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

.

Although elementary quantum rotors do not exist in nature, the model can describe effective

Suppose the n-dimensional position (orientation) vector of the model at a given site ${\displaystyle i}$ is ${\displaystyle \mathbf {n} }$. Then, we can define rotor momentum ${\displaystyle \mathbf {p} }$ by the

commutation relation

of components ${\displaystyle \alpha ,\beta }$

${\displaystyle [n_{\alpha },p_{\beta }]=i\delta _{\alpha \beta }}$

However, it is found convenient[1] to use rotor angular momentum operators ${\displaystyle \mathbf {L} }$ defined (in 3 dimensions) by components ${\displaystyle L_{\alpha }=\varepsilon _{\alpha \beta \gamma }n_{\beta }p_{\gamma }}$

Then, the magnetic interactions between the quantum rotors, and thus their energy states, can be described by the following Hamiltonian:

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

where ${\displaystyle J,{\bar {g}}}$ are constants.. The interaction sum is taken over nearest neighbors, as indicated by the angle brackets. For very small and very large ${\displaystyle {\bar {g}}}$, the Hamiltonian predicts two distinct configurations (ground states), namely "magnetically" ordered rotors and disordered or "paramagnetic" rotors, respectively.^[1]

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.^[2]

Properties

One of the important features of the rotor model is the continuous

${\displaystyle \mathbf {S} _{1i}}$ and ${\displaystyle \mathbf {S} _{2i}}$, the rotor model approximates the low-energy states of a Heisenberg antiferromagnet, with the Hamiltonian

${\displaystyle H_{d}=K\sum _{i}\mathbf {S} _{1i}\cdot \mathbf {S} _{2i}+J\sum _{\langle ij\rangle }\left(\mathbf {S} _{1i}\cdot \mathbf {S} _{1j}+\mathbf {S} _{2i}\cdot \mathbf {S} _{2j}\right)}$

using the correspondence ${\displaystyle \mathbf {L} _{i}=\mathbf {S} _{1i}+\mathbf {S} _{2i}}$^[1]

The particular case of quantum rotor model which has the O(2) symmetry can be used to describe a

See also

• Heisenberg model (quantum)
• Ising model

References