\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}=R$ the critical exponents take the values $\ensuremath{\beta}=\frac{1}{4}$, $\ensuremath{\gamma}=1={\ensuremath{\gamma}}^{\ensuremath{'}}$, $\ensuremath{\delta}=5$, and $0=\ensuremath{\alpha}\ensuremath{\ne}{\ensuremath{\alpha}}^{\ensuremath{'}}=\frac{1}{2}$. For $2<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<R$ the transition in zero field is of first order, and the critical points which occur in finite field are of classical type. For $0<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<2$ there is no phase transition in zero field, although there are classical critical points in a field.

A one-dimensional Ising chain in which nearest neighbour (n.n.) and next nearest neighbour (n.n.n.) exchange interactions depend linearly on the distance between the spins is considered. The magnetic field and temperature dependence of the chain length is exactly calculated by use of a transfer matrix. It is shown that the chain length changes variously because of the competition among the n.n. and n.n.n. exchange interactions and the magnetic field. The zero-temperature phase diagram is also determined, which is composed of four spin states ↑, ↑↓, ↑↑↓ and ↑↑↓↓ (the smallest units of repetitions are shown). Dimerization and trimerization of the lattice are found in the ↑↑↓↓ and ↑↑↓ states respectively.

The one-dimensional Ising model with arbitrary nearest neighbor interactions and a magnetic field is formulated in terms of the spin correlation function relations derived in the preceding article I. The local magnetization and the two-spin correlation functions are shown to satisfy a set of coupled algebraic summation equations. The solution of this set is given in terms of a power series in hyberbolic tangents of the magnetic field strength, for chains of arbitrary length. For cases in which the interaction constants are random variables, explicit expressions are derived for the configurational averaged magnetization and the two-spin correlation functions to the lowest order in the magnetic field.

Introduction to Integrable Many-Body Systems II This is the second part of a three-volume introductory course about integrable systems of interacting bodies. The models of interest are quantum spin chains with nearest-neighbor interactions between spin operators, in particular Heisenberg spin-1/2 models. The Ising model in a transverse field, expressible as a quadratic fermion form by using the Jordan-Wigner transformation, is the subject of Sect. 12. The derivation of the coordinate Bethe ansatz for the XXZ Heisenberg chain and the determination of its absolute ground state in various regions of the anisotropy parameter are presented in Sect. 13. The magnetic properties of the ground state are explained in Sect. 14. Sect. 15 concerns excited states and the zero-temperature thermodynamics of the XXZ model. The thermodynamics of the XXZ Heisenberg chain is derived on the basis of the string hypothesis in Sect. 16; the thermodynamic Bethe ansatz equations are analyzed in high-temperature and low-temperature limits. An alternative derivation of the thermodynamics without using strings, leading to a non-linear integral equation determining the free energy, is the subject of Sect. 17. A nontrivial application of the Quantum Inverse Scattering method to the fully anisotropic XYZ Heisenberg chain is described in Sect. 18. Sect. 19 deals with integrable cases of isotropic spin chains with an arbitrary spin.