The chiral Potts model

In 1987 my coauthors H.Au-Yang, J.H.H. Perk,S. Tang, and Y.M. Lin [27] and I discovered the first example of an integrable model where, in technical language, the spectral variable lies on a curve of genus higher than one. This model is known as the integrable chiral Potts model and is a particular case of a phenomenological model introduced in 1983 by Howes, Kadanoff and den Nijs [28]in their famous study of level crossing transitions.

The Boltzmann weights were subsequently shown by Baxter, Perk and Au-Yang [29] to have the following elegant form  

$${W^h_{p,q}(n)\over W^h_{p,q}(0)}=\prod_{j=1}^{n}({d_pb_q-a_... ...}^n({\omega a_pd_q-d_pa_q\omega^j\over c_pb_q-b_pc_q\omega^j})$$ (5)

where $\omega = e^{2 \pi i/N}.$ The variables a_p,b_p,c_p,d_p and a_q,b_q,c_q,d_q satisfy the equations

 

a^N+kb^N=k'd^N,   ka^N+b^N=k'c^N (6)

with k²+k'²=1 and this specifies a curve of genus N³-2N²+1. When N=2 the Boltzmann weights reduce to those of the Ising model (1) with H=0 and the curve (6) reduces to the elliptic curve of genus 1. However when $N\geq 3$ the curve has genus higher than one. This is the first time that such higher genus curves has arisen in the Boltzmann weights of integrable models.

This model is out of the class of all previously known models and raises a host of unsolved questions which are related to some of the most intractable problems of algebraic geometry which have been with us for 150 years. As an example of these new occurances of ancient problems we can consider the spectrum of the transfer matrix

$$T_{\{l,l'\}}=\prod_{j=1}^{\cal N}W_{p,q}^v(l_j-l'_j)W_{p,q}^h(l_j-l'_{j+1}).$$ (7)

This transfer matrix satisfies the commutation relation

[T(p,q),T(p,q')]=0

and also satisfies functional equations of the Riemann surface at points connected by the automorphism $R(a_q,b_q,c_q,d_q)=(b_q,\omega_q,d_q,c_q).$ For the Ising case N=2 this functional equation reduces to an equation which can be solved using elliptic theta functions. Most unhappily, however, for the higher genus case the analogous solution requires machinery from algebraic geometry which does not exist. For the problem of the free energy Baxter [30] has devised an ingenious method of solution which bypasses algebraic geometry completely but even here some problems remain in extending the method to the complete eigenvalue spectrum [31].

The problem is even more acute for the order parameter of the model. For the N state models there several order parameter parameterized by an integer index n where $1\leq n\leq N-1.$ For these order parameters M_n we conjectured [32] 10 years ago from perturbation theory computations that

M_n=(1-k²)^n(N-n)/2N2.

When N=2 this is exactly the result conjectured by Onsager [33] in 1948 and proven by Yang [2] for the Ising model in 1952. For the Ising model it took only three years to go from conjecture to proof. But for the chiral Potts model a decade has passed and even though Baxter [34]-[35] has produced several elegant formulations of the problem which all lead to the correct answer for the Ising case none of them contains enough information to solve the problem for $N\geq 3.$ In one approach [34]the problem is reduced the the evaluation of a path ordered exponential of nonabelian variables on a Riemann surface. This sounds exactly like problems encountered in nonabelian gauge theory but, unfortunately, there is nothing in the field theory literature that helps. In another approach [35]a major step in the solution involves the explicit reconstruction of a meromorphic function from a knowledge of its zeros and poles. This is a classic problem in algebraic geometry for which in fact no explicit answer is known either. Indeed the unsolved problems arising from the chiral Potts model are so resistant to all known mathematics that I have reduced my frustration to the following epigram;

The nineteenth century saw many brilliant creations of the human mind. Among them are algebraic geometry and Marxism. In the late twentieth century Marxism has been shown to be incapable of solving any practice problem but we still do not know about algebraic geometry.

It must be stressed again that the chiral Potts model was not invented because it was integrable but was found to be integrable after it was introduced to explain experimental data. In a very profound way physics is here far ahead of mathematics.