Quantum rotor model: Difference between revisions

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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 rotor]]s that interact through short-range dipole-dipole magnetic forces originating from their [[magnetic dipole moment]]s (neglecting [[Coulomb force]]s). The model differs from similar spin-models such as the [[Ising model]] and the [[Heisenberg model (quantum)|Heisenberg model]] in that it includes a term analogous to [[kinetic energy]]. See [[Langton'27s_ant#Wave_equation|wave equation model]].

Although elementary quantum rotors do not exist in nature, the model can describe effective [[Degrees of freedom (mechanics)|degrees of freedom]] for a system of sufficiently small number<!-- Is it possible to give an idea of "small number" ?--> of closely coupled [[electrons]] in low-energy states.<ref name="sachdev">{{Cite book|title=Quantum Phase Transitions |last=Sachdev |first=Subir |url=httphttps://books.google.com/books?id=Ih_E05N5TZQC&printsec=frontcover |year=1999 |publisher=[[Cambridge University Press]] |isbn=978-0-521-00454-1 |page= |accessdateaccess-date=2010-07-10}}</ref>

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

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where <math>J,\bar{g}</math> are constants.<!-- corresponding to? J should be proportional to some dipole moment or magneton? Is it possible to discuss the meaning of g? -->. The interaction sum is taken over nearest neighbors, as indicated by the angle brackets. For very small and very large <math>\bar{g}</math>, the Hamiltonian predicts two distinct configurations ([[ground state]]s), namely "magnetically" ordered rotors and disordered or "[[Paramagnetism|paramagnetic]]" rotors, respectively.<ref name="sachdev"/>

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.<ref name="alet">{{Cite journal|last=Alet|first=Fabien|author2=Erik S. Sørensen |title=Cluster Monte Carlo algorithm for the quantum rotor model|journal=Phys. Rev. E|year=2003|volume=67|issue=1|doipages=10.1103/PhysRevE.67.015701|urldoi=http://link.aps.org/doi/10.1103/PhysRevE.67.015701|accessdatepmid=24 July 201012636557|arxiv = cond-mat/0211262 |bibcode = 2003PhRvE..67a5701A |s2cid=25176793}}</ref>

==Properties==

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using the correspondence <math>\mathbf{L}_i=\mathbf{S}_{1i}+\mathbf{S}_{2i}</math><ref name="sachdev"/>

The particular case of quantum rotor model which has the O(2) symmetry can be used to describe a [[superconductor|superconducting]] array of [[Josephson junction]]s or the behavior of [[bosons]] in [[optical lattice]]s.<ref name="vojta"/> Another specific case of O(3) symmetry is equivalent to a system of two layers (bilayer) of a quantum [[Heisenberg model (quantum)|Heisenberg antiferromagnet]]; it can also describe double-layer [[quantum Hall effect|quantum Hall]] ferromagnets.<ref name="vojta">{{Cite arxivjournal|last1=Vojta |first1=Thomas |last2=Sknepnek |first2=Rastko |year=2006 |title=Quantum phase transitions of the diluted O(3) rotor model |eprintarxiv=cond-mat/0606154 |doi=10.1103/PhysRevB.74.094415 |volume=74 |issue=9 |pages=094415 |journal=Physical Review B|bibcode=2006PhRvB..74i4415V |s2cid=119348100 }}</ref> It can also be shown that the [[phase transition]] for the two dimensional rotor model has the same [[Renormalization group|universality class]] as that of [[antiferromagnet]]ic Heisenberg spin models.<ref>{{Cite arxivarXiv |last1=Sachdev |first1=Subir |last2= |first2= |year=1995 |title=Quantum phase transitions in spins systems and the high temperature limit of continuum quantum field theories |eprint=cond-mat/9508080}}</ref>

==See also==

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{{Reflist}}

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