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 (neglecting Coulomb forces). 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 (orientation) vector of the model at a given site i {\displaystyle i} is n {\displaystyle \mathbf {n} } . Then, we can define rotor momentum p {\displaystyle \mathbf {p} } by the commutation relation of components α , β {\displaystyle \alpha ,\beta }
[ n α , p β ] = i δ α β {\displaystyle [n_{\alpha },p_{\beta }]=i\delta _{\alpha \beta }}
However, it is found convenient to use rotor angular momentum operators L {\displaystyle \mathbf {L} } defined (in 3 dimensions) by components L α = ε α β γ n β p γ {\displaystyle L_{\alpha }=\varepsilon _{\alpha \beta \gamma }n_{\beta }p_{\gamma }}
Then, the magnetic interactions between the quantum rotors, and thus their energy states, can be described by the following Hamiltonian:
H R = J g ¯ 2 ∑ i L i 2 − J ∑ ⟨ i j ⟩ n i ⋅ n j {\displaystyle H_{R}={\frac {J{\bar {g}}}{2}}\sum _{i}\mathbf {L} _{i}^{2}-J\sum _{\langle ij\rangle }\mathbf {n} _{i}\cdot \mathbf {n} _{j}}
where J , g ¯ {\displaystyle J,{\bar {g}}} are constants.. The interaction sum is taken over nearest neighbors, as indicated by the angle brackets. For very small and very large g ¯ {\displaystyle {\bar {g}}} , the Hamiltonian predicts two distinct configurations (ground states), namely "magnetically" ordered rotors and disordered or "paramagnetic" rotors, respectively. The interactions between the quantum rotors can be described by another (equivalent) Hamiltonian, which treats the rotors not as magnetic moments but as local electric currents. In higher dimensions, the Hamiltonian can be defined as
H R = J g ¯ 2 ∑ i Δ i − J ∑ ⟨ i j ⟩ n i ⋅ n j {\displaystyle H_{R}={\frac {J{\bar {g}}}{2}}\sum _{i}\Delta _{i}-J\sum _{\langle ij\rangle }\mathbf {n} _{i}\cdot \mathbf {n} _{j}}
where Δ i {\displaystyle \Delta _{i}} is the Laplace-Beltrami operator on the sphere S n {\displaystyle S^{n}} . It is exactly solvable in the large n limit.
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 S 1 i {\displaystyle \mathbf {S} _{1i}} and S 2 i {\displaystyle \mathbf {S} _{2i}} , the rotor model approximates the low-energy states of a Heisenberg antiferromagnet, with the Hamiltonian
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