Estimates of critical lengths and critical temperatures for classical and quantum lattice systems

Abstract

Local Ward identities are derived which lead to the mean-field upper bound for the critical temperature for certain multicomponent classical lattice systems (improving by a factor of two an estimate of Brascamp-Lieb). We develop a method for accurately estimating lattice Green's functionsI _d yielding 0.3069<I ₄<0.3111 and the global bounds (d−1/2)⁻<I _d <(d−1)⁻ for alld⩾4. The estimate forI _d implies the existence of a critical length for classical lattice systems with fixed length spins. Forv-component spins with fixed lengthb on the lattice ℤ^d,v=1, 2, 3, 4, the critical temperature for spontaneous magnetization satisfies

$$\frac{{2Jb^2 }}{k}\frac{{d - 1}}{v}< T{}_c \leqslant \frac{{2Jb^2 }}{k}\frac{d}{v} for d \geqslant 4$$

ford4 Using GHS or generalized Griffiths' inequalities, we find that the upper bounds on the critical temperature extend to certain classical and quantum systems with unbounded spins. Absence of symmetry breakdown at high temperature for quantum lattice fields follows from bounding the energy density by a multiple ofkT. Path space techniques for finite degrees of freedom show that the high-temperature limit is classical.