6.7: Appendix I- Potts Model in One Dimension

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Definition

The Potts model is defined by the Hamiltonian

\[H=-J\sum_{\langle ij\rangle}\delta\nd_{\sigma\ns_i,\sigma\ns_j}-h\sum_i\delta\nd_{\sigma\ns_i,1}\ .\]

Here, the spin variables \(\sigma\ns_i\) take values in the set \(\{1,2,\ldots,q\}\) on each site. The equivalent of an external magnetic field in the Ising case is a field \(h\) which prefers a particular value of \(\sigma\) (\(\sigma=1\) 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

\[\begin{split} Z&={ Tr}\,e^{-\beta H}\\ &=\sum_{\{\sigma\ns_n\}} e^{\beta h\delta_{\sigma\ns_1,1}}\,e^{\beta J\delta_{\sigma\ns_1,\sigma\ns_2}}\,\cdots\, e^{\beta h\delta_{\sigma\ns_N,1}}\,e^{\beta J\delta_{\sigma\ns_N,\sigma\ns_1}}\\ &={ Tr}\,\big(R^N\big)\ , \end{split}\]

where the \(q\times q\) transfer matrix \(R\) is given by

\[R\nd_{\sigma\sigma'}=e^{\beta J\delta_{\sigma\sigma'}}\,e^{1\over 2 \beta h\delta_{\sigma,1}}\,e^{ 1\over 2 \beta h\delta_{\sigma',1}}= \begin{cases} e^{\beta(J+h)} & \hbox{ if $\sigma=\sigma'=1$}\\ e^{\beta J} & \hbox{ if $\sigma=\sigma'\ne 1$}\\ e^{\beta h/2} & \hbox{ if $\sigma=1$ and $\sigma'\ne 1$}\\ e^{\beta h/2} & \hbox{ if $\sigma\ne1$ and $\sigma'= 1$}\\ 1& \hbox{ if $\sigma\ne1$ and $\sigma'\ne 1$ and $\sigma\ne\sigma'$}\ . \end{cases}\]

In matrix form,

\[R=\begin{pmatrix} e^{\beta(J+h)} & e^{\beta h/2} & e^{\beta h/2} & & \cdots & & e^{\beta h/2} \\ e^{\beta h/2} & e^{\beta J} & 1 & & \cdots & & 1 \\ e^{\beta h/2} & 1 & e^{\beta J} & & \cdots & & 1 \\ \vdots & \vdots & \vdots & & \ddots & & \vdots \\ e^{\beta h/2} & 1 & 1 & & \cdots & e^{\beta J} & 1\\ e^{\beta h/2} & 1 & 1 & & \cdots & 1 & e^{\beta J} \end{pmatrix}\]

The matrix \(R\) has \(q\) eigenvalues \(\lambda\ns_j\), with \(j=1,\ldots,q\). The partition function for the Potts chain is then

\[Z=\sum_{j=1}^q \lambda_j^N \ .\]

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

\[\phi=\begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0\end{pmatrix} \qquad,\qquad \psi=\big(q-1+e^{\beta h}\big)^{-1/2} \begin{pmatrix} e^{\beta h/2} \\ 1 \\ \vdots \\ 1 \end{pmatrix}\ .\]

Then \(R\) may be written as

\[R=\big(e^{\beta J}-1\big)\,{\mathbb I} + \big(q-1+e^{\beta h}\big)\,\sket{\psi}\sbra{\psi} + \big(e^{\beta J}-1\big)\big(e^{\beta h}-1\big)\,\sket{\phi}\sbra{\phi}\ ,\]

where \({\mathbb I}\) is the \(q\times q\) identity matrix. When \(h=0\), we have a simpler form,

\[R=\big(e^{\beta J}-1\big)\,{\mathbb I} + q\,\sket{\psi}\sbra{\psi}\ .\]

From this we can read off the eigenvalues:

\[\begin{split} \lambda\ns_1&=e^{\beta J} +q-1 \\ \lambda\ns_j&=e^{\beta J}-1 \quad,\quad j\in\{2,\ldots,q\}\ , \end{split}\]

since \(\sket{\psi}\) is an eigenvector with eigenvalue \(\lambda=e^{\beta J} +q-1\), and any vector orthogonal to \(\sket{\psi}\) has eigenvalue \(\lambda=e^{\beta J}-1\). The partition function is then

\[Z=\big(e^{\beta J} +q-1\big)^N + (q-1)\big(e^{\beta J}-1\big)^N\ .\]

In the thermodynamic limit \(N\to\infty\), only the \(\lambda\nd_1\) eigenvalue contributes, and we have

\[F(T,N,h=0)=-N\kT\ln\big(e^{J/\kT}+q-1\big)\qquad\hbox{ for $N\to\infty$}\ .\]

When \(h\) is nonzero, the calculation becomes somewhat more tedious, but still relatively easy. The problem is that \(\sket{\psi}\) and \(\sket{\phi}\) are not orthogonal, so we define

\[\sket{\xhi}={\sket{\phi}-\sket{\psi}\sbraket{\psi}{\phi}\over \sqrt{1-\sbraket{\phi}{\psi}^2}}\ ,\]

where

\[x\equiv\sbraket{\phi}{\psi}=\bigg( {e^{\beta h}\over q-1+e^{\beta h}} \bigg)^{\!1/2}\ .\]

Now we have \(\sbraket{\xhi}{\psi}=0\), with \(\sbraket{\xhi}{\xhi}=1\) and \(\sbraket{\psi}{\psi}=1\), with

\[\sket{\phi}=\sqrt{1-x^2\>}\,\sket{\chi} + x\,\sket{\psi}\ .\]

and the transfer matrix is then

\[\begin{aligned} R&=\big(e^{\beta J}-1\big)\,{\mathbb I} +\big(q-1+e^{\beta h}\big)\,\sket{\psi}\sbra{\psi}\nonumber\\ &\qquad +\big(e^{\beta J}-1\big)\big(e^{\beta h}-1\big)\,\bigg[(1-x^2)\, \sket{\xhi}\sbra{\xhi} + x^2\>\sket{\psi}\sbra{\psi}+x\,\sqrt{1-x^2}\>\Big(\sket{\xhi}\sbra{\psi} +\sket{\psi}\sbra{\xhi}\Big)\bigg]\nonumber\\ &=\big(e^{\beta J}-1\big)\,{\mathbb I} +\Bigg[ \big(q-1+e^{\beta h}\big) + \big(e^{\beta J}-1\big)\big(e^{\beta h}-1\big)\bigg({e^{\beta h}\over q-1+e^{\beta h}}\bigg)\Bigg] \sket{\psi}\sbra{\psi}\label{RPotts}\\ &\qquad\qquad+\big(e^{\beta J}-1\big)\big(e^{\beta h}-1\big)\, \bigg({q-1\over q-1+e^{\beta h}}\bigg)\, \sket{\xhi}\sbra{\xhi} \nonumber\\ &\qquad\qquad\qquad\qquad+\big(e^{\beta J}-1\big)\big(e^{\beta h}-1\big)\, \bigg({(q-1)\,e^{\beta h}\over q-1+e^{\beta h}}\bigg)^{\!1/2}\,\Big(\sket{\xhi}\sbra{\psi} +\sket{\psi}\sbra{\xhi}\Big)\ ,\nonumber\end{aligned}\]

which in the two-dimensional subspace spanned by \(\sket{\xhi}\) and \(\sket{\psi}\) is of the form

\[R=\begin{pmatrix} a & c \\ c & b\end{pmatrix}\ .\]

Recall that for any \(2\times 2\) Hermitian matrix,

\[\begin{split} M&=a\nd_0\,{\mathbb I} +\Ba\cdot\Btau\\ &=\begin{pmatrix} a\ns_0 + a\ns_3 & a\ns_1 -i a\ns_2 \\ a\ns_1 + i a\ns_2 & a\ns_0-a\ns_3\end{pmatrix}\ , \end{split}\]

the characteristic polynomial is

\[P(\lambda)={ det}\,\big(\lambda\,{\mathbb I}-M\big)=(\lambda-a\ns_0)^2 - a_1^2-a_2^2-a_3^2\ ,\]

and hence the eigenvalues are

\[\lambda\nd_\pm=a\ns_0\pm\sqrt{a_1^2+a_2^2+a_3^2\,}\ .\]

For the transfer matrix of Equation \ref{RPotts}, we obtain, after a little work,

\[\begin{aligned} \lambda\nd_{1,2}&=e^{\beta J}-1+\half\Big[q-1+e^{\beta h} + \big(e^{\beta J}-1\big)\big(e^{\beta h}-1\big)\Big]\\ &\qquad\pm\half\sqrt{\Big[q-1+e^{\beta h} + \big(e^{\beta J}-1\big)\big(e^{\beta h}-1\big)\Big]^{2} -4(q-1) \big(e^{\beta J}-1\big)\big(e^{\beta h}-1\big)\>}\ .\nonumber\end{aligned}\]

There are \(q-2\) other eigenvalues, however, associated with the \((q\!-\!2)\)-dimensional subspace orthogonal to \(\sket{\xhi}\) and \(\sket{\psi}\). Clearly all these eigenvalues are given by

\[\lambda\ns_j=e^{\beta J}-1\qquad,\quad j\in\{3\,,\,\ldots\,,\,q\}\ .\]

The partition function is then

\[Z=\lambda_1^N + \lambda_2^N + (q-2)\,\lambda_3^N\ ,\]

and in the thermodynamic limit \(N\to\infty\) the maximum eigenvalue \(\lambda\ns_1\) dominates. Note that we recover the correct limit as \(h\to 0\).