**In his recently published Paper in Journal of Physics A: Mathematical and Theoretical Loïc Turban from the Université de Lorraine has provided a study of the 1D classical q-state Potts model. Professor Turban talks to JPhys+ to tell us more about this research.**

**Who are you?**

I am Loïc Turban, Emeritus Professor at Lorraine University in Nancy, member of the Statistical Physics Group which I founded at the beginning of the 1980s.

**What prompted you to pursue this field of research? **

It is an old story. My first contribution to the field goes back to 1981 when I introduced a generalization of the quantum Ising chain in a transverse field where the usual first neighbour interaction is replaced by an interaction between *m* successive spins. The interest of this model lies in the richness of its critical behaviour, different values of *m* leading to different two-dimensional (2*D*) universality classes.

**What is this latest paper all about?**

This paper is devoted to the study of the 1*D* *classical q*-state Potts model with *m*-spin interactions *K* in a field *H*. When* H* = 0 an exact solution is obtained by redefining the Potts variables in such a way that the new ones are non-interacting.

The model in a field is self-dual with *K* and *H* exchanging their roles in the duality transformation. Under another change of variables the Potts chain with *N* sites is mapped onto a standard anisotropic Potts model on a cylinder with helical boundary conditions, first-neighbour interactions *K* and *H* in the longitudinal and transverse directions, respectively. Thus the original 1*D* classical system is critical on the selfduality line when both the length* ℓ* = *N*/*m* and the transverse size m of the cylinder go to infinity. The mapping clears up the existence of a unique, *m*-independent self-duality relation for the 1*D* model in a field.

The 1*D* transition is quite unusual in that, *K* and *H* being first-neighbour interactions on the cylinder, the thermal and magnetic sectors of the Potts chain share the same set of critical exponents which are the thermal exponents of the 2*D* Potts model.

**What do you plan to do next?**

It would be interesting to study the finite-size scaling with *m* on the self-duality line, for fixed values of the aspect ratio *ℓ*/*m*, either numerically or analytically in the Ising case, *q* = 2.

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This work is licensed under a Creative Commons Attribution 3.0 Unported License.

Image of Loïc Turban © Bertrand Berche; used with permission; image not covered by CC BY 3.0.

Figure taken from Loïc Turban 2017 *J. Phys. A: Math. Theor.* **50** 205001 © IOP Publishing.

Categories: Journal of Physics A: Mathematical and Theoretical