Spontaneous magnetization of the 0 superintegrable chiral Potts model: 1 0 calculation of the determinant D 2 PQ n a J R.J. Baxter 1 1 Mathematical Sciences Institute, The Australian National ] h University,Canberra, A.C.T. 0200, Australia, e-mail: none c e m 21 December2009 revised 11 January 2010 - t a t s Abstract . t a For the Ising model, thecalculation of thespontaneous magnetization leads to the m problemofevaluatingadeterminant. Yangdidthisbycalculatingtheeigenvaluesinthe - large-lattice limit. Montroll, Potts and Ward expressed it as a Toeplitz determinant d and used Szeg˝o’s theorem: this is almost certainly the route originally travelled by n Onsager. For the corresponding problem in the superintegrable chiral Potts model, o neither approach appears to work: here we show that the determinant D can be c PQ [ expressed as that of a product of two Cauchy-like matrices. One can then use the elementaryexactformulafortheCauchydeterminant. Oneofcourseregainstheknown 2 result, originally conjectured in 1989. v 9 4 KEY WORDS: Statistical mechanics, lattice models, transfer matrices. 5 4 1 Introduction . 2 1 ExtrapolatingfromtheIsingcaseandfromseriesexpansions,Albertinietalconjectured 9 in 1989 [1] that the order parameter or spontaneous magnetization of the solvable N- 0 state chiral Potts model is : v Xi Mr = (1−k′2)r(N−r)/2N2. (1.1) r Here r is an integer, between 0 and N, and k′ is a parameter that is “universal” a in the sense that it is the same for all rows and columns of the lattice, even for an inhomogeneous model where the rapidities vary from row to row and from column to column.[2] Here we consider the ferromagnetically ordered phase of the system, where 0Q and m′ =m+1 (2.4) i = cot(θ /2) if P >Q and m′ =m. i These equations cover all cases. The matrix E(Q,P) is defined similarly, with P,Q interchanged, m,m′ also interchanged, and θ replaced byθ′. i i B is an m by m′ Cauchy-likematrix with elements PQ f f′ (B ) = i j . (2.5) PQ ij c −c′ i j Given c1,...,cm, c′1,...,c′m′ with |m−m′| ≤ 1, there is a unique way of choosing f1,...,fm, f1′,...,fm′ ′ so that BPQ satisfies the orthogonality condition (1.19). The working is given in section 6 of II. We remark in section 4 of III that it is true for arbitrary c ,c′. Let i i ′ m m a = (c −c′), a′ = (c′−c ), (2.6) i i j i i j Yj=1 Yj=1 ′ m m b = (c −c ), b′ = (c′−c′), i i j i i j j=Y1,j6=i j=Y1,j6=i then theresults II.6.8, II.6.13, II.6.16 can bewritten as f2 = ǫa /b , f′2 = −ǫa′/b′, (2.7) i i i i i i where ǫ=±1is independentof i. (Fortheparticularvaluesofc ,c′ givenby(1.21)and(1.22),weobservenumerically i i that f2 and f′2 are positive real if we choose ǫ=1 if P Q.) i i A quantitythat we shall need is (c −c ) (c′ −c′) ∆m,m′(c,c′) = Q1≤iQ, when m≤m′. Thenfrom(1.19)wecanwriteI in(1.18)asB BT . RememberingthatB = m PQ PQ QP −BT , we can then write (1.18) as PQ D (α,β) = det[UBT ], (4.3) PQ PQ where U = B +X (α)E B X (β)E . (4.4) PQ P PQ PQ Q QP Define y =[X (α)] e(P,Q,i), y′ =[X (β)] e(Q,P,j), (4.5) i P i,i j Q j,j then from (2.5) theelements of U are f f′(1+y y′) U = i j i j . (4.6) ij c −c′ i j In general we do not know how to calculate the determinant of such a matrix. However,ifwetakethelimitsα,β→+∞andexpressc ,c′,y ,y′ asrationalfunctions i j i j of λ ,λ′, we findthat a factor λ +λ′ cancels out of theRHS of (4.6). If m′=m,the i j i j result is Cauchy-likematrix, and one can calculate the determinant of U. Hereinafter we take the limitα,β→+∞,so y =x e(P,Q,i), y′ =x′ e(Q,P,j), (4.7) i i j j where xi,x′j aregiven by(3.4). Wewrite DPQ(∞,∞) simply as DPQ. TheintegerL is still finite. The case P > Q, m = m′ The simplest case is when m = m′ and all matrices are square. so from (1.19) and (4.3), D = detU/detB . (4.8) PQ PQ Cauchy-like matrices If A is the m by m matrix with entries 1 A = , (4.9) ij c −c′ i j then it is a Cauchy matrix and its determinant is ∆ (c,c′), using the definition m,m (2.8).[29,eq. 2.7]Anymatrixwithelementsoftheform(2.5)issaidtobeCauchy-like, and has determinant m detB = ∆ (c,c′) f f′ (4.10) PQ m,m i i i=1 Y for all f ,f′. We have in fact chosen the f ,f′ so that B is orthogonal, so has i i i i PQ determinant±1. However,theform(4.10)isconvenienthereasthef ,f′ productswill i i cancel out of (4.8). 8 The determinant of U From (4.8), we still have to calculate the determinant of U. Its elements are given by (4.6), so U is not in general Cauchy-like. However, in the limit α,β→∞ we findthat acommonfactorcancelsfromthenumeratoranddenominatorof(4.6),andU becomes Cauchy-like. Wecan then evaluate its determinant by parallelling (4.9) - (4.10). The case P >Q,m=m′ is the fourth one listed in (2.4), so (E ) =cot(θ /2)δ , (E ) =tan(θ′/2)δ . (4.11) PQ ij i ij QP ij i ij From (2.1), takingthe limit α→+∞, −2k′sinθ xi = [XP(∞)]ii = (1+λ )2−ki′2 . (4.12) i Noting that sinθ cot(θ /2)=1+c =[(1+k′)2−λ 2]/2k′, it follows from (4.7) that i i i i 1+k′−λ y = − i . (4.13) i 1−k′+λ i (A factor 1+k′+λ has cancelled.) i The calculation of y′ is similar, except that now we usesinθ′tan(θ′/2)=1−c′ = j j j j [λ 2−(1−k′)2]/2k′ to obtain j 1−k′−λ′ y′ = j . (4.14) j 1+k′+λ′ j Thus 2(λ +λ′) 1+y y′ = i j . (4.15) i j (1−k′+λ )(1+k′+λ′) i j Also, c −c′ = (λ′2−λ 2)/2k′. (4.16) i j j i We see that the factor λ +λ′ cancels out of (4.6), leaving i j −4k′f f′ U = i j (4.17) ij (1−k′+λ )(1+k′+λ′)(λ −λ′) i j i j soU isaCauchy-likematrix,similartoB ,butwiththedenominatorc −c′ replaced PQ i j by λ −λ′. Analogously to (4.10), its determinant is i j m −4k′f f′ detU = ∆ (λ,λ′) i i . (4.18) m,m (1−k′+λ )(1+k′+λ′) i i i=1 Y Also, from (4.16), we can write (2.5) as −2k′f f′ (B ) = i j (4.19) PQ ij λ2−λ′2 i j so detB is also equal to PQ m detB = ∆ (λ2,λ′2) (−2k′f f′). (4.20) PQ m,m i i i=1 Y The f ,f′ cancel out of theratio (4.8),leaving i i ∆ (λ,λ′) m 2 D = m,m . (4.21) PQ ∆m,m(λ2,λ′2) (1−k′+λi)(1+k′+λ′i) Yi=1 From (1.17), (3.7), (3.8) and (3.9) it follows that ′ R(1+k′) m R(λ′) M(2) 2 = j=1 j . (4.22) r R(1−k′) m R(λ ) Qi=1 i (cid:0) (cid:1) Q 9 All cases When P > Q and m′ = m+1, we can still write D as in (4.3). However, U and PQ B areno longer squarematrices, so wecan longer simply takeproductsor ratios of PQ determinants, as in (4.8). Even so, we can still calculate D (and hence M(2)) by PQ r adding a row to B and U to make them square Cauchy-like matrices. The matrix PQ UBT in (4.3) is then m′ by m′, but is of upper-block-triangular form. The top left PQ block is the original m by m matrix UBT , while the lower-right block is the one by PQ one unit matrix. Hence thedeterminant (4.3) is unchanged and can be evaluated as a product of thetwo m′ bym′ determinants. We do this in the Appendix. The result (A18) is the same as (4.22), except that thefactor R(1+k′) is inverted. From (1.17) and (4.2), M(2) is unchanged by interchanging P with Q, m with m′ r and the λ with the λ′. From (3.8), this inverts the function R. We can use this i j symmetry to calculate M(2) in theother two cases. r For thefour cases (2.4), definethefactor G by G = R(1−k′)R(1+k′) if P Q and m′=m+1 (4.23) = R(1+k′)/R(1−k′) if P >Q and m′=m, then we find that ′ m m M(2) 2 = G R(λ′) R(λ ) (4.24) r j i , (cid:0) (cid:1) Yj=1 Yi=1 for all four cases. 5 The limit L → ∞ Theresult(4.24)isexactforfiniteL. Thelast stepinthecalculation istoletL→∞. For this we shall need for the first time herein the particular definition (1.22) - (1.23) of thepolynomial (1.24). Let c,λ be two variables related as are c ,λ in (3.5). Noting that λ2 −λ2 = i i i 2k′(c−c ),it follows from (3.8) and (1.24) that i R(λ)R(−λ) = (k′/2)m−m′P (c)/P (c). (5.1) P Q Using (1.22), (1.23),this can bewritten as R(λ)R(−λ) = [k′(c+1)/2]m−m′zQ−PW(λ), (5.2) where W(λ) = W (λ)/W (λ), (5.3) P Q N−1 1−z L W (λ) = 1+ ωn(P+L) . (5.4) P ωn−z Xn=1 (cid:16) (cid:17) Wehave cancelled then=0 term in (1.22) from theratio (5.3). These z, w, care related to λ by therelations (1.22), (1.23),(3.5). In particular, c−1 (1−k′)2−λ2 zN =w= = . (5.5) c+1 (1+k′)2−λ2 10

Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant D_PQ

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