Hyperelliptic parametrization of the generalized 6 0 0 order parameter of the N = 3 chiral Potts model 2 n a J R.J. Baxter 6 1 Mathematical Sciences Institute ] h c The Australian National University, Canberra, A.C.T. 0200, Australia e m - 6 January 2006 t a t s . t Abstract a m It has been known for some time that the Boltzmann weights of the chiral - d Potts model can be parametrized in terms of hyperelliptic functions, but as n yet no such parametrization has been applied to the partition and correlation o c functions. Here we show that for N = 3 the function S(tp) that occurs in the [ recent calculation of the order parameters can be expressed quite simply in 1 terms of such a parametrization. v 5 2 1 Introduction 3 1 0 There are a few two-dimensional models (and even fewer three-dimensional 6 models) in equilibrium statistical mechanics that have been solved exactly. 0 / These are lattice models where spins σ are assigned to the sites i of a lattice t i a (usuallythesquarelattice). EachspintakesoneofN possiblevaluesandspins m σ ,σ on adjacent sites i,j interact with a specified positive real Boltzmann i j - d weightfunctionW(σj,σj). Onewantstocalculate thepartitionfunction(also n called the sum-over-states) o c Z = W(σ ,σ ) , (1) : j j v <ij> i X Y X where the sum is over all states of all the spins, and the product is over all r a edges (i,j) of the lattice. If the number of sites is M, we expect the limit κ = lim Z1/M (2) M→∞ to exist and to independent of the shape of the lattice, provided it becomes large in all directions: this is the “thermodynamic limit”, and κ is the expo- nential of the free energy per site. If 1,...,m are sites fixed on the lattice 1 and the limit is taken so they become infinitely deep within it, then we also expect the average hf(σ ,...,σ )i = Z−1 f(σ ,...,σ )W(σ ,σ ) (3) 1 m 1 m j j <ij> X Y to tend to a limit, for any given function f of these m spins. Because spins only interact with their neighbours, one can build up the lattice one row at a time, and associate a row-to-row “transfer matrix” with such an operation. To solve such a model, typically one shows that the Boltzmann weights W satisfy the star-triangle or “Yang-Baxter” relations [4]. These ensure certain commutation relations between the transfer matrices, and this is usually a first step towards calculating κ. The next step is to calculate the order parameters, which are averages of certain functions of a single spin σ deep within the lattice. This is a harder 1 problem than calculating κ. For instance, Onsager [20] calculated κ for the square-lattice Ising model in 1944 , butit was not till 1949 that heannounced at a conference his result for the order parameter (namely the spontaneous magnetization), and not till 1952 before a proof of the result was published by Yang [21] . However, since then the “corner transfer matrix” method has been devel- oped by Baxter [3], and the “broken rapidity line method” by Jimbo, Miwa and Nakayashiki [19]. For many of the solved models (those with the “ra- pidity difference” property), these methods make the calculation of the order parameters comparitively straighforward. Evenso, onemodelhasprovedchallenging, namelythechiralPotts model. This is an N-state modelwhereW(σ ,σ ) dependsonly on thespin difference j j σ −σ , mod N. The Boltzmann weights also depend on two parameters p,q, j j ′ (known as “rapidities”), and on given positive real constants k,k , related by k2+k′2 = 1 . (4) ′ The parameter k plays the role of a temperature, being small at low temper- ′ atures. For 0 < k < 1 the system displays spontaneous ferromagnetic order, ′ becoming critical as k → 1. Its order parameters can be taken to be M = hωrσ1i , (5) r where ω = exp(2πı/N) and r = 1,...,N −1. It was shown in 1988 that its Boltzmann weights satisfy the star-triangle relation [5, 2], and the partition function per site κ was soon calculated [7, 8]. The order parameters were another story. The model had developed from a one-dimensional quantum spin chain, which has the same order parameters. From series expansions it was conjectured [1] in 1989 that M = kr(N−r)/N2 . (6) r Much effort was expended in the ensuing years (certainly by the author) in attempting to derive this result. It was not until 2005 that this was done [17, 16]. 2 ∧∧∧. ∧∧∧. ∧∧∧. ∧∧∧. ∧∧∧. ∧∧∧. . . . . . . .. (cid:0)v@ .. .. (cid:0)v@ .. .. (cid:0)v@ .. .. (cid:0) @ .. .. (cid:0) @ .. .. (cid:0) @ .. ......(cid:0)..........@..................(cid:0)..........@..................(cid:0)....<<.<.....@............. (cid:0)(cid:0).... ....@@ (cid:0)(cid:0).... ....@@ (cid:0)(cid:0).... ....@@ h (cid:0) .. .. @(cid:0) .. .. @(cid:0) .. .. @ v@ .. .. (cid:0)v@ .. .. (cid:0)v@ .. .. (cid:0)v @ .. .. (cid:0) @ .. .. (cid:0) @ .. .. (cid:0) ..@..................(cid:0)..........@..................(cid:0)..........@........<<.<.........(cid:0)......... ....@@ (cid:0)(cid:0).... ....@@ (cid:0)(cid:0).... ....@@ (cid:0)(cid:0).... h .. @(cid:0) .. ..σ @ (cid:0) .. .. @(cid:0) .. ... (cid:0)v@ ... ... 1(cid:0)f@ ... ... (cid:0)v@ ... . (cid:0) @ . . (cid:0) @ . . 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In Figure 1 we show the square lattice L, drawn diagonally, denoting the sites by circles and the edges by solid lines. We also show as dotted (or broken) lines the medial graph of L. Every edge of L is intersected by two dotted lines. With each dotted line we associate a rapidity variable (p, q, h or v). In general these variables may differ from dotted line to dotted line. They must be the same all along the line, except for the horizontal broken line immediately below the central spin σ . We break this below σ and assign a rapidity p to the left of 1 1 the break, a rapidity q to the right. With these choices of rapidities, define F˜ (r) = hωrσ1i . (7) pq In the thermodynamic limit, the star-triangle relations will ensure that F˜ (r) is independent of the “ background” rapidities v,h, because it allows pq us to move any of these dotted lines off to infinity.[6] However, the effect of the break is that we cannot move the broken line p,q away from σ , so F˜ (r) 1 pq will indeed depend on p and q. An important special case is when q = p. Then the p,q rapidity line is not infactbroken,soitcanberemovedtoinfinityandF˜ (r)mustbeindependent pp of p and equal to the order parameter M defined by (5): r M = F˜ (r) . (8) r pp 3 We also define G (r) = F˜ (r)/F˜ (r−1) . (9) pq pq pq The author wrote down [12] functional relations satisfied by G (r) in pq 1998. They do not completely specify G (r), but must be supplemented pq by information on the analyticity properties of G (r). (Just as the relation pq f(z+1) = f(z) only tells us that f(z) is periodic of period 1: however, if we can also show that f(z) is analytic and bounded in the domain 0 ≤ ℜ(z) < 1, then it follows from Liouville’s theorem that f(z) is a constant.) For N = 2 the chiral Potts model reduces to the Ising model and it is quite easy to find the needed analyticity information, to solve the functional relations and obtain the Onsager-Yang result M = k1/4. 1 For N > 2 the problem is much harder. It was not until late 2004 that the authorrealisedthatitisnotactuallynecessarytosolveforthegeneralfunction G (r). It is sufficient to do so for a special “superintegrable” case where q pq is related to p. The function then has quite simple analyticity properties and it quite easy to solve the relations (in fact one does not even need all the relations), to obtain G (r) for this case and to verify the 16–year old pq conjecture (6). For r = 1,...,N, the functions G (r) can all be expressed in pq terms of a single function S(t ) which is defined below. p Even so, it would still be interesting to understand G (r) more gener- pq ally. A fundamental difficulty is that for N > 2 the rapidities p and q are points on an algebraic curve of genus greater than 2, and there is no explicit parametrization of this curve in terms of single-valued functions of a single variable. (There is for N = 2: one can then parametrize in terms of Jacobi ellipticfunctions.) Onecanparametrizeintermsofhyperellipticfunctions[9], but these have N −1 arguments that are related to one another. As yet they have not proved particularly useful, but one lives in hope. The function S(t ) p is a simple example of a thermodynamic property of the chiral Potts model, and has the simplifying feature that it depends on only one rapidity, rather than two. It is an interesting question whether it can be simply expressed in terms of these hyperelliptic functions. For N = 3 these hyperelliptic functions can be expressed in terms of ordinary Jacobi elliptic functions. One still has two related arguments (here termed z and w ), but some of the properties can be expressed as products p p of Jacobi functions, each with an argument z or w , or some combination p p thereof. A number of such results have been obtained.[10], [11, pp. 568, 569] There are two distinct ways of performing the hyperelliptic parametriza- tion. In [9, 18] we used what we shall herein call the “original” parametriza- tion. What we report here is that for N = 3 the function S(t ) can be p expressed quite simply as a productof Jacobi functions of z and w , provided p p we use the second “alternative” parametrization. 2 The function S(t ) p We can take a rapidity p to be a set of variables p = {x ,y ,µ ,t } related to p p p p one another by t = x y , xN +yN = k(1+xNyN) , p p p p p p p 4 (10) kxN = 1−k′/µN , kyN = 1−k′µN . p p p p Thereare various automorphisms or maps thattake one set {x ,y ,µ ,t } p p p p to another set satisfying the same relations (10). Four that we shall use are: R :{x ,y ,µ ,t } = {y ,ωx ,1/µ ,ωt } , Rp Rp Rp Rp p p p p S :{x ,y ,µ ,t } = {y−1,x−1,ω−1/2y /(x µ ),t−1} , Sp Sp Sp Sp p p p p p p V :{x ,y ,µ ,t } = {x ,ωy ,µ ,ωt } , (11) Vp Vp Vp Vp p p p p M :{x ,y ,µ ,t } = {x ,y ,ωµ ,ωt } . Mp Mp Mp Mp p p p p They satisfy RV−1R = V , MRM = R , MSM = S , S2 = VN = MN = 1 . (12) Let q be another rapidity set, related to p by q = Vp, i.e. x = x , y = ωy , µ =µ . (13) q p q p q p We take µ to be outside the unit circle, so p |µ |> 1 . (14) p Then we can specify x uniquely by requiring that p −π/(2N) < arg(x ) < π/(2N) . (15) p We regard x ,y ,µN as functions of t . Then t lies in a complex plane p p p p p containingN branchcutsB0,B1,...,BN−1 onthelinesarg(tp) = 0,2π/N,..., 2π(N −1)/N, as indicated in Fig. 2, while x lies in a near-circular region p roundthepointx = 1, asindicated schematically bytheregion R insidethe p 0 dotted curve of Fig. 2. The variable y can lie anywhere in the complex plane p except in R0 and in N −1 corresponding near-circular regions R1,...,RN−1 round the other branch cuts. With these choices, we say that p lies in the “domain” D. With these choices, we show in [17] that G (r) = k(N+1−2r)/N2 S(t ) , (16) pq p for r = 1,...,N −1, while G (0) = G (N) = k(1−N)/N2 S(t )1−N . (17) pq pq p Hence G (1)···G (N) = 1, in agreement with the definition (9). The func- pq pq tion S =S(t ) is given by p p 2 1 2π k′eıθ logS(t ) = − logk+ log[∆(θ)−t ]dθ , (18) p N2 2Nπ 1−k′eıθ p Z0 where ∆(θ) = [(1−2k′cosθ+k′2)/k2]1/N . (19) 5 B 1 TTT ω TTT TTTx TTT TTT ................ . . . . . R . . 0 . . . . . B . x . 0 . . . . . 1 . . . ................ (cid:20)(cid:20)(cid:20) (cid:20)(cid:20)(cid:20) (cid:20)(cid:20)(cid:20)xω−1 (cid:20)(cid:20)(cid:20) (cid:20)(cid:20)(cid:20) B 2 Figure 2: The cut t -plane for N = 3, showing the three branch cuts B ,B ,B and p 0 1 2 the approximately circular region R in which x lies when p ∈ D. 0 p From [16], particular properties are S(0) = 1 , S(∞) = k−2/N2 , S(t )S(ωt )···S(ωN−1t ) = k−1/N x . (20) p p p p The function S(t ) is single-valued, non-zero and analytic in the cut t plane p p of Figure 2, but only the cut on the positive real axis is necessary: the other cuts can be removed for this function. If S (t ) is the analytic continuation ac p of S(t ) across the branch cut B , then p r S (t ) = S(t ) for r 6= 0 ac p p = (y /x )S(t ) for r = 0 . (21) p p p If we interchange p,q in eqn. 49 of [17], then apply the restriction (13) and use the relation RS = MVRSV together with eqn. 53 of [17], we obtain Gpq(r)Gp′,q′(N −r)= 1 , (22) where p′ = V−1q′ = RSVp. It follows that S(t ) also has the symmetry p S S = S(t )S(1/t ) = k−2/N2 . (23) p RSVp p p 6 3 The Riemann sheets (“domains”) formed by analytic continuation We shall want to consider the analytic continuation of certain functions of t p ontootherRiemannsheets,i.e. beyondthedomainD. Werestrictattentionto functionsthataremeromorphicandsingle-valued inthecutplaneofFigure2, andsimilarlyfortheiranalyticcontinuations. Obviousexamplesarex ,y and p p S(t ). They are therefore meromorphic and single-valued on their Riemann p surfaces, but we need to know what these surfaces are. Westartbyconsideringthemostgeneralsuchsurface. Asafirststep,allow µ to move from outside the unit circle to inside. Then t will cross one of p p the N branch cuts B in Figure 2, moving onto another Riemann sheet, going i back to its original value but now with y in R . Since y is thereby confined p i p to the region near and surrounding ωi, we say that y ≃ ωi. Conversely, by p y ≃ ωi we mean that y ∈R p p i We say that p has moved into the domain D adjacent to D. There are N i such domains D0,D1, ...,DN−1. Now allow µ to become larger than one, so t again crosses one of the p p N branch cuts. Again we require that t returns to its original value. If it p crosses B , then it moves back to the original domain D. However, if it crosses i another cut Bj then xp moves into Rj−i, and we say that p is now in domain Di,j−i. Proceedinginthisway, webuildupaCayleytreeofdomains. Forinstance, the domain D is a third neighbour of D, linked via the first neighbour D ijk i and the second-neighbour D , as indicated in Figure 3. Here x ≃ 1 in D, ij p y ≃ ωi in D , x ≃ ωj in D and y ≃ ωk in D . We reject moves that p i p ij p ijk take p back to the domain immediately before the last, so j 6= 0 and k 6= i. We refer to the sequence {i,j,k,...} that define any domain as a route. We can think of it as a sequence of points, all with the same value of t , on the p successive Reimann sheets or domains. ThedomainsD,D ,D ,... withanevennumberofindices,havex ≃ωℓ, ij ijkℓ where ℓ is the last index. We refer to them as being of even parity and of type ℓ. The domains D , D ,... have y ≃ ωℓ and are of odd parity and type ℓ. i ijk D D D D i ij ijk Figure 3: A sequence of adjacent domains D,D ,D ,D . i ij ijk The automorphism that takes a point p in D to a point in D , respectively, i is the mapping A = Vi−1RV−i . (24) i 7 If q = A p, then i x =ω−iy , y = ωix , t = t . (25) q p q p q p Because of (12), A = A , so there are N such automorphisms. i+N i We can use these maps to generate all the sheets in the full Cayley tree. Supposewe have a domain with route {i,j,k,...} and we apply the automor- phism A to all points on the route. From (25) this will generate a new route α {α,i−α,j+α,k−α,...}. For instance, if we apply the map A to the route α {m} from D to Dm, we obtain the route {α,m−α} to the domain Dα,m−α. Thus the map that takes D to D is A A . ij i i+j Iterating, we find that the map that takes D to D is ijk...mn A A A ···A . (26) i i+j j+k m+n We must have A2 = 1 , (27) i 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 G. 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 willhave a Riemann surfacethat can beobtained from G by identifying certain sites with one another, thereby creating circuits and usually reducing the graph to one of finite dimensionality. From (25), the maps A0,A1,...,AN−1 leave tp unchanged. We shall often findithelpfultoregardt asafixedcomplexnumber,thesameinalldomains, p and to consider the corresponding values of x ,y (and the hyperelliptic vari- p p ables z ,w ) in the various domains. To within factors of ω, the variables x p p p and y will be the same as those for D in even domains, while they will be p interchanged on odd domains. Analytic continuation of S(t ) p Now return to considering the function S(t ). It is sometimes helpful to write p this more explicitly as S(x ,y ). Then from (21) the map that takes S(t ) p p p from domain D to D is i q = A p : S(x ,y ) = (y /x )−δiS(x ,y ) , (28) i q q q q p p where x ,y are given by (25) and δ = 1 if i = 0, mod N; otherwise δ = 0. q q i i Note that x ,y are obtained by interchanging x ,y and multiplying them q q p p by powers of ω. For given t , letS (t )bethevalueof S(t )in thecentral domain D,given p 0 p p by the formula (18). Iterating the mappings (28) from domain to domain, in any domain we must have S(x ,y ) = ωα(y /x )rS (t ) , (29) p p p p 0 p 8 where α, r are integers. Note that in this equation x ,y are the values for p p the domain being considered: they are not the corresponding initial values of the central domain D. In particular, in the domain D we obtain ijk r = −δ +δ −δ . (30) i i+j j+k 4 The original hyperelliptic parametriza- tion for N = 3 Hereinafter we restrict our attention to the case N = 3 and use the hyperel- liptic parametrization and notation of previous papers.[9, 10, 11, 13] We use only formulae that involve ordinary Jacobi elliptic (or similar) functions of one variable. ′ Given k,k , we define a “nome” x by ∞ 12 1−x3n (k′/k)2 = 27x . (31) 1−xn ! n=1 Y We regard x as a given constant, not the same as the rapidity variable x . It p ′ is small at low temperatures (k small), and increases to unity at criticality ′ (k = 1). We introduce two elliptic-type functions ∞ (1−ωxn−1z)(1−ω2xn/z) h(z) = ω2h(xz) = , (32) (1−ω2xn−1z)(1−ωxn/z) n=1 Y ∞ (1−x3n−2/z)(1−x3n−1z) φ(z) = z1/3 . (33) (1−x3n−2z)(1−x3n−1/z) n=1 Y We then define two further variables z ,w by p p t = x y = ωh(z ) = h(−1/w ) =ω2h(−w /z ) , (34) p p p p p p p Thesearetherelations (27)of[10]. Therelations (32)of[10]arealsosatisfied: x−3y3µ−6 = φ(xz /w2)3 = φ(−xz w )3 = φ(−xw /z2)3 , (35) p p p p p p p p p as are the relations (4.5), (4.6) of [11], in particular ∞ (1−x2n−1z/w)(1−x2n−1w/z)(1−x6n−5zw)(1−x6n−1z−1w−1) w = (1−x2n−2z/w)(1−x2nw/z)(1−x6n−2zw)(1−x6n−4z−1w−1) n=1 Y (36) writing z ,w here simply as z,w. p p The z ,w variables satisfy the automorphisms p p z = xz , z = 1/(xz ) , z = −1/w , Rp p Sp p Vp p w = z /w , w = 1/(xw ) , w = z /w . (37) Rp p p Sp p Vp p p 9 The operation p → Mp multiplies (z w )1/3 by ω, but does not change z ,w p p p p themselves. ′ The variables z ,w are of order unity when k ,x are small, µ is of order p p p ′ 1/k , and x ≃ 1. This is the low-temperature limiting case of p ∈ D. It is p convenient to define u = {z ,−1/w ,−w /z } . (38) p p p p p The three automorphisms that leave t unchanged, while taking D to p D ,D ,D , respectively, are 0 1 2 A = V2R , A = RV2 , A = VRV . (39) 0 1 2 If q = A p, then i x = ω−iy , y = ωix , t = t , (40) q p q p q p and u = A u , where A ,A ,A are the three-by-three matrices q i p 0 1 1 0 x−1 0 0 0 x 1 0 0 A0 =  x 0 0  , A1 =  0 1 0  , A2 =  0 0 x−1         0 0 1   x−1 0 0   0 x 0                    They satisfy the identities A A A = A A A (41) i j i j i j for all i,j. They permute the three elements z ,−1/w ,−w /z of u and multiply p p p p p them by powers of x, the product of the elements remaining unity. Let z0,w0 p p be the values of z ,w on the central sheet D. Then it follows that on any p p sheet, for the same common value of t , p z = xmα , w = xnβ , (42) p p p p where {α ,−1/β ,−β /α } is a permutation of {z0,−1/w0,−w0/z0}. p p p p p p p p Repeated applications of the three automorphisms will therefore generate a two-dimensional set of permutations and multiplications of the elements of u . Each member of the set corresponds to a site on the honeycomb lattice of p Figure4. Adjacent Riemannsheets correspondto adjacent sites of thelattice. Sheets of even parity correspond to sites represented by circles, those of odd parity arerepresented by squares. if i is theinteger insidethecircle or square, then for even sites x ≃ ω−i, while on odd sites y ≃ ωi. The numbers shown p p in brackets alongside each site are the integers m,n of (42). Thusfor thefunctionsz andw of t , thegraphG of theRiemann surface p p p reduces to this two-dimensional honeycomb lattice. Note that the sites X,Y,Z in the figureare third neighbours of the central site D, and each can be reached from D in two three-step ways. For instance, Y is both D and D .1 021 211 1 Note that for D we here take the intermediate site j to be represented Figure 4 by the ijk integer −j, mod 3. This is changed in the next section to +j. 10