Quantum rotor model

The quantum rotor model is a mathematical model for a quantum system. 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.In particular, O(2) quantum rotor models can be used to describe superconducting Josephson junction arrays or the behavior of bosons in optical lattices. Also, it can be shown that O(3) rotor models are equivalent to bilayer quantum Heisenberg antiferromagnet and can also describe double-layer quantum Hall ferromagnets.

The model can be visualized as an array of rotating electrons behaving as rigid rotors with dipole-dipole interactions. The Hamiltonian for this model can be given as:

$H_{R}={\frac {J{\bar {g}}}{2}}\sum _{i}\mathbf {L} _{i}^{2}-J\sum _{\langle ij\rangle }\mathbf {n} _{i}\cdot \mathbf {n} _{j}$

where $\mathbf {n} _{i}$ are n-dimensional position vector operators, $\mathbf {L} _{i}$ are the corresponding conjugate (angular) momentum operators and $J,{\bar {g}}$ are constants. In the limit of very small and very large ${\bar {g}}$, we obtain two distinct ground states, "magnetically" ordered, and disordered or "paramagnetic" ground states respectively.

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 $\mathbf {S} _{1i}$ and $\mathbf {S} _{2i}$, the rotor model approximates the low energy states of a Heisenberg antiferromagnet, with Hamiltonian $H_{d}=K\sum _{i}\mathbf {S} _{1i}\cdot \mathbf {S} _{2i}+J\sum _{\langle ij\rangle }\left(\mathbf {S} _{1i}\cdot \mathbf {S} _{1j}+\mathbf {S} _{2i}\cdot \mathbf {S} _{2j}\right)$ using the correspondence $\mathbf {L} _{i}=\mathbf {S} _{1i}+\mathbf {S} _{2i}$

It can also be shown that the phase transition for the two dimensional rotor model has the same universality class as that of antiferromagnetic Heisenberg spin models.

See also

• Heisenberg model (quantum)
• Ising model