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Acta Physica Sinica

ON THE ABSENCE OF THE PHASE TRANSITION IN A LOW-DIMENSIONAL MAGNETIC SYSTEM WITH MULTI-QUADRATIC EXCHANGE INTERACTIONS

Ruibao Tao, Fu‐Cho Pu, (1)复旦大学物理系; (2)中国科学院物理研究所

Published Jan 1, 1980• 0 citations
Publisher

Abstract

It is proved rigrously that the long range order for isotropic multi-quadratic Heisenberg model is absent in the 1 and 2-dimensional systems.
DOI
10.7498/aps.29.658

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It is rigorously proved that at any nonzero temperature, a one- or two-dimensional isotropic spin-SSS Heisenberg model with finite-range exchange interaction can be neither ferromagnetic nor antiferromagnetic. The method of proof is capable of excluding a variety of types of ordering in one and two dimensions.
The method of Mermin and Wagner [Phys. Rev. Lett. 17, 1133 (1966)] is used to show that one- and two-dimensional spin systems interacting with a general isotropic interaction H=12 ∑ ijn Iij(n) (Si·Sj)n,where the exchange interactionsIij(n) are of finite range, cannot order in the sense that 〈Oi〉=0for all traceless operators Oi defined at a single site i. Mermin and Wagner have proved the above for the case n = 1 with Oi = Si, i.e., for the Heisenberg Hamiltonian. The proof allows us to rule out the possibility that a small isotropic biquadratic exchange (Si·Sj)2 could induce ferromagnetism or antiferromagnetism in a two-dimensional Heisenberg system.
We have considered a classical spin system, consisting of 3-component unit vectors, associated with a two-dimensional lattice {uk, k ∈ Z2}, and interacting via a translationally and rotationally invariant pair potential, of the long-range ferromagnetic form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), and xj denotes dimensionless coordinates of the lattice sites. This potential model is known rigorously to possess an ordering transition at finite temperature, and has been characterized quantitatively by Monte Carlo simulation, whose results suggest a continuous transition taking place at [Formula: see text]; comparison is also reported with other theoretical treatments, such as Spherical Model, Molecular Field and Two Site Cluster approximations.