An essentially geometrically frustrated magnetic object: an exact solution

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.

9 pages, 3 figures