PHYSICA d ELSEVIER Physica A 228 ( 1996) 1-32

Interacting dimers on the honeycomb lattice: an exact solution of the five-vertex model

H.Y. Huang a, F.Y. Wu a, H. Kunz b, D. Kim” a Depurtment of Physics, Northeastern University, Boston, MA 02115. USA

’ lnstitut de Physique ThPorique. EC& Polytechnique F~dtkle, Lnusunne, Switzerland

’ Center for Theoreticnl Physics, Seoul National Universi~, Seoul, South Koretr I51 -742

Abstract

The problem of close-packed dimers on the honeycomb lattice was solved by Kasteleyn in 1963. Here we extend the solution to include interactions between neighboring dimers in two spatial lattice directions. The solution is obtained by using the method of Bethe ansatz and by converting the dimer problem into a five-vertex problem. The complete phase diagram is obtained and it is found that a new frozen phase, in which the attracting dimers prevail, arises when the interaction is attractive. For repulsive dimer interactions a new first-order line separating two frozen phases occurs. The transitions are continuous and the critical behavior in the disorder regime is found to be the same as in the case of noninteracting dimers characterized by a specific heat exponent CY = l/2.

1. Introduction

An important milestone of the modern theory of lattice statistics is the exact solution

of the dimer problem obtained by Kasteleyn [ l] and by Fisher [ 21. Kasteleyn and

Fisher considered the problem of close-packed dimers on the simple quartic lattice and

succeeded in evaluating its generating function in a closed-form expression. While the

solution shows that close-packed dimers on the square lattice do not exhibit a phase

transition, Kasteleyn [ 31 later pointed out that dimers on the honeycomb lattice do

possess phase changes, and that the transitions are accompanied by frozen ordered

states. The solution, which has since been analyzed by one of us [ 4,5], can be used to

describe domain walls in two dimensions [6,7].

In this paper we consider once again close-packed dimers on the honeycomb lattice,

but now with the introduction of interactions between neighboring dimers along two

0378-4371/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved PII SO378-437 I (96)00057-X

2 H.Y Huang et ul./Phgsicu A 228 (1996) 1-32

lattice directions ’ . We show that, with the onset of dimer-dimer interactions, a new

ordered phase emerges if the interaction is attractive. For repulsive interactions the phase

diagram is drastically changed and a tricritical point emerges. We deduce locations of

all phase boundaries and study its critical behavior.

We analyze interacting dimers by first converting the problem into a five-vertex

model. For noninteracting dimers this leads to a free-fermion model [S] which can

be solved by using the method of Pfaffians [ 1,4,5]. But when interactions are present

the five-vertex model has general vertex weights and the method of Pfaffians is no

longer applicable. While its solution is in principle obtainable from that of the six-

vertex model by Sutherland, Yang and Yang announced in [9], details of [ 91 have

not yet been published. Likewise, recently published analyses of the general six-vertex

model in the regime A < 1 by Nolden [lo] and in the regime A > I by Bukman

and Shore [ 11,121, where A is a parameter occurring in the six-vertex model, do not

readily translate into the five-vertex problem since the five-vertex model corresponds to

taking the IAl -+ cx limit. In fact, it is precisely because of this special situation that the

analysis of the five-vertex model as a limit of the six-vertex model requires special care.

To bc sure, several authors [7,13] have recently studied the five-vertex model. But the

live-vertex model considered in [ 131 is confined to a special regime of the parameter

space which dots not yield the complete complexity of the system. The treatment in

[ 71, which was aimed to studying domain walls, is more complete but analyzes the

Bethe ansatz solution along a line somewhat different from what WC shall present, and

is not transparent in extracting relevant information on the dimer system. It is therefore

useful to have an alternate and self-contained analysis of the five-vertex model in the

language of dimer statistics.

We take up this subject matter in the present paper. Our approach is essentially

that of [ 91, by considering the solution of the Bethe ansatz equations in the complex

plane. However, we follow the Bethe ansatz solutions closely and explicitly carry out

all relevant contour integrations as dictated by relevant physical considerations. This

leads to a complete and clear picture of the phase diagram and critical behavior of the

interacting dimer system. Particularly, we find the emergence of a new ordered phase

for attractive dimer-dimer interactions, and the existence of a first-order lint terminating

at a new kind of tricritical point, when the interactions are repulsive.

The organization of our paper is as follows. The problem of interacting dimers is

defined in Section I and mapped into a five-vertex model in Section 2. The Bethe ansatz

equation is set up in Section 3, and solved in Section 4 in the case of noninteracting

dimers. In Section 5 we analyze the general Bethe ansatz equation, obtaining expressions

for the free energy and its derivative. This leads to the determination of the contour of

integrations in Section 6 and the complete phase diagram in Section 7. Finally, the

critical behavior is determined in Section 8 by applying perturbation calculations to the

’ Some thirty-five years after the publication of Kasteleyn’s original work I I I in Physica A, it appears fitting

to dedicate this paper to Hans W. Capel. Editor of Physica A since 1974, on the occasion of his sixtieth

birthday.

H. E Huang et al. /Physica A 228 (I 996) 1-32 3

Fig. I. The honeycomb lattice drawn as a brick-wall lattice showing relative positionings of the u, P. and w dimers. The dotted boxes correspond to vertices of the square lattice.

WI Q2 w3 w4 us Qa

Fig. 2. The six vertex model and the associated weights.

free energy.

2. The five-vertex model

Consider close-packed dimers on an honeycomb lattice L: which we draw as a “brick-

wall” shown in Fig. 1. To each dimer along the three edges incident at a vertex, one

associates a fugacity, or weight, U, U, or w. A vertical u dimer and a horizontal u dimer

are said to be neighbors if they happen to occupy two neighboring sites in the same row.

Let two neighboring u and u dimers interact with an energy --E and thus possessing a

Boltzmann factor

Jsi = l@T, (2.1)

with A > 1 (A < 1) denoting attractive (repulsive) interactions. Other pairs of dimers

such as U-U, U-W, etc., are not interacting in our model. Then, by replacing the two sites

inside each dotted box containing a w-edge in Fig. 1 by a single vertex, and regarding

a dimer incident to this vertex as a bond covering the corresponding lattice edge, the

honeycomb lattice C reduces to a simple quartic lattice, and dimer coverings on L lead

to vertex configurations of a five-vertex model [4]. Configurations of the five-vertex

model are shown in Fig. 2 in the context of a six-vertex model. It is straightforward to

verify that we have the correspondence

{WI, ‘3-‘2,6?3. w4, US, 0-‘6} = (0, w, u, u, &, Jhuo}. (2.2)

Here, for definiteness, we assume {u,u, w} > 0. Note that (2.2) is the most general

five-vertex model, since one can always take & = m, if wg $ w,5.

The partition function of a vertex model is defined as

z = c n ‘-k(v),

config vertices

(2.3)

H. Y Huung et al. /Physiw A 228 (1996) i-32

(a) (b)

Cc) Cd)

Fig. 3. The four possible ordered states. (a) The U phase, with II = 0 or cy = 0 and II dominating. (b) The

1’ phase, with II = N or CY = I and I’ dominating. (c) The IV phase, with II = N or (Y = I and II’ dominating.

(d) The ,I phase, with II = N/2 or u = l/2 and A dominating.

where the summation is taken over all allowed vertex conligurations, the product is over

all vertices of the square lattice and wkCl,) stands for the weight of a vertex C. The case

of E = 0 or A = I leads to the free-fermion model satisfying the free-fermion condition

WIW2 + w3w4 = w5wfl [81. For a simple quartic lattice of size M x N, one defines the per-site free energy

~(u,L~,MJ;~) = limcu(MN)-‘InZ, (2.4) . -

for the five-vertex model. It follows that the per-site generating function for the dimer

problem defined in a similar way is f/2.

Ordered states. It is instructive to examine the possible ordered states of the dimer lattice.

When 11, ~1, or w dominates, the ordered states are those shown respectively in Figs. 3a,

3b. and 3c, where the lattice C is completely covered by 11, I!, or nl dimers. These arc

the ordered states occurring in the free-fermion case [4]. But when A dominates (large

positive E), a new ordered state can materialize as shown in Fig. 3d. It is this ordering

that adds to new features to the interacting dimer system.

3. The Bethe ansatz equation

To begin with, consider the general six-vertex model on a simple quartic lattice of A4

rows and N columns with periodic boundary conditions in both directions. Applying a

transfer matrix in the vertical direction and using the fact that ~1, the number of empty

edges (those not covered by bonds) in a row of vertical edges, is conserved, one can

evaluate the partition function (2.3) using the Bethe ansatz [ 141. The Bcthe ansatz

formulation for the general six-vertex model has been given in [ 9,151 in a ferroelectric

language from which the five-vertex limit does not follow straightforwardly. Here, for

completeness, we state the Bethe ansatz equation for the six-vertex model in terms of

the vertex weights [ 161.

H.l: Huang et al./Physica A 228 (1996) 1-32

In the limit of large M, N, one finds

with

n

A,(n) =liq--n n( W3@4 - @SW6 - wl W3 Z,i

,j= I W4 - WI Z.j It

n N--n

h.(n) = w4 n(

-1 WI W2 - WgW6 - w2w4Z,i

-1 .+I @I - w4Z,j I

s

(3.1)

(3.2)

for II 5 N, where the II complex numbers ZJ, j = 1,2, . . . . n are the solution of the Bethe

ansatz equation

II

z.,! = (-1y+’ n( B(Zi3 Z,j)

) ;=, B(Z,j* Zi) ’ j= 1,2 ,..., II, (3.3)

with

B(z,z’) =w2w4+w,w3zz’ - (WlW2 + w3w4 - w$og)z’. (3.4)

Note that for fixed 1 < n 5 N, one has generally (:) different n,(n) and A,(n). It is

understood that it is the largest ones for each n that are used in (3.1). We remark that

a useful parameter occurring in the analysis of the six-vertex model is

A = (~16~2 + ~3~4 - wswg)/2&,W@3w4.

It is then clear that /AI + cc in the limit of wt + 0, making the five-vertex model a

very special limit.

Specializing (3.2),(3.3) to the five-vertex model weights (2.2), one obtains

AR(n) = (pdN&,N,

A/*(n) =LiN fi(-Xl +X2Z.j), (3.5) i=l

where the ii’s are to be determined from the Bethe ansatz equation

z,,! = (-I)“+’ fi(s), j= l,Z...,n, (3.6)

with

W XI = -, x2 = iA, p= $1 --A). (3.7)

11

It is clear that AR does not contribute unless n = N. But for n = N the partition function

can be trivially evaluated. In this case there are no u dimers and hence each row of C

is covered completed by u or w dimers and one has

H.Y. Huang et al./Phy~i~u A 22X (1996) 1-32

z = (w” + IINy, f( I(, 13, w; A) = max{In n’, In u}. (3.8)

Alternatively, one can show from (3.5) and (3.6) that &(N) + A,,(N) = nlN + 11”

from which (3.8) also follows. Hence from here on we consider AL only.

Combining (2.4) and (3. I ), one has

(3.9)

where

,f(n) = lim N-’ InAL( (3.10) N+CX

The prescription of the Bethe ansatz is that. for each fixed II, one solves (3.6) fat

L,. This leads to generally many sets of solutions. One next picks the set of solution

which gives the largest .f(n) for each II. Then, the free energy (3.9) is given by the

largest ,f(n) among all II. We shall refer to the set of Z, which gives rise to the final

expression of the partition function (3.9) as the maximal set, and to the prescription of

maximization as the maximal principle.

We first point out some immediate consequences of (3.6). First, it is clear that if :,

is a solution of (3.6), then its complex conjugate zf is also a solution’ so that the zj’s

arc distributed symmetrically with respect to the real axis. Secondly, for N = even at

least. the negations of p and ij leave (3.6) unchanged. Thus, if z,i is a solution of (3.6).

then -:, is the solution when p is replaced by -/?. Combining these two observations,

we lind the solutions for p and -p related by a simple reflection about the imaginary

axis. Finally, multiplying the II equations in (3.6), one obtains the identity

[ 1 fJz, N=l. FI

(3.1 I)

4. Noninteracting dimers

It is instructive to apply the Bethe ansatz consideration to the free-fermion case ol

141. In this free-fermion case we have {xi ,x2} = {~v/u,~J/u}. A = I, p = 0, and (3.6)

becomes

‘, N = (-I)‘+‘, j= I,2 ,..., II. (4.1)

Thus the :i’s are on the unit circle 1: 1 = I, and can take on any II of the N roots of

(4. I ). This gives (r) eigenvalues n,(n) as expected.

’ This has the consequence that one must write ; = 1; /& with -r 5 q$ ( ?r, implying that all branch cuts

muqt be taking along the negative real axis. This observation plays a major role in ensuing considerations.

H.l! Huung et ul./Physica A 228 (1996) 1-32

Fig. 4. The maximal contour (heavy curve) for noninteracting dimers

For fixed XI, x2, the maximal set of z,i which gives the largest ,4,(n) for each n is

obtained by choosing the II largest 1x1 + xzzjl. Now for any IZ write

zi = c’@J, (Y = n/N. (4.2)

In the limit of large M, N, the zj’s are distributed continuously on the unit circle with

a uniform angular density N/27r. For fixed LY write za = eia”, the n largest 1x1 + xzzil

are those given by the z,i’s on the arc of the circle extending from z,* to in as shown in

Fig. 4. One then obtains from (3.5) and (3.10) after replacing II by LY in the argument,

In

f(a) =lnu+& ln(xt +.x~z)$. J * 7” (4.3)

The maximal free energy is therefore, after using (3.9),

f(z*, u, w; 1) = ?a; f(a) a

= f(ao> aor

=lnu+ & J ln(xi + x@)dB, (4.4)

-*olr

where LYO is determined by

f’(a0) = lnjxt + .xzz~~] = 0. (4.5)

If 1 ,ni, x2, or equivalently u, u, w, form a triangle, we have 0 < LYO < 1 and f =

f( aa) analytic in U, u, w. If 1, x1, x2, or equivalently u, u, w, do not form a triangle, then

there are two possibilities. For xi +x2 < 1 or w + u < u, we have 1x1 + XIzjl < 1 for

all z,i, and as a consequence the maximal set is the empty set or, equivalently, CYO = 0.

This leads to

f(u,u,w;l) =Inu, n> W+U. (4.6)

For 1x1 - x21 > 1 or 1~ - cl > U, we have 1x1 + x2z,il > 1 for all z.i, so that we take

the maximal set CYO = 1 and zno = eirr. This leads to, after carrying out the integration in

(4.4))

H. Y Huang ef al. /Physica A 228 (I 996) 1-32

f(U,u, w; I) =Inu + maxln{x2,xl}

=max {lnc, In w}, II < Iw - ~‘1. (4.7)

Thus, the phase boundary is

/s, f x21 = I or Iw f ~1 = II. (4.8)

These results are in agreement with [4]. The ordered states (4.6) and (4.7) arc the

frozen states shown in Figs. 3a-c in which the dimer lattice C is completed covered

by II,~‘, or w dimers. The transitions arc of second order. We note, in particular, that

the phase boundary (4.8) is determined by setting f’( cua) = 0 at LYO = 0, or I, the two

points where the path of integration in (4.3) either just emerges or completes a closed

contour. The observation of this mechanism underlining the onset of phase transitions

proves useful in later considerations.

5. Analysis of the Bethe ansatz equation

We now return to the Bethe ansatz equation (3.6). Define a constant C ( LY, p) by

[ 1 N

C(Q) = (-I)"" fi(l - PZj),

;=I (5.1)

with +!I~) E argC lying in interval (-n-/N, r/N]. Then the Bethe ansatz equation (3.6)

becomes

[- 1 N

C(a,/?) ‘7; = (I - &YN, (5.2)

and the Nth root of (5.2) gives a trajectory r on which all solutions X, must reside,

C&’ =, = ( I - &)Q?,, di = 2n-j/N, j = 1.2 ,..., N, (5.3)

where C = C(a, 0) s Ic(~,p)I. The trajectory f is a curve in the complex ; plane

which is symmetric with respect to the real axis and is given by the equation

c/z1 = jl -Pzl” (0. (5.4)

However, by diagonalizing the transfer matrix explicitly for N < 18, Noh and Kim

[ 171 have found that for p > I the largest eigenvalue assumes the “bounded magnon”

ansatz of Noh and Kim [7] in the form of

I Z, = - p , ,j=2,3 t... >?I, (5.5)

H. I: Huang et al. /Physica A 228 (1996) 1-32 9

(4 (b) (cl

Fig. 5. Possible contours and the corresponding solutions of (5.4) for p > 0. (a) rl > d,. (b) d = c&.. (c)

tl < &

where j.?Jij remains finite as p + 00. This says that in the thermodynamic limit of

N + cc and (Y # 0, 1, one root ~1 resides at infinity while all other roots converge to

I/p. Using this ansatz, one obtains from (5.2)

c =o, a#O,l, p> 1, (5.6)

and from (3.5) and (3.10)

f(a) =(1 -a)Inu+culnu, ~y#O,l, /3> 1. (5.7)

While we shall make use of (5.7) to determine the phase boundary, however, to make

our presentation self-contained we shall proceed for the time being without using the

ansatz (5.5) and the result (5.7). It will be seen that one is led to the same phase

boundary for fl > 1.

Define

d E C/ljil, d, 3 a”( 1 - a)‘--a. (5.8)

Then, by examining solutions of (5.4) on the real axis, it is straightforward to verify that

for p > 0, r assumes the topology shown in Figs. 5a-c, respectively, for d > d,, d = d,,

and d < d,. The topology of r for p < 0 is deduced from those in Fig. 5 by applying a

reflection about the imaginary axis, and is shown in Fig. 6. Note that the constant C can

be determined once one point on r is known. Particularly, if r intersects the negative

real axis at .X = -R, where R > 0, then we have

C = h(a, R) = / 1 + PRI”‘IIRI. (5.9)

Note that, since contours can be deformed as long as they do not cross poles and move

along branch cuts, it is the topology of the contours that is important.

In the limit of large N, the distribution of zeroes on r becomes continuous. Let p(z)

be the density of zeroes so that N s p(z)dz over any interval of f gives the number of

zi’s in that interval. Then, using (5.3), one finds

IO H.): Huang et ul./Pky~icu A 228 (1996) l-32

(a) (b) (c)

Fig. 6. Possible contours and the corresponding solutions of (-5.4) for fi < 0. (a) tl > tlC. (b) rl = tl, (c)

tl < &

p(z)=& .!- ( z z -*p-t 1 . (5.10)

Let ra be the segment(s) of r symmetric with respect to the real axis and on which

the maximal set of z,i’s resides. For fixed CY, we find as in (4.3)-(4.5),

1 .i( 1 ff ,f(cu) =Inu+- - - ~ In(xt + x2z)dz,

25-i 7. 2 -0-l ) I ‘II

(5.11)

f(u,r:, w A) = f(LYO), (5.12)

where q is the value of LY which maximizes f(a).

Note that the a-dependence of the free energy (5. I 1) for fixed cy now enters through

both ro and p( 7,). However, they obey two constraints. First, the fact that there are II

zi’s implies

(5.13)

In addition, taking the absolute value of (3.1 I), one obtains

n-.- 1, ‘,.I - 1‘0

(5.14)

which leads to, in the N ----t cc limit,

(5.15)

For each fixed LY, the two constraints (5.13) and (5.15) together with the maximal

principle of the free energy are sufficient to determine To. Once TO is known, the free

energy can be evaluated using (5.1 I ) and (5.12). In carrying out integrations along

1’0, one is aided by the fact that the path of integration can be deformed as long as

it does not cross poles nor run along branch cuts. Care must be taken, however, when

TO intersects branch cuts. When this happens, integration along TO can be computed by

completing the contour into a closed loop and using the Cauchy residue theorem. Details

of relevant integrations are straightforward, and several useful formulas are collected in

Appendix A for ready references.

H.Y Huang et al./Physica A 228 (1996) l-32 I I

In our discussions below we shall also need to evaluate f’(a). Using (5.11) for

f(a), generally the LY dependence comes in through both the path rn and the explicit

dependence of p( 2) on cr. Let I-0 consist of an open path running continuously from z;

to zo and, in addition to the open path, possibly another closed contour intersecting the

real axis. Since TO can be freely deformed except the terminal points and the intersection

points with the branch cut, the derivative of (5.11) with respect to (Y is derived from

three kinds of contributions. First, the two terminal points ~0 and z; will move with LY.

We write

ia I Raei’, y = p(z0) (dzo/da) 3 (5.16)

and, due to the fact that the expression (5.10) for p(z) contains a factor 1/2rri, we

have

Y* = -p(7$)(&/da). (5.17)

Then, the contribution to f’(a) due to the a-dependence of the terminal points gives

rise to yln(xt + ~2~0) + y* In(xt + x26). Secondly, if TO consists of another closed

contour intersecting the branch cut in (5.11) at one or two points zr < 0, then since the

integration path can be deformed, the a-dependence is through the intercepts zr only

(which now moves along the branch cut), the contribution due to zr can be treated as

in the above. Namely, we regard two points just above and below zr as two terminal

points. This leads to a contribution of 21ri C,. y,, where

yr = p( z,) (dz,/dcu) = pure imaginary = -y,*, (5.18)

and this contribution is the same for all branch cuts. Thirdly, there is a contribution due

to the second term in p( z ) as shown in (5.10). Combining the three, one obtains

f’(a) = y ln(xl + ~2~0) + y* ln(xt + x2$)

(5.19)

where the 3~ sign is determined by the orientation of TO at the intercepts.

Finally, it is clear from (5.1 I) and (5.19) that, in carrying out the contour integrations

for f(a) and f’( (u), it is important to determine the location of the branch point -xi /x2

relative to the contour TO. First, from the readily verified identity

one locates the branch point -xl/x2 by

x,/x:!>(l -p>-‘, w>u

<(l-p)-‘, w<u,

We remark that from the inequality

XI/X2 < IPI-‘? P < 0,

(5.20)

(5.21)

(5.22)

12 H. Y Huang er 01. /Physicu A 228 (I 996) I-32

one has (see below) for Fig. 6c the inequality R > RI > I/Z-’ > x1/x2, implying that

the points x = -R, -RI are both on the branch cut. We further relate c, yr, y, and y*

by taking the derivatives of (5.13) and (5.15) with respective to cx, obtaining

yInza+.v*lnzo* +27riC(ty,) = & J Adz.

r z - P-’ I’ ”

(5.23)

(5.24)

It now follows from (5.19)-(5.24) that we have

f’(a) = yln (:+x2) +y*In(%+.xZ) +&/‘nz-zln(~:X2z)dz,

Ii)

(5.25)

provided that y,.‘s in (5.19) and (5.24) are the same, namely, ra cuts both branch cuts

at the same points. In Section 7 we shall compute f’(a) using (5.25) which applies

to all cases and all cr, including the case that Ta consists of an open path as well as a

closed contour. The derivative f’(a) near the phase boundaries will also be computed

in Section 8 by analyzing small perturbations of the free energy.

6. The contour r at (Y = O,l--, 1, and l/2

In our discussions we shall need to evaluate integrals at aa = 0, I -, I, and I /2. It

turns out that the contours r for (Y = 1 and cx = I- should be considered with care. In

this section we consider the contour f at these special points.

(a) (Y = 0. This is the case that TO begins to emerge with very few zi’s. The constraint

(5.14) then dictates that za m I and, using (5.4) and (5.14), one finds c(O, /3) = I

and r the unit circle.

(b) Q = I. This is the case of II = N when one picks all N z,‘s and hence TO = r. One

can use (5.2) to obtain

[ CN(l$-(-P)N 1 z;y+...- I =o. (6.1)

Using (6. I ), the constraint (5.14) leads to the relation

CN(I$) = I -PN, ~ I

(6.2)

or, in the thermodynamic limit,

C(l,P) = I, IPI < I,

=lPI? IPI > 1. (6.3)

H.Z Huang et ul./Ph.ysica A 228 (1996) 1-32 13

One also finds from (5.8) that d, = 1 for (Y = I, and hence

d>d,, IPI < 1,

= d,., IpI > 1.

The trajectory TO is, from (5.4) with C = C( 1, ,B),

(6.4)

(6.5)

~ I i-P =I, IPI<l, (6.6)

1 I I - - 1 = 1, Pz

IpI > 1. (6.7)

The contour (6.6) for I/? < 1 is a circle of radius ( 1 - p*)-’ centered at x =

-p/( 1 - /3*) on the real axis, where x = Re( z). Particularly, the circle intersects the

real axis at x = -R2, RI, where

R2=(l-P)-‘, R,=(l+p)-‘. (6.8)

The contour (6.7) for IpI > 1 is the vertical straight line x = (2p) -‘.

It is readily verified that by integrating along the contours (6.6) and (6.7), one has

1 _ Jp(z) lnzdz =O, 2ri

I’

(6.9)

=-InlPI, IPI > 1. (6.10)

Thus, the constraint (5.13) is not satisfied for IpI > I, indicating that the cy = 1 solution

is spurious, a conclusion not surprising since (3.8) implies that one cannot consider ilk

alone when n = N. This leads us to consider instead the LY = l- solution as a limit to

oJ= 1.

(c) Ly= l-.

We consider the cases p > 0 and /3 < 0 separately.

For p > 0, the contours are those shown in Fig. 5. In all cases, since the branch cuts

are on the negative axis, the contour can be deformed to form a single closed contour

intersecting the real axis at two points and enclosing the origin. Let the intersecting

point on the negative real axis be x = -R. Using (A.5) and (A.6) in Appendix A, one

finds from (5.15) the relation

In h( LY, R) = 0, (6.1 I)

where (Y = I-, h(a, R) is defined in (5.9) and is equal to C. Thus, one obtains C = 1.

In addition, one can solve for R from (6.11) and obtains

R= (1 -p)-‘, p< 1,

R = Ipln’c’-n) -+ co, ,f3 > 1. (6.12)

Thus, the intercept with the negative axis R for p < 1 is the same as that given in (6.8).

Once C = 1 is known, one can compute the location of other intercept(s) with the real

14 H. I! Humg et cd. /Physica A 228 (I 996) 1-32

axis which will bc all positive including the intercept ( 1 + ,0-’ given by (6.8). Note

that the second line of (6.12) (and what follows) is what one would obtain without

making use of the bounded magnon ansatz (5.5). In this case the precise location 01

the other two intercepts (for p > 1, see below) does not concern us since the contour

can be deformed freely in the x > 0 plane as long as it does not cross the pole at p-‘.

For /3 < 0 the contours are those shown in Fig. 6. Consider first Fig. 6a where the

contour intersects the negative real axis at one point at x = -R, one finds again C = 1

and the two intercepts (6.8) with R = ( I - p)-‘. In the cast of Fig. 6c where I’

intersects the real axis at four points as shown with -R < -RI < -R* < R3, generally

the contour cannot be deformed into a single loop due to the presence of the branch

cuts. But the integration (5.15) can be carried out as in the above, yielding

ln]I(Q’, RI) + Inh(cu, R2) + lnh(cu, R) = 0, (6.13)

where

h(cu, R) = h(a, RI) = h(a, R2) = h(a, -R3) = C. (6.14)

The identities (6.13) and (6.14) can hold only for C = I,

&=(JPI-I)-‘, &=(JPJ+l)-‘3 (6.15)

which is the same as (6.8)) and both R and R3 diverging as in the second line of (6.12).

Note in particular that R2 coincides with R = (1 - p)-’ in (6.8) of Fig. 5a. Now WC

have again from (5.8) cl, = I for LY = I-. Thus, in contradistinction to (6.3), one finds

C( I-, /3) = I for all p and therefore

d = IPI-’ > cl0 IPI < 1, (6.16)

cd,, IPI > I. (6.17)

Thus, the contour r consists of two loops when IpI > I, with the outer loop residing

in the inlinite regime.

We note that while (6.16) is the same as (6.4), (6.17) is different from (6.5).

Finally, one verifies that, by deforming r into a closed contour enclosing the origin but

not p-‘, the constraint (5.13) is identically satisfied for all p.

(d) cy= l/Z.

We are primarily interested in the free energy and its derivatives near a phase transition

point which, as in the nonintersecting case, occurs when the maximal contour either

closes or just begin to emerge. Thus, for a = l/2, we consider the cases shown in Figs.

5c and 6c when the contour r consists of two loops and TO is one of the two loops. But

this cannot happen for 0 > 0. When p > 0, the inner loop and portion of the outer loop

arc in the x > 0 half plane. Then TO in the maximal solution of the free energy (5.11 )

cannot be either of the two loops, since some points on the other loop will have larger

values of ln(x’ + x22). Hence (Y = l/2 can occur only for p < 0. This conclusion is

also expected on physical grounds, that the LY = l/2 ordered state in Fig. 3d dominates

H. Y Huang et al. / Physica A 228 (I 996) I-32 IS

only when the interaction between neighboring u and u dimers is sufficiently attractive,

namely, when A is sufficiently large or p sufficiently negative.

For j3 < 0 the contour r intersects the negative real axis (and the branch cut of In z ) at three points. One verifies after some algebra the identity

s p(z)dz = aln(z0( #a,

inner loop

for any LY. Therefore TO must be the outer loop.

Taking the outer loop as ru, we have firstly from the constraint (5.13),

1

S(

1 Ly --~ dz=l-(Y, cr=G z z-P-’ )

r0

leading to the correct value (Y = l/2. The constraint (5.15) now yields

o=& J( 1 112 --~ lnzdz 2 Z-P-’ >

lb

=lnR- iln(R+P-‘),

or, equivalently,

R*-&fl-‘=O.

This yields the solutions

(6.18)

(6.19)

(6.20)

Thus, we see that, as expected, (6.20) has a solution only for sufficiently negative

p < -4.

Furthermore, when ru is a closed contour, one has from (5.9), (5.8) and using

(6.20),

C(&P)=JIpI, dc=& (6.21)

and

d=-+zgd,. PC-4. (6.22)

Thus, r consists of two loops only when /3 < -4. In this case r intercepts the real axis

at the four points -R < -RI < -Rz < R3 as shown in Fig. 6c and determined from

C1.r = dm. This leads to

(6.23)

I6 H. K Huang et al. /Physica A 22X (I 996) l-32

In summary, we have found that, for LY = I-, we have C = I and I- consists of one

loop for IpI < 1 which intersects the real axis at (6.8), and two loops for 101 > I

intersecting the real axis at 4 points. In the latter case when p > I, the contour can be

deformed into one loop enclosing the origin and therefore the situation is the same as for

IpI < I. In the case of p < - I the outer loop resides in the infinite regime. For p < I,

the intercept of r on the negative axis and closest to the origin is at x = -( I - /3) _I.

For (Y = l/2, one finds C = m and that, for fl < -4, I- consists of two loops

intersecting the real axis at the four points given by (6.23).

7. The phase diagram

Ideally one would like to proceed at this point to compute the free energy (5.12) from

which the complete thermodynamics of the dimer system can be determined. However,

as this evaluation involves path integrations which generally cannot be put into closed

forms, we shall in the next section apply small perturbations to the free energy near

the phase boundaries. We proceed here to first determine the phase boundaries and the

phase diagram.

Guided by the analysis of the A = I solution of Section 4, we expect singularities

of the fret energy (5.12) to occur when either the maximal path fo contains a small

emerging segment or when it completes a closed contour. As in the case of A = I, this

will happen at (~0 = 0 (segment emerging) and LUO = 1 (closed contour) leading to the

ordered states shown in Figs. 3a-c. In addition, we expect another singularity of the free

energy to occur at LXY~ = l/2 for p < -4 as TO completes the outer loop of two closed

contours, leading to the order state shown in Fig. 3d. In all these cases the free energy

,f( CJ) should be the maximal solution at CY = cyo = 0, I, or I /2, and the phase boundary

is given by f’(q) = 0 provided that f(a) is concave at eye. Once the contours To

is known, both the free energy (5.1 I) and its derivative (5.25) can be evaluated at

cy = 0, I, I /2.

Consider first cy = 0. This is the case that re begins to emerge at :a = Roe” - I and

?‘+.v* = 1. It then follows from (5.1 1) and (5.25) that we have, for 0 < 1 at least,

,f(O) =lnn,

f’(O)=ln !i!f-?% ( > 14 (7.1)

For cy = I, (or more precisely cy = I -) and l/2, both integrals (5.1 I ) and (5.25)

can bc evaluated using the contours TO determined in the preceding section. We leave

details of the evaluations elsewhere, and collect here the results.

For LY = I one finds, for /? > 1,

f(1) =lnl,,

,f’(l) =In(L,/N), (7.2)

H.Y Huang et al./Physica A 228 (1996) 1-32 17

and, for p < 1,

f(I) =max {lnw,lno},

f’( 1) =ln (

w(w - u)

WU - UU( 1 - A) > ’ w>u

v-w =ln -

( > Au ’ v > w.

For CY = l/2 one finds, for p < -4,

f(i*) =iln(Auu),

f’({*) =ln[c(l - $)*I,

(7.3)

(7.4)

where Ri is given by (6.20).

To ensure that the free energy f( cq) is indeed the maximal solution and that f’( aa) =

0 is a phase boundary, we need to ascertain that f(q) is a maximum. This turns out

to be a delicate matter for ‘~0 = 0, 1, as detailed calculations show that f”(q) = 0.

However, one can proceed as follows.

First consider the case p > 1. As discussed in Section 5 and guided by numerical

evidence, the ground state is given by the bounded magnon solutions (5.5) with the

free

and

energy f( cu) given by (5.7). It follows that the maximal free energy is

~f(r4,~,~;h)=max{lnu,lnu}, w<u(l-A),

the phase boundary for p > 1 is f(O+) = f( 1 -), or

u = 0 for w < u( 1 - A).

(7.5)

(7.6)

Alternately, one can also arrive at the same phase boundary (7.6) without invoking

(5.5) and (5.7): assuming that f(a) is either concave or convex in 0 < (Y 5 1, it is

easy to see that f’( 0) = 0 and f’( 1) = 0 cannot be the phase boundary for /? > 1. This

follows from the observations that, using (7.1))

f’(0) 2 0 -+ w + ho > u

-+u>u (using p > 1 or w + Au < 0)

-f’(l) > 0,

and using (7.2),

,f’(l) <o+u<u

+u>w+Au

-+ f’(0) < 0.

Thus, for p > 1, f’(0) = 0

shown in Figs. 7a and 7b. In

(again using p > 1)

and f’( 1) = 0 correspond to, respectively, situations

the former case, f’(0) = 0 is not a phase boundary

(7.7)

(7.8)

H.K Huung et al./Physicu A 228 (1996) 1-32

(a) (b) Cc) (4 Fig. 7. Behavior of .~‘(cu). The case of p > I is shown in (a),(b) indicating a first order transition. and the

case of 0 < I in (c) and (d) indicating a continuous transition.

since f(0) < f( 1 ), and in the latter case ,f’( 1) = 0, is not a phase boundary since

,f( 1 ) < f(0). Furthermore, relations (7.7) and (7.8) are consistent with the fact that

the maximal free energy occurs at, respectively, the frozen states CQ = 1 and cyu = 0. It

follows that the accompanying transition is between the two frozen phases and is thus

of first order.

However, the phase boundary are given correctly by S’(O) = 0 and f’( 1) = 0 for

/3 < 1. This can be seen as follows.

For f(0) to be a maximum we have always f’(0) 5 0, and for f( 1) a maximum

we have f’( 1) > 0. We shall establish in Appendix C that, for p < I or ( I - A)[: < w,

one has

f’(O) I O-t.f’(l) < 0, (7.9)

.f’(l) > O+f’(O) > 0. (7.10)

Thus, under the same convexity and concavity assumption, f( 0) and S( I ) can indeed be

the maximum of the free energy. Thus, the phase boundaries are f’( 0) = 0 and ,f’( 1) = 0

and the accompanying transition is continuous. Explicitly, the phase boundaries are, at

q = 0,

w+k=u for (I-_)v<w, (7.1 I)

and, at cq) = I,

1=(:-1)(:-l) forw>u,

i: = w + /\u for w + ho > u > w.

(7.12)

(7.13)

Finally, for the phase boundary at ‘~0 = l/2, we observe from (7.4) that, for p < -4,

,f’( LY) is discontinuous at CY = I /2 with

j-c;-) > .f(++). (7.14)

Then f(a) is a maximum at cy = I /2, provided that we have

,f(i-) 20 and f(i+) 50. (7.15)

H. IT Huang et al. /Physica A 228 (I 996) I-32 19

r 0 3 1 0 + 1

a a

(4 (b)

Fig. 8. Behavior of f(a) at (Y = l/2. (a) f’( f-) = 0. (b) f’( i+) = 0.

The phase boundary is therefore the borderline cases f( $-) = 0 and f( &+) = 0 when

f( $) begins to exhibit a maximum as shown in Fig. 8. Now, for p < -4 and writing

uA - lplw = u, it can be readily verified that we have

“A-W>u>O. Rh -

(7.16)

Thus, the phase boundary at LYO = l/2 is, from (7.4))

“A-“=&. R*

(7.17)

Explicitly, (7.17) can be written as

llL’A3-[w*+2W(U+U)+U2+U~]A*+[2W~+2W(U+U)+UU]h-w2=0,

(7.18)

which reflects its full symmetry with respect to u and u.

The phase diagram. Since the vertex weights (2.2) are arbitrary to an overall constant

and since there exists an expected U, u symmetry, it is convenient to consider the phase

diagram in the parameter space u/w, u/w, A. We have found the existence of five regimes

W (a0 = 0), U (a0 = l), V (a0 = l), A (a0 = l/2) and D (disordered). The regimes

U, Y W, A are phases in which dimers are frozen in respective configurations of Figs.

3a-d with values of “0 fixed as indicated. The phase boundaries are given by (7.6),

(7. I I ) , (7.12)) (7.13), and (7.6)) leading to the phase diagrams shown in Fig. 9.

For A = 1, the noninteracting case, the phase diagram is shown in Fig. 9a, and the

boundaries separating regimes W/D, U/D, and VD are given respectively by (7.11) , (7.13), (7.12).

For A > 1, corresponding to attractive interactions between u and u dimers, a new

ordered phase A with LYO = l/2 arises. A typical phase diagram is shown in Fig. 9b,

where, in addition to those phase boundaries already present in Fig. 9a, a new phase

boundary (7.18) separates regimes D and A.

For A < 1, corresponding to repulsive interactions between U, u dimers, a typical

phase diagram is shown in Fig. 9c. In addition to the boundaries already present in Fig.

9a, regimes U and V now share a boundary given by (7.6), namely u = u > w/( 1 - A),

20

ib3.5

w

(a>

3

v I\

2 jlizl D

W U 0 2 3

”

w

(b) Fig. 9. Phase diagrams for fixed A. (a) A = I. the noninteracting case. (b) A > I, the case of attractive

interactions between u. 13 dimcrn. A new ordered phase n arises for /3 < -4, or I./M. > 4/( A - I ). (c) A < I,

the case of repulsive interactions between N and I’ dimers. The U and V regimes share a tirst-order boundary

denoted by the heavy line, the circle denotes a tricritical point.

across which there is a first-order transition. This leads to the existence of a special

transition point at II = I: = w/( I - A). It is a point where two lines of continuous

transition merge into a first order line and may be called a kind of tricritical point.

However, it is different from ordinary tricritical points in that the discontinuity along

the first order line does not vanish at that point.

8. The critical behavior and expansions of the free energy near phase boundaries

In this section WC derive expansions of the free energy (5.1 1) for small deviations

near phase boundaries, and use the expansions to obtain the critical behavior in the

disorder regime.

The phase boundaries are characterized by cy N 0, I and l/2 and, in all cases except

for a = I-, /? < -I for which TO consists of double loops, ra is a a single trajectory

extending from a point Z$ to its complex conjugation ~0. However, as discussed in

Appendix B, the two loops in the case of LY = I-, /3 < -1 can be deformed into a

single closed loop in the computation of the free energy. Therefore, without loss 01

generality, we consider TO a continuous curve extending from Z$ to ~0. Thus, we write

~0 = Roe”, z. - p-’ = Ae’+, (8.1)

where Ro > 0, A > 0,O < {/3, c$} < T. We consider 0, q5 N 0 or rr, and there are three

cases to consider.

(a) cy = 0. In this cast ra is a small arc of radius Ro extending from angle -B to 8.

Then, depending on whether {0,4} N 0 or N rr, we have three cases as shown in Fig.

IO to consider. In all cases, (5.13) can be used to relate cy to 0 and 4.

For 0 < ,BRo < I, the situation shown in Fig. IOa, (5.13) can be written as

H. I! Huang et al. /Physica A 228 (I 996) I-32 21

O<PR< I bR>l P<O

Fig. 10. The three possible arrangements of (8. I ) for (Y = 0.

a 1 1 a

Cy=-

277i ./( --- dz Z z-P-' > *

70

=& j(dO-ad&

f3 =-- _ae2 77 VT ’

Here, we have used the fact that, for 6’ or C$ near 7rTT, we must write

(8.2)

(8.3)

Similarly, for the cases PRO > 1 and j? < 0 shown in Figs. lob and 1Oc and for which

(8.3) is not needed, we obtain

(8.4)

Furthermore, in all cases Ro, A and IpI-’ form a triangle implying the relation

Ra:A:IPI-‘=sin~#~:sin8:Isin(q!1-0)1. (8.5)

Thus, one obtains

R. = sin C#I

A= sin 0

psin(4 - 0) ’ psin(4 - 0) (8.6)

where p always has the same sign as 4 - 8. For given (Y and p, either (8.2) or (8.4)

and the expression of RO in (8.6) relates RO to 8. To determine RO and 13 individually,

another relation connecting Ro and 0 is needed. This is provided by (5.15).

For 0 < PRO < 1, (5.15) can be written as

7.0

o=& J(

I a - - ___ In zdz Z z-P-' 1 *

70

22 H. Y Hunn~ et ul. /Physical A 228 (I 996) l-32

0 (--@ 4-r = In Ro - a: In IRoe” - p-’ 1 7T 7T 7T

i’ In IROe’p - p-’ l’dp,

b

which can be written compactly as

H Roe’P _ p-1 ’

Roei0 _ p-’ dq.

0

(8.7)

(8.8)

It can be verified that (8.8) also holds for PRO > I and p < 0. Thus, Ro and 0 can be

generally determined, although implicitly.

Now we specialize the above consideration to small cy. When LY is small, 0 is also

small. Then, expanding (8. I ) and (KS), one obtains Ro = I + O( es), A = 1 I - ,f-’ / +

O(6”) and

CY

‘+ (PRO)-’ _, =@=’ I (8.9)

establishing that 0 - cyrr.

We can now expand f( cr) at cy = 0. First, we have the following integration formulas.

Consider integrations along an arc extending from z$ to ~0, where, for 0,d - n-, it is

again understood that the arc should be considered as consisting of two pieces as shown

in (8.3). Then one finds

3,

In(xl + x2z) dz = i lnlx’ + XZROI - 0-O

,’ 5;

dz = $ In/x’ + x2p-’ + ~2Al

(XI + x2p-’ + x2A12 + A(Ro)> 4 N 0,

In/x’ + x2p-’ - x2AJ - (r ~ 413 (XI + x2p-’ )x2A

6~ (xl + xzp-’ - xzA)* >

+ A(Ro), 4 c-u T,

A(Rn) =InA-InIRo+p-‘I, /!I-rr. X’ -x2R0<0,

= 0, otherwise. (8.1 I)

Using (8. IO), (8. I I ) and the small angle expansion (8.9), one finds after some algebra

that in all cases the expansion of the free energy (5. I 1) at cy = 0 is

H.Y Hunng et al./Physico A 22% (1996) l-32 23

&+ A0 0 B’ B’ 0

(4 (b) Fig. 11. The two possible arrangements of (8.1) for a = I.

f(a) =lnu+Czln(x, +.X2) - @r~( (X;;Q)‘) +c@>. (8.12)

Note that the corresponding expression (34) of [ 71 contains a typographical error.

(b) (Y = 1. In this case the contour TO is almost a closed loop, and can be considered as

a closed loop r’ intercepting the negative real axis at -Rn plus a small arc extending

from z$ to zo. Now we have always 19 N 7~ and, depending on whether 4 N r (p > 0)

or q5 N 0 (p < 0)) we have the two cases shown in Fig. 11. Thus, (5.13) leads to

zo I

a=iGi 2 J(

1

I-’

-+&it& ./(t-+)dz

zo*

0-?r 4-T = 1+0+- -ff------, p>o %- 7r

=1+0-t 0-r 4 --iY;, p<o.

7T

We again find (8.6) hold for all /?, and that, for p > 0, (5.15) leads to

ff o=-& --~ SC

1

z-P-' >

zo

In zdz + & J( i - --!?--- Z-P-' > In zdz

Z I“ *

20

r-0

=a In&--InIP+ & ln [ J I i?Oei'D+p-' 2 Roei(r-o) +0-l I I dv

0

(8.13)

(8.14)

A similar calculation also produces (8.14) for p < 0.

We now specialize to CY = 1 - and 0 = P--. Expanding (8.14) and (8.13), one obtains

R~=(l-~)-‘+~[(~-~)*l,A=)(l-~)-‘+~-’~+~[(~-~)~] and

(T-0) l- [

(PROP_, + 1 1 = (1 - a)r. (8.15)

It then follows that, after using (5.20), (8. lo), (8.11), (8.15) and some lengthy algebra,

one arrives at the expansion valid for all p,

f(a)=lno+(a-1) In .~-~j(l-p) [( > (12

+ln Sp+l )I

24 H. Y Huang et al. /Phy.yica A 228 (I 996) I-32

k. p q

0

Fig. 12. Plot of (8.1) for CY = l/2.

+L(a - l)3%-2 X2[XI - PtxlP + x2)1

6 (1 -P12[x2- (1 -P)x,12

for R0 N (1 -P)_t > x,/x2,

for R,J - (I -p)-’ < x,/q. (8.16)

(c) LY = l/2. In this case the contour re (after some deformation in the case of CE = $+)

is a closed loop plus a small arc running from z$ to ~0, both intercepting the negative axis at -Ro for (Y N f*. Since -Rh < p < 0, we have always 0, C#J N r as shown in

Fig. 12. Therefore, (5.13) yields

o-97 +?r cr=l--_+--a----

T 7T . (8.17)

One also finds Ro, A given by (8.6). In addition, (5.15) now leads to

m--H

0=2InRa-InA- & In .I’ !

Roefp + p-’ 2 ROei(~--H) + p-l dpo.

0

a (PRO)-’ + 1 1

= 2(& - Cry)%-,

from which one deduces after some lengthy algebra the expansion

where

D(Xl,X2,P) =p-' + 2x,x2R4f (XI + ~2P-')x2R$

(XI -~2&)~ - (XI - x2Rr)*

(8.18)

(8.19)

(8.20)

(8.21)

Here, the upper (lower) sign pertains to cy > l/2 ( CY < I /2). This is an extension of

the corresponding expressions (58)) (68) and (69) of [ 71.

The critical behavior. We have obtained expansions of the free energy (5.1 I ) in the

disorder regime near the phase boundaries LYO = 0, I, I /2 to be given by, respectively,

H.E Huuq et al./Physica A 228 (1996) 1-32 2s

(8.12), (8.16), and (8.20). It is now a simple matter to verify that, in all cases, the

maximal free energy assumes the form

f[ao(t) 1 = f[LYO(O) 1 + c(u, u, w, A) t3’2, (8.22)

where t is some measure of a small deviation from the phase boundary in the parameter

space, c( U, U, w, A) is a function regular in t, and aa( t) is the value of CY determined

from the maximal principle. Considered as a vertex model [4], for example, t can

be IT - T, /, where T, is the critical temperature. It then follows from (8.22) that the

transition is of second order (continuous) and the specific exponent is u = I /2. In order

to check the internal consistency of our results, however, we shall define f > 0 by writing

in respective equations for the phase boundary w + w(1 +t), or w + w(l -t) to

ensure in the disorder regime. We then expect the resulting expression for C(U, c, w, A)

to reflect a {IA, U} and LYO = (0, 1) symmetry.

To verify (8.22)) we apply the maximal principle to the free energy (8.12), (8.16),

and (8.20). Consider first (8.12)) the expansion of f( cu) at CY = 0, for which the phase

boundary is xi +x2 = 1 or u = w + Au. Near the phase boundary we write w = w( 1 + t),

where t is small, and determine LYO( t) from

W f’[ao(t)] =ln 1 + -t ( 1 I4

- $[ao(t)]V ( (x:;xz2,J = O.

Substituting this LYO( t) into (8.12) and expanding for small t, one obtains

(8.23)

2w 2 f[cwo(t)] = lnr*+ - -

37r J uu/\ t312. (8.24)

This leads to (8.22).

Consider next (8.16), the expansion of f( (u) at (Y = 1. For the first line of (8.16),

the phase boundary is (7.13) or v = w + Au. Near the phase boundary we define t by

writing w ---t w( 1 + t) in (7.13) and obtain

~‘]cYo(~)] =ln(l -Et) + i[(~g(t) - 1127r2 (

(1~[l;;2~~(x;I:~~~*1112)

=o. (8.25)

This leads to

2w f[ao(t)l =lnv+ -

$_ A- t3/2.

3r UVA (8.26)

Note that (8.24) and (8.26) reflect the expected {u, v} and (~0 symmetry.

For the second line of (8.16), the phase boundary is XI -x2/( 1 -p) = 1 or (7.12).

Near the phase boundary we define t by writing w -+ w( 1 - t) in (7.12) and obtain

f’[c~o(t)] =ln 1 - ( 2w-U-V t

W-U > + @o(t) - 1127r2

(

XI (XIP fx2)

1x2 - (1 - P)Xl12 > =o. (8.27)

26

This lcads to

H. E Huang et cd. /Physicu A 228 (1996) 1-32

2 .f[ae(t)] =InM:--t-t

3%- li

2( 2MJ - u - 0)3 $,X

Auu(u+u) . (8.28)

Here the term linear in t comes from the first term in (8.28) and does not contribute to

the “specific heat” exponent.

Finally, consider (8.20)) the expansion of the free energy at Q = l/2. Near the phase

boundary uA- w/R+ = v’?& or (7.18), we define t by writing w = rv( I - I) in (7.18).

Then cye( t) is determined from

=o, (8.29)

where D (xl, x2,f3) is given by (8.2 1) . This yields the maximal free energy

which reflects the proper {u,u} symmetry. Results (X.24), (8.26), (8.28) and (8.30)

now confirm (8.22) (note however the extra term linear in t in (8.28) which does not

contribute to the “specific heat”). In writing down (X.24), (8.26), (8.28) and (X.30),

we have used the respective critical conditions to simplify the expressions.

9. Summary

We have solved the problem of interacting dimers on the honeycomb lattice by

solving the equivalent five-vertex model using the method of Bcthe ansatz. The free

energy is given by (5.1 I) and the maximal free energy by (5.12) with TO, the contour

of integrations, subject to constraints (5.13) and (5.15). Phase transitions are then

associated with contours either just emerging or completing a closed loop. This lcads

to the determination of the phase boundaries (7.6)) (7. I I ), (7.12), (7.13) and (7. I8),

and the phase diagrams shown in Fig. 9. We find the occurrence of a new frozen ordered

phase for attractive dimer interactions, and a new first-order line ending at a tricritical

point for repulsive dimer interactions. We also find, at a = I, ra consist of one loop for

Ipi < I and two loops for IpI > I with the outer loop residing in the infinite regime.

But in the latter case /-a can always be deformed into a single loop in computations of

the free energy and its derivative with respective to LY, much simplifying the algebra.

At cy = l/2, Z-0 is found to be the outer loop of two loops, both of which in the linitc

regime, and this occurs only for p < -4. We have also evaluated the fret energy in

perturbative expansions near the phase boundary. This leads to the determination of the

critical behavior in the disorder regime with the specific heat exponent cy = l/2.

H. I! Hunng et al. /Physica A 228 (I 996) 1-32 21

Acknowledgements

One of us (FYW) would like to thank R.J. Baxter for a useful conversation; DK

would like to thank M. den Nijs for comments and support during his visit to the

University of Washington. Work by HYH and FYW is supported in part by NSF grants

DMR-9313648 and INT-9207261, and work by DK is supported in part by KOSEF

through CTP, by MOE of Korea and by NSF grant DMR-9205 125.

Appendix A. Evaluation of contour integrals

In this appendix we evaluate contour integrals occurring in our analysis of the free

energy and its derivative. Consider the integral

I=& s

ln(xt + ;2z) dz, (A.1)

i- 1 iI

where re consists of a single closed path enclosing the origin and cutting the negative

real axis at x = -R. The constants XI, x2 are positive and the branch cut is taken along

the negative real axis extending from the branch point x = -xi/x2 to x = --cxj. The

simple pole b on the real axis can be either positive or negative.

The integral (A.l) can be evaluated by extending To along the branch cut (if TIJ

intersects the branch cut) into a closed contour so that

I=& .i’

ln(xl + x2z >

z-b dz - contributions along the branch cut. (A.2)

clmed contour

The integral of the closed contour can be evaluated using the Cauchy residue theorem

and the contributions along the branch cut can be straightforwardly evaluated. The result

will now depend on whether fe encloses the simple pole 0. Since the evaluation of each

of the contributions in (A.2) is standard, we omit the details and give here the results.

We have the following situations:

( I ) The contour Te does not intersect the branch cut, the situation shown in Fig. A. I a.

In this case the contour integral (A. 1) is evaluated directly using the Cauchy theorem,

yielding

I = lo(xl ,x2; b) = 0, TO does not enclose b

= II (XI, x2; b) = In(xt + x2b), To encloses b.

(A.3)

(A.4)

(2) The contour TO intersects the branch cut at x = -R as shown in Fig. A.lb. In this

case we find

f=f2(il,x2:b.R)=In[x~~R+:2~], Tedoesnotencloseb

=13(xI.x2;b,R) =In[x2(R+b)], TO encloses b. (A.6)

2x H. I! Huung et nl. /Physica A 228 (I 996) I-32

Fig. A. I. The two distinct cases in the evaluation of the integral (A. I ). (a) The contour /‘,I does not intersect

the branch cut of In(.xl + _r?;). (b) I’,, intersects the branch cut.

Appendix B. Proof of (7.2)-(7.4)

In this appendix we present a detailed derivation of (7.2), (7.3) and (7.4). We use

(5. I 1) for f(a) and (5.25) for f’( cu), but first we need to determine the location 01

-0 and $, the terminal points of Tn.

Consider first the case of cy = I - for which Tn is derived by opening r a little at

one of its intercepts with the real axis, say zn. To determine at which intercept it opens

up, we expand ,f(cu) = ,f( I ) - ( I - a)f’( I ). It follows from the maximal principle

that ~0 is the intercept where J“( 1) is numerically the smallest. It can be shown that if

T(J intersects the axis at two points, one on the positive and one on the negative axis,

then ,f’( 1) is smaller if the intersection is on the negative axis, and if I-0 intersects the

negative axis at two or three points, then za is the intercept which is the closest to the

origin. With this understanding in mind, we can now evaluate f( I -) and f’( I ~1. It is

useful to recall from (5.9) that, in all cases, we have

In R - LY In 1 I + PRI = - In C.

In addition, if TO is a single closed contour we have from (5.23)

(B.1)

?‘ + J’* = I if /-n does not enclose p-’

=2 if 1-0 encloses 8-l.

We have the following cases to consider.

(B.2)

(a) cy = I -, ,B > I. In this case we have C = I, and the contour TO consists of

two loops as shown in Fig. 5c with the outer loop intersecting the negative real axis at -R = _@+fl) and the three other intercepts on the positive axis. Then, the contour

I.0 can be deformed into a single closed loop enclosing the origin but not the point p-‘.

Therefore using (5.23), (A.6) and (AS) in Appendix A, we find from (B.l). (B.2),

(5.1 I). and (5.25),

v+?‘* =I, (B.3)

.f‘( I-) =Inrr + 1s(xt,x2;0, R) - LU~~(,XI,X~;~-‘, RI

=lnlr+ln(x;?R) -culn xs(R+P-‘)

Xl + xzp-’ 1

H.Y Huang er al./Phyrica A 228 (1996) l-32

= In ~1,

f’(l-)=ln 3+x2 +Z~(O,~;O,R)-Z~(X~,X~;~-~,R) [ I

=ln(L:/zd).

29

(B.4)

(B-5)

This establishes (7.2).

(b) (Y = I, I/?[ < I. The contour consists of one loop enclosing the origin but not p-t,

the situations shown in Figs. 5a and 6a. Thus we have from (B.2) y + y* = 1. The

intercept with the negative real axis is at -R = - ( 1 - p) -I. Thus we take za = -R.

We have also C = 1 and there is no need to distinguish era = 1 and aa = 1-. We have

two cases to consider.

For M/ > u or, equivalently from the identity (5.21), R < XI/X:! or TO not intersecting

the branch cut of In(x) + x~z). We have

.f(l) =lnu+It(xt,x2;0) -

=lnu+Inxt -0

= In w,

.f’(I)=ln %--x2 +13(0,1 [ I

RR) - lo(xm;P-‘)

(B.6)

=In $-x2 +lnR-0 [ I

=ln w(w- 0) I wu-UU(l -A) . (B.7)

For u > w or R > x1/x2, and the contour TO intersecting the branch cut of ln(xt +

x2 z ) , we have

=lnl* + ln(x2R) - In ~2(R+/3-‘)

XI + x2p-’ I = Inu,

f’(I)=ln %+., +I,(0,1;0,R)-12(~,,X2;~-‘,R) [ I G-W

=ln - [ I Au .

(B.8)

(B.9)

H. E Hucmg et al. /PI~~~siccc A 228 (I 996) 1-32

Fig. R.1. The completion of l.0 given by two loops into n single closed loop for CY = I -, /II d -I, Contributions from the two line integrals along the cut cancel each other.

(c) a = I-, p < -I. The contour consists of two loops intersecting the negative real

axis at -R, -RI and -R:, as shown in Fig. 6c. Here we have

R = Ip1”” 1 -W) > R, = (IpI - I)-’ > IpI-’ > R2 = (I -p)-‘. (B.10)

Therefore WC take ZQ = -R2. In addition, we have C = I and from (B.2) y + y* = I.

Now the branch cut of In z intersects To at two points -R, and -RI, and from the

identity (5.21) the branch cut ofln(xl +x~z) also intersects To at -Rand -RI. Hence

WC can use (5.25) to evaluate ,f’( I -).

It is now easy to see that both .f( I -) and S’( I -) arc the same as those obtained

in (b). namely, given by (B.8) and (B.9), respectively. The key lies in the fact that,

as WC now show, the two integrals can be computed by expanding the two loops into

a single closed loop as shown in Fig. B. I. Topologically. this single closed loop is the

same as that in (b), namely, it encloses the origin and excludes the point p-‘. WC can

then use results of (b) provided that the difference of the integrals due to the branch

cuts is zro.

To see that the two loops can be expanded into a single closed loop, WC observe from

Fig. B.1 that the difference is due to the line integrations along the cut between -R

and -RI. (Recall from (5.22) that for /3 < 0 the two points -R and -RI arc always

on the branch cut.) Now for f(a) given by (5.1 I ) this diffcrcnce is In( R/R, ) for the

integrand having a simple pole at 0, and ln[ ( I + PR)/( ( I + PRl)J for the intcgrand

having a simple pole at ,E-‘. Using (B.1) and C = I one verifies that this difference

vanishes in f( cu). For ,f’( I ) given by (5.25), the corrections due to the two branch cuts

arc both In[(l +PR)/((I +/?RI)] and cancel each other since the two lint integrals

run in opposite directions. Hence we obtain (B.8) and (B.9) for ,f( I ) and ,f’( I ). This

established (7.3).

(d) N = I /2. /3 < -4. In this case 1’0 is the outer loop shown in Fig. 6c, enclosing both

the origin and the point p-‘. Hence from (B.2) we have y + .v* = 2. Now, for N = A-

the opening of I‘ is at :a = -R+ Similarly, it can be shown that, for cy = i-t, r-0 begins

to grow along the inner loop at ~0 = -R_. Thus one obtains from (5. I I ) and (5.25))

f‘( $5) =lnir + f~(-u~,_~~;O, R,) - $I~(x~,_Y~;~-‘, R,)

=Inlr+In(.qR,) -~In[~~(R++fl-‘)J

f’(

H. I! Huang et al. /Physica A 228 (I 996) 1-32

= iln(huu),

;*)=2h[$+r2] +13(0J;/3-9?+) -13(X,J2;P-9+

=2h[%+x2] +ln(R++P-‘1 -In[x2(R++P-‘)I

+(I - -&J2].

31

(B.ll)

)

(B.12)

(B.13)

This establishes (7.4).

Appendix C. Proof of (7.9) and (7.10)

In this appendix we establish relations (7.9) and (7.10) for p < 1. First, for w > u,

we have, using (7.1),

f’(0) < 0 -+ w + Au < u

+w<u

-f’(l) < 0, (C.1)

where the last step follows from the observation that in the right-hand side of (7.3) we

have

w(w-u) W ( I( wu - uu

wu - uu( 1 - A) = ii wu - uu + Auu > < w.

U (C.2)

For w < u, we use p < 1 or, equivalently, w > ( 1 - A)u to see that

f’(0) 5 0 --f u > w + Au

-rw+Au > w+A(w+Au)

=w(l+A)+A2u

> (I -A)(1 +A)u+A2u (usingp< 1)

= L:

-“P(l) < 0, (C.3)

where the last step follows from the last line of (7.3). Thus, we have established (7.9).

To establish (7.10), we have first for w > u,

f’(1) ro+y (

wu - uu

> >l

I* wu - uu + Auv

dW>U

+w+Ac>u

32 H. l! Hucrtq PI nl. /Pl?gsicn A 228 (1996) i-32

+ .f’(O) > 0.

We have also, for \t’ < c,

But we have also

It follows from the last two inequalities that we have

II < I: < M’ + AL,.

Thus, from (7.2), we have ,f’(O) > 0 which establishes (7. IO).

(C.4)

(C.5)

(C.6)

(C.7)

References

1 I 1 P.W. Kasteleyn, Physica 27 ( 1961) 1209.

121 M.E. Fisher. Phys. Rev. 124 (1961) 1664.

13 1 P.W. Kasteleyn, J. Math. Phys. 4 ( 1963) 287.

141 F.Y. Wu, Phys. Rev. 168 (1968) S39.

151 F.Y. Wu, Phys. Rev. Lett. 18 (1967) 60.5.

16 1 V.L. Pokrovsky and A.L. Talapov, Phys. Rev. Lett. 42 ( 1979) 65.

17 I J.D. Noh and D. Kim, Phys. Rev. E 49 (1994) 1943.

I 8 I C. Fan and EY. Wu, Phys. Rev. Phys. Rev. B 2 ( 1970) 723.

19 I R. Sutherland, C.N. Yang and C.P. Yang, Phys. Rev. L&t. I9 ( 1967) 588.

I IO] 1.M. Nolden, J. Stat. Phys. 67 (1992) 155.

I I I ] D.J. Bukman and J.D. Shore, J. Stat. Phys. 78 (1995) 1277.

I I2 I J.D. Shore and D.J. Bukman. Phys. Rev. Lett. 72 ( 1994) 604.

I I.? I M. Guhicsi, H.V. Bcijeren and A.C. Levi, Phys. Rev. E 47 ( 1997) 2473.

1 I4 1 E.H. Lieb, Phys. Rev. Lett. 18 ( 1967) 692.

I IS I E.H. Licb and EY. Wu, in: Phase Transitions and Critical Phenomena, Vol. I, C. Domb and M.S. Green.

eds. (Academic Press. New York, 1972).

I I6 I M. Gaudin, La fonction d’ondc de Bethe (Masson. Paris, 1983).

I 17 I J.D. Noh and D. Kim, unpublished.