## Abstract

We show that the free energy of the three-state model can be expressed as products of Jacobi elliptic functions, the arguments being those of an hyperelliptic parametrization of the associated chiral Potts model. This is the first application of such a parametrization to the *N*-state chiral Potts free-energy problem for *N*>2.

## 1. Introduction

In the field of solvable models in statistical mechanics, the *N*-state two-dimensional chiral Potts model has proved particularly challenging. This is despite the fact that for *N*=2 it reduces to the Ising model, the free energy of which was calculated by Onsager (1944).

The model was developed by Howes *et al*. (1983), von Gehlen & Rittenberg (1985), Au-Yang *et al*. (1987) and McCoy *et al*. (1987). It was first fully defined as a general *N*-state lattice model by Baxter *et al*. (1988), who showed that it satisfied the star–triangle relations. These relations define ‘rapidity’ variables *p*, *q*, such that the Boltzmann weights are functions of *p* and *q*.

The model is also contained in the solution by Korepanov (1986) of the operator form of the star–triangle relations. This solution was later found more explicitly by Bazhanov & Stroganov (1990).

Many of the citations herein are to earlier papers by the author; in these we shall abbreviate ‘Baxter’ to ‘B’.

With previously solved models, such as the hard-hexagon model (B 1980), the calculation of the free energy was straightforward once one had obtained the star–triangle relation and rapidities for that model. Typically, it turned out that there was a transformation in terms of Jacobi elliptic functions, such that depended on *p*, *q* only via their difference . The free energy is , where is the partition function per site. This must also be a function of *u*, and there were always ‘rotation’ and ‘inversion’ relations (B 1982*a*,*b*) of the form(1.1)where *λ* is a positive real constant (the ‘crossing parameter’) and a known meromorphic function. For *u* real and , the Boltzmann weights are real and positive, and one can develop low-temperature series expansions in the usual way that indicate that is analytic, non-zero and bounded in the vertical strip . Equations (1.1) then determine and one can solve them by Fourier transforms or other methods.

The *N*>2 chiral Potts model, however, does *not* have the ‘rapidity difference property’, i.e. there is no transformation that takes to a function only of *q*−*p*. This makes the calculation of much more difficult, and it was not till 1990 that an explicit result was obtained (B 1990, 1991*a*). The calculation of the spontaneous magnetizations (order parameters) is even harder, and was not accomplished until 2005 (B 2005*a*,*b*).

The calculation of proceeds in two stages. First, one calculates the partition function per site of an associated ‘ model’, which is closely related to the superintegrable case of the chiral Potts model. Then, one uses this result to calculate .

In the calculation of the order parameters, we used the fact that certain functions , are similar to . In fact, they are ratios of special cases of .

Although the main task, namely the calculation of the free energy and order parameters of the chiral Potts model, has been accomplished, it remains disappointing that one has no elegant parametrization in terms of elliptic functions, as one has for models with the difference property. These parametrizations explicitly exhibit the poles and zeros of on an extended *p*, *q* Riemann surface. There is a parametrization of the rapidities and Boltzmann weights of the *N*-state chiral Potts model in terms of hyperelliptic functions (B 1991*b*), but these have arguments that are related to one another in a complicated way, and until recently they have not been found to be particularly useful.

However, we have now found that for *N*=3, the function mentioned above can be expressed as a ratio of generalized elliptic functions of these arguments (B 2006). The obvious question is whether can be similarly expressed. We show here that the answer is yes. The functions that occur are the same as those that appear in other solvable models, notably the Ising model.

## 2. The function *τ*_{2}(*p*,*q*)

For the chiral Potts model, the rapidity *p* can be thought of as the set of variables , related to one another by(2.1)Here, *k*, are real positive constants, satisfying(2.2)In terms of the of B *et al*. (1988) and B (1991*b*), .

There are various automorphisms or maps that take one set to another set satisfying the same relations (2.1). Four that we shall use are:(2.3)whereThey satisfy(2.4)Here, *U*=*RSV*, *S* being the operator *S* used in B *et al*. (1988) and B (2006).

We take to be outside the unit circle, so(2.5)Then we can specify uniquely by requiring that(2.6)

We regard as functions of . Then lies in a complex plane containing *N* branch cuts on the lines , as indicated in figure 1, while lies in a near-circular region round the point , as indicated schematically by the region inside the dotted curve of figure 1. The variable can lie anywhere in the complex plane, *except* in and in corresponding near-circular regions round the other branch cuts. With these choices, we say that *p* lies in the ‘domain’ .

It can be helpful to consider the low-temperature limit, when . The branch cuts and the regions then shrink to the points . If is held fixed, not at , then , and .

Define *q* similarly to *p*, also lying in . Then from equation (39) of B (1991*a*) and equation (54) of B (2003*a*), the partition function per site of the model is given by(2.7)where(2.8)We have omitted a factor from , and to ensure that this formula is correct for , we invert in B (1991*a*, 2003*a*).

This function satisfies the relations(2.9)and(2.10)where(2.11)

## 3. The Riemann sheets (‘domains’) formed by analytic continuation

We shall consider the analytic continuation of certain functions of onto other Riemann sheets, i.e. beyond the domain . We restrict attention to functions that are meromorphic and single-valued in the cut plane of figure 1, and similarly for their analytic continuations. Obvious examples are and . They are therefore meromorphic and single-valued on their Riemann surfaces, but we need to know what these surfaces are.

We start by considering the most general such surface. As a first step, allow to move from outside the unit circle to inside. Then will cross one of the *N* branch cuts in figure 1, moving onto another Riemann sheet, going back to its original value but now with in . Since is thereby confined to the region near and surrounding , we say that . Conversely, by we mean that .

We say that *p* has moved into the *domain* *adjacent* to . There are *N* such domains .

Now allow to become larger than 1, so again crosses one of the *N* branch cuts. Again we require that returns to its original value. If it crosses , then it moves back to the original domain . However, if it crosses another cut , then moves into , and we say that *p* is now in domain .

Proceeding in this way, we build up a Cayley tree of domains. For instance, the domain is a third neighbour of , linked via the first neighbour and the second neighbour , as indicated in figure 2. Here, in , in , in and in . We reject moves that take *p* back to the domain immediately before the last, so and . We refer to the sequence that define any domain as a *route*. We can think of it as a sequence of points, all with the same value of , on the successive Reimann sheets or domains.

The domains , , , with an even number of indices, have , where is the last index. We refer to them as being of even *parity* and of *type* . The domains , have and are of odd parity and type .

The automorphism that takes a point *p* in to a point in , respectively, is the mapping(3.1)If , then(3.2)Owing to (2.4), , so there are *N* such automorphisms.

We can use these maps to generate all the sheets in the full Cayley tree. Suppose we have a domain with route and we apply the automorphism to all points on the route. From (3.2), this will generate a new route . For instance, if we apply the map to the route from to , we obtain the route to the domain . Thus, the map that takes to is .

Iterating, we find that the map that takes to is(3.3)We must have(3.4)since applying the same map twice merely returns *p* to the previous domain.

Let us refer to the general Riemann surface we have just described as . It consists of infinitely many Riemann sheets, each sheet corresponding to a site on a Cayley tree, adjacent sheets corresponding to adjacent points on the tree. A Cayley tree is a huge graph; it contains no circuits and is infinitely dimensional, needing infinitely many integers to specify all its sites.

Any given function will have a Riemann surface that can be obtained from by identifying certain sites with one another, thereby creating circuits and usually reducing the graph to one of finite dimensionality.

From (3.2), the maps leave unchanged. We shall often find it helpful to regard as a fixed complex number, the same in all domains, and to consider the corresponding values of (and the hyperelliptic variables ) in the various domains. To within factors of *ω*, the variables and will be the same as those for in even domains, while they will be interchanged on odd domains.

### (a) Analytic continuation of

Consider as a function of *q* (so replace *p* by *q* in the previous discussion of the domains). More specifically, think of it as a function of the complex variable . For (and ), it is apparent from (2.7) that is an analytic function of , except for the *single* branch cut , being single-valued across the other cuts. Let , so , and define(3.5)Then it follows from (2.10) that(3.6)

for , where(3.7)and .

Equation (3.6) defines the mapping applied to the function of *q*. Iterating, it follows that if , then(3.8)where .

We can also keep *q* fixed in and consider the analytic continuation of as *p* moves from sheet to sheet. If , so , , we can verify from (2.7) that(3.9)Also, from (2.11),(3.10)

Equation (3.9) defines the for the function of *p*. Iterating, it follows that if , then(3.11)

Note that for both and , it is true that .

## 4. Hyperelliptic parametrization for *N*=3

Hereinafter we restrict our attention to the case *N*=3. A parametrization of was developed in previous papers, in terms of a ‘nome’ *x* and two related parameters (B 1991*b*, 1993*a*,*b*, 1998). The nome *x* is like *k* and in that it is a constant; it is not to be confused with the rapidity variable .

However, in B (2006), we showed that the function could not be expressed as a single-valued function of these original parameters . This is because have the same values (for given ) in the domains and , whereas has different values therein. These domains are obtained by the maps , , respectively, and indeed we see from (3.8) and (3.9) that these maps give different results for for both the *q* and the *p* variables. Thus, cannot be a single-valued function, either of for fixed *p*, or of for fixed *q*.

The same problem occurs with the domains , and the corresponding maps , .

The situation is not lost. We also showed in B (2006) that there is another way of parametrizing . This alternative way preserves the property that the nome *x* is small at low temperatures ( small). It can be obtained from the original parametrization by leaving unchanged and transforming according to the rule(4.1)(taking ). This mapping leaves the relations (2.1) and (2.2) unchanged.

Doing this, eqn (21) of B (1993*a*) becomes(4.2)while the two equivalent relations (4.5) and (4.6) of B (1993*b*) remain unchanged(4.3)

These *z*, *w* are related to by various elliptic-type equations that we shall give below, being the arguments (more precisely the exponentials of i times the arguments) of Jacobi elliptic functions of nome *x*. Thus, *x*, like , is a constant, while *z*, *w* are two more rapidity variables, dependent on *p*. We shall also refer to them as .

First, we introduce elliptic-type functions(4.4)so that , and define two sets of parameters *p*, in terms of ,(4.5)extending , to all *j* by(4.6)Thus,(4.7)We shall also need certain cube roots of , choosing them so that(4.8)

Then, after applying the mapping (4.1), equations (27) and (32) of B (1993*a*) become(4.9)(4.10)for . The second form of (4.10) is to be preferred as it fixes the choice of the leading cube root factor in the function , so fixing .

Similarly, replacing *p* by *q* in the above equations, we define , and hence , in terms of two related variables .

### (a) The low-temperature limit

At low temperatures, are small. For (and not close to a cube root of unity), we can choose to tend to non-zero limits as . Then (4.3) both give . Also, , so (2.1) and (4.9) determine uniquely (with the same value for ). Then can be calculated from the last part of equation (2.1), and itself from (4.10). Hence, when are small,for . Note that are of order unity, but the are of order , while is of order .

### (b) Mappings

The effect of the automorphisms on is(4.11)The mappings take a point *p* within to another point within ,(4.12)and if , then .

Also, if , then(4.13)for . We see that in this new parametrization does *not* have the same effect on as (nor do and ), so we no longer have the problem referred to at the beginning of this section.

If we identify Riemann sheets that have the same values of for given , then ceases to be a Cayley tree and becomes the two-dimensional honeycomb lattice of figure 3.

To see this, note that if are the values of on the central sheet , then it follows from (4.13) that on any Riemann sheet the analytic continuations of (for a given value of ) are(4.14)choosing the upper (lower) signs on sheets of even (odd) parity. Here, *m*, *n* are integers satisfying(4.15)The Riemann surface for therefore corresponds to a two-dimensional graph , each site of being specified by the two integers *m*, *n* and corresponding to a Riemann sheet of the surface.

This is indeed the honeycomb lattice shown in figure 3. Adjacent sites correspond to adjacent Riemann sheets. Sheets of even parity correspond to sites represented by circles, those of odd parity are represented by squares. If *i* is the integer shown inside the circle or square, then on even sites , and on odd sites . The numbers shown in brackets alongside each site are the integers *m*, *n* of (4.14); we refer to the corresponding sheet as ‘the sheet ’.

We can trace this reduction of to the fact that the automorphisms applied to satisfy(4.16)For instance, the relation implies that , and indeed we see from figure 3 that these are the two three-step routes from to *Y*. Similarly for the two routes to *Z*, or the two routes to *X*.

We noted in §3 that the function , considered as a function of either *p* or *q*, also satisfies (4.16). It follows that its Riemann surface can be embedded in that of the reduced graph .

We focus on the *q* dependence of . Let its value when (given by equation (2.7)) be . Then by iterating (3.6) we find that on the Riemann sheet (4.17)where(4.18)

### (c) The function

We can eliminate the distinction between even and odd sheets by working, not with , but instead with the closely related function(4.19)(using equation (2.10)). Then (3.6) becomes(4.20)for . If is the value of in , then it follows that for a given value of ,(4.21)on all sheets .

It follows that is a single-valued meromorphic function on the Riemann surface , and that the orders of its zeros and poles are linear in the integers *m*, *n* of (4.14) (with *p* replaced by *q* therein). This suggests that it may be possible to write as a product of functions of , and indeed we shall find that this is the case.

We shall explicitly exhibit the dependence of on by writing it as . Then the three relations (4.20) become(4.22)

If *p* is fixed within , then from (2.7) and (4.19), is bounded and has no zeros or poles for . (Both and *α* become infinite as , but their ratio remains finite.) Together with this restriction, the relations (4.22) define to within a multiplicative factor independent of *q*. To see this, suppose we had two such solutions, then (4.22) would imply that their ratio was unchanged by , for . This would mean that the ratio, considered as a functions of , did not have the branch cuts , , . It would therefore be an entire bounded function of , and hence by Liouville's theorem a constant (independent of *q*). This constant could be determined from the product relation (2.10), which now takes the simple form(4.23)

We shall also use the *p*-relation (3.9). Together with (3.10), this implies that(4.24)

## 5. Various *p*, *q* relations

First, we present various relations that enable one to express certain rational functions (including ) of as products of elliptic functions of .

Some can be obtained by applying the rule (4.1) to equation (34) of B (1993*a*) or to equation (4.9) of B (1993*b*). In particular, we obtain(5.1)

Applying the automorphism *U* to the *q* variable in (5.1), and then using the *V* and *R* automorphisms, we obtain the two sets of relations(5.2)true for all integers

The function has a leading factor , which gives a contribution to the right-hand side (RHS) of (5.1). This cube root can and should be chosen to be . The corresponding contributions in (5.2) are and .

We define the elliptic function(5.3)for integer *r* (in particular *r*=1 and 3), satisfyingThen and , so (4.9) and (4.10) can be used to relate various and functions. In particular, from (4.8) and (4.10),(5.4)for any cyclic permutation of . We note that (4.2) can be written as(5.5)and set(5.6)It follows from (5.3) and (5.5) that these various expressions for *γ* are all equal.

We also define(5.7)and note that(5.8)

We find the following six identities and indicate below the method of their proof. The first two can also be derived from (5.2) by applying the duality/conjugate modulus mapping of appendix A.

We have also checked all the identities of this section numerically, for arbitrarily chosen , to 25 digits of accuracy,(5.9)(5.10)(5.11)(5.12)for .

Letting , we see that (5.10) remains true if therein is replaced by or by . Also, (5.11) is true if is replaced by or by .

### (a) Proof of the identities

We have proved the identities (5.9)–(5.12). Let be the ratio of the LHS to the RHS of any of these identities.

First note that *a priori* we expect to be a single-valued function of only if we introduce the branch cuts of figure 1 into the complex -plane.

Indeed, the identities (5.9) involve and , and these will need additional cuts linking the in order to completely fix the choice of the cube root in (4.8).

The map that takes *q* from one such choice to another is the map . However, from (2.3) and (4.12), this merely increases (decreases) by one in the first (second) set of the identities (5.9). If we take some symmetric function of all the (for all ), then will be invariant under , which means that these extra cuts are not needed for this function.

The next step is to use (3.2) and (4.13) to show that the mapping merely permutes, and possibly inverts, the of each set of identities. This is true for each of the sets of identities (5.10)–(5.12), but for (5.9) the two sets are interchanged by this mapping.

Again, let be some symmetric function of all the (and if necessary their inverses) within a set (e.g. the sum of the fifth powers of every in one of equations (5.12), summed over *i* and *j*), now regarding the two identities (5.9) as forming one combined set. Then will be unchanged by each of the three mappings , for . This means that it has the same value on either side of any of the branch cuts . The cuts are therefore unnecessary: is a *single-valued function* of the complex variable .

The only possible singularities of are therefore poles, arising from poles, and possibly zeros, in the . The only places these can occur are when , , and . We can restrict our attention to , since is unchanged (for a given value of ) by crossing the . Thus, and are both greater than 1, and each lie in a region near the points , . This means that the only points to consider are(5.13)for all *i* and *m*.

Each can be written as a ratio of pole-free functions. In every case, if the numerator (denominator) has a zero at one of the points (5.13), then so does the denominator (numerator), and both zeros are simple. Thus, no has a pole or zero at any of the above points.

The function is therefore a single-valued and analytic function in the complete -plane, including the point at infinity (this is the last of the points listed above). By Liouville's theorem it is therefore a constant.

One can write down a polynomial of finite degree whose roots are the and whose coefficients are symmetric functions . Since each such coefficient is a constant, so are the roots. Thus, every is a constant. By looking at special values of *q*, e.g. one of the values above, in each case we can show that the constant is unity, and hence prove the identities (5.9)–(5.12). In the last two identities, one needs to take the limit as or , using (4.2), (4.9), (4.10) and (5.4)–(5.6).

(One can streamline the procedure: for instance, each of the 27 identities in the first set of relations (5.9) can be obtained from one of them by using the mappings , , , , so it is sufficient to do the last step for just one of the equations, say .)

## 6. Calculation of *T*(*p*,*q*)

We now look for solutions of (4.22), using the identities (5.8)–(5.11). This leads us to define, similarly to §6 of B (2006), the function(6.1)It satisfies(6.2)and is a natural extension of the elliptic function .

We further define the functionssuppressing their dependence on *p*. They have been constructed so that(6.3)for and all integers *m*. Because they are products over (or ), they, like , , , are unchanged by . They only have zeros or poles when , have zeros or poles, i.e. when or equals or . None of these zeros or poles occur when .

The factors , in (6.3) are irritating as they do not occur in (4.22). However, they are independent of *p*, so we may hope to remove them by introducing some additional simple factor that is also independent of *p*. Also, (4.22) is unchanged by multiplying by any function of *p* only.

We therefore try the ansatz(6.4)where are integers to be determined. We first seek to satisfy the relations (4.22), and find that we can match the powers of therein by taking(6.5)

At this stage we are free to choose , which corresponds to multiplying by a factor . This factor has no effect on the relations (4.22) and is bounded, with no zeros or poles, for . From the argument after (4.22), it must therefore be independent of *q*. Taking and , or and , we obtain the identity(6.6)for all *q*.

Without loss of generality we can therefore choose , giving , . Substituting the ansatz (6.4) into (4.23), the and functions cancel out, leaving(6.7)Since *p* and *q* are independent variables, must be a constant (independent of both *p* and *q*). We can absorb this constant into the factor in (6.4), so we can set(6.8)

We can calculate from (4.24). Take *i*=1 therein, so and and for . Let be the function defined above, but with *p* replaced by . From the above definitions and properties we can verify thatand(6.9)Using these, we find (after some work) from (5.12) and (6.4) that (4.24) is satisfied for *i*=1 iff(6.10)Taking *i*=2 or 3 in (4.24) merely permutes in the working, which leaves the *p*-independent result (6.10) unchanged.

We can use (4.10) to eliminate the factors in favour of . Using also (6.7), we obtain(6.11)

Writing as and using (4.10), we find that(6.12)

Our ansatz (6.4) is now(6.13)Using (6.3) and (6.12), we find that this expression does indeed satisfy the relations (4.22), so from the argument following (4.22) we see that the expression (6.13) must be correct to within a factor that is independent of *q*. It also satisfies (4.23), so this factor must be cube root of unity. Since (4.24) is satisfied, this root must be unity itself. This completes the proof of (6.13).

### (a) Relation to the order parameter function

In an earlier paper (B 2006), the author considered the function that occurs in the derivation of the order parameter of the chiral Potts model. This is given by(6.14) being defined by (2.8). This is very similar to the function of this paper, in fact if is the point (iii) of equation (5.13), with and , and is the point (iv), with , then(6.15)

For *N*=3, we can use our results (4.19) and (6.13) to express the RHS of (6.15) in terms of our elliptic-type functions. Because for both points (iii) and (iv), the factors cancel. The factors reinforce, giving(6.16)whereThis is indeed the result given in equations (57) and (62) of B (2006) (with *p* therein replaced by *q*). Because the functions have cancelled, it contains no elliptic-type functions with arguments , or .

## 7. Summary

For *N*=3, we have written the integral expression (2.7) as a product of generalized elliptic functions, with arguments that are ratios and products of the variables defined in (4.5) and (4.7). These are the variables of the alternative hyperelliptic parametrization of the chiral Potts model.

Is this progress? To the author it seems that the answer is indeed yes: functions such as the of (6.1) occur naturally in the free energy of other models, such as the Ising, six- and eight-vertex models, so it is interesting to see them occurring again in the three-state chiral Potts model. Indeed, it may indicate some intriguing relations between such models. Certainly, it provides an explicit formulation of the meromorphic structure of on its Riemann surface.

Unfortunately, it is not clear how one could proceed to *N*>3. There seems then to be no reason to expect to be able to express quantities, such as as products of single-argument Jacobi elliptic functions. The best one can hope for is to write them as ratios of hyperelliptic theta functions, as in B (1991*b*). These are entire functions of two or more related variables, all of whose zeros are simple. To write in such a parametrization, one would need to generalize the hyperelliptic theta functions to make the order of the zeros increase linearly with the distance of the zero from some origin. The author knows of no such generalization. Indeed, while Jacobi's triple product identity enables one to write Jacobi theta functions as either products or sums, there seems no reason to expect the same of functions, such as the numerators or denominators in (6.1).

There is also a problem with extending our working to the free energy of the full *N*=3 chiral Potts, which is given by the double integral in equation (46) of B (1991*a*) and equation (61) of B (2003*a*). The author showed in B (2003*b*) that the graph of the Riemann surface of this function has one more dimension than . If we fix *p* and consider the free energy as a function of *q*, then we do not have the relation (4.16) and we need a three-dimensional lattice to represent the Riemann surface, rather than the honeycomb lattice of figure 3. This implies the need for one more ‘hyperelliptic’ variable, in addition to and , and it is quite unclear what this may be. (No such extra variable is needed for the *N*=2 Ising case.)

## Footnotes

- Received February 6, 2006.
- Accepted April 10, 2006.

- © 2006 The Royal Society