In a previous paper, we have discussed how the Landau potential (entering in Landau theory of phase transitions) can be simplified using the Poincaré normalization procedure. Here, we apply this approach to the Landau-deGennes functional for the isotropic-nematic transitions, and transitions between different nematic phases, in liquid crystals. We give special attention to applying our method in the region near the main transition point, showing in full detail how this can be done via a suitable simple modification of our Poincaré-like method. We also consider the question if biaxial phases can branch directly off the fully symmetric state; some partial results in this direction are presented.

1.
L. D.
Landau
,
Nature
138
,
840
-
841
(
1936
);
L. D.
Landau
,
Zh. Exsp. Teor. Fiz.
7
,
19
-
32
& 627-637 (
1937
), http://www.ujp.bitp.kiev.ua/files/journals/53/si/53SI08p.pdf.
2.
L. D.
Landau
and
E. M.
Lifshitz
,
Statistical Physics
(
Pergamon Press
,
1958
).
3.
Yu. M.
Gufan
,
Structural Phase Transitions
(
Nauka
,
1982
).
4.
I. A.
Sergienko
,
Yu. M.
Gufan
, and
S.
Urazhdin
,
Phys. Rev. B
65
,
144104
(
2002
).
5.
G.
Gaeta
,
Ann. Phys.
312
,
511
-
540
(
2004
).
6.
G.
Gaeta
, “
Poincaré-like approach to Landau theory. I. General theory
,”
J. Math. Phys.
56
,
083504
(
2015
).
7.
V. I.
Arnold
,
Geometrical Methods in the Theory of Ordinary Differential Equations
(
Springer
,
1983
).
8.
C.
Elphick
 et al,
Physica D
29
,
95
-
127
(
1987
);
C.
Elphick
, et al,
Physica D
32
,
488
(
1988
).
9.
S.
Walcher
,
Math. Ann.
291
,
293
-
314
(
1991
);
S.
Walcher
,
J. Math. Anal. Appl.
180
,
617
-
632
(
1993
).
10.
G.
Cicogna
and
G.
Gaeta
,
Symmetry and Perturbation Theory in Nonlinear Dynamics
(
Springer
,
1999
).
11.
G.
Cicogna
and
S.
Walcher
,
Acta Appl. Math.
70
,
95
-
111
(
2002
).
12.
S.
Walcher
, “
Convergence of perturbative expansions
,” in
Encyclopedia of Complexity and Systems Science
(
Springer
,
2009
), pp.
6760
-
6771
;
S.
Walcher
, reprinted in
Mathematics of Complexity and Dynamical Systems
(
Springer
,
2012
), pp.
1389
-
1399
.
13.
G.
Gaeta
,
Acta Appl. Math.
70
,
113
-
131
(
2002
).
G.
Gaeta
,
Lett. Math. Phys.
42
,
103
-
114
(
1997
);
G.
Gaeta
,
Lett. Math. Phys.
57
,
41
-
60
(
2001
).
15.
P. G.
de Gennes
and
J.
Prost
,
The Physics of Liquid Crystals
(
Oxford UP
,
1993
).
16.
E.
Virga
,
Variational Theories for Liquid Crystals
(
Chapman & Hall
,
1995
).
17.
E. F.
Gramsbergen
,
L.
Longa
, and
W. H.
de Jeu
,
Phys. Rep.
135
,
195
-
257
(
1986
).
18.
D.
Allender
and
L.
Longa
,
Phys. Rev. E
78
,
011704
(
2008
).
19.
M. J.
Freiser
,
Phys. Rev. Lett.
24
,
1041
-
1043
(
1970
).
20.
M. V.
Jaric
and
J. L.
Birman
,
J. Math. Phys.
18
,
1459
-
1465
(
1977
), 2085.
21.
D. H.
Sattinger
,
J. Math. Phys.
19
,
1720
-
1732
(
1978
).
22.
L.
Michel
,
Rev. Mod. Phys.
52
,
617
-
651
(
1980
).
23.
L.
Michel
,
J. S.
Kim
,
J.
Zak
, and
B.
Zhilinskii
,
Phys. Rep.
341
,
1
-
395
(
2001
), http://www.sciencedirect.com/science/article/pii/S0370157300000880.
24.
P.
Chossat
and
R.
Lauterbach
,
Methods in Equivariant Bifurcation Theory and Dynamical Systems
(
World Scientific
,
2000
).
25.
P.
Chossat
,
Acta Appl. Math.
70
,
71
-
94
(
2002
).
26.
M.
Abud
and
G.
Sartori
,
Ann. Phys.
150
,
307
-
372
(
1983
).
27.
G.
Sartori
,
La Rivista del Nuovo Cimento
14
,
1
(
1991
);
G.
Sartori
,
Acta Appl. Math.
70
,
183
-
207
(
2002
).
28.
H.
Broer
, in
Lecture Notes in Mathematics
(
Springer
,
1981
), Vol.
898
.
29.
G.
Benettin
,
L.
Galgani
,
A.
Giorgilli
, and
J. M.
Strelcyn
,
Nuovo Cimento B
79
,
201
-
223
(
1984
).
30.
A.
Giorgilli
, “
On the representation of maps by Lie transforms
,”
Istsituto Lombardo (Rend. Scienze)
146
,
251
-
277
(
2012
); preprint arXiv:1211.5674 (2012).
31.
M.
Joyeux
,
J. Chem. Phys.
109
,
2111
-
2122
(
1998
).
32.
D.
Sugny
and
M.
Joyeux
,
J. Chem. Phys.
112
,
31
-
39
(
2000
).
33.

By this we mean of course the degree of the polynomial; it is customary to use the word “order” in this context since the degree of the Landau polynomial corresponds to the order in perturbation theory we are considering.

34.

It should also be recalled that Poincaré-Birkhoff normal forms have been used to compute quantum spectra with a remarkable level of accuracy up to near dissociation threshold, see, e.g., Refs. 31 and 32.

35.

Which is implemented by an algebraic manipulation language, given the considerable complexity of the algebra involved.

36.

Note the effective order parameters reduce to the two independent eigenvalues λ1, λ2 of Q, as different Q’s related by a similarity transformation are equivalent.

37.

It should be noted this will be non-homogeneous; thus, the disappearing terms are actually just recombined to a simpler expression in the new coordinates.

38.

As for the determinant of Q and its iterates, these are of course also invariant but are expressed as a polynomial in terms of the traces, e.g., for a three-dimensional matrix A, one has Det(A) = (1/3)Tr(A3) − (1/2)Tr(A2)Tr(A) + (1/6)[Tr(A)]3.

39.

This would actually find some relevant obstacles when dealing with the physically significant problem discussed in Secs. VI and VII; what we actually (implicitly) use there is a more refined version of the normalization algorithm,14 and resorting to explicit allows to avoid discussing the general mathematical theory, for which the reader is referred to Refs. 10, 13, and 14.

40.

In the mathematical literature, these are also mentioned as 2-covariants, where the “2” specifies the SO(3) representation they follow; we recall vector SO(3) representations have dimension d = 2ℓ + 1 with ℓ ∈ Z+, and in this case ℓ = 2.

41.

Homogeneous covariants are determined up to linear combinations; thus, one could choose expressions with different overall constants and, in the case where more covariants of the same order exist, different linear combinations of them.

42.

It should be noted that, albeit we computed all the low order covariants, such a complete knowledge is not always needed: in fact, covariants index the (covariant) changes of variables, and knowing only some of them means that the considered changes of variable would not be the most general ones. E.g., in the following, we will see that disregarding one of the fourth order covariants, e.g., F4(2), would not harm our procedure.

43.

In mathematically more precise terms, we should actually require that c1 is large enough, with c1>ε with ε depending on the region of the phase space we want to study.

44.

Dealing with still higher orders does not present any new conceptual difficulty, but the algebraic manipulations become rapidly quite involved.

45.

Note however that η = 0 leaves us with T3 fully undetermined (in fact, in (55)η and T3 always appear together), and T22=q2=λ/4.

46.

We stress we have only studied local stability, determined by the Hessian; our analysis does not exclude the possibility that Φ has also different local minima, maybe with a lower value.

47.

It should be stressed again that the xi appearing above are those obtained after the normalization steps have been carried out; this has to be taken into account in trying to compute this solution branch in explicit terms. However, our choice of one-dimensional representative has been guided by Remark 1, so that these are embedded in invariant subspaces for the full Landau theory.

48.

The conceptual problems related to inversion are radically solved by considering more refined changes of variables, corresponding to Lie series.10,28–30 In this way, the map yx corresponds to the time-one flow of a vector field X = hi(x)(∂/∂xi), and inversion corresponds to time-reversed flow. The computational details to actually obtain the inversion at finite orders are however essentially the same. See Refs. 11 and 12 for other details on convergence issues.

49.

So called after the Latvian-born Russian mathematician Theodor Molien (Riga 1861–Tomsk 1941), the relevant work for our topic dates back to 1897.

You do not currently have access to this content.