Quantum rotor model
The quantum rotor model is a mathematical model for a quantum system. It can be visualized as an array of rotating electrons which behave as rigid rotors that interact through short-range dipole-dipole magnetic forces originating from their magnetic dipole moments. The model differs from similar spin-models such as the Ising model and the Heisenberg model in that it includes a term analogous to kinetic energy.
Although elementary quantum rotors do not exist in nature, the model can describe effective degrees of freedom for a system of sufficiently small number of closely coupled electrons in low-energy states.
Formulation
Suppose the n-dimensional position vector of the model at a given site is. Then, we can define rotor momentum by the commutation relation of componentsHowever, it is found convenient to use rotor angular momentum operators defined by components
Then, the magnetic interactions between the quantum rotors, and thus their energy states, can be described by the following Hamiltonian:
where are constants.. The interaction sum is taken over nearest neighbors, as indicated by the angle brackets. For very small and very large, the Hamiltonian predicts two distinct configurations, namely "magnetically" ordered rotors and disordered or "paramagnetic" rotors, respectively.
The interactions between the quantum rotors can be described by another Hamiltonian, which treats the rotors not as magnetic moments but as local electric currents.
Properties
One of the important features of the rotor model is the continuous O(N) symmetry, and hence the corresponding continuous symmetry breaking in the magnetically ordered state. In a system with two layers of Heisenberg spins and, the rotor model approximates the low-energy states of a Heisenberg antiferromagnet, with the Hamiltonianusing the correspondence