Chiral Potts model

The model is called self-dual, if the Fourier transform of the weight is equal to the weight. A special (genus 1) case had been solved in 1982 by Fateev and Zamolodchikov.[7] By removing certain restrictions of the work of Alcaraz and Santos,[8] a more general self-dual case of the integrable chiral Potts model was discovered.[1] The weight are given in product form[9][10] and the parameters in the weight are shown to be on the Fermat curve, with genus greater than 1.

In Canberra, the general solution for all k (the temperature variable) was found.[2] The weights were also given in product form and it was tested by Fortran that they satisfy the star–triangle relation. The proof was published later.[11]

From the series[5][12] the order parameter is conjectured[13] to have the simple form

${\displaystyle \langle \sigma ^{n}\rangle =(1-k'^{2})^{\beta },\quad \beta =n(N-n)/2N^{2}.}$

It took many years to prove this conjecture, as the usual corner transfer matrix technique could not be used, because of the higher genus curve. This conjecture was finally proven by Baxter in 2005[14][15] using functional equations and the "broken rapidity line" technique of Jimbo et al.[16] assuming two rather mild analyticity conditions of the type commonly used in the field of Yang–Baxter integrable models. Most recently, in a series of papers[17][18][19][20][21][22][23] an algebraic (Ising-like) way of obtaining order parameter has been given, giving more insight into the algebraic structure.

In 1990 Bazhanov and Stroganov[24] show that there exist an 2 × N L-operators, which satisfy the Yang–Baxter equation

${\displaystyle L_{i_{1}\alpha }^{j_{1}\beta }(x)L_{i_{2}\beta }^{j_{2}\gamma }(y)R_{j_{1}j_{2}}^{k_{1}k_{2}}(y/x)=R_{i_{1}i_{2}}^{j_{1}j_{2}}(y/x)L_{j_{2}\alpha }^{k_{2}\beta }(y)L_{j_{1}\beta }^{k_{1}\gamma }(x),\quad 0<i,j,k\leq 1,\quad 0\leq \alpha ,\beta ,\gamma \leq N-1.}$

where 2 × 2 R-operator is the 6-vertex R-matrix (see Vertex model). The product of four chiral Potts weight S was shown to intertwine two L-operators as

${\displaystyle L_{i_{1}\alpha _{1}}^{i_{2}\alpha _{2}}{\hat {L}}_{i_{2}\beta _{1}}^{i_{3}\beta _{2}}S_{\alpha _{2}\beta _{2}}^{\alpha _{3}\beta _{3}}=S_{\alpha _{1}\beta _{1}}^{\alpha _{2}\beta _{2}}{\hat {L}}_{i_{1}\beta _{2}}^{i_{2}\beta _{3}}L_{i_{2}\alpha _{2}}^{i_{3}\alpha _{3}},\quad 0<i_{i}\leq 1,\quad 0\leq \alpha _{i},\beta _{i}\leq N-1.}$

This inspired a most important breakthrough, namely the functional relations for the transfer matrices of the chiral Potts models are discovered.[25]

Using these functional relation, Baxter was able to calculate the eigenvalues of the transfer matrix of the chiral Potts model,[26] and obtained the critical exponent for the specific heat α=1-2/N, which was also conjectured in reference 12. The interfacial tensions are also calculated by him with the exponent μ=1/2+1/N.[27][28]

The integrable chiral Potts weights are given in product form [2] as

${\displaystyle W_{pq}(n)\!=\!{\Big (}{\mu _{p} \over \mu _{q}}{\Big )}^{\!\!n}\prod _{j=1}^{n}{y_{q}-x_{p}\omega ^{j} \over y_{p}-x_{q}\omega ^{j}},\quad {\overline {W}}_{pq}(n)\!=\!{\big (}{\mu _{p}\mu _{q}}{\big )}^{\!n}\!\prod _{j=1}^{n}{\omega x_{p}\!-\!x_{q}\omega ^{j} \over y_{q}\!-\!y_{p}\omega ^{j}},}$

where ω^N=1 and we associate with each of the rapidity variable p with three variables (x_p,y_p,μ_p) satisfying

${\displaystyle x_{p}^{N}+y_{p}^{N}=k(1+x_{p}^{N}y_{p}^{N}),\quad k'^{2}+k^{2}=1,\quad k'\mu _{p}^{N}=1-ky_{p}^{N},\quad k'\mu _{p}^{-N}=1-kx_{p}^{N}.}$

It is easy to see that

${\displaystyle W_{pp}(a-b)=1,\quad {\overline {W}}_{pp}(a-b)=\delta _{a,b}}$

which is similar to Reidemeister move I. It was also known that the weights satisfying the inversion relation,

${\displaystyle W_{pq}(a-b)W_{qp}(a-b)=1,\quad \sum _{d=0}^{N-1}{\overline {W}}_{pq}(a-d){\overline {W}}_{qp}(d-a')=r_{pq}\delta _{a,a'}.}$

This is equivalent to Reidemeister move II. The star-triangle relation

${\displaystyle \sum _{d=1}^{N}\,{\overline {W}}_{pr}(a-d)\,W_{pq}(d-c)\,{\overline {W}}_{rq}(d-b)=R_{pqr}\,{\overline {W}}_{pq}(a-b)\,{W}_{pr}(b-c)\,W_{rq}(a-c)}$

is equivalent to Reidemeister move III. These are shown in the figure shown here. [29]

Weights of the Integrable chiral Potts models

Property of the Weights: Reidemeister Move I

Inversion Relation of the Weights: Reidemeister Move II

Star-triangle relation: Reidemeister Move III