Image of the week: Potts-model critical manifolds

By Rebecca McCutcheon on February 15, 2016

Potts-model critical manifolds revisited: Christian R Scullard and Jesper Lykke Jacobsen

This paper, which the authors say could be considered an addendum to their recent work: High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials [1], is part of a series of work to compute percolation and Potts-model thresholds using their radically different technique: the method of critical polynomials.

The advance made in [1] allowed the polynomial to be calculated on graphs with hundreds of edges in some cases, producing estimates of Potts and percolation thresholds far exceeding that possible with traditional techniques, such as Monte Carlo. In this paper the authors compute critical polynomials for the q-state Potts model on the Archimedean lattices, using a parallel implementation of the algorithm in [1].

These critical manifolds reveal many interesting features in the antiferromagnetic region of the Potts model, and determine accurately the extent of the Berker–Kadanoff phase for the lattices studied.

The image shows the eleven Archimedean lattices.

By definition, an Archimedean lattice is such that each vertex is surrounded by the same types of faces, appearing in the same cyclic order.

The Potts model is exactly solvable on the square, triangular and hexagonal lattices. In their work the authors are focusing on the remaining eight cases which are unsolved.

This work has thrown up further puzzling issues and problems to be investigated.

This work is licensed under a Creative Commons Attribution 3.0 Unported License. Main image and thumbnail (adapted): Christian R Scullard and Jesper Lykke Jacobsen 2016 J. Phys. A: Math. Theor. 49 125003