Flux representation for the vector Potts model

One of the models that have been most widely studied for gaining insight
on the ferromagnetic phase transitions is the so-called \textit{vector
Potts model} (VPM) with $N$ states, a straightforward generalization
of the Ising model~{[}1{]}. It is well known the fact that, except
for some very particular values of $N$ and of the dimension $d$
of the corresponding lattice, this model can not be studied in an
entirely analytical manner. Therefore, one can say that our understanding
about the VPM in the general case is based on results of computer
simulations that employ \textit{Monte Carlo methods}. A class of such
methods that received considerable attention during the last decade
is that of the \textit{worm algorithms}~{[}2{]}. Devising these algorithms
involves changing the original variables of the problem, the ``Potts
spins'', to new ones that are known as \textit{flux variables}. One
says that the introduction of these variables corresponds to utilizing
a \textit{flux representation} (FR) for the VPM~{[}3,4{]}. First
we discuss how the description of the model can be reformulated in
terms of the flux variables. Then we explain how the FR allows us
to conceive new Monte Carlo methods for studying the VPM. Moreover,
we consider a second application of the FR to the analysis of the
VPM within the \textit{mean field approximation}. Preliminary results
-- valid in this approximation -- for observables such as the magnetization
are presented.

{[}1{]} F. Y. Wu, \textit{Rev. Mod. Phys.} \textbf{54}, 235 (1982).

\noindent {[}2{]} N. Prokof'ev and B. Svistunov, \textit{Phys. Rev.
Lett.} \textbf{87}, 160601 (2001).

\noindent {[}3{]} Y. D. Mercado, H. G. Evertz and C. Gattringer, \textit{Phys.
Rev. Lett.} \textbf{106}, 222001 (2011).

\noindent {[}4{]} Y. Mercado, H. Evertz and C. Gattringer, \textit{Computer
Physics Communications} \textbf{183}, 1920 (2012).