An essentially geometrically frustrated magnetic object: an exact solution

TL;DR

This paper introduces an exact solution for a hypothetical high-dimensional tetrahedral magnetic system, revealing key features of geometric frustration such as degeneracy and lack of phase transition, applicable to real systems at high dimensions.

Contribution

It provides an exact analytical solution for a multidimensional tetrahedral spin system, illustrating fundamental properties of geometric frustration in magnetic materials.

Findings

High degeneracy of ground state

No magnetic phase transition observed

Curie-Weiss behavior persists down to zero temperature

Abstract

Starting from the archetypical geometrically frustrated magnetic objects -- equilateral triangle and tetrahedron -- we consider an imaginary object: a multidimensional tetrahedron with spins $1/2$ in the each vertex and equal Heisenberg magnetic exchange along each edge. Many-particle case is obtained by setting dimensionality $d$ to high numbers, hence providing likely the most geometrically frustrated magnetic system ever possible. This problem has an exact solution for each $d$, obtained by simple student-level approach. As a result, this imaginary object clearly demonstrates at $d\to{}\infty$ all the features characteristic for the real geometrically frustrated magnetic systems: highly-degenerate ground state; absence of the magnetic phase transition; perfect Curie-Weiss behavior down to $T\to{}0$; and vanishingly small exchange energy per one spin.