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 <a href="/facts/Rigid_rotor/Ue8nrySx">rigid rotors</a> that interact through short-range dipole-dipole magnetic forces originating from their <a href="/facts/Magnetic_dipole_moment/VpAQKJtr">magnetic dipole moments</a> (neglecting <a href="/facts/Coulomb_force/V0lR5NKv">Coulomb forces</a>). The model differs from similar spin-models such as the <a href="/facts/Ising_model/lvBpj2h1">Ising model</a> and the <a href="/facts/Heisenberg_model_(quantum)/8RPu8n5n">Heisenberg model</a> in that it includes a term analogous to <a href="/facts/Kinetic_energy/aJRxUd4p">kinetic energy</a>.
Although elementary quantum rotors do not exist in nature, the model can describe effective <a href="/facts/Degrees_of_freedom_(mechanics)/2BqcdnLA">degrees of freedom</a> for a system of sufficiently small number of closely coupled <a href="/facts/Electrons/L0KBo5cW">electrons</a> in low-energy states.
Suppose the n-dimensional position (orientation) vector of the model at a given site 
 
 
 
 i
 
 
 {\displaystyle i}
 
 is 
 
 
 
 
 n
 
 
 
 {\displaystyle \mathbf {n} }
 
. Then, we can define rotor momentum 
 
 
 
 
 p
 
 
 
 {\displaystyle \mathbf {p} }
 
 by the <a href="/facts/Commutation_relation/eYxzFmFY">commutation relation</a> of components 
 
 
 
 α
 ,
 β
 
 
 {\displaystyle \alpha ,\beta }

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

However, it is found convenient to use rotor <a href="/facts/Angular_momentum/7TuSVFxq">angular momentum</a> operators 
 
 
 
 
 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 }}

Then, the magnetic interactions between the quantum rotors, and thus their energy states, can be described by the following <a href="/facts/Hamiltonian_mechanics/2ReerlNA">Hamiltonian</a>:

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}}

where 
 
 
 
 J
 ,
 
 
 
 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}}}
 
, the Hamiltonian predicts two distinct configurations (<a href="/facts/Ground_state/ML7cfCug">ground states</a>), namely "magnetically" ordered rotors and disordered or "<a href="/facts/Paramagnetism/vj1SMbbp">paramagnetic</a>" rotors, respectively.
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.

Quantum rotor model open-in-new

Quantum rotor model