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6.7: Appendix I- Potts Model in One Dimension

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Definition

The Potts model is defined by the Hamiltonian

H=Jijδσi,σjhiδσi,1 .

Here, the spin variables σi take values in the set {1,2,,q} on each site. The equivalent of an external magnetic field in the Ising case is a field h which prefers a particular value of σ (σ=1 in the above Hamiltonian). Once again, it is not possible to compute the partition function on general lattices, however in one dimension we may once again find Z using the transfer matrix method.

Transfer matrix

On a ring of N sites, we have

Z=TreβH={σn}eβhδσ1,1eβJδσ1,σ2eβhδσN,1eβJδσN,σ1=Tr(RN) ,

where the q×q transfer matrix R is given by

Rσσ=eβJδσσe12βhδσ,1e12βhδσ,1={eβ(J+h) if σ=σ=1eβJ if σ=σ1eβh/2 if σ=1 and σ1eβh/2 if σ1 and σ=11 if σ1 and σ1 and σσ .

In matrix form,

R=(eβ(J+h)eβh/2eβh/2eβh/2eβh/2eβJ11eβh/21eβJ1eβh/211eβJ1eβh/2111eβJ)

The matrix R has q eigenvalues λj, with j=1,,q. The partition function for the Potts chain is then

Z=qj=1λNj .

We can actually find the eigenvalues of R analytically. To this end, consider the vectors

ϕ=(100),ψ=(q1+eβh)1/2(eβh/211) .

Then R may be written as

R=(eβJ1)I+(q1+eβh)|ψψ|+(eβJ1)(eβh1)|ϕϕ| ,

where I is the q×q identity matrix. When h=0, we have a simpler form,

R=(eβJ1)I+q|ψψ| .

From this we can read off the eigenvalues:

λ1=eβJ+q1λj=eβJ1,j{2,,q} ,

since |ψ is an eigenvector with eigenvalue λ=eβJ+q1, and any vector orthogonal to |ψ has eigenvalue λ=eβJ1. The partition function is then

Z=(eβJ+q1)N+(q1)(eβJ1)N .

In the thermodynamic limit N, only the λ1 eigenvalue contributes, and we have

F(T,N,h=0)=NkBTln(eJ/kBT+q1) for N .

When h is nonzero, the calculation becomes somewhat more tedious, but still relatively easy. The problem is that |ψ and |ϕ are not orthogonal, so we define

|χ=|ϕ|ψψ|ϕ1ϕ|ψ2 ,

where

xϕ|ψ=(eβhq1+eβh)1/2 .

Now we have χ|ψ=0, with χ|χ=1 and ψ|ψ=1, with

|ϕ=1x2|χ+x|ψ .

and the transfer matrix is then

R=(eβJ1)I+(q1+eβh)|ψψ|+(eβJ1)(eβh1)[(1x2)|χχ|+x2|ψψ|+x1x2(|χψ|+|ψχ|)]=(eβJ1)I+[(q1+eβh)+(eβJ1)(eβh1)(eβhq1+eβh)]|ψψ|+(eβJ1)(eβh1)(q1q1+eβh)|χχ|+(eβJ1)(eβh1)((q1)eβhq1+eβh)1/2(|χψ|+|ψχ|) ,

which in the two-dimensional subspace spanned by |χ and |ψ is of the form

R=(accb) .

Recall that for any 2×2 Hermitian matrix,

M=a0I+aτ=(a0+a3a1ia2a1+ia2a0a3) ,

the characteristic polynomial is

P(λ)=det(λIM)=(λa0)2a21a22a23 ,

and hence the eigenvalues are

λ±=a0±a21+a22+a23 .

For the transfer matrix of Equation ???, we obtain, after a little work,

λ1,2=eβJ1+12[q1+eβh+(eβJ1)(eβh1)]±12[q1+eβh+(eβJ1)(eβh1)]24(q1)(eβJ1)(eβh1) .

There are q2 other eigenvalues, however, associated with the (q2)-dimensional subspace orthogonal to |χ and |ψ. Clearly all these eigenvalues are given by

λj=eβJ1,j{3,,q} .

The partition function is then

Z=λN1+λN2+(q2)λN3 ,

and in the thermodynamic limit N the maximum eigenvalue λ1 dominates. Note that we recover the correct limit as h0.


This page titled 6.7: Appendix I- Potts Model in One Dimension is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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