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Boundary polarization in the six-vertex model N. M. Bogoliubov, A. V. Kitaev, and M. B. Zvonarev Steklov Institute of Mathematics at St. Petersburg, Fontanka 27, St. Petersburg 191011, Russia (February 1, 2008) 2 0 Vertical-arrow fluctuations near the boundaries in the six-vertex model on the two-dimensional 0 N ×N square lattice with the domain wall boundary conditions are considered. The one-point 2 correlation function (“boundary polarization”) is expressed via the partition function of the model n on a sublattice. The partition function is represented in terms of standard objects in the theory of a orthogonal polynomials. Thisrepresentation isusedtostudythelargeN limit: thepresenceofthe J boundary affects themacroscopic quantitiesof themodel evenin this limit. Thelogarithmic terms 1 obtained are compared with predictions from conformal field theory. 2 PACS numbers: 05.50.+q, 05.70.Np, 02.30.Ik ] h I. THE MODEL all arrows on the left and right boundaries are pointing c e outward (see, Fig. 2). m Inthispaper,weshallconsiderthesix-vertexmodelon - a square lattice. Originally, this model was introduced ? ? ? ? ? t (cid:27) - a as a model describing the ferroelectric properties of the t hydrogen-bondedplanarcrystals[1]. Thehydrogenatom (cid:27) - s . positions are specified by attaching arrows to the lattice t (cid:27) - a edges. In the six-vertex case, the arrows are arranged m in such a way that there are always two arrows point- (cid:27) - - ingawayfrom,andtwoarrowspointinginto,eachlattice d (cid:27) - vertex (the so-called “ice rule”); thus, there are six pos- n sible states at each vertex (see, Fig. 1). The statistical 6 6 6 6 6 o weights, a, b and c, of the allowed states are invariant c [ under the simultaneous reversal of all arrows: FIG.2. The domain wall boundary conditions. 3 v a b c ThismodelwasintroducedinRef.[5]inconnectionwith 6 the calculation of the correlation functions for exactly 4 ? ? 6 solvable 1+1 dimensional models [6]. It appears that 1 (cid:27) (cid:27) - - - (cid:27) some problems from the theory of alternating sign ma- 7 ? ? ? trices[7,8]anddominotilings[9]maybereformulatedin 0 terms of this model. 1 6 6 ? Aperiodic boundary conditions are of interest since 0 - - (cid:27) (cid:27) (cid:27) - / theydemonstratethe influenceoftheboundariesandin- t 6 6 6 a ternaldefectsofrealphysicalsystemsontheirbulkprop- m FIG.1. Thevertices ofthesix-vertexmodel andtheirsta- erties. By renormalization group method, it was shown tistical weights. that the behavior of the correlation functions near the - d surfaces and defects is quite different from the bulk be- n The partitionfunction of the model onanN N square havior [10]. The exactly solvable six-vertex model with o × lattice is obtained by summing over all possible arrow theDWBCprovidesustheopportunitytostudythesur- c configurations C , face phenomena beyond renormalizationgroup scheme. v: { } i X Z = an1bn2cn3, N II. THE PARTITION FUNCTION AND THE r XC a { } BOUNDARY POLARIZATION wheren1,n2,andn3arethenumberofverticesoftypea, b, andc in configurationC, respectively (n1+n2+n3 = The partition function Z of the model with the N2). DWBC may be representedNas the determinant of an The six-vertex model was studied for both periodic N N matrix [11,12]. Though there exist several com- (PBC) [2,3] and fixed boundary conditions [4]. In this bin×atorialrepresentationsfor this determinant[13,14],it paper,weareconcernedexclusivelywiththedomainwall hassofaronlybeencalculatedexplicitlyforsomespecial boundaryconditions(DWBC),namely,allarrowsonthe cases [7]. Significantprogresshas recently been achieved top and bottom of the lattice are pointing inward while 1 in studying the asymptotics of Z as N . Un- Theboundarypolarizationonthearbitraryedgeisex- N → ∞ der some special restrictions on the values of the vertex pressed in terms of linear combination of partition func- weights, the bulk free energy was calculated in Ref. [15] tions on sublattices. by using the Toda equation [16]. A more general result was obtained in Ref. [17] by a reformulation of the six-vertex model as a Hermitian matrix model to which III. THE CONNECTION WITH ORTHOGONAL saddlepointintegrationmethodwasapplied. Arepresen- POLYNOMIALS tation convenient for the largeN analysis was suggested also in Ref. [18]. Due to the multiplicativity of the partition function, Less is known about the correlation functions of this one can set c = 1 without loss of generality. Hence, the model for at least two reasons. First, the calculation of model is characterized by only two parameters, a and thecorrelators,ingeneral,isamorecomplicatedproblem b. In Fig. 4 the phase diagram on the (a,b) plane for than the calculationofthe correspondingpartitionfunc- the model with PBC is plotted (cf., Fig. 8.5 of Ref. [3]). tion. Second, the lack of translation invariance caused It may be regarded as the phase diagram for the model bythespecialboundaryconditionsintroducesadditional withtheDWBC,inthesensethatthefreeenergytakesa difficulties. Somecorrelationfunctionsfortheinhomoge- differentanalyticformintheregionsdividedbythesolid neous model (the model with the statistical weights de- and dashed lines (see Ref. [17] for details). Naturally, pendingonthepositionofthevertex)withspecialchoice the ground state and low temperature behavior do not of the weights were considered in Ref. [19]. coincide with those for the model with PBC. Theboundarypolarizationistheone-pointcorrelation function that describes the probability for the arrow on b 6 the fixed lattice edge on the boundary to be pointing in (cid:0) eitherdirection. Thesymmetryofthemodelallowsusto (cid:0) consider only vertical arrows. Let us denote by χ the N probability for the vertical arrow to be pointing down. Note that χN for PBC is independent of the position 1 (cid:0)@@q q q qqqqqq of the edge and is just a spontaneous polarization of the @@ qq (cid:0) q system[3]. Here,weshalldiscussχ fortheedgelocated @@ q N q atthelower-rightcornerofthelattice(inFig.3thisedge @@ qq (cid:0) is dotted): @@ qq @@(cid:0)q - ? ? ? ? ? 1 a (cid:27) - FIG.4. The phasediagram in terms of theweights a and (cid:27) - b. (cid:27) - In this and next section, we shall consider the model (cid:27) - on the solid line in Fig. 4. It is convenient to use the (cid:27) - following parametrization of the vertex weights on this 6 6 6 6 6 line: a=1/2 x FIG.3. χN is calculated for thearrow on thedotted edge. (cid:26)b=1/2−+x , −1/2<x<1/2. Itturnsoutthatifthearrowonthedottededgeispoint- It follows from Ref. [11] that the partition function of ing down,thendue to the imposedboundaryconditions, themodelisrepresentedasthedeterminantofanN N the allowed vertex configuration at the bottom and the matrix: × right boundaries is determined in a unique way. Actu- ally, in the lower-right corner one has the vertex of type detN Z = M , (2) c; thus, the rest of the 2N 2 vertices are of type b, N N 1 2 and the DWBC are valid for−the residual N 1 N 1 [φ(x)]N2 − k! sublattice. Thus, − × − (cid:18)kQ=1 (cid:19) where the matrix elements of are given by χN =cb2N−2ZZNN−1, (1) dα+k M = φ(x), α,k =0,1,...,N 1, (3) Mαk dxα+k − andtheproblemofstudyingthecorrelationfunction,χ , N is reduced to the analysis of the ratio of the partition and φ(x)= (1 x)(1 +x) −1. functions ZN 1/ZN. 2 − 2 − (cid:2) (cid:3) 2 Matrix is a Hankel matrix, that is, it has constant suggestedinRef.[22]andworkedoutfororthogonalpoly- M entries along the antidiagonals. Its determinant can be nomialswithnonanalyticweightsinRef.[23]. Usingthis expressed in terms of objects related to the theory of technique, we obtain the main result of the paper: orthogonalpolynomials. Let p (ξ), n 0, be a sequence n of monic orthogonalpolynomials with≥weight ρ(ξ): πe−1 lnh =2NlnN +2Nln +lnN N (cid:20)cos(πx)(cid:21) ∞ p (ξ)p (ξ)ρ(ξ)dξ =δ h , n,m 0. (4) 2π2 1 ϕ(x,N) Z n m mn n ≥ +ln + + + , (8) −∞ (cid:20)cos(πx)(cid:21) 4N 2N(lnN)2 ··· Then the determinant of the Hankel matrix with the el- where ements ϕ(x,N)=( 1)Ncos[2πx(N +1/2)] = ∞ ξα+kρ(ξ)dξ, α,k =0,1,...,N 1, (5) − αk H Z − −∞ and the omitted terms are of the order 1/[N(lnN)3]. iHsereequha0listode[t1e4r]mdineteNdHfrom=EhqN0.b(N14)−,1abnN2d−b2n··a·rbe2Nt−h2ebNco−e1f-. NobottaeintehdatinfoRrexf. [=23]0. this result coincides with the one Having expansion (8), it is easy to find the expansion ficients of the corresponding three-term recurrence rela- for χ : tion, N lnχ = pn+1(ξ)=(an+ξ)pn(ξ) bnpn 1(ξ), n 1, N − − ≥ π(1/2 x) π 1/2 x 2Nln − +ln − with the initial conditions p0(ξ)=1 and p1(ξ)=ξ+a0. − (cid:20) cos(πx) (cid:21) (cid:20)cos(πx)1/2+x(cid:21) The function φ(x) has the following integral represen- 1 ϕ(x,N 1) tation: − +.... (9) −12N − 2N(lnN)2 φ(x)= ∞ e xξe ξ/2dξ. − −| | FromEq.(9),wegetallincreasingtermsforthepartition Z −∞ function ZN, Combining this representation with Eq. (3) and Eq. (5), oneshowsthatdetNM=detNH,wheretheweightρ(ξ) ZNN∼ Cexp f0N2+f1N +f2lnN , (10) is equal to →∞ (cid:0) (cid:1) where C >0 is a bounded function of N, and ρ(ξ)=e xξe ξ/2. (6) − −| | π(1/4 x2) 1 Then, since the coefficients b and h satisfy the well- f0 =ln − , f1 =0, f2 = . n n (cid:20) cos(πx) (cid:21) 12 knownrelationbn =hn/hn 1,n 1,wehavedetN = h0h1 hN 1, and the desi−red re≥presentation for prMoba- bility·(·1·) is−[20] V. RESULTS AT OTHER POINTS (N!)2 χN+1 =a−2N−1b−1 . (7) At present there are several points on the phase di- h N agram, where the determinant has been calculated in Now let us discuss briefly the generalcase when a and closed form. b are arbitrary positive constants. The partition func- (i) The “free-fermion case”, a2+b2 =1 (dotted circu- tion is given by formulas (2) and (3), while φ(x) should lar quadrant on the phase diagram). On this circle, one be changed (see, for example, [6,11,12]), the weight ρ(ξ) has a very simple result ZN =1. This result can be ob- is obtained by straightforwardcalculations [17], and the tained by an appropriate limit from the inhomogeneous corresponding expression for χN+1 differs from Eq. (7) partition function [6,12]. Therefore, one has by a constant. lnχ =2(N 1)lnb. (11) N − IV. THE LARGE N LIMIT (ii) The “ice point”, a =b =1. All weights are equal, and the partition function is just the number of allowed configurations C (Sec. I). In this case one has Ref. [7] Equation(7) reducesthe problemof calculationofthe { } probabilityχN+1tothecalculationofthenormalizingco- N (3j 2)! efficient hN. At present there exists a powerful method ZN = − ; forstudyingtheN →∞behaviorofhN. Thismethodis jY=1(N −1+j)! based on the matrix Riemann-Hilbert conjugation prob- lem [21]. The corresponding asymptotic technique was thus, 3 16 1 16 5 1 5 1 lnχ =Nln ln + + + . (12) [1] E.H. Lieb, F.Y. Wu, in Phase Transitions and Critical N 27 − 2 27 36N 72N2 ··· Phenomena, edited by C. Domb and M.S. Green (Aca- demic Press, London, 1972), Vol. 1, p.321. For ZN we have asymptotic expansion (10), where f0 = [2] E.H. Lieb, Phys. Rev. 162, 162 (1967); Phys. Rev. ln3√43, f1 = 0 and f2 = −356. Note that for the model Lett. 18, 1046 (1967); Phys. Rev. Lett. 19, 108 (1967); with PBC [3], f0PBC = ln3√83 > f0, that is, for large N B. Sutherland,Phys.Rev.Lett. 19, 103 (1967). the number of allowed configurations for the DWBC is [3] R.G. Baxter, Exactly Solved Models in Statistical Me- less than for periodic ones. chanics (Academic press, San Diego, 1982). (iii) The point a=b=1/√3. In this case [7] [4] R.J. Baxter and A.L. Owczarek, J. Phys. A 22, 1141 (1989);M.T.Batchelor,R.J.Baxter,M.J.O’Rourke,and 2 C.M. Yung, J. Phys. A 28, 2759 (1995); K. Eloranta, J. 1 N (3j 1)! Stat. Phys. 96, 1091 (1999); C.B. Thorn, Phys. Rev. D Z2N+1 = 3N2  (N +−j)! , 63, 105009 (2001). jY=1 [5] V.E. Korepin,Commun. Math. Phys. 86, 391 (1982).   [6] V.E.Korepin,N.M.Bogoliubov,andA.G.Izergin,Quan- tum Inverse Scattering Method and Correlation Func- (3N 1)!(N 1)! Z2N = 3N[−(2N 1−)!]2 Z2N−1, [7] Gtio.nKsu(pCearmbebrrgi,dgInetU.Mniavtehrs.itRyesP.rNesost,icCeasm3b,r1id39ge(,11999963)).. − [8] D.M. Bressoud, Proofs and Confirmations; The Story of and we immediately get the Alternating SignMatrix Conjecture(CambridgeUni- versity Press, Cambridge, 1999). 4 1 27 1 1 lnχ =Nln + ln + ; (13) [9] H. Cohn, N. Elkies, and J. Propp, Duke Math. J. 85, N 9 2 4 − 18N ··· 117 (1996); W. Jockush, J. Propp, and P. Shor, e-print math.CO/9801068. thus, f0 = ln√23, f1 = 0 and f2 = 118. The first omitted [10] A. Hanke and M. Kardar, Phys. Rev. Lett. 86, 4596 terminEq.(13)isoftheorderN−2,anddependsonthe (2001); A. Hanke, Phys. Rev. Lett. 84, 2180 (2000); parity of N. A.Hanke,M.Krech,F.Schlesener,andS.Dietrich,Phys. We complete this paper with the following statement. Rev. E60, 5163 (1999). One can see that in expansion (9), which was obtained [11] A.G. Izergin,Dokl.Akad.Nauk(SSSR)297, 331 (1987) for a+b = 1, there exists a logarithmic term. For arbi- [Sov. Phys. Dokl. 32, 878 (1987)]. trarya andb, we state that the logarithmic terms in the [12] A.G. Izergin, D.A. Coker, and V.E. Korepin, J. Phys. A largeN expansionoflnχ willtakeplaceonlyunderthe 25, 4315 (1992). N following conditions: weights a and b are either on the [13] A. Lascoux, Séminaire Lotharingien Combin. 42 Art. solid line a+b = 1, or on the dashed lines a b = 1 B42p, 15pp. (1999). of the phase diagram(Fig. 4). For the model|wi−th P| BC, [14] C. Krattenthaler, Séminaire Lotharingien Combin. 42 the logarithmic terms could be explained via conformal Art. B42q, 67pp. (1999). field theory [6]. Similar arguments are valid for the dis- [15] V.E. Korepin and P. Zinn-Justin, J. Phys. A 33, 7053 (2000). cussed model. The influence of the boundary condition [16] K.Sogo,J.Phys.Soc.Jpn.62,1887(1993);A.G.Izergin, changing operators [24] should also now be taken into E.Karjalainen,N.A.Kitanine,J.Math.Sci.(NewYork) consideration. If we move awayfrom the solid or dashed 100, 2141 (2000). lines, the logarithmic terms vanish and the expansion of [17] P. Zinn-Justin, Phys.Rev.E 62, 3411 (2000). lnχ will be in integer powers of N. Expansions (11), N [18] N.A.Slavnov,Zap.Nauchn.Sem.POMI269,308(2000) (12) and (13) indeed contain only integer powers of N, (available at cond-mat/0005298). thus confirming the above statement. [19] V. Korepin and P. Zinn-Justin, e-print nlin.SI/0008030; Finally, we would like to mention that the six-vertex J. deGier and V.Korepin, e-print math-ph/0101036. model with any boundary conditions can be considered [20] N.M.Bogoliubov,A.V.Kitaev,andM.B.Zvonarev,Zap. as a model for a description of interface roughening of a Nauchn.Sem. POMI 269, 136 (2000). crystal surface [25]. An important point in these stud- [21] A.S. Fokas, A.R. Its, and A.V. Kitaev, Commun. Math. ies is the existence of exact analytical results, which are Phys. 142, 313 (1991); 147, 395 (1992). known for the six-vertex model with PBC [1–3] . We [22] P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Ve- believe that our analytical results for the model with nakides, and X. Zhou, Int. Math. Res. Notices 16, 759 DWBC provide one more basis for the experiments and (1997); Commun. PureAppl. Math. 52, 1491 (1999). simulations in this direction. [23] T. Kriecherbauer and K. T-R McLaughlin, Int. Math. We thank A. H. Vartanian for bringing Ref. [23] to Res. Notices 6, 299 (1999). our attention. This work was partially supported by the [24] J.L. Cardy, Nucl.Phys. B 342, 581 (1989). RFBR Grant No. 01-01-01045. [25] H.vanBeijeren,Phys.Rev.Lett.38,993(1977);E.Car- lon, G. Mazzeo, and H. van Beijeren, Phys. Rev. B 55, 757 (1997); 4