Kramers' opacity law

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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 is . Then, we can define rotor momentum by the commutation relation of components

However, it is found convenient[1] to use rotor angular momentum operators defined (in 3 dimensions) by components

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

where are constants.. The interaction sum is taken over nearest neighbors, as indicated by the angle brackets. For very small and very large , 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 and , the rotor model approximates the low-energy states of a Heisenberg antiferromagnet, with the Hamiltonian

using the correspondence [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

References

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