Eight-vertex model and Painlevé VI equation. II. Eigenvector results
aa r X i v : . [ m a t h - ph ] D ec Eight-vertex model and Painlev´e VI equationII. Eigenvector results
Vladimir V. Mangazeev and Vladimir V. Bazhanov Department of Theoretical Physics,Research School of Physical Sciences and Engineering,Australian National University, Canberra, ACT 0200, Australia.
Abstract
We study a special anisotropic
XYZ -model on a periodic chain of an odd length andconjecture exact expressions for certain components of the ground state eigenvectors. Theresults are written in terms of tau-functions associated with Picard’s elliptic solutions of thePainlev´e VI equation. Connections with other problems related to the eight-vertex modelare briefly discussed. email: [email protected] email: [email protected] Introduction
This is a sequel to our paper [1] devoted to connections of the eight-vertex model of statisticalmechanics [2] with the theory of Painlev´e transcendents. Here we study a related case of theanisotropic
XYZ -model on a periodic chain of an odd length, N = 2 n + 1.Let σ ( j ) x , σ ( j ) y and σ ( j ) z , j = 1 , . . . , N , denote usual Pauli matrices acting at the j -th site ofthe chain. Consider a particular XYZ -Hamiltonian H XYZ = − N X j =1 ( J x σ ( j ) x σ ( j +1) x + J y σ ( j ) y σ ( j +1) y + J z σ ( j ) z σ ( j +1) z ) , (1)where the coefficients J x = 2 (1 + ζ ) ζ + 3 , J y = 2 (1 − ζ ) ζ + 3 , J z = ζ − ζ + 3 , (2)are specific rational functions of a single parameter ζ , which satisfy the relation J x J y + J y J z + J z J x = 0 . (3)Baxter [3] proved that for an infinitely large chain, N → ∞ , the ground state eigenvalue of (1)in this case has a very simple form E ( ζ ) N = − J x + J y + J z − . (4)Later on it was conjectured [4] that this expression is exact for all finite odd values of N .The Hamiltonian (1) commutes with the transfer matrix of the eight-vertex model (8V-model), where the Boltzmann weights a , b , c , d (we use standard notations of [2], see Sect. 2below for further details) are constrained as( a + ab ) ( b + ab ) = ( c + ab ) ( d + ab ) , (5)and the variable ζ in (2) is given by ζ = cdab , γ = ( a − b + c − d )( a − b − c + d )( a + b + c + d )( a + b − c − d ) . (6)Additional variable γ , introduced for later convenience, is connected to ζ by a simple self-reciprocal rational substitution ζ = γ + 3 γ − , γ = ζ + 3 ζ − . (7)The spectrum of (1) possesses an S symmetry group with respect to permutations of theconstants J x , J y and J z . Indeed, any such permutation can be compensated by a linear transfor-mation which acts on the eigenvectors and does not affect the spectrum. For the parametrization(2) this group is generated by two substitutions of the variable ζ , s xy : ζ → − ζ = ⇒ J x ↔ J y , J z → J z , s xz : ζ → ζ +3 ζ − = ⇒ J x ↔ J z , J y → J y . (8)2he largest eigenvalue of the transfer matrix, corresponding to (4), also has a remarkablysimple conjectured form [4] Λ = ( a + b ) N , N = 2 n + 1 , (9)which is expected to hold for finite chains.In [5] we studied Baxter’s famous TQ -equation for this simple eigenvalue (9) and foundcorresponding eigenvalues of the Q -operator. With an appropriate normalization they can beexpressed through certain polynomials P n ( x, z ) = n X k =0 r ( n ) k ( z ) x k , z = γ − , n = 0 , , , . . . , (10)of the variable x , defined by the following quadratic equation (see also (23)), (cid:16) √ x − γ √ x (cid:17) = −
16 ( a − b ) c d ( c + d ) ( a + b + c + d )( a + b − c − d ) . (11)The coefficients r ( n ) i ( z ), i = 0 , . . . , n , appearing in (10), are polynomials in the variable z = γ − with positive integer coefficients. Detailed definitions of the polynomials (10) are presentedin Sect. 2. Here we want to illustrate their connection to the Painlev´e VI equation. Thisconnection manifests itself in some specific properties of the coefficients r ( n ) i ( z ). In particular,let s n ( z ) ≡ r ( n ) n ( z ) be a coefficient in front of the leading power of x in (10). In [5] we conjecturedthe following recurrence relation2 z ( z − z − ∂ z log s n ( z ) + 2(3 z − (9 z − ∂ z log s n ( z )++8(2 n + 1) s n +1 ( z ) s n − ( z ) s n ( z ) − [4(3 n + 1)(3 n + 2) + (9 z − n (5 n + 3)] = 0 , (12)where s ( z ) = s ( z ) ≡
1, which uniquely determines the polynomials s n ( z ), for all n ∈ Z . Lateron we proved [1] that Eq.(12) exactly coincides with the recurrence relation for the tau-functionsassociated with special elliptic solutions of the Painlev´e VI equation. In this letter we extendthese connections to study ground state eigenvectors of H XYZ , corresponding to the eigenvalue(4).For odd N all eigenvalues of (1) are double degenerate. Thus, there are two ground stateeigenvectors H XYZ Ψ ± = E Ψ ± , S Ψ ± = ± Ψ ± , R Ψ ± = Ψ ∓ , (13)where S = σ (1) z ⊗ σ (2) z ⊗ · · · ⊗ σ ( N ) z , R = σ (1) x ⊗ σ (2) x ⊗ · · · ⊗ σ ( N ) x , (14)and [ H XYZ , S ] = [ H XYZ , R ] = 0 , R S = ( − N S R . (15)Due to the spin reversal symmetry, generated by the operator R , it is enough to consider oneof these vectors. For definiteness, consider the vector Ψ − . Omitting the suffix “ − ”, we denoteits components as Ψ i ,i ,...,i N , where i , i , . . . , i N ∈ { , } and assume an orthonormal basis | i i , i = 0 ,
1, for each spin σ z | i = + | i , σ z | i = −| i . (16)The ground state eigenvectors are translationally invariant and possess a left-right reflectionsymmetry. Taking this into account we will give only one non-vanishing representative from3ach symmetry class. Note that for non-vanishing components Ψ i ,i ,...,i N of Ψ − the number of“down-spins” in the set { i , i , . . . , i N } is odd, while for vanishing components it is even,Ψ i ,i ,...,i N ≡ , if i + i + · · · + i N = 0 (mod 2) . (17)The fact that both the coefficients (2) and the eigenvalue (4) are rational functions in ζ withinteger coefficients implies that with a suitable normalization one can make all componentsof the eigenvector Ψ − to be polynomials in ζ with integer coefficients [6] (such that there areno polynomial factors common for all components). This choice is unique up to a numericalnormalization. The later is fixed by the requirementsΨ , . . . , | {z } n +1 , . . . , | {z } n (cid:12)(cid:12)(cid:12) ζ =0 = 1 , for odd n ; Ψ , . . . , | {z } n , . . . , | {z } n +1 (cid:12)(cid:12)(cid:12) ζ =0 = 1 , for even n. (18)Note that in the case ζ = 0 the Hamiltonian (1) reduces to that of the XXZ -model with theparameter ∆ = − /
2. From this point of view the normalization (18) is identical to that usedin [7], where this particular
XXZ -model was studied.We have calculated all components of the eigenvectors directly from the definition (13) for N ≤
17 (and some particular components for N ≤
25) and made several interesting observationswhich we formulate as conjectures valid for all N = 2 n + 1. As an example we present here Conjecture 1.
The norm of the eigenvector Ψ − is given by | Ψ − | = X i ,i ,...,i N ∈{ , } Ψ i ,i ,...,i N = (4 / n ζ n ( n +1) s n ( ζ − ) s − n − ( ζ − ) , (19) where s n ( ζ − ) , n ∈ Z , are defined by the recurrence relation (12) with z = ζ − and s ( z ) = s ( z ) ≡ . Other conjectures on the properties of eigenvectors require additional notations; they arepresented in Sect. 3. Basic definitions for the 8V-model, a brief review of some of our previousresults [5] and some new results on the eigenvalues of Baxter’s Q -operators are given in Sect. 2.In Conclusion we discuss some unresolved questions and connections of our results to otherproblems related with the eight-vertex model, in particular, to the eight-vertex solid-on-solidmodel [8] with the domain wall boundary condition [9] and the three-coloring problem [10]. TQ -equation We consider the eight-vertex model on the N -column square lattice with the periodic (cylindri-cal) boundary conditions and assume that N is an odd integer N = 2 n + 1. Following [2] weparametrize the Boltzmann weights a , b , c , d of the model as a = ρ ϑ (2 η | q ) ϑ ( u − η | q ) ϑ ( u + η | q ) , We use the notation of [11] for theta-functions ϑ k ( u | q ), k = 1 , , ,
4, of the periods π and πτ , q = e iπτ ,Im τ >
0. The theta-functions H ( v ), Θ ( v ) of the nome q B used in [2] are given by q B = q , H ( v ) = ϑ ( πv K B | q ) , Θ ( v ) = ϑ ( πv K B | q ) , where K B ( k ) is the complete elliptic integral of the first kind with the elliptic modulus k = ϑ (0 | q B ) /ϑ (0 | q B ). = ρ ϑ (2 η | q ) ϑ ( u − η | q ) ϑ ( u + η | q ) ,c = ρ ϑ (2 η | q ) ϑ ( u − η | q ) ϑ ( u + η | q ) , (20) d = ρ ϑ (2 η | q ) ϑ ( u − η | q ) ϑ ( u + η | q ) , and fix the normalization factor, ρ = 2 ϑ (0 | q ) − ϑ (0 | q ) − . (21)With this parametrization the constraint (5) is equivalent to the condition η = π/ , (22)which will be always assumed throughout this paper. This still leaves two arbitrary parameters:the (spectral) parameter u and the elliptic nome q = e iπτ , Im τ >
0. The variables ζ , γ and x ,defined in (6), (7) and (11), can be written as ζ = (cid:20) ϑ ( π | q ) ϑ ( π | q ) (cid:21) , γ = − (cid:20) ϑ ( π | q / ) ϑ ( π | q / ) (cid:21) , x = γ (cid:20) ϑ ( u | q / ) ϑ ( u | q / ) (cid:21) , z = γ − . (23)Note that the last expression for x determines our choice of a particular root of the quadraticequation (11). TQ -equation Any eigenvalue, T ( u ), of the row-to-row transfer matrix of the 8V-model satisfies Baxter’sfamous TQ -equation [2], T ( u ) Q ( u ) = φ ( u − η ) Q ( u + 2 η ) + φ ( u + η ) Q ( u − η ) , (24)where, with an account of (21), φ ( u ) = ϑ N ( u | q ) . (25)With the parametrization (20), (21) the eigenvalue (9) takes the form T ( u ) = ( a + b ) N = φ ( u ) , η = π/ , N = 2 n + 1 . (26)Equation (24) for this eigenvalue, has been studied in [5]. It has two different solutions [12, 13],denoted Q ± ( u ) ≡ Q ± ( u, q , n ), which are entire functions of the variable u and obey the followingperiodicity conditions [2, 14], Q ± ( u + π ) = ± ( − n Q ± ( u ) , Q ± ( u + πτ ) = q − N/ e − iNu Q ∓ ( u ) , Q ± ( − u ) = Q ± ( u ) . (27)The above requirements uniquely determine Q ± ( u ) to within u -independent normalization fac-tors. The solutions Q ± ( u ) satisfy the quantum Wronskian relation [12, 15] Q + ( u + η ) Q − ( u − η ) − Q + ( u − η ) Q − ( u + η ) = 2 iφ ( u ) W ( q , n ) , (28)where W ( q , n ) is a function of q and n only (The fact that W ( q , n ) does not depend on thevariable u follows from (24) and (27)). Note that, taking into account the periodicity (27), onecan bring Eq.(24) to the form φ ( u ) Q ( u ) + φ ( u + 2 π/ Q ( u + 2 π/
3) + φ ( u + 4 π/ Q ( u + 4 π/
3) = 0 . (29) We use exactly the same definition of the transfer matrix as in [2]. Q ( u ) = ( Q + ( u ) + Q − ( u )) / , Q ( u ) = ( Q + ( u ) − Q − ( u )) / , (30)which are simply related by the periodicity relation Q ( n )1 , ( u + π ) = ( − n Q ( n )2 , ( u ) . (31)Bearing this in mind we will only quote results for Q ( u ), writing it as Q ( n )1 ( u ) to indicate the n -dependence. Introduce new functions P n ( u ) instead of Q ( n )1 ( u ), Q ( n )1 ( u ) = N ( q , n ) ϑ ( u/ | q / ) ϑ n ( u/ | q / ) P n ( u ) , (32)where N ( q , n ) is an arbitrary normalization factor. The analytic properties of P n ( u ) are de-termined by the periodicity relations (27) and the fact that the eigenvalues Q ( n )1 , ( u ) are entirefunctions of the variable u . A simple analysis shows that P n ( u ) is an even doubly periodicfunction of u , P n ( u ) = P n ( u + 2 π ) = P n ( u + πτ ) , P n ( u ) = P n ( − u ) , (33)with all its poles (of the order 2 n and lower) located at the point u = πτ /
2. Every suchfunction is an n -th degree polynomial in the variable x , given by (23) (see § q . Let us nowchange independent variables from u and q to the variables x and z = γ − , defined in (23), and(with a slight abuse of notations) write P n ( u ) as P n ( x, z ) = n X k =0 r ( n ) k ( z ) x k . (34)The TQ -equation (29) can be re-written in terms of the polynomials P n ( x, z ). For a fixedvalue of the nome q , the variable x in (23) is a function of u , so we can write it as x = x ( u ).Introduce two new variables x ± = x ( u ± π ) = γ /x ( u ∓ π ) . (35)They satisfy the relations x + x − = ( x − ( x z − , x + + x − = 2 z ( x z + 1) − x ( z + 4 z − z ( x z − , (36)which can be easily solved for x ± in terms of x and z . The resulting expressions involve a squareroot from a third order polynomial in x . It is convenient to define f ± = 12 ± x ( z − x − z + 1]2 z ( x − − x + ) ( x z − , ρ ± = x ± − − z x ± ) x . (37)With all these new notations the TQ -equation (29) can now be transformed to its algebraic form, P n ( x, z ) = ρ + f n +1 − P n ( z − x − − , z ) + ρ − f n +1+ P n ( z − x − , z ) . (38)Substituting (34) into the last equation, and expanding it near the point x = 0, one imme-diately obtains a simple relation, r (0) n ( z ) ≡ P n (0 , z ) = 4 − n z − ( z + n (3 z − P n ( z − , z ) − − n z − ( z − ∂ P n ( x, z ) ∂x (cid:12)(cid:12)(cid:12) x = z − , (39)quoted here for future references. An apparent pole at u = π + πτ / Q ( n )1 ( u ) vanishes at this point as a consequence of theperiodicity conditions (27). .3 Polynomials P n ( x, z ) Let us now substitute polynomials (34) into the TQ -equation (38). Excluding x + and x − withthe help of (36), one can readily see that the RHS of (38) is a rational function of x (indeed, itis a symmetric function of x + and x − and, therefore, can be expressed through two elementarycombinations (36)). Writing Eq.(38) as a polynomial in x and equating its coefficients to zeroone obtains an (overdetermined) system of homogeneous linear equations for n + 1 unknowns r ( n )0 ( z ) , r ( n )1 ( z ) , . . . , r ( n ) n ( z ). All elements of the coefficient matrix for this system are rationalfunctions of the variable z with integer coefficients. This means that with a suitable normal-ization all r ( n ) k ( z ), k = 0 , , . . . , n , can be made polynomials in z with integer coefficients (suchthat there are no polynomial factors common for all r ( n ) k ( z )). Thus, P n ( x, z ) are two-variablepolynomials in x and z with integer coefficients. The first few of them are given in (46) andAppendix A below.Originally, we have calculated [5] these polynomials for n ≤
10 by directly solving Eqs.(29)and (38) by a combination of analytical and numerical techniques. For larger n this did notappear be to practical due to complexity of intermediate expressions. Subsequently, in the samepaper [5], we found a more efficient method for the calculation of P n ( x, z ), based on the partialdifferential equation (44), discussed below. We have observed that all coefficients of P n ( x, z )are, in fact, positive integers for all n ≤
100 and suggested that these coefficients might have acombinatorial interpretation (which is yet to be found).Below we summarize all important properties of P n ( x, z ) discovered in our previous works[1, 5]. Conjecture A ( [1, 5]) . (a) The degrees of the polynomials r ( n ) i ( z ) , i = 0 , . . . , n , appearing as coefficients in the expansion(34), are given by deg[ r ( n ) k ( z )] ≤ ⌊ n ( n − / k/ ⌋ , (40) where ⌊ x ⌋ denotes the largest integer not exceeding x .(b) if the normalization of P n ( x, z ) is fixed by the requirement r ( n ) n (0) = 1 , (41) then all polynomials r ( n ) k ( z ) , k = 0 , , . . . , n , have positive integers coefficients in theirexpansions in powers of z . The normalization (41) will be implicitly assumed throughout the rest of the paper. Themost important property of the polynomials P n ( x, z ) is that they satisfy a remarkable linearpartial differential equation. This equation can be written in different forms, depending on thechoice of independent variables and unknown function. First consider the case of the originalvariables u and q of the 8V-model. Introduce the functionsΦ ± ( u, q , n ) = ϑ n +11 ( u | q ) ϑ n (3 u | q ) Q ± ( u, q , n ) , (42)where Q ± ( u, q , n ) are eigenvalues of the Q -operators, defined in Sect.2.2. The analytic propertiesof Φ ± ( u, q , n ) in the variable u are determined by (27).7 onjecture B ( [5]) . The functions Φ ± ( u, q , n ) , defined by (42), satisfy the non-stationarySchr¨odinger equation q ∂∂q Φ( u, q, n ) = n − ∂ ∂u + 9 n ( n + 1) ℘ (3 u | q ) + c ( q, n ) o Φ( u, q, n ) . (43)Here the modular parameter τ plays the role of the (imaginary) time and the time-dependentpotential is defined through the elliptic Weierstrass ℘ -function [11] (our function ℘ ( v | e iπǫ ) hasthe periods π and πǫ ). The constant c ( q, n ) appearing in (43) is totally controlled by thenormalization of Q ± ( u ) and can be explicitly determined once this normalization is fixed (seeEqs.(37) and (38) in [5]). Equation (43) is obviously related to the Lam´e differential equationand could be naturally called the “non-stationary Lam´e equation”. This equation arises invarious contexts [16, 17] which are not immediately related to this paper.The differential equation (43) can be equivalently rewritten in an algebraic form for thepolynomials P n ( x, z ). n A ( x, z ) ∂ x + B n ( x, z ) ∂ x + C n ( x, z ) + T ( x, z ) ∂ z o P n ( x, z ) = 0 , (44)where A ( x, z ) = 2 x (1 + x − xz + x z )( x + 4 z − xz − xz + 4 x z ) ,B n ( x, z ) = 4(1 + x − xz + x z )( x + 3 z − xz + 3 x z )++2 nx (1 − z + 21 z − x z + 3 x z (3 z + 6 z − − x (1 − z + 23 z + 9 z )) ,C n ( x, z ) = n [ z (9 z −
5) + x z (3 z + 11 z −
2) + x (9 z − z + 19 z − −− x z + nz (1 − z − x (9 z − z + 3) + x (3 z − z + 4) + 8 x z )] ,T ( x, z ) = − z (1 − z )(1 − z )(1 + x − xz + x z ) . (45)It is fairly easy to prove [5] that the differential equation (43), restricted to a class of functionsΦ( u, q , n ) with suitable analytic properties in the variable u , implies the functional equation(29). The non-trivial part of the Conjecture B is the fact of existence of solutions of (43) withthese analytic properties. For Eq.(44) this translates into a question of existence of solutions,which are polynomials in the variable x .Equation (44) is extremely useful for finding polynomial solutions, even though the coeffi-cients therein look very complicated. The first polynomials P n ( x, z ) read P ( x, z ) = 1 , P ( x, z ) = x + 3 , P ( x, z ) = x (1 + z ) + 5 x (1 + 3 z ) + 10 , P ( x, z ) = x (1 + 3 z + 4 z ) + 7 x (1 + 5 z + 18 z ) + 7 x (3 + 19 z + 18 z ) + 35 + 21 z, , (46)the next one is given in Appendix A. The constant term and leading coefficient in these poly-nomials (with respect to the variable x ) are determined by the following Conjecture C ( [1, 5]) . The coefficients for the lowest and highest powers of x in P n ( x, z ) ,corresponding to k = 0 and k = n in (34), read s n ( z ) ≡ r ( n )0 ( z ) = τ n ( z, − / , s n ( z ) ≡ r ( n ) n ( z ) = τ n +1 ( z, / , (47)8 here the functions τ n ( z, ξ ) (for each fixed value of the their second argument ξ ) are determinedby the recurrence relation z ( z − z − [log τ n ( z )] ′′ z + 2(3 z − (9 z − τ n ( z )] ′ z ++8 h n − ξ − i τ n +1 ( z ) τ n − ( z ) τ n ( z ) −− [12(3 n − ξ − n − ξ ) + (9 z − n − n − ξ )] = 0 , (48) with the initial condition τ ( z, ξ ) = 1 , τ ( z, ξ ) = − ξ + 5 / . (49) The functions τ n ( z, ξ ) are polynomials in z for all n = 0 , , , . . . , ∞ . As explained in [5], the partial differential equation (44) leads to a descending recurrencerelations for the coefficients in (34), in the sense that each coefficient r ( n ) k ( z ) with k < n canbe recursively calculated in terms of r ( n ) m ( z ), with m = k + 1 , . . . , n and, therefore, can beeventually expressed through the coefficient r ( n ) n ( z ) of the leading power of x . Conditions thatthis procedure truncates (and thus defines a polynomial, but not an infinite series in negativepowers of x ) completely determine the starting leading coefficient as a function of z . The aboveconjecture implies that these truncation conditions are equivalent to the recurrence relation (12)which is a particular case of (49) for ξ = 1 /
6. Similar reasonings apply to the coefficient r ( n )0 ( z )in (34) (the constant term with respect to the variable x ).Note, that Eq.(48) exactly coincides [1] with the recurrence relation for the tau-functionsassociated with special elliptic solutions of the Painlev´e VI equation. We conclude this section with a short analysis of the algebraic form x n − − x + x − (1 − xz ) P n ( x + , z ) P n ( z − x − − , z ) + x n + − x + x + (1 − xz ) P n ( x − , z ) P n ( z − x − , z )= 1( x − (cid:18) z ( xz − ( x + − x − ) x ( z − (cid:19) n W n ( z ) (50)of the quantum Wronskian relation (28). Here W n ( z ) is related to W ( q , n ) in (28), W ( q , n ) = ( − n i [2 ϑ ( π/ | q )] n +1 N ( q , n ) W n ( z ) . (51)Equation (50) is an algebraic identity valid for arbitrary values of x . Expanding this identityaround x = z − , one obtains W n ( z ) = − s n ( z ) P n ( z − , z ) , n ≥ , (52)where s n ( z ) is defined by (12) (it coincides with the Painlev´e VI tau-function s n ( z ) = τ n +1 ( z, / ξ = 1 / x = 0 and using (52), one obtains P n (1 , z ) = 4 n s n ( z ) , n ≥ . (53)Interestingly the quantity P n ( z − , z ), entering (52) is also determined by the Painlev´e VI recur-rence relation (48). 9 onjecture D. The value P n ( z − , z ) is determined by recurrence relation (48) with ξ = 2 / , P n ( z − , z ) = ( − / n z − n τ n +2 ( z, / , n ≥ . (54)Combining the above formulae one obtains the following expression for the quantum Wron-skian, W n ( z ) = − ( − / n z − n τ n +1 ( z, / τ n +2 ( z, / , n ≥ . (55) To formulate our results for the eigenvectors (13) we need to define an additional set of poly-nomials p n ( y ) and q n ( y ), n ∈ Z . In principle, these polynomials can be defined by yet anotherrecurrence relation of the Painlev´e VI type (though more complicated than (48)) which will bepresented elsewhere. For our purposes here it is much simpler to define these new polynomials p n ( y ) and q n ( y ) as subfactors of already introduced polynomials s n ( z ). We will do this by meansof the Conjecture E, given below. Remind that s n ( z ), n ∈ Z are defined by Eq.(12) with theinitial conditions s ( z ) = s ( z ) ≡ Conjecture E. (a) The polynomials s k +1 ( y ) factorize over the integers, s k +1 ( y ) = s k +1 (0) p k ( y ) p k ( − y ) , p k (0) = 1 , k ∈ Z , (56) where p k ( y ) are polynomials in y with integer coefficients, deg p k ( y ) = k ( k + 1) , such that p ′ k (0) > , k ≥ and p ′ k (0) < , k ≤ − , where p ′ k ( y ) = dp k ( y ) /dy denotes the derivative in y . Note that p − ( y ) = p ( y ) ≡ .(b) the polynomials p k ( y ) possess the symmetry p k ( y ) = (cid:16) y (cid:17) k ( k +1) p k (cid:16) − y y (cid:17) , k ∈ Z , (57) (c) polynomials s k ( y ) factorize over the integers, s k ( y ) = c k (1 + 3 y ) k ( k +1) p − k − (cid:16) y −
11 + 3 y (cid:17) q k − ( y ) , k ∈ Z , (58) where q k ( y ) are polynomials in y with integer coefficients, deg q k ( y ) = k ( k + 1) , q k (0) = 1 and c k = 2 − k ( k +2) , k ≥ c k = 2 − k (2 / k +1 , k < . (59) (d) polynomials q k ( y ) possess the symmetry q k ( y ) = (cid:16) y (cid:17) k ( k +1) q k (cid:16) y −
11 + 3 y (cid:17) , k ∈ Z , (60)A few first polynomials s n ( z ), p n ( y ) and q n ( y ) is listed in Appendix A. Let us mention onesimple, but important corollary of the above conjecture. The LHS of (58) is an even functionsof the variable y . Combining this fact with the symmetry relation (57), one immediately deducethat q k ( y ) is also an even function, q k ( − y ) = q k ( y ) , k ∈ Z . (61)10n Conjecture 1, given in the Introduction, we have stated an explicit expression (19) for thenorm | Ψ − | of the eigenvector Ψ − as a function of the parameter ζ entering the Hamiltonian (1).Note, that using the factorization and symmetry properties (56)-(61), one can show that therescaled norm N n ( ζ ) = ( ζ + 3) − n ( n +1) / | Ψ − | (62)is invariant with respect to the full S symmetry group generated by the substitutions (8),which is a well expected result. Further, is easy to see, that modulo a trivial numerical factor,the expression for the norm remains unchanged upon the replacement n → − n −
1, whichcorresponds to a negation of the length of the chain, N → − N . In other words, the norm is aneven function of the length of the chain. It would be interesting to understand a reason of thisphenomenon.We are now ready to present further conjectures on the properties of the eigenvectors. Conjecture 2.
The component of the eigenvector Ψ − with one spin down is given by ψ ... = 1 N ζ n ( n − / s n ( ζ − ) , N = 2 n + 1 , (63) where s n ( z ) is defined by (47). A few first polynomials s n ( z ) and s n ( z ) is listed in Appendix A. Conjecture 3.
The component of the eigenvector Ψ − with all spins down is given by ψ ... = ζ n ( n +1) / s n ( ζ − ) , N = 2 n + 1 , (64) where s n ( z ) is defined by (47). It is interesting to note that to within a simple power of ζ the above two components of theeigenvector precisely coincide with the constant term and leading coefficients of the polynomial P n ( x, z ), which is simply connected (32) with the corresponding eigenvalue of the Q -operator.We believe that this fact certainly deserves further studies.Finally, consider components of Ψ − with alternating (up and down) spins in the chain, A n ( ζ ) = Ψ ... , for odd n ; A n ( ζ ) = Ψ ... for even n . (65)In the case ζ = 0 these are largest components of the eigenvector. Conjecture 4.
The components of Ψ − with alternative spins are given by A k ( ζ ) = 2 k (2 − k ) (3 + ζ ) k ( k − ζ k ( k − p k − (cid:16) − ζ ζ (cid:17) q k − ( ζ − ) A k +1 ( ζ ) = 2 − k (3 + ζ ) k ( k +1) ζ k ( k − p k (cid:16) − ζ ζ (cid:17) q k − ( ζ − ) (66)A few polynomials p n ( y ), q n ( y ) and A n ( ζ ) is listed in Appendix A. As noted before, thecase ζ = 0 corresponds to the XXZ -model with ∆ = − /
2. It is known [7, 18–20], that in thisparticular case, the values of the components (65), normalized by (18), coincide with the numberof alternating sign matrices A n (0) = A n = n − Y k =0 (3 k + 1)!( n + k )! (67)11alculated in [21]. Using this result in (66) one easily obtains for n ≥ p n ( ) = (cid:18) (cid:19) n ( n +1) n Y k =0 (2 k )!(6 k + 1)!(4 k )!(4 k + 1)! , (68) ζ n ( n +1) q n ( ζ − ) | ζ =0 = 2 − n − n Y k =0 (2 k + 1)!(6 k + 4)!(4 k + 2)!(4 k + 3)! . (69)Apparently one can derive these expressions directly from the definitions of the polynomials p n ( y ) and q n ( y ), given in Conjecture E, however, we postpone this to a future publication.Finally, mention one amusing observation connected with the expressions (66). It is notdifficult to analytically derive an asymptotic expansion A ( asymp ) n ( ζ ) which correctly reproducefirst terms of the expansion of A n ( ζ ) for small ζ up to the order O ( ζ n ), A n ( ζ ) = A ( asymp ) n ( ζ ) + O ( ζ n ) , ζ → . (70)Analytically continuing this asymptotic expansion to n = 0, A ( asymp ) n ( ζ ) (cid:12)(cid:12)(cid:12) n =0 = 1 − ζ − ζ − ζ − ζ − ζ − ζ − . . . (71)and plugging its coefficients into Sloane’s integer sequences database (in a search for a discovery)we found that they only “slightly” mismatched numbers of lattice animals made of n three-dimensional cubes [22], which are 1 , , , , , , . . . . Of course, it would be extremelyweird if they matched. In this paper we have demonstrated that a particular anisotropic
XYZ -model, defined by (1)and (2), is deeply related with the theory of Painlev´e VI equation. We have proposed exactexpressions for the norm (Conjecture 1) and certain components of the ground state eigenvectors(Conjectures 2,3,4). The results are expressed in terms of the tau-functions associated with thespecial elliptic solutions of the Painlev´e VI equation [1].In this connection, it is useful to mention other celebrated appearances of Painlev´e transcen-dents in mathematical physics. The most prominent examples include the two-dimensional Isingmodel [23], the problem of isomonodromic deformations of the second order differential equa-tions [24] and the field theory approach to dilute self-avoiding polymers on a cylinder [25–29].The latter problem is connected with the massive sine-Gordon model at the supersymmetricpoint (where the ground state energy vanishes exactly due to supersymmetry). Our previouswork [5] grew up from attempts to develop an alternative approach to this polymer problembased on the lattice theory. It turns out that all non-trivial information about dilute polymerloops is contained in the ground state eigenvalues [5] of the Q -operator for the 8V-model on aperiodic chain of an odd length, connected with the special XYZ -model, considered in this paper.In [5] we have found that these eigenvalues can be uniquely determined as certain polynomialsolutions P n ( x, z ) of the partial differential equation (44) (remind that the variable x and z areconnected to the original spectral parameter u and the elliptic nome q , respectively, see (23)).So far we have not ultimately understood the role of this equation in the Painlev´e VI theory, butthere is no doubt that there are profound connections. For example, one-variable specializationof P n ( x, z ) at particular values of x (which remain polynomials in the variable z ) are connected12ith the tau-functions associated with the Picard solutions of the Painlev´e VI equation (seeEqs.(53) and (54)). The same property is enjoyed also by the coefficients in the expansion of P n ( x, z ) in powers of x . Note, that namely these coefficients provide “construction materials” inthe expression for the ground state eigenvectors (see Eqs.(19), (56)-(66)). Most of our results areconjectures and it is, of course, desirable to obtain their proofs. Another outstanding problemis an algebraic construction of the Q -matrix. As noted in [30], the method used in [2] for theconstruction of the Q -matrix cannot be executed in its full strength for η = π/
3, since someaxillary Q -matrix, Q R ( u ), in [2] is not invertible in the full 2 N -dimensional space of states ofthe model. Apparently, the construction of [2] could be modified to resolve this difficulty. Wehope to address this question in the future.It is reasonable to expect that mathematical structures, similar to those described above(namely, the partial differential equations and Painlev´e type recurrence relations), should mani-fest itself in other problems, closely related to the 8V-model with η = π/
3. The most immediatecandidate is the corresponding “eight-vertex solid-on-solid” (8VSOS) model which belongs to arich variety of algebraic constructions associated with the 8V-model [8]. Recently, Rosengren [9],motivated by considerations of the 3-coloring problem [10], studied precisely this 8VSOS-modelwith η = π/ Acknowledgments
The authors thank B.M.McCoy for valuable comments and H.Rosengren for sending us thepreprint of his recent paper [9] and interesting correspondence. After completion of this manuscriptwe received the preprint [31] on the same subject, but without essential overlaps with the presentpaper. We thank A.V. Rasumov for sending us their preprint [31].
Appendix A. Polynomials P n ( x, z ) , s n ( z ) , s n ( z ) , p n ( z ) and q n ( z ) . In this Appendix we present explicit expressions for the polynomials P n ( x, z ), s n ( z ), s n ( z ), p n ( z ) and q n ( z ) for small values of the their index n .The two-variable polynomials P n ( x, z ), represented by Eq.(34), are defined as solutions ofthe TQ -equation (38) normalized by (41). These polynomials can be efficiently calculated from13he differential equation (44), P ( x, z ) = 1 , P ( x, z ) = x + 3 , P ( x, z ) = x (1 + z ) + 5 x (1 + 3 z ) + 10 , P ( x, z ) = x (1 + 3 z + 4 z ) + 7 x (1 + 5 z + 18 z ) + 7 x (3 + 19 z + 18 z ) + 35 + 21 z, P ( x, z ) = x (1 + 6 z + 18 z + 30 z + 9 z ) + 9 x (1 + 8 z + 38 z + 152 z + 57 z )+ 18 x (2 + 19 z + 111 z + 217 z + 99 z ) + 12 x (7 + 72 z + 171 z + 198 z )+ 18(7 + 14 z + 11 z ) . (A.1)The polynomials s n ( z ) and s n ( z ), where n ∈ Z , are defined by Eq.(47) and the recurrencerelation (48), (49). For non-negative n they coincide with the coefficients of the highest andlowest powers of x in P n ( x, z ), corresponding to k = n and k = 0 in the expansion (34), s − = (81 + 1215 z + 10206 z + 64638 z + 353565 z + 544563 z + 352836 z ) ,s − = (27 + 270 z + 1620 z + 7938 z + 3969 z ) ,s − ( z ) = (9 + 54 z + 225 z ) ,s − ( z ) = (3 + 9 z ) ,s − ( z ) = s ( z ) = s ( z ) = 1 ,s ( z ) = 1 + z,s ( z ) = 1 + 3 z + 4 z ,s ( z ) = 1 + 6 z + 18 z + 30 z + 9 z ,s ( z ) = 1 + 10 z + 51 z + 168 z + 355 z + 318 z + 121 z . (A.2)and s ( z ) = 1 , s ( z ) = 3 , s ( z ) = 10 ,s ( z ) = 35 + 21 z,s ( z ) = 126 + 252 z + 198 z ,s ( z ) = 462 + 1980 z + 3960 z + 4004 z + 858 z ,s ( z ) = 1716 + 12870 z + 47190 z + 105820 z + 143520 z + 90558 z + 24310 z . (A.3)The polynomials p n ( y ) and q n ( y ) are defined by the factorization relations (56) and (58) in14onjecture E, p − ( y ) = 1 − y + 12 y − y + 81 y − y + 66 y ,p − ( y ) = 1 − y + 5 y ,p − ( y ) = p ( y ) = 1 ,p ( y ) = 1 + y + 2 y ,p ( y ) = 1 + 2 y + 7 y + 10 y + 21 y + 12 y + 11 y ,p ( y ) = 1 + 3 y + 15 y + 35 y + 105 y + 195 y + 435 y + 555 y + 840 y + 710 y + 738 y + 294 y + 170 y . (A.4)and q − ( y ) = 1 + 3 y + 39 y + 21 y ,q − ( y ) = 1 + 3 y ,q − ( y ) = q ( y ) = 1 ,q ( y ) = 1 + 3 y ,q ( y ) = 1 + 8 y + 29 y + 26 y ,q ( y ) = 1 + 15 y + 112 y + 518 y + 1257 y + 1547 y + 646 y . (A.5)Finally, we list polynomials A n ( ζ ) from the expressions (66) for the alternative spin compo-nents (65), A ( ζ ) = 1 , A ( ζ ) = 2 , A ( ζ ) = 7 + ζ , A ( ζ ) = 2(3 + ζ )(7 + ζ ) ,A ( ζ ) = (3 + ζ )(143 + 99 ζ + 13 ζ + ζ ) ,A ( ζ ) = 2(26 + 29 ζ + 8 ζ + ζ )(143 + 99 ζ + 13 ζ + ζ ) ,A ( ζ ) = (26 + 29 ζ + 8 ζ + ζ ) × (8398 + 14433 ζ + 7665 ζ + 2010 ζ + 240 ζ + 21 ζ + ζ ) ,A ( ζ ) = 2(646 + 1547 ζ + 1257 ζ + 518 ζ + 112 ζ + 15 ζ + ζ ) × (8398 + 14433 ζ + 7665 ζ + 2010 ζ + 240 ζ + 21 ζ + ζ ) A ( ζ ) = 2(646 + 1547 ζ + 1257 ζ + 518 ζ + 112 ζ + 15 ζ + ζ ) × (1411510 + 4598551 ζ + 5518417 ζ + 3530124 ζ + 1331064 ζ ++ 327810 ζ + 53382 ζ + 5820 ζ + 506 ζ + 31 ζ + ζ ) . (A.6) Appendix B. Comments on the 8VSOS-model
Recently, Rosengren [9], motivated by considerations of the 3-coloring problem [10], studiedthe 8VSOS-model with η = π/ P ( SOS ) n ( t, s ), addingthe superscript “SOS” to indicate their relevance to the 8VSOS-model . A detailed inspectionof these polynomials suggests that Conjecture 5.
The polynomials P ( SOS ) n ( t, s ) , for even values of n = 0 , , . . . , are uniquelydetermined (up to a numerical normalization) by the the following partial differential equationin the variables x and s , n A ( SOS ) ( t, s ) ∂ t + B ( SOS ) n ( t, s ) ∂ t + C ( SOS ) n ( t, s ) + T ( SOS ) ( t, s ) ∂ s o P ( SOS ) n ( t, s ) = 0 , (B.1) where A ( SOS ) ( t, s ) = 2 t (1 − t ) (1 + 2 s − t ) (cid:0) s + 2 s − t − st (cid:1) (2 t + st − s ) ,B ( SOS ) n ( t, s ) = − s ) t − s ) (cid:0) − s + s n + 3 sn − s − n (cid:1) t + (cid:0) sn + 8 s n + 40 s n + 60 s n + 8 n − s − s − − s − s (cid:1) t − s ) s (cid:0) − s + 3 s n + 5 sn − s + 3 n − (cid:1) t + 4 ns (1 + 2 s ) ,C ( SOS ) n ( t, s ) = 2 n (2 + s ) (1 + n ) t − n (2 + s ) (1 + s ) (1 + n ) t + ns (cid:0) s + 4 s n + 26 s + 15 s n + 18 s + 18 sn + 2 + 5 n (cid:1) t − n (1 + 2 s ) s ( s + sn + 1) ,T ( SOS ) ( t, s ) = 4 (cid:0) − s (cid:1) s (2 + s ) (1 + 2 s ) t . (B.2)A similar differential equation exists for odd values of n . Obviously, the above property is acounterpart of the partial differential equation (44) in the main text of this paper.Next, define one-variable polynomials p ( SOS ) n ( s ) = (cid:16) s (cid:17) [ n ] − [ n ] (cid:16) s (cid:17) [ ( n − ] P ( SOS ) n (1 + 2 s, s ) (B.3)where [ x ] denotes the integer part of x . We suggest that Here we use the variables t and s instead of x and ζ used in [9]. These variables are similar, but not identical ,to our variables x and z in (23). In particular, the variable t (which corresponds to x in [9]) is also connectedwith the spectral parameter, while the variable s (denoted as ζ in [9]) is related to the elliptic nome q . These polynomials are simply related to those introduced in [9], p ( SOS ) n ( s ) = “ s ” [ n ] “ s ” [ ( n − ] p ( R ) n ( s )where p ( R ) n ( s ) are defined by the first unnumbered equation after the Proposition 3.12 in [9]. We thanksH.Rosengren for sending us the modified definition (B.3). onjecture 6. The polynomials p ( SOS ) n ( s ) satisfy the following recurrence relations s ( s − ( s + 2)(2 s + 1) ∂ s log p ( SOS ) n ( s ) + 2( s − s − s − s − ∂ s log p ( SOS ) n ( s ) − n + 1)(2 n + 3) p ( SOS ) n +1 ( s ) p ( SOS ) n − ( s )( p ( SOS ) n ( s )) + (22 n + 35 n + 18) s + (46 n + 98 n + 42) s + 13 n + 29 n + 12 = 0 (B.4) with the initial condition p ( s ) = 1 , p ( s ) = 1 + 3 s . The structure of the relation (B.4) is very similar to that of the recurrence relation (48) forthe tau-functions of Painlev´e VI equation. It should be noted that Eq.(B.4) is not written in acanonical form for such recurrence relations. Nevertheless, we expect that it could be broughtto such form by a suitable change of variables. We also expect that (B.4) can be connected withthe Picard elliptic solutions of the Painlev´e VI following the method of our previous paper [1].
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