Order parameters in quasi-1D spin systems

In this work we extend the notion of what is meant by a meanfield. Meanfields are approximately maps - through some self consistency relation - of a complex, usually manybody, problem to a simpler more readily solvable problem. This mapping can then be solved to represent properties of the complex many body problem using some self consistency relations. Prototypical examples of simpler meanfield problems (meanfield systems) are the single site and free particle problems. Here we propose a new class of simple meanfield systems where the simple problem to be solved is a 1D spin chain. These meanfields are particularly useful for studying quasi-1D models, where there is a 3D system composed of weakly coupled 1D spin chains with the coupling in the transverse direction weaker than in the 1D direction. We illustrate this idea by considering meanfields for the Ising (of any coupling sign) and ferromagnetic Heisenberg models with one direction coupled much more strongly then the other directions (quasi-1D systems) which map at meanfield level onto the 1D Ising and 1D ferromagnetic Heisenberg models. We also consider more exotic models to illustrate other methods of solving 1D systems, namely the $N$-state Potts model. Magnetic phase transition temperatures and are obtained for all three models, we see that they significantly differ from the usual meanfield estimates. Indeed if the 1D direction has coupling $\Gamma$ and the transverse directions have coupling $J$ with $\lambda\sim\frac{\Gamma}{J}\gg1$ then regular meanfield would predict the transition temperature to be $k_{B}T_{c}\sim\Gamma$ for all three models while 1D meanfield predicts temperatures of $k_{B}T_{c}\sim\frac{\Gamma}{\log\left(\lambda\right)}$ for the Ising and Potts models and $k_{B}T_{c}\sim\frac{\Gamma}{\sqrt{\lambda}}$ for the ferromagnetic Heisenberg model. Cluster 1D meanfield extensions are also proposed.