Berry phase in the rigid rotor: Emergent physics of odd antiferromagnets

Abstract

The rigid rotor is a classic problem in quantum mechanics, describing the dynamics of a rigid body with its center of mass held fixed. It is well known to describe the rotational spectra of molecules. The configuration space of this problem is SO(3), the space of all rotations in three dimensions. This is a topological space with two types of closed loops: trivial loops that can be adiabatically shrunk to a point and nontrivial loops that cannot. In the traditional formulation of the problem, stationary states are periodic over both types of closed loops. However, periodicity conditions may change if Berry phases are introduced. We argue that time-reversal symmetry allows for only one new possibility---a Berry phase of $\ensuremath{\pi}$ attached to all nontrivial loops. We derive the corresponding stationary states by exploiting the connection between SO(3) and SU(2) spaces. The solutions are antiperiodic over any nontrivial loop; i.e., stationary states reverse sign under a $2\ensuremath{\pi}$ rotation about any axis. Remarkably, this framework is realized in the low-energy physics of certain quantum magnets. The magnets must satisfy the following conditions: (a) the classical ground states are unpolarized, carrying no net magnetization, (b) the set of classical ground states is indexed by SO(3), and (c) the product $N\ifmmode\times\else\texttimes\fi{}S$ is a half-integer, where $N$ is the number of spins and $S$ is the spin quantum number. We demonstrate this result in a family of Heisenberg antiferromagnets defined on polygons with an odd number of vertices. At each vertex, we have a spin-$S$ moment that is coupled to its nearest neighbors. In the classical limit, these magnets have coplanar ground states. Their quantum spectra, at low energies, correspond to ``spherical top'' and ``symmetric top'' rigid rotors. For integer values of $S$, we recover traditional rigid-rotor spectra. With half-integer $S$, we obtain rotor spectra with a Berry phase of $\ensuremath{\pi}$.