Table Of ContentBoundary 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 re-
sult 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.
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(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
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for a+b = 1, there exists a logarithmic term. For arbi- [Sov. Phys. Dokl. 32, 878 (1987)].
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N
following conditions: weights a and b are either on the [13] A. Lascoux, S´eminaire 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, S´eminaire 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).
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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);
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