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 (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.^[1]

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

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

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

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

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

where $J,{\bar {g}}$ are constants.. The interaction sum is taken over nearest neighbors, as indicated by the angle brackets. For very small and very large $\bar{g}$, 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 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 the 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 ${\mathbf {L}}_{i}={\mathbf {S}}_{{1i}}+{\mathbf {S}}_{{2i}}$^[1]

The particular case of quantum rotor model which has the O(2) symmetry can be used to describe a superconducting array of Josephson junctions or the behavior of bosons in optical lattices.^[3] Another specific case of O(3) symmetry is equivalent to a system of two layers (bilayer) of a quantum Heisenberg antiferromagnet; it can also describe double-layer quantum Hall ferromagnets.^[3] 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.^[4]

See also

• Heisenberg model (quantum)
• Ising model

References

1. Sachdev, Subir (1999). Quantum Phase Transitions. Cambridge University Press. ISBN 978-0-521-00454-1. Retrieved 2010-07-10.
2. Alet, Fabien; Erik S. Sørensen (2003). "Cluster Monte Carlo algorithm for the quantum rotor model". Phys. Rev. E. 67 (1). arXiv:cond-mat/0211262. Bibcode:2003PhRvE..67a5701A. doi:10.1103/PhysRevE.67.015701. Retrieved 24 July 2010.
3. Vojta, Thomas; Sknepnek, Rastko (2006). "Quantum phase transitions of the diluted O(3) rotor model". arXiv:cond-mat/0606154.
4. Sachdev, Subir (1995). "Quantum phase transitions in spins systems and the high temperature limit of continuum quantum field theories". arXiv:cond-mat/9508080.