Heisenberg Spin Fluid in an External Magnetic Field

Abstract

We develop a general method to study inhomogeneous liquids in an external field using orthogonal polynomials tailored to the one-body density. The procedure makes integral equation calculations of these systems no more difficult than those of ordinary homogeneous molecular fluids. We apply this method to the ferromagnetic Heisenberg spin fluid in a magnetic field using a “reference” version of the Zerah-Hansen closure, with no further approximations. Comparison with simulation shows this integral equation procedure yielding nearly exact results. [S0031-9007(98)05886-4] The Gibbsian N-body density function of a Hamiltonian HN that is rotationally and translationally invariant must itself be rotationally and translationally invariant, as must then also be all reduced n-body density functions of this Hamiltonian. In particular, the one-body density is a constant. An external field destroys this homogeneity, producing anisotropy or nonuniformity in the one-body density [1,2], and so makes necessary the joint calculation of the coupled one-body and two-body density functions. A striking if familiar example of the response of a bulk system to an external field is ferromagnetism. In this paper, we shall use this particular case to present a general procedure to compute the coupled one-body and two-body density functions of an inhomogeneous classical fluid in an external field. Remarkably, the procedure is no more difficult to carry through than similar calculations for ordinary homogeneous systems. Perhaps the simplest model of a disordered continuum system exhibiting ferromagnetic behavior is a fluid of hard spheres with embedded Heisenberg spins described using classical statistical mechanics [3 ‐6]. This interaction potential is clearly inadequate to model ferrofluids, in particular at low concetrations, where the dipole-dipole interaction is dominant, but it is however relevant for a description of ferromagnetism in undercooled liquid metal alloys [7]. The Heisenberg spin fluid in an external magnetic field B0 is succinctly defined by the canonical excess partition function Z ex › 1 s4p V d N Z N Y j›1 fdrj dvj f0svj dg