High-Temperature series for the $RP^{n-1}$ lattice spin model (generalized Maier-Saupe model of nematic liquid crystals) in two space dimensions and with general spin dimensionality n

Abstract

High temperature series expansions of the spin-spin correlation functions of the RP^{n-1} spin model on the square lattice are computed through order beta^{8} for general spin dimensionality n. Tables are reported for the expansion coefficients of the energy per site, the susceptibility and the second correlation moment. Comment: 6 pages, revtex, IFUM 419/FT, 2 figures not included

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arXiv:hep-lat/9205027v1 27 May 1992
High-Temperature series for the RPn−1lattice spin model
(generalized Maier-Saupe model of nematic liquid crystals)
in two space dimensions and with general spin dimensionality n
P. Butera and M. Comi
Istituto Nazionale di Fisica Nucleare
Dipartimento di Fisica, Universit` a di Milano
Via Celoria 16, 20133 Milano, Italy
High temperature series expansions of the spin-spin correlation functions of the RPn−1spin model
on the square lattice are computed through order β8for general spin dimensionality n.
Tables are reported for the expansion coefficients of the energy per site, the susceptibility and the
second correlation moment.
PACS numbers: 05.50.+q,64.60.Cn, 64.70.Md, 75.10.Hk
I. INTRODUCTION
Interest in the classical RPn−1spin systems [ [1]] on a two-dimensional lattice has been revived recently by the
results of a MonteCarlo simulation[ [2]] interpreted as evidence of a second order ”topological” phase transition, taking
place for values of the spin dimensionality n ≥ 3. This is unexpected according to renormalization group ideas. Indeed
the RPn−1models have the same formal continuum limit as the conventional O(n) symmetric n−vector spin models,
therefore they should belong to the same universality class and should not behave differently for n ≥ 3 ( when n = 2
the RPn−1model trivially reduces to the n−vector model). However the global topologies of the spin manifolds: the
hypersphere Sn−1with antipodal points identified in the case of the RPn−1model and simply Sn−1in the case of
the n-vector model, are different and it has long been known that this might be a reason for different phase diagrams[
[3]].
MonteCarlo studies of these systems, mainly in the n = 3 case, sometimes with conflicting or not completely
convincing results are by now numerous [ [4,5,6,7,8,9,10]], and they have been augmented by recent more extensive
simulations on large lattices [ [2,11,12]] using cluster algorithms[ [13]] in order to reduce the critical slowing-down.
On the other hand, high temperature expansion (HTE) studies are still practically absent, the only exceptions being,
to the best of our knowledge, a series through order β9for the internal energy and a series for the mass gap through
order β5in the n = 3 case [ [7]]. These expansions have been helpful for a first check of MonteCarlo simulation codes,
and series for other quantities and for other values of n would be equally welcome.
We have extended to every value of the spin dimensionality n through order β8the computation of the internal
energy , and for the first time we have computed series for the susceptibility and the second correlation moment.
These series are probably not long enough to provide, by their own, convincing evidence about the existence, the
location and the nature of a possible critical point, but we believe it is useful to make them promptly available so that
they can serve not only to check MonteCarlo data, but also for future more extensive high temperature calculations.
We shall explain later why our computational method, based on the Schwinger-Dyson recursion equations[ [14]],
although very transparent, becomes rapidly cumbersome and therefore is unable, in its present form, to produce
substantially longer series.
II. THE HIGH TEMPERATURE SERIES
Let us briefly describe the model and fix our conventions.
The partition function of the model is
Z =
??
x
ds(x)δ(s(x)2− 1)exp[β
2
?
x
?
µ=1,2
(s(x) · s(x + eµ))2] (1)
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The variables of the model are n-component classical spins s(x) of unit length associated to each site x = x1e1+x2e2
of a 2-dimensional square lattice, e1and e2are the two elementary lattice vectors.
The Hamiltonian and the integration measure have a global O(n)/Z2 and a local Z2 invariance. Since in two
dimensions continuous symmetries are unbroken[ [15]], the most general correlation function < φ(C) > can be written
as
< φ(C) >=< φ(x1,x2,...,xn;{bi,j}) >=<
?
1≤i<j≤n
(s(xi) · s(xj))bij> (2)
with integer bi,j ≥ 0. The local invariance under Z2, which also cannot break[ [16]], implies the further restriction
that each s(xi) has to appear in φ(C) an even number of times.
The correlation function (2) may be represented graphically as follows: the lattice points x1,x2,...,xn are taken
as vertices and a line connecting the vertices xi and xj is associated to each factor s(xi) · s(xj) in φ(C). In terms
of graphs the local Z2invariance requires that the degree of each vertex be even. Thus, for instance, the correlation
< s(x1) · s(x2) > vanishes trivially.
The fundamental two-spin correlation is then G(x2− x1;β,n) =< (s(x1) · s(x2))2>.
In particular we have −G(e1;β,n) = E, the energy per site.
We also have computed the moments m(l)(β,n) of the connected correlations
C(x2− x1;β,n) =
?
a,b
<
?
sa(x1)sb(x1)− < sa(x1)sb(x1) >
??
sa(x2)sb(x2)− < sa(x2)sb(x2) >
?
>= G(x2− x1;β,n) − 1/n
(3)
which are defined as follows
m(l)(β,n) =
?
x
| x |lC(x;β,n) =
?
r
a(l)
rβr
(4)
The HTE coefficients for G(e1;β,n) =?
rgr(n)βrare:
g0(n) =1
n
g1(n) =
n − 1
n2(n + 2)
g2(n) =
(n − 1)(n − 2)
n3(n + 2)(n + 4)
g3(n) =(n − 1)(72 + 18n − 11n2− n3+ n4)
n4(n + 2)3(n + 4)(n + 6)
g4(n) =(n − 1)(n − 2)(528 + 130n − 17n2− 3n3+ n4)
n5(n + 2)3(n + 4)(n + 6)(n + 8)
g5(n) = ((n − 1)(284160+ 130496n− 104032n2− 53344n3+ 6888n4+ 5496n5+ 474n6− 56n7− 2n8+ n9)
/(n6(n + 2)5(n + 4)2(n + 6)(n + 8)(n + 10))
g6(n) = (n − 1)(n − 2)(11704320+ 8093952n− 1233088n2− 1863104n3− 200776n4+ 103840n5
+26210n6+ 1386n7− 100n8− 2n9+ n10)/(n7(n + 2)5(n + 4)3(n + 6)(n + 8)(n + 10)(n + 12))
g7(n) = (n − 1)(341118812160+ 428301582336n+ 17644511232n2− 191549657088n3− 76694446080n4
+17276826240n5+ 16424658272n6+ 1926697808n7− 951227456n8− 295105184n9
−5505626n10+ 10001781n11+ 1876337n12+ 133277n13+ 1527n14− 171n15+ 9n16+ n17)
/(n8(n + 2)7(n + 4)4(n + 6)3(n + 8)(n + 10)(n + 12)(n + 14))
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g8(n) = (n − 1)(n − 2)(1271577968640+ 1237547925504n− 87404783616n2− 441393059328n3
−107082739328n4+ 37546256480n5+ 17834481104n6+ 440575008n7− 777645296n8
−105547274n9+ 10134853n10+ 3591697n11+ 316891n12+ 7749n13− 289n14− n15+ n16)
/(n9(n + 2)7(n + 4)3(n + 6)3(n + 8)(n + 10)(n + 12)(n + 14)(n + 16))
For n = 3 we have (compare with Ref.[ [7]]):
G(e1;β,3) =1
3+
2
45β +
2
945β2+
2
7875β3+
34
467775β4+
12179386
142468185234375β7+
13402
2280403125β5+
10702
47888465625β6
33996598
4872411935015625β8+ ...+
The HTE coefficients for m(0)(β,n), also called the susceptibility, are:
a(0)
0(n) =n − 1
n
a(0)
1(n) =
4(n − 1)
n2(2 + n)
a(0)
2(n) =4(n − 1)(8 + 3n + n2)
n3(2 + n)2(4 + n)
a(0)
3(n) =4(n − 1)(96 + 64n + 32n2+ 5n3+ n4)
n4(2 + n)3(4 + n)(6 + n)
a(0)
4(n) =4(n − 1)(1 + n)(3456+ 1968n + 570n2+ 89n3+ 9n4+ n5)
n4(2 + n)4(4 + n)2(6 + n)(8 + n)
a(0)
5(n) = (4(n − 1)(122880+ 101888n+ 40640n2+ 42528n3+ 35696n4+ 11094n5
+1807n6+ 162n7+ 10n8+ n9))/(n6(2 + n)5(4 + n)2(6 + n)(8 + n)(10 + n))
a(0)
6(n) = (4(n − 1)(−115015680− 79331328n+ 74609664n2+ 96772864n3+ 44006080n4
+15702208n5+ 7513312n6+ 2862016n7+ 648560n8+ 87178n9+ 7048n10+ 364n11+ 19n12+ n13))
/(n7(2 + n)6(4 + n)3(6 + n)2(8 + n)(10 + n)(12 + n))
a(0)
7(n) = (4(n − 1)(43104337920+ 43866980352n− 5407064064n2− 15002345472n3+ 3765867520n4
+8878097920n5+ 4282305280n6+ 1196842912n7+ 326380672n8+ 97376320n9
+22123168n10+ 3228422n11+ 292472n12+ 16058n13+ 566n14+ 23n15+ n16))
/(n8(2 + n)7(4 + n)3(6 + n)3(8 + n)(10 + n)(12 + n)(14 + n))
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a(0)
8(n) = (4(n − 1)(−94746307461120− 126660674322432n− 5226623926272n2+ 66792515567616n3
+32795171340288n4+ 48194863104n5− 862398014464n6+ 2921457334912n7
+2239266005664n8+ 790758440112n9+ 185029551696n10+ 37818452512n11+ 7776875970n12
+1395247971n13+ 184588028n14+ 16678488n15+ 985722n16+ 36650n17+ 952n18+ 32n19+ n20))
/(n9(2 + n)8(4 + n)4(6 + n)3(8 + n)2(10 + n)(12 + n)(14 + n)(16 + n))
For n = 3 these formulae give:
m(0)(β,3) =2
3+
8
45β +208
4725β2+
10273872032
427404555703125β7+
704
70875β3+
12704
5457375β4+
934133719808
183909666174609375β8+ ...
8254816
15962821875β5
+
37545856
335219259375β6+
The HTE coefficients for m(2)(β,n), the second correlation moment, are:
a(2)
1(n) =
4(n − 1)
n2(2 + n)
a(2)
2(n) =4(n − 1)(28 + 8n + n2)
n3(2 + n)2(4 + n)
a(2)
3(n) =4(n − 1)(624 + 344n + 124n2+ 15n3+ n4)
n4(2 + n)3(4 + n)(6 + n)
a(2)
4(n) =4(n − 1)(52224+ 57856n+ 37760n2+ 13200n3+ 2844n4+ 376n5+ 25n6+ n7)
n5(2 + n)4(4 + n)2(6 + n)(8 + n)
a(2)
5(n) = (4(n − 1)(1044480+ 1553408n+ 1394176n2+ 692768n3+ 237584n4+ 50998n5
+7107n6+ 642n7+ 30n8+ n9))/(n6(2 + n)5(4 + n)2(6 + n)(8 + n)(10 + n))
a(2)
6(n) = (4(n − 1)(420249600+ 1092464640n+ 1523555328n2+ 1255855616n3+ 710497728n4
+285572064n5+ 84264528n6+ 18404144n7+ 2909092n8+ 327054n9+ 25803n10+ 1354n11+ 44n12+ n13))
/(n7(2 + n)6(4 + n)3(6 + n)2(8 + n)(10 + n)(12 + n))
a(2)
+232510854656n4+ 131053062400n5+ 54800469376n6+ 17324337248n7+ 4212618016n8
+794772080n9+ 114865432n10+ 12396858n11+ 977532n12+ 55028n13+ 2106n14+ 53n15+ n16))
/(n8(2 + n)7(4 + n)3(6 + n)3(8 + n)(10 + n)(12 + n)(14 + n))
7(n) = (4(n − 1)(83979141120+ 204309430272n+ 291728203776n2+ 300109848576n3
a(2)
8(n) = (4(n − 1)(−58788371496960+ 8364704661504n+ 280171118592000n2+ 479504520511488n3
+464202911416320n4+ 329227765829632n5+ 188047485044736n6+ 87029638424064n7
+32173730443520n8+ 9456558685824n9+ 2219800018368n10+ 419325652576n11+ 63930454192n12
+7800108776n13+ 746995212n14+ 54841620n15+ 3013992n16+ 120454n17+ 3374n18+ 67n19+ n20))
/(n9(2 + n)8(4 + n)4(6 + n)3(8 + n)2(10 + n)(12 + n)(14 + n)(16 + n))
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In particular, the HT expansion of m(2)(β,3) is:
m(2)(β,3) =
8
45β +
488
4725β2+
2896
70875β3+
1123712
81860625β4+
12824336768
38854959609375β7+
67018144
15962821875β5+
2023066384
1676096296875β6
18110407484144
208430954997890625β8+ ...+
A correlation length may be defined, as usual, in terms of the ratio of m(2)(β,n) and m(0)(β,n).
Let us notice that a few simple checks of the formulae are possible: all HTE of the connected correlations have
to vanish for n = 1 because of the triviality of the RP0model. For n = 2, the expansions should reduce to the
corresponding ones for the O(2) (or XY-) vector model. Finally, for n = 3 the HTE of C(e1;β,n) agrees with the
calculation of Ref.[ [7]].
Our HTE have been computed from the Schwinger-Dyson equations of the model, an infinite system of linear
equations among the correlation functions. The generic equation, which may be deduced following closely Ref.[ [14]],
has the structure
< φ(C) >=
1
n + g1− 2
?
β
?
µ
(< φ(C−
µ) > − < φ(C+
µ) >) + (b12− 1) < φ(C12,12) > −
n
?
j=3
b1j< φ(C2j
12,1j) >
?
(5)
Here we have assumed that the vertices x1and x2are connected by one line at least, g1is the degree of the vertex
x1, bij the number of lines connecting the vertices xiand xj, < φ(C−
from < φ(C) > by removing a factor s(x1) · s(x2) and replacing it by s(x1) · s(x1+µ)s(x2) · s(x1+µ), namely
µ) > denotes the correlation function obtained
φ(C−
µ) = φ(C)s(x1) · s(x1+µ)s(x2) · s(x1+µ)
s(x1) · s(x2)
(6)
and analogously
φ(C+
µ)= φ(C)((s(x1) · s(x1+µ))2
φ(C)
(s(x1) · s(x2))2
φ(C12,12)=
φ(C2j
12,1j)= φ(C)
s(x2) · s(xj)
s(x1) · s(x2)s(x1) · s(xj).
The HTE of the correlation < φ(C) > is obtained solving iteratively eqs.(5) by the same procedure as in the case of
the n−vector model[ [14]]. Here however, a difficulty is met: while in the case of the n−vector model a large fraction
of the graphs generated after the first few iterations can be neglected, in this case, due to the local Z2 symmetry,
all graphs contribute nontrivially to the final results and therefore must be recorded. Thus the required computer
memory rapidly becomes exceedingly large and it is difficult to push the expansion beyond the 8-th order. However
not all the blame should be laid upon the computational technique since the combinatorial complexity of the expansion
is really higher and of a faster growth with the order than in the n−vector case. It is also interesting to recall that
analogous difficulties were met when performing strong coupling expansions in the Hamiltonian formalism[ [6]].
A simple analysis of the series by ratio and Pad´ e approximants methods[ [17]] (see Fig.1 and Fig.2 ) suggests
the existence of a critical point when n ≈ 2, but, unfortunately, the series seem to be not long enough to warrant
any reasonably safe conclusion when n = 3 or greater. To be sure, for various values of n there are some Pad´ e
approximants of the susceptibility having a real positive singularity or a complex conjugate pair of singularities
nearby the real positive β axis and in the expected position. The same happens for the logarithmic derivative of the
susceptibility. These poles however, at this order of approximation, are not stable enough to enable us to exclude
the possibility of an artifact of low order approximants to mimic the steep increase of the susceptibility. Thus some
completely different scenarios are still compatible with our series, for instance:
a) in analogy with the behavior of the n−vector model[ [14]] a critical point exist for n ≈ 2. As n is increased and
varied through some ˜ n ≤ 3, the critical point might split into an unphysical pair of complex conjugate singularities
so that the model becomes asymptotically free for n ≥ 3. This conjecture might be supported both by the alternate
ratios plots of Fig. 1, which seem to show the onset of an oscillatory trend [ [17]] and by some Pad´ e approximants to
the susceptibility or its log-derivative whose nearest singularities in the right half β plane are complex.
b) a critical point exists for all n as suggested by Ref.[ [2]].
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ACKNOWLEDGMENTS
Our thanks are due to Sergio Caracciolo for suggesting to undertake this computation and to Alan Sokal for
further encouragement and useful discussions. We also are indebted to U. Wolff for a useful discussion and for kindly
permitting us to use his unpublished MonteCarlo data in Fig.2. Finally we thank A. J. Guttmann and G. Marchesini
for carefully reading a draft of this note. Our work has been partially supported by MURST.
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FIG. 1. Alternate ratios ¯ rs(n) = (a(0)
the expansion coefficients of the susceptibility for various val-
ues of the spin dimensionality n are plotted versus 1/s. Going
from the lower plot to the upper, we have n = 2,3,..,8.
s−1(n)/a(0)
s+1(n))1/2of
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FIG. 2. The susceptibility for n = 3 at order β8vs. β. The
continuous curve shows the [4/4] Pad´ e approximant (which
is singular at β ≈ 5.175 ± 0.315i). The dashed curve shows
the sum of the susceptibility series truncated at order β8.
The squares represent data from the MonteCarlo simulation
of Ref. [ [7]]. The triangles represent unpublished data from
a MonteCarlo cluster simulation performed by U. Wolff[ [11]]
on lattices of size up to 2562.
7