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 L {\displaystyle \mathbf {L} } \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
Sachdev, Subir (1999). Quantum Phase Transitions. Cambridge University Press. ISBN 978-0-521-00454-1. Retrieved 10 July 2010.
Alet, Fabien; Erik S. Sørensen (2003). "Cluster Monte Carlo algorithm for the quantum rotor model". Physical Review E. 67 (1): 015701. arXiv:cond-mat/0211262. Bibcode:2003PhRvE..67a5701A. doi:10.1103/PhysRevE.67.015701. PMID 12636557.
Vojta, Thomas; Sknepnek, Rastko (2006). "Quantum phase transitions of the diluted O(3) rotor model". Physical Review B. 74 (9): 094415. arXiv:cond-mat/0606154. Bibcode:2006PhRvB..74i4415V. doi:10.1103/PhysRevB.74.094415.
Sachdev, Subir (1995). "Quantum phase transitions in spins systems and the high temperature limit of continuum quantum field theories". arXiv:cond-mat/9508080.
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