Universit´esParis 6 et Paris-Sud, Ecole´ Normale Sup´erieure, Ecole´ Polytechnique M2 CFP - Parcours Physique Macroscopique et Complexit´e

Some properties of the Potts model

This part should be written on a separate paper. It will be graded over 10 points. The three parts I, II et III are widely independent.

The Potts model is defined as follows. Each node l of a d dimensional hypercubic lattice contains a spin σl that can take s different values, σl ∈ {1, ..., s} where s is an integer greater or equal to 2. The energy of a configuration C of these spins is given by :

1 X X E(C) = − J 0 (sδ − 1) − h (sδ − 1) (1) 2 l,l σl,σl0 l σl,1 l,l0 l where the sums go over all N nodes of a d-dimensional hypercubic lattice, Jl,l0 = J > 0 for nearest-neighbors and 0 otherwise is a coupling constant and hl > 0 a magnetic field favoring the occupancy of state σl = 1 at site l.

I. General considerations.

In this part, the magnetic field hl is taken equal to zero.

1) For which configurations of the spins is the energy minimized ?

2) What is the degeneracy of this ground state ?

3) For each value of σ ∈ {1, ..., s}, we introduce the quantity : 1 φσ = (sδ − 1) (2) l s − 1 σl,σ where l is a given site of the lattice. σ What is the value of the average hφl i in the low temperature limit ? In the high temperature limit ?

1 4) Why ml ≡ hφl i can be considered an order parameter associated to the condensa- tion in the state 1 of the spins ?

II. Case of the dimension d = 1.

In this part, the magnetic field hl is taken equal to zero. Periodic boundary conditions σN+1 = σ1 are adopted. Up to an irrelevant additive constant, the energy can finally be written N E X = −K δσlσl+1 , (3) kBT l=1 where K ≡ J/kBT .

1) Write the definition of the partition function Z corresponding to this energy.

2) Rewrite this partition function as a function of a s × s transfer matrix T to be defined. Write explicitly T for the particular value s = 3.

3) By relying on the symmetry of this transfer matrix, show that it admits a simple symmetric eigenvector, associated to an eigenvalue to be determined. Find the other eigenvectors by noting that they are orthogonal to this first eigenvector. Show that they all correspond to a same eigenvalue. What is the degeneracy of this second eigenvalue ?

4) Deduce the explicit expression of the partition function.

5) Determine the free energy in the thermodynamic limit.

6) By assuming that, as in the case of the 1D Ising model, the thermodynamic limit of the correlation function is given by :  s λ2 hσrσr+si = , (4) λ1

where λ1 > λ2 are the two largest eigenvalues of the transfer matrix, determine the explicit expression of the correlation length ξ. Is there a phase transition in this model ?

III. Variational mean-field theory.

The goal of this part is to determine a mean-field free energy of the Potts model for any dimension d, by relying on a variational approach.

1) Show that, for any choice of a normalized probability distribution ρ, i.e. satisfying X ρ(C) = 1, (5) C the free energy F of the Potts model is such that :

F ≤ Fρ ≡ hEiρ + kBT hln ρiρ, (6) where X hAiρ ≡ ρ(C)A(C) (7) C stands for the average of any observable A with the weight ρ and C ≡ {σ1, σ2, ..., σN } defines a configuration of the system.

2) In the following, the probability distribution ρ is chosen of the form : Y ρ(σ1, σ2, ..., σN ) = ρl(σl), where ρl = al + blδσl,1 (8) l is a one-site normalized distribution satisfying :

s X ρl(σl) = 1. (9)

σl=1

By using the normalization of ρl and the definition of ml, express al and bl as functions of ml and s.

3) Determine the averages hδσl,1iρ, hδσl,σl0 iρ and hln ρiρ as functions of ml and s. 4) Deduce that

Fρ 1 X X NkBT = − J 0 m m 0 − h m − ln s s − 1 2 l,l l l l l s − 1 l,l0 l   kBT X 1 + [1 + (s − 1)m ] ln(1 + (s − 1)m ) + (1 − m ) ln(1 − m ) s s − 1 l l l l l (10)

5) From now on, we focus on the particular case of a uniform magnetic field hl ≡ h, ml ≡ m. How is equation (10) simplified in this case ?

MF 6) The mean-field free energy F is defined as Fρ taken at the value of m that renders Fρ stationary. Show that this value of m satisfies the implicit equation :

e esh /kB T − 1 m = e , (11) esh /kB T + s − 1 where he is an effective magnetic field, to be determined as a function of h, m, J and the dimension d.

7) What is the solution of Eq.(11) when T → ∞ ? T → 0 ?

8) Expand the thermodynamic potential ΓMF ≡ F MF + Nhm at small m up to third order in m. What main difference with an Ising type system is found for s > 2 ? This feature can be shown to be characteristic of a first-order phase transition.

9) In this question, h = 0. Within the mean-field approximation considered here, the free energy can be shown to have near the transition two minima, one at m = 0 and the other at some m 6= 0. At the transition temperature Tc, the values of the free energy at the two minima become equal. Thus, in addition to Eq. (11), the critical temperature Tc and critical magnetization mc satisfy the second equation

F (mc,Tc) = F (m = 0,Tc). (12)

Determine mc and Tc.