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 [math]\displaystyle{ i }[/math] is [math]\displaystyle{ \mathbf{n} }[/math]. Then, we can define rotor momentum [math]\displaystyle{ \mathbf{p} }[/math] by the commutation relation of components [math]\displaystyle{ \alpha,\beta }[/math]

[math]\displaystyle{ [n_{\alpha},p_{\beta}]=i\delta_{\alpha\beta} }[/math]

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

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

[math]\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 }[/math]

where [math]\displaystyle{ J,\bar{g} }[/math] are constants.. The interaction sum is taken over nearest neighbors, as indicated by the angle brackets. For very small and very large [math]\displaystyle{ \bar{g} }[/math], 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 [math]\displaystyle{ \mathbf{S}_{1i} }[/math] and [math]\displaystyle{ \mathbf{S}_{2i} }[/math], the rotor model approximates the low-energy states of a Heisenberg antiferromagnet, with the Hamiltonian

[math]\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) }[/math]

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

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