Lower Bounds on the Ground State Entropy of the Potts Antiferromagnet on Slabs of the Simple Cubic Lattice

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by  Robert Shrock

Lower Bounds on the Ground State Entropy of the Potts Antiferromagnet on Slabs of
the Simple Cubic Lattice
Robert Shrock and Yan Xu
C. N. Yang Institute for Theoretical Physics
State University of New York
Stony Brook, NY 11794
We calculate rigorous lower bounds for the ground state degeneracy per site, W, of the q-state
0
Pottsantiferromagnetonslabsofthesimplecubiclatticethatareinfiniteintwodirectionsandfinite
1
in the third and that thus interpolate between the square (sq) and simple cubic (sc) lattices. We
0
giveacomparisonwithlarge-qseriesexpansionsforthesqandsclatticesandalsopresentnumerical
2
comparisons.

n
a PACSnumbers:
J

2
1 I. INTRODUCTION thatonetakesindefiningW( G ,q). HerebyW( G ,q)
{ } { }
we mean the function obtained by setting q to the given

ch] is Nanonizmerpoorgtraonutndsusbtajetectenintrosptayt(ispteicrallatmtieccehsaitnei)c,sS, 0as6=a0n, ovaflauebfiiprasrttaitnedgtrhaepnhtGakbiipn.g, ann→ele∞m.enFtoarrythleowne→r b∞ounlidmiits
e exceptiontothethirdlawofthermodynamicsandaphe- W( Gbip. ,q) √q 1,sothatforq >2,thePottsanti-
m nomenon involving large disorder even at zero tempera- ferr{omagn}etha≥s a no−nzero groundstate entropy on such
at- tnurdee.nSoitnecsetSh0e=nukmBblenrWof,wlahtteirceeWsit=es,limS0n→=∞0Wist1o/etn.quainvd- aWl(astqti,cqe).≥A(qb2e−tt3eqr+lo3w)/e(rqb−o1u)n[d11f]o.rInthperesvqiuoaursewloartktic[1e2i]s-
t 6 [15]oneofusandTsaiderivedlowerandupperboundson
s alent to W > 1, i.e., a total ground state degeneracy
t. Wtot. that grows exponentially rapidly as a function of W for a variety of different two-dimensional lattices. It
a was found that these lower bounds are quite close to the
n. One physical example is provided by H O ice, for
m 2 actual values as determined with reasonably good accu-
which the residual entropy per site (at 1 atm. pressure)
- is measured to be S = (0.41 0.03)k , or equivalently, racy from large-q series expansions and/or Monte Carlo
d 0 ± B measurements.
W =1.51 0.05 [1]-[3]. A salient property of ice is that
n
± In the present paper we generalize these lower bounds
o the groundstateentropyoccurswithoutfrustration;i.e.,
on two-dimensional lattice graphs by deriving lower
c each of the ground state configurations of the hydrogen
[ atomsonthebondsbetweenoxygenatomsminimizesthe bounds onW( G ,q) for sections of a three-dimensional
{ }
internal energy of the crystal [4]. lattice,namelythesimple cubiclattice,whichareofinfi-
1
niteextentintwodirections(takentoliealongthexand
v A model that also exhibits groundstate entropy with-
y axes) and finite in the third direction, z. By compar-
8 outfrustrationandhenceprovidesa usefulframeworkin
5 which to study the properties of this phenomenon is the ison with large-q expansions and numerical evaluations,
9 q-state Potts antiferromagnet [5]-[7] on a given lattice Λ we show how the lower bounds for the W functions for
1 these slabs interpolate between the values for the (re-
or, more generally, a graph G, for sufficiently large q.
1. Consider a graphG=(V,E), defined by its vertex (site) spectivethermodynamiclimitsofthe)squareandsimple
0 and edge (bond) sets V and E. Denote the cardinalities cubic lattices. These bounds are of interest partly be-
0 cause one does not know the exact functions W(sq,q) or
of these sets as n(G) = V n and e(G) = E , and
1 | | ≡ | | W(sc,q) for general q.
: let {G} ≡ limn(G)→∞G. An important connection with
v graph theory is the fact that the zero-temperature par-
i tition function of the q-state Potts antiferromagnet on
X
the graph G satisfies Z(G,q,T = 0) = P(G,q), where II. CALCULATIONAL METHOD
r
a P(G,q)isthechromaticpolynomialexpressingthenum-
ber of ways of coloring the vertices of G with q colors
Let us consider a section (slab) of the simple cubic
such that no two adjacent vertices have the same color
latticeofdimensionsL L L vertices,whichwede-
(called a proper q-coloring of G) [8, 9]. Thus, x× y× z
notesc[(L ) (L ) (L ) ],wherethebound-
x BCx y BCy z BCz
× ×
W( G ,q)= lim P(G,q)1/n . (1.1) ary conditions (BC) in each direction are indicated by
{ } n→∞ the subscripts. The chromatic polynomial of this lattice
Ingeneral,forcertainspecialvaluesofq,denotedqs,one will be denoted P(sc[(Lx)BCx×(Ly)BCy×(Lz)BCy],q).
has the following noncommutativity of limits [10] We will calculate lower bounds for W(sc[(Lx)BCx
×
(L ) (L ) ],q)inthelimitL andL
y BCy z BCz x y
lim lim P(G,q)1/n = lim lim P(G,q)1/n , (1.2) with L ×fixed. These are independe→nt∞of the bo→un∞d-
n→∞q→qs 6 q→qsn→∞ ary conzditions imposed in the directions in which the
and hence it is necessary to specify which order of limits slab is of infinite extent, and hence, for brevity of nota-
2
tion, we will denote the limit limLx, Ly→∞sc[(Lx)BCx× trix, as explained below) we will use periodic bound-
(L ) (L ) ] simply as S , where S stands ary conditions in the x direction. Note that the proper
y BCy× z BCz (Lz)BCz
for “slab”. We will consider both free (F) and periodic q-coloring constraint implies that FBC and PBC are
z z
(P) boundary conditions in the z direction, and thus equivalent if L = 2. The number of vertices for G =
z
slabs such as S , S , etc. For technical reasons (to sc[(L ) (L ) (L ) ] is n = L L L . The
3F 3P x BCx × y BCy × z BCz x y z
get an expression involving a trace of a coloring ma- specific form of Eq. (1.1) for our calculation is
W(S ,q)= lim lim [P(sc[(L ) (L ) (L ) ],q)]1/n . (2.1)
(Lz)BCz Ly→∞Lx→∞ x P × y BCy × z BCz
To derive a lower bound on W(S ,q), we gen- λ . It follows that
(Lz)BCz T,max
eralize the method of Refs. [11]-[14] from two to three
dofimtheenssiloanbso.rtWhoegcoonnaslitdoerthtewxodaidrejacctieonnt,twraitnhsvxevrsaeluselsicxes Lxli→m∞[P(sc[(Lx)P×(Ly)BCy×(Lz)BCz],q)]1/Lx =λT,max.
0 (2.4)
andx +1. Thesearethussectionsofthesquarelatticeof
0 Now for the transverse slices G and G , denoted
dimensionL L ,whichwedenoteG =sq[(L ) x0 x0+1
(L ) ] ayn×dGz =sq[(L ) x(0L ) ] y B.CWy×e genericallyasts((Lz)BCz),thechromaticpolynomialhas
z BCz x0 x0+1 y BCy× z BCz x0+1 the form
labela particularcolorassignmentto the vertices ofG
x0
that is a proper q-coloring of these vertices as C(Gx0) P(Gx0,q)=P(Gx0+1,q)= cj(λts((Lz)BCz),j)Ly
and similarly for Gx0+1. The total number of proper Xj
q-colorings of G is (2.5)
x0
where the c are coefficients whose precise form is not
j
needed here. The set of λ ’s is indepen-
N =P(Gx0,q)=P(Gx0+1,q) . (2.2) dent of the length L and althtso(u(Lgzh)BtChzis),jset depends on
y
BC , the maximal one (having the largest magnitude),
Now let us add the edges in the x direction that join y
λ , is independent of BC (e.g., [16] and
these two adjacent transverse slices of the slab together. ts((Lz)BCz),max y
references therein). Hence,
Among the 2 color configurations that yield proper q-
N
coloringsofthese twoseparateyz transverseslices,some lim [P(G ,q)]1/Ly lim ( )1/Ly
will continue to be proper q-coloringsafter we add these Ly→∞ x0 ≡ Ly→∞ N
edges that join them in the x direction, while others will = λ . (2.6)
not. We define an -dimensional coloring compat- ts((Lz)BCz),max
ibility matrix T wNith×eNntries T equal to The two adjacent slices together with the edges in
(i) 1 if the color assignments C(CG(Gx0)),aCn(dGxC0+(1G) ) are the x direction that join them constitute the graph
properq-coloringsafterthe edgesixn0the xdirecxti0o+n1have sc[2F ×(Ly)BCy ×(Lz)BCz]. We denote the chromatic
been added joining G and G , i.e., if the color as- polynomial for this section (tube) of the sc lattice as
atshingednec(doiil)otro0aeifsastcihhgenvecedorlttoeorxatxvhs0(sexigv0ne,mryt,eeznx)xt0vsi+n(Cx1G0(Gx+01i),syad,nizffd)eCrine(nGGtxfr0+om1); PPq-((cssocclo[[22rPFin×g×c(o(LLnydy))iBtBiCoCyny×).×(TL(hLzi)szB)hCBazCs]z,t]q,h)qe)bfeo(crwamuhsicehofistheeqpuraolpteor
x0 x0+1
are not proper q-colorings after the edges in the x di-
P(sc[2 (L ) (L ) ],q)
rection have been added, i.e., there exists some color as- F × y BCy × z BCz
saisgsnigendetdotaovtehrteexvevr(txe0x,yv,(xz)i+nG1,xy0,tzh)aitnisGequal.toCalecaorlloyr, = c′j(λtube((Lz)BCz),j)Ly (2.7)
0 x0+1 Xj
T =T . The chromaticpolynomial for the slabis then
ij ji
given by the trace where c′ are coefficients analogous to those in (2.5).
j
Therefore,
P(sc[(Lx)P ×(Ly)BCy×(Lz)BCz],q)=Tr(TLx). (2.3) Lyli→m∞ [P(sc[2F ×(Ly)BCy×(Lz)BCz],q)]1/Ly =
Since T is a realsymmetric matrix,there exists an or- = λ . (2.8)
thogonalmatrixAthatdiagonalizesT: ATA−1 =T . tube((Lz)BCz),max
diag.
Now let us denote the column sum (CS)
Letusdenotethe eigenvaluesofT asλ ,1 j .
T,j
N ≤ ≤N
Since T is a real non-negative matrix, we can apply the N
generalized Perron-Frobenius theorem [17, 18] to infer CS (T)= T , (2.9)
j ij
that T has a real maximal eigenvalue, which we denote Xi=1
3
which is equal to the row sum N T , since TT = T. metric matrix T and r N [19]
j=1 ij ∈ +
We also define the sum of all enPtries (SE) of T as
N SE(Tr) 1/r
λ (T) . (2.12)
SE(T)= T . (2.10) max
ij ≥(cid:20) (cid:21)
iX,j=1 N
NotethatSE(T)/ istheaveragerow(=column)sum. The lower bound is then
N
Next, we observe that
SE(T)=P(sc[2F ×(Ly)BCy×(Lz)BCz],q) . (2.11) W(S(Lz)BCz,q)≥W(S(Lz)BCz,q)ℓ (2.13)
To obtain our lower bound, we then use the r = 1
special case of the theorem that for a non-negative sym- where
SE(T) 1/(LyLz)
W(S ,q) = lim
(Lz)BCz ℓ Ly→∞(cid:18) (cid:19)
N
P(sc[2 (L ) (L ) ],q) 1/(LyLz)
= lim F × y BCy× z BCz
Ly→∞(cid:20) P(sq[(Ly)BCy×(Lz)BCz],q) (cid:21)
λ 1/Lz
= tube((Lz)BCz),max . (2.14)
(cid:20) λ (cid:21)
ts((Lz)BCz),max
III. RESULTS FOR SLAB OF THICKNESS be
Lz =2 WITH FBCz 1
λ = q4 8q3+29q2 55q+46+ R
tube(2),max 2(cid:20) − − 22 (cid:21)
We now evaluate our general lower bound in Eqs. p
(3.4)
(2.13) and (2.14) for a slab of the simple cubic lattice where
with thickness L = 2 and FBC , denoted S . In this
case the transverzse slice is the gzraph sq[2F ×2F(Ly)BCy]. R22 = q8−16q7+118q6−526q5+1569q4
For FBC , an elementary calculation yields
y
3250q3+4617q2 4136q+1776 . (3.5)
− −
P(sq[2 (L ) ],q)=q(q 1)(q2 3q+3)Ly−1 (3.1)
F y F We then substitute these results for λ and
× − − ts(2),max
λ into the L = 2 special case of (2.14) to
tube(2),max z
with a single λts(2F) = λts(2P) ≡ λts(2), and this is also obtain W(S2,q)ℓ, and thus the resultant lower bound
the maximal λ for PBCy [10, 20], so that on W(S2F,q) = W(S2P,q) ≡ W(S2,q): W(S2,q) ≥
W(S ,q) .
2 ℓ
λ =q2 3q+3 . (3.2)
ts(2),max
−
We next use the calculation of IV. COMPARISON WITH LARGE-q SERIES
EXPANSIONS
P(sc[2 (L ) 2 ],q) = P(sc[2 2 (L ) ],q)
F y F F F F y F
× × × ×
One way to elucidate how this lower bound W(S ,q)
2 ℓ
= P(sq[4 (L ) ],q) compareswiththeexactW(sq,q)andW(sc,q)istocom-
P y F
×
pare the large-q series expansions for these three func-
(3.3) tions. For this purpose, it is first appropriate to give
somerelevantbackgroundonlarge-qseriesexpansionsfor
in Ref. [34] (where each of the 2 BC’s is equivalent to W( G ,q) functions. Since there are qn possible color-
F
{ }
2 ),fromwhichwecalculatethemaximalλ to ings of the vertices of an n-vertex graph G with q colors
P tube(2),max
4
if no conditions are imposed, an obvious upper bound As noted above, lower bounds on W(Λ,q) obtained
on the number of proper q-colorings of the vertices of G from the inequality (2.12) for two-dimensionallattices Λ
is P(G,q) qn. This yields the corresponding upper were found to be quite close to the actual values of the
≤
bound W( G ,q) < q. Hence, it is natural to define a respective W(Λ,q) for a large range of values of q. This
{ }
reduced function that has a finite limit as q , canbeunderstoodforlargevaluesofq fromthefactthat
→∞
theycoincidewiththelarge-qexpansionstomanyorders,
Wr( G ,q)=q−1W( G ,q) . (4.1) and the agreement actually extends to values of q only
{ } { }
moderately above q = 2. For example, the lower bound
For a lattice or, more generally, a graph whose vertices on W(sq,q) in [11] is equivalent to W(sq,q) (1+y3).
haveboundeddegree,Wr( G ,q)isanalyticabout1/q= This agreeswith the small-y series up to orde≥rO(y6), as
{ }
0. (W ( G ,q) is non-analytic at 1/q = 0 for certain
r is evident from comparison with Eq. (4.6). This lower
{ }
families of graphsthat containone or more verticeswith
boundalsoagreesquitecloselywiththevalueofW(sq,q)
unbounded degree as n , although the presence of
determined by Monte Carlo simulations in [10, 12, 13]
→ ∞
a vertex with unbounded degree in this limit does not
(see Table 1 of [10] and Table 1 of [12]). We include
necessarilyimply non-analyticityofW ( G ,q)at1/q=
r this comparison here in Table I. For our purposes, it
{ }
0[21,22].) Itisconventionaltoexpressthelarge-qTaylor
is sufficient to quote the results from Ref. [10] only to
series for a function that has some factors removed from
threesignificantfigures. Since weareusinglarge-q series
W , since this function yields a simpler expansion. A
r for this comparison, we list the results in Table I for
chromatic polynomial has the general form
a set of values q 4. As another example, the lower
≥
bound obtained for the honeycomb lattice in Ref. [13],
n−k(G)
P(G,q)= (−1)jan−jqn−j , (4.2) Wfor(hWc,(qh)c,≥q)(t1o+Oy(y5)110/)2.,Tahgruese,ditwwitahs tfohuensdmtahlal-tyfoserriaelsl
Xj=0
of the cases studied, W(Λ,q) provides not only a lower
ℓ
bound on W(Λ,q), but a rather good approximation to
wherethean−j >0andk(G)isthenumberofconnected
the latter function. It is thus reasonable to expect that
components of G (taken here to be k(G) = 1 without
thiswillalsobetrueforthelowerboundsW(S ,q)
loss of generality). One has an = 1, an−1 = e(G), and, (Lz)BCz ℓ
for the slabs S of the simple cubic lattice considered
providedthatthegirthg(G)>3[23],asisthecasehere, Lz
here, of infinite extent in the x and y directions and of
an−2 = e(2G) . A κ-regular graph is a graph such that thickness L in the z direction.
each ver(cid:0)tex h(cid:1)as degree (coordination number) κ. For a z
FromingredientsgiveninRef. [25],wehavecalculated
κ-regular graph, e(G) = κn/2. The coefficients of the
a large-q expansion of the W(sc,q) for the simple cubic
three terms of highest degree in q in P(G,q) for a κ-
(sc) lattice and obtain
regular graph are precisely the terms that would result
from the expansion of [q(1−q−1)κ/2]n. Hence, for a κ- W(sc,q)=1+3y3+22y5+31y6+O(y7) . (4.7)
regular graph or lattice, one usually displays the large-q
series expansions for the reduced function
In Table I we list the corresponding values of W(sc,q)
obtained from this large-q series, denoted W(sc,q) ,
W(Λ,q) ser.
W(Λ,q)= . (4.3) for q 4. We also list estimates of W(sc,q), denoted
q(1 q−1)κ/2 ≥
W(sc,q) , for 4 q 6 from the Monte Carlo calcu-
− MC
≤ ≤
lationsinRef. [29]. Oneseesthattheapproximatevalues
The large-q Taylorseriesforthis function canbe written
obtainedfromthelarge-qseriesareclosetotheestimates
in the form
fromMonteCarlosimulationsevenforq valuesaslowas
∞ q =4.
W(Λ,q)=1+Xj=1wΛ,jyj , (4.4) pleTchuebcicoolardttinicaeti(oonf ninufimnbiteerefxotretnhteinS2tFheslxabanodf tyhedisriemc--
tions) is κ(S )=5. We thus analyze the reduced func-
where tionW(S ,q2)F=W(S ,q) /[q(1 q−1)5/2]. Thishasthe
2 ℓ 2 ℓ
−
1 large-q (small-y) expansion
y = . (4.5)
q 1
− W(S ,q) =1+2y3+2y5+9y6+O(y7) . (4.8)
2 ℓ
The two results that we shall need here are the large-
q (i.e., small-y) Taylor series for W(sq,q) and W(sc,q). As this shows, W(S2,q)ℓ provides an interpolation be-
The large-q series for W(sq,q) was calculated to succes- tweenW(sq,q)andW(sc,q);forexample,thecoefficient
sively higher orders in [24]-[28]. Here we only quote the of the y3 term is 1 for Λ = sq, 2 for Λ = S2, and 3 for
terms to O(y11): Λ = sc. Furthermore, the coefficient of the y5 term is
0 for Λ = sq, 2 for Λ = S , and 22 for Λ = sc. This
2
W(sq,q) = 1+y3+y7+3y8+4y9+3y10 is in agreement with the fact that the exact functions
W(S ,q) interpolate between W(sc,q) and W(sc,q)
(Lz)F
+ 3y11+O(y12) . (4.6) as L increases from 1 to [30] and the expectation,
z
∞
5
asdiscussedabove,thatW(S ,q) shouldbecloseto One then needs λ . The relevant (4 4 di-
(Lz)F ℓ tube(3P),max ×
W(S ,q). mensional) transfer matrix for this tube graph was cal-
(Lz)F
culated for Ref. [34], and we have used this to compute
λ numerically. The results for W(S ,q) are
tube(3P),max 3P
given in Table I.
V. RESULTS FOR SLABS OF THICKNESS
Lz =3, 4 WITH FBCz
VII. DISCUSSION
For the slab of the simple cubic lattice with thick-
ness L = 3 and FBC , denoted S , the transverse
slice iszthe graphsq[3 z(L ) ]. T3hFe chromaticpoly- Since the slabs of infinite extent in the x and y direc-
F y BCy
nomials P(sq[3F (Ly)×F],q), P(sq[3F (Ly)P],q), and tions and of finite thickness Lz geometrically interpolate
P(sq[3 (L ) ×],q) (where TP deno×tes twisted peri- between the square and simple cubic lattices, it follows
F y TP
odic, i.e.,×Mo¨bius BC) were computed for arbitrary L that the resultant W functions for these slabs interpo-
y
in Refs. [31], [32], and [33], respectively, and the maxi- late between W(sq,q) and W(sc,q) [30]. Given that it
mal λ was shown to be the same for all of these bound- was shown previously that the lower bounds W(Λ,q)ℓ
ary conditions. For the reader’sconvenience, we list this obtained by the coloring matrix method are quite close
λ inEqs. (9.2) and(9.3)ofthe Appendix. The to the actualvalues ofthe respective W(Λ,q)for a num-
ottsh(e3rF)i,nmpauxt that is needed to obtain the lower bound in ber of two-dimensional lattices, this is also expected to
Eq. (2.14)isthemaximalλforthechromaticpolynomial be true for the W(S(Lz)BCz,q)ℓ bounds. We have shown
roefltehveanstc[2trFa×ns3fFer×mLayt]rtiuxbtehgartapdhe,tei.rem.,iλnetusbet(h3eF)c,mharoxm. Tathiec Wab(osvce,qh)owviWa a(Sc2o,mq)pℓairnitseornpoolfattehsebleatrwgeee-qn sWer(isesq,eqx)paannd-
polynomial for this tube graph was given with Ref. [34]. sions for these three functions. Table I provides a fur-
Because it is 13 13 dimensional, one cannot solve the ther numerical comparison for W(S3F,q)ℓ, W(S4F,q),
corresponding ch×aracteristic polynomial analytically to andW(S3P,q)withW(sq,q)andW(sc,q),thelatterbe-
obtain λ for general q. However,one can cal- ing determined to reasonably good accuracy from large-
culate λtube(3F),maxnumerically, and we have done this. q series expansions and, where available, Monte Carlo
Combinitnugbe(t3hFe)s,mearxesults with Eqs. (9.2) and (9.3), we measurements. As noted, the lower end of the range of
then evaluate the lower bound W(S ,q) by evaluating q values for the comparisonis chosen as q =4 in view of
the L =3 special case of (2.14). 3F ℓ the use of large-q series.
z
For sections of lattices, and, more generally, graphs
For the slab of the simple cubic lattice with thickness
that are not κ-regular,one can define an effective vertex
L =4 and FBC , S , the transverse slice is the graph
z z 4F degree (coordination number) as [14]
sq[4 (L ) ]. Herethemaximalλ istheso-
F× y BCy ts(4F),max
lutionof the cubic equation(9.5) givenin the Appendix.
2e(G)
One also needs the maximal λ for the chromatic polyno- κ = . (7.1)
eff
mialofthesc[2 4 L ]tubegraph,i.e.,λ . n(G)
F× F× y tube(4F),max
The relevant(136 136 dimensional)transfer matrix for
thistubegraphwa×scalculatedforRef. [34],andwehave For 3 Lz < , the slab of the simple cubic lattice (of
≤ ∞
uobsetdainthtishetolocwoemrpbuotuenλdtuWbe((4SF),m,qa)x nfurommertihcaellLy. W=e4tshpeen- ninofitnκit-ereegxutleanr,tbinutthhaesxthaenedffyectdiivreecctoioonrds)inwatitiohnFnBuCmzbeirs
4F ℓ z
cial case of Eq. (2.14). The results for W(S ,q) and
3F 1
W(S ,q) are listed in Table I.
4F κeff(S(Lz)F)=2(cid:18)3− L (cid:19) . (7.2)
z
We observe that for the q values considered in Table
VI. RESULT FOR SLAB OF THICKNESS Lz =3 I, W(sq,q) > W(S ,q) > W(S ,q) > W(S ,q) >
WITH PBC 2 ℓ 3F ℓ 4F
z W(sc,q). The fact that for fixed q, the exact function
W(S ,q) is a non-increasing function of L , and, for
(Lz)F z
It is also of interest to obtain a lower bound for W q >2,a monotonicallydecreasingfunctionofL ,follows
z
for a slab with periodic boundary conditions in the z from a theorem proved in Ref. [30]. To the extent that
direction, since these minimize finite-volume effects. For thelowerboundsW(S ,q) lieclosetotheactualval-
(Lz)F ℓ
this purpose we consider the slab of the simple cubic ues of W(S ,q), it is understandable that they also
(Lz)F
lattice with thickness Lz = 3 and PBCz, S3P. In this exhibit the same strict monotonicity. As was noted in
case the transverse slice is the graph sq[3P (Ly)BCy]. Ref. [30], the reason for the monotonicity of the exact
×
ForFBCy the chromaticpolynomialinvolvesonlyone λ, values is that the number of proper q-colorings per ver-
andthisisalsothemaximalλforPBCy andTPBCy [36], tex of a lattice graph is more highly constrained as one
viz., increasesthe effective coordinationnumber ofthe lattice
section. (This is also evident in Fig. 5 of [10].) In the
λ =q3 6q2+14q 13 . (6.1) present case, the monotonicity can be seen as a result of
ts(3P),max − −
6
the fact that the effective coordinationnumber increases with
monotonically as a function of L .
z
Theuseofperiodicboundaryconditionsinthez direc- R3 =(q2 5q+7)(q4 5q3+11q2 12q+8) . (9.3)
− − −
tion minimizes finite-size effects, so that for a given L ,
z
For the infinite-length strip of the square lattice with
W(S ,q) would be expected to be closer to W(sc,q)
(Lz)P width4andfreetransverseboundaryconditions,sq[4
than W(S ,q) [30]. Again, to the extent that the F
(Lz)F ] [31, 35] ×
lower bounds are close to the actual W functions for
∞
these respective slabs, one would expect W(S ,q)
(Lz)P ℓ W(sq[4 ],q)=(λ )1/4 (9.4)
to be closer than W(S(Lz)F,q)ℓ to W(sc,q). Our results F ×∞ 4F,max
agreewiththisexpectation. Incontrastto W(S ,q),
(Lz)F where λ is the largest root of the cubic equation
W(S(Lz)P,q)isnot,ingeneral,anon-increasingfunction 4F,max
of L , as was discussed in general in [30] (see Fig. 1
z x3+b x2+b x+b =0 (9.5)
therein). Thus, values ofW(S ,q), andhence, a for- 4F,1 4F,2 4F,3
(Lz)P
tiori,W(S ,q) ,mayactuallylieslightlybelowthose
(Lz)P ℓ with
for W(sc,q), as is evident for the W(S ,q) entries in
3P ℓ
Table I. b = q4+7q3 23q2+41q 33 (9.6)
4F,1 − − −
VIII. CONCLUSIONS b = 2q6 23q5+116q4 329q3+553q2 517q+207
4F,2 − − −
Inthispaperwehavecalculatedrigorouslowerbounds (9.7)
for the ground state degeneracy per site W, equivalent
to the groundstate entropy S =k lnW, of the q-state and
0 B
Pottsantiferromagnetonslabsofthesimplecubiclattice
b = q8+16q7 112q6+449q5 1130q4+1829q3
that are infinite in two directions and finite in the third. 4F,3 − − −
Via comparison with large-q expansions and numerical
1858q2+1084q 279 . (9.8)
evaluations, we have shown how the results interpolate − −
between the square (sq) and simple cubic (sc) lattices.
Acknowedgments: Thisresearchwassupportedinpart
by the NSF grantPHY-06-53342. RS expresses his grat-
itude to coauthors on previous related works, in partic-
ular, S.-H. Tsai and J. Salas, as well as N. Biggs, S.-C.
Chang, and M. Roˇcek.
IX. APPENDIX
We note the following results on Ed lattices and lat-
tice sections: W(Λ ,2) = 1 for any bipartite lattice;
bip.
W(sq,3)=(4/3)3/2 [37]; and W( L ,q)=W( C ,q)=
{ } { }
q 1,where L andC denote the n-vertexline andcir-
n n
−
cuit graphs. For the infinite-length square-lattice strip
of width 2, W(sq[2 ],q) = W(sq[2 ],q) =
F P
× ∞ × ∞
q2 3q+3, where, as in the text, the subscripts F
−
pand P denote free and periodic boundary conditions in
the directionin whichthe stripis finite. For the infinite-
length strip of the square lattice with (transverse)width
3 and free transverse boundary conditions, sq[3 ]
F
×∞
[31–33]
W(sq[3 ],q)=(λ )1/3 (9.1)
F ×∞ 3F,max
where
1
λ = (q 2)(q2 3q+5)+ R (9.2)
3F,max 2(cid:20) − − 3 (cid:21)
p
7
TABLE I: Comparison of lower bounds W(S(Lz)BCz,q)ℓ for
(Lz)BCz = 2F = 2P, 3F, 4F, 3P with approximate values of
W(Λ,q)forthesquare(sq)andsimplecubic(sc)latticesΛ,asde-
terminedfromlarge-q seriesexpansions, denoted W(Λ,q)ser. and,
where available, Monte Carlo simulations, denoted W(Λ,q)MC.
WealsolistW(sq,q)ℓ forreference. Seetextforfurtherdetails.
q W(sq,q)MC W(sq,q)ser. W(sq,q)ℓ W(S2F,q)ℓ W(S3F,q)ℓ W(S4F,q)ℓ W(sc,q)ser. W(sc,q)MC W(S3P,q)ℓ
4 2.34 2.34 2.33 2.13 2.07 2.04 2.06 1.9 1.78
5 3.25 3.25 3.25 2.96 2.875 2.83 2.75 2.7 2.62
6 4.20 4.20 4.20 3.87 3.765 3.71 3.58 3.6 3.51
7 5.17 5.17 5.17 4.81 4.69 4.64 4.48 − 4.43
8 6.14 6.14 6.14 5.76 5.64 5.58 5.41 − 5.37
9 7.125 7.125 7.125 6.73 6.605 6.54 6.36 − 6.325
10 8.11 8.11 8.11 7.71 7.58 7.51 7.32 − 7.29
100 − 98.0 98.0 97.5 97.4 97.3 97.0 − 97.0
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