I am also interested in the q-state Potts model, but I see here neither a question nor an aspect of it to be discussed.

Thanks for your time

Jose Gaite Snehadri Bihari Ota The standard Potts model is a lattice statistical model with pair interactions betweenq-state variables attached to neighbouring site. The q-state Potts model consists of a lattice of spins sigma_i, which can take q different values, and whose Hamiltonian is ,

H = sum K delta_sigmai delta_sigmaj.

For q = 2, it is exactly the ising model with some scaling.

Here are some good pdfs :

http://academics.smcvt.edu/jellis-monaghan/Talks/Intro%20to%20Potts.pdf

Graner, François; Glazier, James (1992). "Simulation of biological cell sorting using a two-dimensional extended Potts model" (PDF). Phys. Rev. Lett. 69 (13): 2013–7. Bibcode:1992PhRvL..69.2013G

Glassiness in cellular Potts model of biological tissue is controlled by disordered energy landscape : Preprint Glassiness in cellular Potts model of biological tissue is c...

Potts model code

http://www.new-npac.org/projects/cdroms/cewes-1998-05/reports/hpfforspmd/html/node64.html

Thanks. The states correspond to the orientations of the microscopic domains.

The standard tutorial review on the standard Potts model is by F.Y. Wu, Rev.Mod.Phys. 54, 235 (1982). Of course, there are numerous more recent publications. Nevertheless, this review may be recommended as a good starting point.

The Potts model shows phase transition and has been studied in two-dimensions (2D) and 3D.

When the transition is weakly first-order, a good equilibration in the computer simulation is needed to conclude the order of the transition. In finite systems such as that used in microcanonical Monte Carlo the specific-heat becomes negative at the transition when it is first-order. In this sense microcanonical Monte Carlo gives good equilibration.

Snehadri Bihari Ota:

See the discussion by D.H.E.Gross, A.Ecker, and X.Z. Zhang, Ann.Physik 5 (1996), 446, simulating the 2d Potts model for q=10.

I am happy to see that this question is gaining momentum. Walter Selke has mentioned Wu's review, which is still useful. It leaves open the interesting question of the nature of the phase transition of the 3-state model in 3d, namely, whether it is first or second order. I do not know if it has already been settled. Any input?

Jose Gaite,

Indeed, it looks like quite elaborate numerical methods (especially, series expansions and MC) favor, for the pure, non-dilute 3-state Potts model in d=3, a transition of weak first order. But it seems that an exact, analytic proof is still missing.

In continuation with earlier reply (Walter Selke):

The 2D 3-state Potts model shows a second-order transition. Therefore, one can expects that by increasing the dimension to 3D the nature of the transition is boosted and also of second-order.

Can I know the papers which suggest that 3D 3-state Potts model gives a weak first-order transition?

Snehadri Bihari Ota,

In fact, one expects that higher dimension will stabilize a phase transition. However, the order of the transition may not be clear.

In case of the 3d 3-state Potts model, the following articles may be of interest:

(i) High temperature series expansions: M. Hellmund and W. Janke, PRE 74, 051113 (2006)

(ii) MC simulations: W. Janke and R. Villanova, Nucl. Phys.B 489, 679 (1997)

(iii) Analytic theories: H. Duminil-Copin, Current Developments in Mathematics, p.27-71 (2015)

Of course, references in those papers and later citations are of interest, as well, favoring a weak first-order transition.

Thanks for the references

Weak first-order is that which has small latent heat and small discontinuity in the magnetization. Monte Carlo simulations always have the disadvantage of determining small discontinuity at a transition due to discrete steps and numerical accuracy in the calculation and new techniques can prove to be useful. Theoretical calculation should not have such disadvantage, but there can be the limitations due to assumptions and approximations.

In two-dimensions, the rough demarcation for the order of the transition is that the transition is first-order when q>4 and higher order for q≤4. What is that demarcation for 3D?

If we consider the consequences of RG theory such as upper critical dimension, then the order of transition is retained in a model as the dimension is increased.

What are the reasons for a change in the order of a transition as the dimension is increased? One answer can be that, if the spin configurations are symmetric with respect to (the increase in) dimension then one can expect that the order of the transition is retained. A counter-example can be the XY-model where vortex-strings appear in 3D.

Recent Monte Carlo simulations suggest that 2D 4-state Potts model has a first-order transition in stead of higher-order transition. The transition is weak and occurs at 0.92 with a latent heat of 0.18 (Journal of Modern Physics, 8 (2017) 602-606). It is based on the observation that a finite isolated system at the first-order transition shows negative specific heat which is an equilibrium property.

SB Ota,

I am afraid, your own article, in JMP (2017), needs a critical comment. As you stated there, you simulated a rather small lattice size. Furthermore, the conclusion contradicts reliable and well established results on the 2d 4-state Potts model, using various approaches, including, e.g., conformal field theory and extensive MC simulations; see, for instance, J. Salas and A.D. Sokal, J. Stat. Phys. 88, 67 (1997), and references therein. My conclusion is, that much care is needed in doing and interpreting MC simulations.

In continuation with earlier reply (Walter Selke)

A small correction: The page number is 567 for J.Stat.Phys. article

Thanks for your reply

We have simulated 30x30 lattice. In microcanonical Monte Carlo (our article), at a first-order transition, there is a 'S'-bend for T vs E, which tends to a discontinuity as the system size is increased (in the thermodynamic limit). Therefore, 'small-size' is not always a draw-back for detecting a first-order transition in this simulation. But for smaller lattice sizes in Potts model there is some lack-of-ergodicity at below the transition (J.phys.: Condens. Matter 12 (2000) 2233-2243).

Numerical methods such as series expansion, renormalization group and computer simulations have contributed to the understanding of phase transitions in Potts model. Some amount of work is left when the transition is weak.

Since there is no posting for more than 15 days we can close the discussion. Thanks for your time.

Ähem...

series expansions and the renormalization group are by no means numerical methods...

Regards

Hannes

I will correct the earlier answer which mentions 'numerical method'.

A new renormalization group study shows that there is a second-order phase transition in 3 dimensions with the symmetry of the 3-state Potts model.

Preprint New renormalization group study of the 3-state Potts model a...