Appendix 1: Calculation of the Metropolis and the Glauber Transition Probabilities for the Ising Model and for the q-state Potts Model

(i) Transition Probabilities in the Ising Model

Consider a spin model in a specific state, a particular spin S_i and the set { S_r : S_r is a nearest neighbour of S_i }. The energy E_i contributed by the spin S_i is the sum of its interaction energies with the S_r and so

E_i = -J.Σ_r(S_i.S_r) = -J.S_i.Σ_rS_r

In the Ising model there are only two possible spin values, +1 and -1, so the only possible change to the spin S_i is for it to take the opposite value, namely, -S_i. Call this spin S_j . In this case the energy E_j associated with the new spin value is the sum of the interaction energies of S_j with the S_r and so

E_j = -J.Σ_r(S_j.S_r) = -J.Σ_r(-S_i.S_r) = -J.-S_i.Σ_rS_r = J.S_i.Σ_rS_r

Thus the difference in the "before" and "after" energies is

ΔE = E_j - E_i = ( J.S_i.Σ_rS_r ) - ( -J.S_i.Σ_rS_r ) = 2.J.S_i.Σ_rS_r                  (1)

Suppose that there is a maximum number of nearest neighbours which can be possessed by any spin, and let this number be denoted by n. Then the maximum value for Σ_rS_r is +n and the minimum value is -n. Thus there are, at most, 2n+1 possible values for ΔE, corresponding to the possible values for Σ_rS_r , namely:

-n, -n+1, -n+2, …, -2, -1, 0, 1, 2, …, n-1, n

although in a particular Ising spin model not all of these values may in fact be possible.

The transition probabilities for the Metropolis algorithm and for the Glauber algorithm were given in Section 1.7 as follows:

Metropolis algorithm: W(Si → Sj)	= 1 if ΔEji ≤ 0
	= e-ΔEji/(kBT) otherwise.
Glauber algorithm: W(Si → Sj)	= 1 / [ 1 + e ΔEji/(kBT) ]

where T is the temperature and k_B is Boltzmann's constant. From equation (1) we obtain:

Metropolis: W(Si → Sj)	= 1 if 2.J.Si.ΣrSr ≤ 0
	= e -2JSi.SrSr/kBT otherwise.
Glauber: W(Si → Sj)	= 1 / [ 1 + e 2JSi.SrSr/(kBT) ]

Taking J = k_B = 1 the algorithms become:

Metropolis: W(Si → Sj)	= 1 if 2.Si.ΣrSr ≤ 0
	= e -2.Si.SrSr/T otherwise.
Glauber: W(Si → Sj)	= 1 / [ 1 + e 2.Si.SrSr/T) ]

Let S₂ = 2.S_i.Σ_rS_r then we have:

Metropolis: W(Si → Sj)	= 1 if S2 ≤ 0
	= e -S2/T otherwise.
Glauber: W(Si → Sj)	= 1 / ( 1 + eS2/T )

There are at most 2n+1 possible values for S₂ (assuming a maximum of n nearest neighbours) as follows:

-2n, -2n+2, -2n+4, …, -2, 0, 2, …, 2n-2, 2n

so the transition probabilities that we seek are:

Metropolis:
S2:    -2n    -2n+2   …    -2    0      2     4     …     2n
W:      1       1     …     1    1    e-2/T  e-4/T   …    e-2n/T

Glauber:
S2:      -2n        …        -2          0        2       …      2n
W:    1/(1+e-2n/T)   …    1/(1+e-2/T)    1/2    1/(1+e2/T)  …    1/(1+e2n/T)

(ii) Transition Probabilities in the Potts Model, Version A.

Consider a spin model in a specific state, a particular spin S_i (1 ≤ S_i ≤ q) and the set { S_r : S_r is a nearest neighbour of S_i }. The energy E_i associated with the spin S_i is the sum of its interaction energies with the S_r and so E_i = -J.Σ_r[δ(S_i,S_r)].

In the q-state Potts model there are q possible spin values, 1, …, q, so the spin S_i may change to any of q-1 possible new values. We select one at random; call this spin S_j. In this case the energy E_j associated with the new spin value is the sum of the interaction energies of S_j with the S_r and so E_j = -J.Σ_r[δ(S_j,S_r)], where δ() is the Kronecker delta function: δ(x,y) = 1 if and only if x = y (otherwise 0). Thus the transition energy, ΔE, is

E_j - E_i = -J.Σ_r[δ(S_j,S_r)] - -J.Σ_r[δ(S_i,S_r)]
= -J.{ Σ_r[δ(S_j,S_r)] - Σ_r[δ(S_i,S_r)] }
so ΔE = J.{ Σ_r[δ(S_i,S_r)] - Σ_r[δ(S_j,S_r)] }                  (2)

Suppose that there is a maximum number of nearest neighbours which can be possessed by any spin, and let this number be denoted by n. Then the maximum value for each of Σ_r[δ(S_i,S_r)] and Σ_r[δ(S_j,S_r)] is +n and the minimum value is 0. Thus the maximum value for

Σ_r[δ(S_i,S_r)] - Σ_r[δ(S_j,S_r)]                  (3)

is +n and the minimum is -n, so (as with the Ising model) there are, at most, 2n+1 possible values for ΔE (and thus for the transition probabilities) depending on the possible values for (3), namely:

-n, -n+1, -n+2, …, -2, -1, 0, 1, 2, …, n-1, n

although in a particular q-state Potts spin model not all of these values may in fact be possible.

The Metropolis and the Glauber algorithms for the q-state Potts model, version A, are obtained by substituting the RHS of (2) for ΔE into the definitions given in Section (i) above. As before we take J = k_B = 1, so the algorithms become:

Metropolis: W(S_i → S_j) = 1 if Σ_r[δ(S_i,S_r)] - Σ_r[δ(S_j,S_r)] ≤ 0,

= e^-{Sr^[δ(Si^,Sr^{)] - S}r^[δ(Sj^,Sr^)]}/T otherwise,

that is, W(S_i → S_j) = 1 if Σ_r[δ(S_i,S_r)] ≤ Σ_r[δ(S_j,S_r)],

= e^-{Sr^[δ(Si^,Sr^{)] - S}r^[δ(Sj^,Sr^)]}/T otherwise.

Glauber: W(S_i → S_j) = 1 / ( 1 + e^{Sr^[δ(Si^,Sr^{)] - S}r^[δ(Sj^,Sr^)]}/T )

Let S₃ = Σ_r[δ(S_i,S_r)] - Σ_r[δ(S_j,S_r)] then we have:

Metropolis: W(S_i → S_j) = 1 if S₃ ≤ 0,

= e ^-S3^/T otherwise.

Glauber: W(S_i → S_j) = 1 / ( 1 + e^S3^/T )

There are at most 2n+1 possible values for S₃ (assuming a maximum of n nearest neighbours) as follows:

-n, -n+1, -n+2, …, -2, -1, 0, 1, 2, …, n-1, n

so the transition probabilities that we seek are:

Metropolis:
S3:  -n    -n+1   …    -1    0      1       2    …     n
W:    1     1     …     1    1    e-1/T    e-2/T   …    e-n/T

Glauber:
S3:      -n       …        -1         0        1       …       n
W:    1/(1+e-n/T)  …    1/(1+e-1/T)   1/2    1/(1+e1/T)  …    1/(1+en/T)

It will be noted that these are not the same transition probabilities as in the Ising case.

(iii) Transition Probabilities in the Potts Model, Version B.

We follow the same reasoning as in the previous section.

Consider a spin model in a specific state, a particular spin S_i (1 ≤ S_i ≤ q) and the set { S_r : S_r is a nearest neighbour of S_i }. The energy E_i associated with the spin S_i is the sum of its interaction energies with the S_r and so E_i = -J.Σ_r[2.δ(S_i,S_r)-1].

In the q-state Potts model there are q possible spin values, 1, …, q, so the spin S_i may change to any of q-1 possible new values. We select one at random; call this spin S_j. In this case the energy E_j associated with the new spin value is the sum of the interaction energies of S_j with the S_r, so E_j = -J.Σ_r[2.δ(S_j,S_r)-1]. Thus the transition energy, ΔE, is

E_j - E_i = -J.Σ_r[2.δ(S_j,S_r)-1] - -J.Σ_r[2.δ(S_i,S_r)-1]
= -J.{ Σ_r[2.δ(S_j,S_r)] - Σ_r[2.δ(S_i,S_r)] }
so ΔE = 2.J.{ Σ_r[δ(S_i,S_r)] - Σ_r[δ(S_j,S_r)] }                  (4)

Suppose that there is a maximum number of nearest neighbours which can be possessed by any spin, and let this number be denoted by n. Then the maximum value for each of Σ_r[δ(S_i,S_r)] and Σ_r[δ(S_j,S_r)] is +n and the minimum value is 0. Thus the maximum value for

Σ_r[δ(S_i,S_r)] - Σ_r[δ(S_j,S_r)]                  (5)

is +n and the minimum is -n, so (as with the Ising model and the Potts model, Version A) there are, at most, 2n+1 possible values for ΔE (and thus for the transition probabilities) depending on the possible values for (5), namely:

-n, -n+1, -n+2, …, -2, -1, 0, 1, 2, …, n-1, n

although in a particular q-state Potts spin model not all of these values may in fact be possible.

The Metropolis and the Glauber algorithms for the q-state Potts model, Version B, are obtained by substituting the RHS of (4) for ΔE into the definitions given in Section (i) above. As before we take J/k_B = 1, so the algorithms become:

Metropolis: W(S_i → S_j) = 1 if 2.{ Σ_r[δ(S_i,S_r)] - Σ_r[δ(S_j,S_r)] } ≤ 0,

= e^-2.{Sr^[δ(Si^,Sr^{)] - S}r^[δ(Sj^,Sr^)]}/T otherwise,

that is, W(S_i → S_j) = 1 if Σ_r[δ(S_i,S_r)] ≤ Σ_r[δ(S_j,S_r)],

= e^-2.{Sr^[δ(Si^,Sr^{)] - S}r^[δ(Sj^,Sr^)]}/T otherwise.

Glauber: W(S_i → S_j) = 1 / ( 1 + e^2.{Sr^[δ(Si^,Sr^{)] - S}r^[δ(Sj^,Sr^)]}/T )

Let S₄ = 2.{Σ_r[δ(S_i,S_r)] - Σ_r[δ(S_j,S_r)]} then we have:

Metropolis: W(S_i → S_j) = 1 if S₄ ≤ 0,

= e ^-S4^/T otherwise.

Glauber: W(S_i → S_j) = 1 / ( 1 + e^S4^/T )

There are at most 2n+1 possible values for S₄ (assuming a maximum of n nearest neighbours) as follows:

-2n, -2n+2, -2n+4, …, -2, 0, 2, …, 2n-2, 2n

so the transition probabilities that we seek are:

Metropolis:
S4:   -2n   -2n+2    …    -2    0      2       4     …    2n
W:     1       1     …     1    1    e-2/T    e-4/T    …    e-2n/T

Glauber:
S4:      -2n         …        -2         0         2        …       2n
W:    1/(1+e-2n/T)    …    1/(1+e-2/T)    1/2   1/(1+e2/T)    …    1/(1+e2n/T)

It will be noted that these are not the same transition probabilities as for Version A of the Potts Model, but they are the same as those for the Ising model.

(iv) The Equivalence of the 2-state Potts Model and the Ising Model.

It is commonly said that the 2-state Potts model is equivalent to the Ising model. This is true if Version B of the Potts model is meant, because then the transition probabilities are the same and so the dynamics of the two models are the same. Usually, however, when the Potts model is discussed it is Version A which is meant, and in this case, even for q=2, the transition probabilities are not the same. Thus at the microscopic level there are differences between the Ising model and the 2-state Potts model, Version A, so we are led to ask whether the 2-state Potts model, Version A, is strictly equivalent to the Ising model.

The transition probabilities (in both the Metropolis algorithm and the Glauber algorithm) are determined by (i) the transition energies, i.e., the energy difference between the initial state and a possible new state, (ii) the value of J/k_B and (iii) the temperature. Since (ii) and (iii) are not properties of the spin model itself the question becomes whether (i) is the same regardless of whether the spins are described in terms of one model definition (Ising) or in terms of the other (2-state Potts, Version A). We shall first consider the 2-state Potts model, Version B.

Version B:

Consider a spin S_i and the set S = { S_r : S_r is a nearest neighbour of S_i }. S may be partitioned into S₁ = { S_r : S_r = S_i } and S₂ = { S_r : S_r ≠ S_i }.

Suppose S_i and the S_r are viewed as Ising spins. Suppose S_i = +1, then (from section (i) above and the definition of the Ising model in Section 1.5):

ΔE_Ising = 2.J_Ising.(+1).Σ_rS_r = 2.J_Ising.( |S₁| - |S₂| )

Suppose S_i = -1, then

ΔE_Ising = 2.J_Ising.(-1).Σ_rS_r = -2.J_Ising.( -|S₁| + |S₂| ) = 2.J_Ising.( |S₁| - |S₂| )

Thus in each case ΔE_Ising = 2.J_Ising.( |S₁| - |S₂| ).

Now suppose S_i and the S_r are viewed as spins in the 2-state Potts model, Version B. Then (from the definition of the Potts model, Version B, in Section 1.5)

ΔE_Potts,B = -J_Potts,B.2.{ Σ_r[δ(S_j,S_r)] - Σ_r[δ(S_i,S_r)] }

where S_j is some spin value other than S_i, so

ΔE_Potts,B = -J_Potts,B.2.( |S₂| - |S₁| ) = J_Potts,B.2.( |S₁| - |S₂| )

Since ΔE_Ising = ΔE_Potts,B we may conclude that the Ising model is strictly equivalent to the 2-state Potts model, Version B.

Version A:

As above, consider a spin S_i and the set S = { S_r : S_r is a nearest neighbour of S_i }. S may be partitioned into S₁ = { S_r : S_r = S_i } and S₂ = { S_r : S_r ≠ S_i }. As before, ΔE_Ising = 2.J_Ising.( |S₁| - |S₂| ).

Now suppose S_i and the S_r are viewed as spins in the 2-state Potts model, Version A. Then (from the definition of the Potts model, Version A, in Section 1.5)

ΔE_Potts,A = -J_Potts,A.{ Σ_r[δ(S_j,S_r)] - Σ_r[δ(S_i,S_r)] }

where S_j is some spin value other than S_i, so

ΔE_Potts,A = -J_Potts,A.( |S₂| - |S₁| ) = J_Potts,A.( |S₁| - |S₂| )

Since ΔE_Ising = 2.ΔE_Potts,A the transition energies (and thus the transition probabilities) are different.

Okano et al. (1997, p.738) state: "It is known that the critical points [of the q-state Potts model] locate at J_c = log(1+√q)." Taking T_c = 1/J_c and q = 2 we obtain

T_c = 1/ln(1+√2) = 1.134593

This is exactly one-half of the value usually given for the Ising model, namely, 2/ln(1+√2) = 2.269185 (see e.g. Stinchcombe 1983, p.177, taking J = 1 = k_B), which value is obtained from simulations (see Section 3.3(a)). Thus although the Ising model and the 2-state Potts model are commonly said to be equivalent, the values of the critical temperatures are not the same when Version A of the 2-state Potts model is used.

Nevertheless, the critical exponents of the Ising model and the 2-state Potts model, Version A, are found to be the same, the only difference being the value of the critical temperature. Thus the difference between the two models is really only one of the scale used for the temperature.

Consider the Potts model, Version A, in which the interaction energy is J_Potts,A = 2.J_Ising then

ΔE_Potts,A = 2.J_Ising.( |S₂| - |S₁| ) = ΔE_Ising

so we can also say that the 2-state Potts model, Version A, is strictly equivalent to the Ising model provided the interaction energy of the 2-state Potts model is double that of the Ising model. Since the units of the critical temperature are related to units of J, and a J-unit in this modified Potts model, Version A, denotes twice the energy of a J-unit in the Ising model, the numerical value of the critical temperature in the former model is thus one-half of its value in the latter model, thus explaining the difference in the numerical values of the critical temperature in the Ising model and in the 2-state Potts model, Version A.

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