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Journal of Mathematical Physics

Correlation Inequalities for Heisenberg Ferromagnets

C. A. Hurst, S. Sherman

Published Aug 1, 1970• 8 citations
Publisher

Abstract

First, for a classical discrete approximation to the Heisenberg ferromagnet, in a Euclidean space of dimension 2k where k is a positive integer, the correlation is nonnegative and a monotonic increasing function of interactions. Second, for the quantum mechanical Heisenberg ferromagnet itself, the correlation is nonnegative but not necessarily a monotonic increasing function of interaction. Both results allow for long-range interactions. Third, a family of inequalities on the number of connected components in graphs is deduced.

DOI
10.1063/1.1665412

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First, for a classical discrete approximation to the Heisenberg ferromagnet, in a Euclidean space of dimension 2k where k is a positive integer, the correlation is nonnegative and a monotonic increasing function of interactions. Second, for the quantum mechanical Heisenberg ferromagnet itself, the correlation is nonnegative but not necessarily a monotonic increasing function of interaction. Both results allow for long-range interactions. Third, a family of inequalities on the number of connected components in graphs is deduced.
Within a random phase approximation,the quantum Heisenberg ferromagnetic chain with long-range interaction proportional to r-p was studied by Green-function method.The results indicated that for exchange interaction decaying as r-p for 1p2,there existed a phase transition at finite temperature,and that no transition to a ferromagnetic phase at any finite temperature occurs for p≥2.The results are reasonable agreement with those of the Monte Carlo calculations and the other theoretical approaches.
Simple but rigorous derivations are presented for the inequalities2−η≤dγ′/(2β+γ′)≤dγ′/(2−α′),2−η≤d(δ−1)/(δ+1),γ≤(2−η)ν.(Proofs of the first two inequalities have been given by Buckingham and Gunton.1) Here d is the dimensionality and α′, β, γ, γ′ and δ are the exponents characterizing a ferromagnet near its critical point,2 while η and ν describe the decay of the corresponding spin-spin correlation functions2 Γ2(r)=〈S0zSrz〉, according to ∑ |r|<RΓ2(r)∼R2−η(T=Tc,H=0,R→∞),κ1∼(T−Tc)ν,(T→Tc,H=0),where the inverse range of correlation κ1 may be defined by κ1−2=[Σrr2Γ2(r)]/[ΣrΓ2(r)]. The corresponding exponent ηE for the energy-energy correlation function (essentially Γ4(r, a)−[Γ(a)]2, where Γ4(r, a)=〈S0zSazSrzSr+az〉) is proved to satisfy 2−ηE≤dα′,2−ηE≤dαc/(1+ζ+αc),where the specific heat CM at T=Tc diverges with the magnetization as M−αc while the energy derivative | ∂U/∂M |Tc varies as Mζ. (Mean Field or classical values are αc=0, ζ=1.) The proofs are based on the ``intuitively obvious'' properties: (a) positivity, namely, Γ1=〈S0z〉≥0,Γ(r)−Γ12≥0,Γ4(r, a)−[Γ2(a)]2≥0, and (b) monotonic increase of Γ1, Γ2 and Γ4 under increase of magnetic field and decrease of temperature. These properties are known3 to be rigorously valid for Ising models of arbitrary spin, lattice structure and ferromagnetic coupling (Jij≥0). Their validity for real magnets together with the experimental observation δ≤4.7 leads to the significant conclusion η≥0.05.
The thermodynamic properties and critical behavior of a Heisenberg ferromagnet in two dimensions are studied using the Handscomb quantum Monte Carlo method. The spins (with quantum number S=12) are arranged on a square lattice and have a long-range interaction with a power-law decrease, J(r)∝r−p, with distance r. The parameter p is taken to be in the range 2<p⩽6. The system has an ordering transition at a nonzero temperature Tc when 2<p<4. The dependence of Tc on p and the critical behavior of the susceptibility are investigated. Properties of spin correlation functions in different regimes (of p and T) are discussed. The results are compared with previous approximate analytic quantum theories.
A Green-function method is employed within a random phase approximation to study the one-dimensional quantum Heisenberg model with long-range ferromagnetic interaction proportional to ${r}^{\ensuremath{-}p}.$ We have shown that for 1<p<21<p<21<p<2 there exists a phase transition at finite temperatures and estimate its critical temperature.