Quantum rotor model

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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]

Formulation

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

[ n α , p β ] = i δ α β {\displaystyle [n_{\alpha },p_{\beta }]=i\delta _{\alpha \beta }} {\displaystyle [n_{\alpha },p_{\beta }]=i\delta _{\alpha \beta }}

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

H R = J g ¯ 2 ∑ i L i 2 − J ∑ ⟨ i j ⟩ n i ⋅ n j {\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}} {\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 J , g ¯ {\displaystyle J,{\bar {g}}} {\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 g ¯ {\displaystyle {\bar {g}}} {\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]

In higher dimensions, the Hamiltonian can be defined as

H R = J g ¯ 2 ∑ i Δ i − J ∑ ⟨ i j ⟩ n i ⋅ n j {\displaystyle H_{R}={\frac {J{\bar {g}}}{2}}\sum _{i}\Delta _{i}-J\sum _{\langle ij\rangle }\mathbf {n} _{i}\cdot \mathbf {n} _{j}} {\displaystyle H_{R}={\frac {J{\bar {g}}}{2}}\sum _{i}\Delta _{i}-J\sum _{\langle ij\rangle }\mathbf {n} _{i}\cdot \mathbf {n} _{j}}

where Δ i {\displaystyle \Delta _{i}} {\displaystyle \Delta _{i}} is the Laplace-Beltrami operator on the sphere S n {\displaystyle S^{n}} {\displaystyle S^{n}}. It is exactly solvable in the large n limit.[1][3]

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

H d = K ∑ i S 1 i ⋅ S 2 i + J ∑ ⟨ i j ⟩ ( S 1 i ⋅ S 1 j + S 2 i ⋅ S 2 j ) {\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)} {\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 L i = S 1 i + S 2 i {\displaystyle \mathbf {L} _{i}=\mathbf {S} _{1i}+\mathbf {S} _{2i}} {\displaystyle \mathbf {L} _{i}=\mathbf {S} _{1i}+\mathbf {S} _{2i}}[1]

Applications

The particular case of the quantum rotor model which has O(2) symmetry can be used to describe a superconducting array of Josephson junctions or the behavior of bosons in optical lattices.[4] 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.[4] 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.[5]

See also

References

  1. Sachdev, Subir (1999). Quantum Phase Transitions. Cambridge University Press. ISBN 978-0-521-00454-1. Retrieved 10 July 2010.
  2. Alet, Fabien; Erik S. Sørensen (2003). "Cluster Monte Carlo algorithm for the quantum rotor model". Phys. Rev. E. 67 (1) 015701. arXiv:cond-mat/0211262. Bibcode:2003PhRvE..67a5701A. doi:10.1103/PhysRevE.67.015701. PMID 12636557. S2CID 25176793.
  3. Zinn-Justin, Jean (23 October 1998). "Vector models in the large $N$ limit: a few applications". arXiv.org.
  4. 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. S2CID 119348100.
  5. Sachdev, Subir (1995). "Quantum phase transitions in spins systems and the high temperature limit of continuum quantum field theories". arXiv:cond-mat/9508080.