
\sect{Chain of coupled rotors}

A one-degree-of-freedom rotor is described by the Hamiltonian $p^2/(2\mass a^2)$ 
where  $\mass a^2$ is the moment of inertia. Its interaction with an external field 
is described by ${- h \cos(\varphi)}$. 
The magnetization is ${M \equiv \braket{\cos(\varphi)}}$.  
In thermal equilibrium at temperature $T$, it can be expressed as  ${M = R(\chi h)}$ 
where $R()$ is some nonlinear function that is defined such that $R'(0)=1$, 
and $\chi$ is the  susceptibility,

\Dn 

We consider a chain of such rotors.  
Near neighbor rotors are coupled by the interaction ${-\varepsilon \cos(\phi_i)}$, 
where ${\phi_i = \varphi_{i}-\varphi_{i-1}}$. 
We define the correlation function ${C(r) = \braket{\cos(\varphi_r)\cos(\varphi_0)}}$.  

\Dn

(1) For a single classical rotor at temperature $T$ determine the susceptibility $\chi$.   
Additionally write what is the function $R(x)$. The latter expression can 
be written in terms of definite integrals. Care about the $R'(0)=1$ convention.

\Dn

(2) For a chain of  classical rotors, determined what is $T_c$ and what 
is $\chi$ for ${T>T_c}$ on the basis of mean field theory. 

\Dn

(3) For a chain of  classical rotors, calculate the correlation function ${C(r)}$, 
and the implied correlation scale $\xi$. Express the result using the function $R()$.   

\Dn 

(4) Determine what is $\chi$ on the basis of an exact calculation.
Express your result in terms of $\xi$.

\Dn 
 
(5) Considering a single quantum rotor at ${T=0}$, 
use dimensional analysis in order to guess what is $\chi$. 
If an actual calculation is done the numerical prefactor is~$2$.

\Dn

(6) For a chain of quantum  rotors, 
determine, via same mean-field reasoning as in item 2, 
the condition on $\mass a^2$ for observing a phase transition.
Additionally, plot a schematic ${(\mass a^2, T)}$ phase diagram  
with the following lines:
phase transition according to mean field theory (dotted);
phase transition according to the exact analysis (solid).



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