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THE N →∞LIMIT OF THE CHIRAL POTTS MODEL

arXiv:hep-th/9305171v2 25 Jun 2006hep-th/9305171OSU-SM-93-01THE N →∞LIMIT OF THE CHIRAL POTTS MODELHELEN AU-YANG and JACQUES H.H. PERK∗Department of Physics, Oklahoma State UniversityStillwater, OK 74078-0444, USA†ABSTRACTWe consider the N →∞limits of the N-state chiral Potts model.

We find newweights that satisfy the star-triangle relations with spin variables either taking allthe integer values or having values from a continous interval. The models providechiral generalizations of Zamolodchikov’s Fishnet Model.1.

IntroductionIn the integrable N-state chiral Potts model1 the Boltzmann weights W(a−b)and W(a −b) — corresponding to the horizontal and vertical pair interactionsbetween spins in states a and b — satisfy the “star-triangle” equation2,3NXd=1W qr(b−d) Wpr(a−d) W pq(d−c) = Rpqr Wpq(a−b) W pr(b−c) Wqr(a−c). (1)The resulting weights have a product form.2,3 Here we rewrite4 these results asWpq(n)Wpq(0) =sin θp sin φqsin θq sin φpn/2NnYj=1sin[π(j −12)/N −(θq −φp)/2N]sin[π(j −12)/N + (φq −θp)/2N],(2)W pq(n)W pq(0) = sin θp sin θqsin φp sin φqn/2NnYj=1sin[π(j −1)/N + (φq −φp)/2N]sin[πj/N −(θq −θp)/2N],(3)using the substitutionsbpcp= ω12eiθp/N,apdp= eiφp/N,ω = e2πi/N,cpdp=eiφp sin φpeiθp sin θp1/2N,(4)where θp and φp differ by a factor N from those of Baxter.4 The last equality in(4) follows from the integrability conditions2,3 ork = sin 12(θp −φp)sin 12(θp + φp) = sin 12(θq −φq)sin 12(θq + φq).

(5)∗Supported in part by NSF Grants Nos. DMS 91-06521 and INT 91-12563†Work done at RIMS, Kyoto University, Sakyo-ku, Kyoto 606, Japan

2Helen Au-Yang and Jacques H.H. Perk——The N→∞Limit of the Chiral Potts ModelThe Fourier transforms, which are the weights after duality transformation,2,3W (f)pq (n) = N −1N−1Xj=0ω−jn Wpq(j),W (f)pq (n)) = N −1N−1Xj=0ω−jn W pq(j),(6)can be rewritten asW (f)pq (n)W (f)pq (0)= ein(θq−φq+θp−φp)/2NnYj=1sin[π(j −1)/N + (¯θq −¯θp)/2N]sin[πj/N −(¯φq −¯φp)/2N],(7)W (f)pq (n)W (f)pq (0)= ein(θq−φq−θp+φp)/2NnYj=1sin[π(j −12)/N −(¯θq −¯φp)/2N]sin[π(j −12)/N + (¯φq −¯θp)/2N],(8)where¯φp = 12(θp + φp) + 12i log sin φpsin θp,¯θp = 12(θp + φp) −12i log sin φpsin θp.

(9)By direct substitution we can show that if the weights satisfy the star-triangleequation (1) then their Fourier transforms satisfy the star-triangle equationNRpqrW (f)qr (a) W (f)pr (b) W (f)pq (a+b) =N−1Xd=0W (f)pq (b−d) W (f)pr (a+b−d) W (f)qr (d). (10)This equation has the exact same form as equation (1), as can be seen replacinga →a −b, b →b −c, a + b →a −c, and c + d →d.For θp = φp, θq = φq we recover the self-dual Fateev and Zamolodchikov5solution with W (f)(n)/W (f)(0) = W(n)/W(0), W (f)(n)/W (f)(0) = W(n)/W(0).2.

The N→∞Limit of the Boltzmann WeightsWe shall now obtain the N →∞limit of the Boltzmann weights (2) and (3)or their dual weights (7) and (8). Note that these all have the product formW(n)W(0) = An/NnYj=1sinπ(j + α −1)/Nsinπ(j + β −1)/N,A = sin(πβ)sin(πα),(11)with α and β given constants depending on parameters θp, θq, φp, and φq satisfying(5).

Also, the condition on A guarantees that W(n + N) = W(n). We can rewriteformula (11) in terms of a convergent series in powers of 1/N, i.e.logW(n)W(0) = logAn/NnYj=1j + α −1j + β −1+∞Xl=0Bl+1(α) −Bl+1(β)(l + 1)!2πNl ddzllogsin zzz=πn/N,(12)

Helen Au-Yang and Jacques H.H. Perk——The N→∞Limit of the Chiral Potts Model3where 0 < n < N and we used the functional relations of Bernoulli polynomials6Ppm=0 pmBm(x)yp−m = Bp(x + y) and Bp(x + 1) −Bp(x) = pxp−1.There are three regimes for the limit, the first having N →∞, while nremains finite.

In this case, (12) results inI.W(n)W(0) =nYj=1j + α −1j + β −1 =−nYj=1j −βj −α = Γ(n + α)Γ(β)Γ(n + β)Γ(α),−∞< n < ∞. (13)The second regime has N, n →∞such that 2πn/N →x remains finite.Consequently, the weights W(n) in (11), which originally took N different valuesand which were periodic modulo N, now depend on the continuous spin values xand they are periodic modulo 2π.

Using the asymptotic formula7 for log Γ(x), andB1(x) = x −12 we find that (12) in this limit givesII.W(x) = C Ax2π − x2π sin 12xα−β,(14)where [x] stands for integral part of x andC = W(0) (N/π)α−β Γ(β) /Γ(α). (15)The results in Regimes I and II are consistent with duality transformation.

Moreprecisely, if the limiting weights are in Regime II, their Fourier transforms are inRegime I, and vice versa. Indeed, the infinite sum in the Fourier transform can besummed using the formula8∞Xn=−∞einxnYj=1j + α −1j + β −1 = 2β−α−1 Γ(1 −α)Γ(β)Γ(β −α)e12i(1−α−β)(x−π) | sin 12x|β−α−1,(16)for 0 < x < 2π (and periodically extended) and with α and β in (13) calculatedfrom (2) or (3), to give an identical formula for the weights as in (14) for Regime II,with new values of α and β corresponding to (7)-(9).

This generalizes the large-Nlimit of Fateev and Zamolodchikov.5A third intermediate regime can also appear with N, n →∞such thatn/f(N) →x for some function f(N) that blows up slower than N. We haveIII.W(x) = D A−12sign(x) |x|α−β,−∞< x < ∞,(17)which is a chiral generalization of Zamolodchikov’s Fishnet Model.10 In (17), D =C A12, with C given by (16). The sign function in (17) emerges if we rewrite (11) fornegative n in the form (12), see also (13), and compare constants in (17) for x > 0and x < 0; then we can use Γ(1 −α)/Γ(1 −β) = A Γ(β)/Γ(α), which follows fromΓ(x)Γ(1−x) = π/ sin(πx), leading to an extra factor A for x < 0.

The sign functionrelates directly to the effect of the integer part in (14) near x = 0; in fact, by theidentical reasoning for Regime II, we need the same extra factor A for −2π < x < 0in (14).We note that in the previously known cases5,10 α + β = 1 and A = 1. Wehave only one condition (11) on A allowing the deformations (13), (14), and (17)to provide integrable field theories with chirality.

4Helen Au-Yang and Jacques H.H. Perk——The N→∞Limit of the Chiral Potts Model3.

The N→∞Limit of the Chiral Potts ModelHaving obtained explicit prescriptions on how to take the N →∞limit ofthe Boltzmann weights of the N-state chiral Potts model, we are now in a positionto examine the various limits of the star-triangle equation (1). The summation overd must be split in several pieces as we must choose from which regimes to takethe three weights in the summand.

As N →∞, we can rigorously show that mostpieces can be ignored.The least complicated (but also a most interesting) case occurs when all theβ −α in the formulae of section 2 are between 0 and 1, defining a principal domainfor the spectral parameters. In this case, the three types of large-N behavior donot mix.

More precisely, if we take the three spin states a, b, and c in (1) mutuallyseparated according to one and the same regime in the sense of the previous section— so that we are taking the same type of limit for all three weights in the right-handside of (1) — the dominant part of the sum over spin state d comes from the piecewith all three weights in the left-hand side of (1) from the same regime. This iseasily proved as the formulae in section 2 give full control over the leading term andthe corrections in the large-N limit.For example, taking Regime II, the star-triangle equations becomeZ 2π0dw W qr(y −w) Wpr(x −w) W pq(w −z) = R Wpq(x −y) W pr(y −z) Wqr(x −z),(18)after a suitable renormalization of R. This has the solutionWpq(x) = Cpq e(γp−γq)( x2π − x2π) sin 12xλp−λq,W pq(x) = Cpq e(γp+γq)( x2π − x2π) sin 12xλq−λp−1,(19)withλp = 1π arctansin θpcos θp + k,γp = 12 log1 + 2k cos θp + k21 −k2,(20)and similar formulae for λq, γq, λr, and γr, as follows from the prescriptions givenin the previous two sections.

If λp < λq < λr < 1 + λp all six Boltzmann weights in(18) are real and positive and the parameters are in the principal domain.We remark that it is straightforward to show that the weights obtained byjust dropping the integral part [x/2π] in (19) also satisfy the same star-triangleequation (18). This solution can be viewed as the Fateev-Zamolodchikov solution5with a site-dependent gauge transformation.We emphasize that we have bothnumerically verified and analytically proved that the “chiral” weights (19) alsosatisfy (18).Since Wpq(x) and W pq(x) are now functions of x modulo 2π, their FouriertransformsW (f)pq (j) = 12πZ 2π0dx e−ijx Wpq(x),W (f)pq (j) = 12πZ 2π0dx e−ijx W pq(x), (21)

Helen Au-Yang and Jacques H.H. Perk——The N→∞Limit of the Chiral Potts Model5are over all the integer value j, ranging from −∞to ∞.

Substituting (21) into (18),we find that the Fourier transforms satisfy different star-triangle equations,2πR W (f)qr (a) W (f)pr (b) W (f)pq (a+b) =∞Xd=−∞W (f)pq (b−d) W (f)pr (a+b−d) W (f)qr (d), (22)in which the sum is over all integer values of d.Taking Regime I instead of Regime II, leads to the proof of a star-triangleequation of the form (21) instead of (18). This is an interesting identity in the theoryof generalized hypergeometric series generalizing the Dougall-Ramanujan identity.9Choosing Regime III, we can prove a star-triangle equation of the form (18),but with integration over (−∞, +∞).

This generalizes Symanzik’s integral,11 whichZamolodchikov used to prove the star-triangle equation for the Fishnet Model10and which further provided the proof for the Fateev-Zamolodchikov model5 via aconformal transformation, ξ = tan 12w.Finally, as in our joint work with Baxter,2 we can let the R-matrix be theproduct of four weights of any of the above types I, II, or III, i.e.R(a, b, c, d) = W p1q1(a −c) Wp1q2(c −b) W p2q2(d −b) Wp2q1(a −d). (23)Then any such infinite-dimensional R-matrix satisfies the Yang-Baxter equation.But our solutions are very different from those of Gaudin,12 Shibukawa and Ueno.13References1.

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A.Erd´elyi et al., (McGraw-Hill, New York, 1953), eqs. 1.13 (2), (6).7.

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1.18 (1).8. Ref.

6, eqs. 2.1.2 (6), 2.9 (1), (13), (22), (27).9.

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97B (1980) 63.11. K. Symanzik, Lett.

Nuovo Cim. 3 (1972) 734.12.

M. Gaudin, J. Phys. France 49 (1988) 1857.13.

Y. Shibukawa and K. Ueno, Lett. Math.

Phys. 25 (1992) 239; Waseda Universitypreprint.

6Helen Au-Yang and Jacques H.H. Perk——The N→∞Limit of the Chiral Potts ModelErrataFormula (12) has the higher orders misprinted and is only correct to theorder needed in the actual N →∞limits presented in this work.

The result correctin higher order can be found in (3.15) with (3.13) of math.QA/9906029 which hasbeen published as Physica A 268 (1999) 175–206. That work goes also into far moredetail in the subject matter.

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