One-dimensional q-state Potts model with multi-site interactions: An interview with Loïc Turban

By Phil Brown on April 27, 2017

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.

Loic Turban © Bertrand Berche; used with permission

Loïc Turban © Bertrand Berche; used with permission

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 (2D) universality classes.

What is this latest paper all about?

This paper is devoted to the study of the 1D 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 1D 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 1D model in a field.

(a) Under the change of spin variables the 1D Potts model with m-spin interactions in an external field is mapped onto a 2D standard Potts model on a cylinder with helical boundary conditions. H and K are the first-neighbour interactions along the helix and parallel to its axis, respectively. The helix has a length N/m and m spins per turn. The helicity factor is 1/m. (b) The same lattice in a rectangular representation. Loïc Turban 2017 J. Phys. A: Math. Theor. 50 205001 © IOP Publishing

The 1D 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 2D 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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