Berry phase in the rigid rotor: the 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 centre of mass held fixed. 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 non-trivial 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 $\pi$ attached to all non-trivial loops. We derive the corresponding stationary states by exploiting the connection between $SO(3)$ and $SU(2)$ spaces. The solutions are anti-periodic over any non-trivial loop, i.e., stationary states reverse sign under a $2\pi$ 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 $N\times S$ 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 neighbours. 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 $\pi$.

arXiv comment14 pages, 12 figures

References in corpus (15)

• Exact spectra, spin susceptibilities and order parameter of the quantum Heisenberg antiferromagnet on the triangular lattice
• Geometric phase from Aharonov-Bohm to Pancharatnam-Berry and beyond
• Z_2-vortex ordering of the triangular-lattice Heisenberg antiferromagnet
• Quantum numbers for relative ground states of antiferromagnetic Heisenberg spin rings
• Ground state properties of antiferromagnetic Heisenberg spin rings
• Structure and relevant dimension of the Heisenberg model and applications to spin rings
• Unexpected properties of the first excited state of non-bipartite Heisenberg spin rings
• The Tower of States and the Entanglement Spectrum in a Coplanar Antiferromagnet
• From skyrmions to Z2 vortices in distorted chiral antiferromagnets
• Anyons in integer quantum Hall magnets
• Spin-Momentum Locked Topological Surface States, non-trivial Berry's phase and magnetoelectric quantization in topological insulators
• Studying Continuous Symmetry Breaking using Energy Level Spectroscopy
• Dirac cone in iron-based high Tc superconductors
• Effective theories for quantum spin clusters: Geometric phases and state selection by singularity
• Semiclassical analysis of two- and three-spin antiferromagnets and anyons on a sphere

Cited by in corpus (2)

• Topological charges of periodically kicked molecules
• Finite-Size Spectral Signatures of Order by Quantum Disorder: A Perspective from Anderson's Tower of States