Octupolar Tensors for Liquid Crystals

0.36 MB·

by  Yannan Chen

Octupolar Tensors for Liquid Crystals
Yannan Chen Liqun Qi Epifanio G. Virga
∗ † ‡
April 4, 2017
7
1
Abstract
0
2
A third-order three-dimensional symmetric traceless tensor, called the octupolar tensor,

r
p has been introduced to study tetrahedratic nematic phases in liquid crystals. The octupolar
A potential, a scalar-valuedfunction generatedon the unit sphere by that tensor, should ideally
have four maxima capturing the most probable molecular orientations (on the vertices of
2
a tetrahedron), but it was recently found to possess an equally generic variant with three

]
h maxima instead of four. It was also shown that the irreducible admissible region for the
p octupolartensorinathree-dimensionalparameterspaceisboundedbyadome-shapedsurface,
-
h beneath which is a separatrix surface connecting the two generic octupolar states. The latter
t
a surface, which was obtained through numerical continuation, may be physically interpreted
m
as marking a possible intra-octupolar transition. In this paper, by using the resultant theory
[
of algebraic geometry and the E-characteristic polynomial of spectral theory of tensors, we

2 give a closed-form, algebraic expression for both the dome-shaped surface and the separatrix
v
surface. This turns the envisaged intra-octupolar transition into a quantitative, possibly
1
6 observable prediction. Some other properties of octupolar tensors are also studied.
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Keywords. Octupolarordertensors;Resultants;Tracelesstensors;Liquidcrystas;Intra-
6
0 octupolar transitions.
.
1
0
7
1
:
v
i
X
r
a
∗SchoolofMathematicsandStatistics,ZhengzhouUniversity,Zhengzhou450001, China([email protected]).
This author was supported by the National Natural Science Foundation of China (Grant No. 11401539), the
Development Foundation for Excellent Youth Scholars of Zhengzhou University (Grant No. 1421315070), and the
Hong KongPolytechnic University Postdoctoral Fellowship.
†Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong
Kong ([email protected]). This author’s work was partially supported by the Hong Kong Research Grant
Council (Grant No. PolyU 501913, 15302114, 15300715 and 15301716).
‡Mathematical Institute, University of Oxford, UK ([email protected]), on leave from Dipartimento di
Matematica, Universit`a di Pavia, via Ferrata 5, I-27100 Pavia, Italy ([email protected]). The work of this
author was done while he was visiting the Oxford Centre for Nonlinear PDE at the University of Oxford, whose
kind hospitality hegratefully acknowledges.
1
1 Introduction
Liquid crystals are well-known for visualization applications in flat panel electronic displays. But
beyond that, various optical and electronic devices, such as laser printers, light-emitting diodes,
field-effect transistors, andholographicdatastorage, wereinvented withthedevelopment of bent-
core (banana-shaped) liquid crystals [8, 9]. A third-order three dimensional symmetric traceless
tensor was introduced in [6] to characterize condensed phases exhibited by bent-core molecules
[16, 11]. Based on such a tensor, the orientationally ordered octupolar (tetrahedratic) phase
has been both predicted theoretically [13, 2] and confirmed experimentally [22]. After that, the
octupolar order parameters of liquid crystals have been widely studied [1, 20, 7]. Generalized
liquid crystal phases are also considered in [17, 12], which feature octupolar order tensors among
so many others.
Ideally, the octupolar order in a molecular ensemble of generalized liquid crystals is expected
to exhibit four directions in space pointing towards the vertices of a tetrahedron, along which
specific molecular axes are most likely to be oriented. Fel [6] first proposed to use a third-order
three dimensional symmetric tensor A, which here is called the octupolar tensor, to describe the
tetrahedraticsymmetryoftheoctupolarorder. AccordingtotheBuckingham’sformula[3,20]for
the expansion in Cartesian tensors of the orientational probability density function for uniaxial
nematics, the octupolar tensor A is traceless, i.e., the trace of each slice matrix of A vanishes.
As shown in [7], by a judicious choice of the Cartesian coordinate system, such a tensor can be
represented by three independent parameters, which there were called α ,β and α .
0 3 2
Virga [21] and Gaeta and Virga [7], in their studies of third-order octupolar tensors in two
and three space dimensions, respectively, also introduced the octupolar potential, a scalar-valued
functionontheunitsphereobtainedfromtheoctupolartensor. Inparticular,GaetaandVirga[7]
showed that the irreducible admissible region for the octupolar potential is bounded by a surface
in the three-dimensional parameter space which has the form of a dome and, more importantly,
that there are indeed two generic octupolar states, divided by a separatrix surface in paramter
space. Physically, the latter surface was interpreted as representing a possible intra-octupolar
transition.
In this paper, by using the resultant theory of algebraic geometry and the E-characteristic
polynomial of the spectral theory of tensors, we give a closed-form, algebraic expression for both
the dome and the separatrix. This turns the intra-octupolar transition envisioned in [7] into a
quantitative, possibly observable prediction. Some other properties of octupolar tensors are also
studied.
In Section 2, we prove that the traceless property of octupolar tensors is preserved under
orthogonal transformations. This property motives us to choose a proper Cartesian coordinate
system to reduce the independent elements of the octupolar tensor A from seven to three. By
assuming that the North pole (0,0,1) is a maximum point of the octupolar potential on the
⊤
unit sphere, we identify an irreducible, admissible region for only three independent parameters
of A, which we shall also call α ,β and α to ease comparison with [7].
0 3 2
Using the multi-polynomial resultant from algebraic geometry [5], we derive the resultant
2
Res(Ax2) in Section 3. Res(Ax2) = 0 implies that A has zero E-eigenvalues. After that, we
construct the E-characteristic polynomial φA(λ) of A; this latter is a fourteen-degree polynomial
withonlyevendegreetermsanditsconstanttermisthesquareofRes(Ax2). WhenRes(Ax2) = 0,
6
φA(λ) = 0 if and only if λ is an E-eigenvalue of A.
In Section 4, by assuming that the North pole (0,0,1) is the global maximum point of the
⊤
octupolar potential on the unit sphere, the admissible region is further reduced. The boundary
of such a reduced admissible region was referred to as the dome in [7]. By exploring the E-
characteristic polynomial φA(λ) of A, we give the algebraic expression for the dome explicitly.
We said already that two generic octupolar states were identified in in [7] on the two sides of
a separatrix surface in parameter space. It should beadded here that in in [7] both the dome and
the separatix were determined by numerical continuation: no closed-form was given for either of
them. In Section 5, we also obtain the explicit, algebraic expression for the separatrix.
Finally, some concluding remarks are drawn in Section 6.
2 Preliminaries
The octupolar tensor in liquid crystals is a traceless tensor. Now, we give the formal definition of
traceless tensors and prove that the traceless property of a symmetric tensor are invariant under
orthogonal transformations [7]. For convenience, wedenote as R[m,n] thereal-valued space of mth
order n dimensional symmetric tensors.
Definition 2.1. Let T = [t ] R[m,n]. If
i1i2···im ∈
n
t = 0 for all i ,...,i = 1,2,...,n,
iii3 im 3 m
···
i=1
X
then T is called a traceless tensor.
Theorem 2.2. Let T = [t ] R[m,n] be a traceless tensor and Q = [q ] Rn n be an
i1i2···im ∈ ij ∈ ×
orthogonal matrix. We denote by TQm R[m,n] the symmetric tensor whose elements are
∈
n n n
[TQm] = t q q q .
i1i2···im ··· j1j2···jm i1j1 i2j2··· imjm
jX1=1jX2=1 jXm=1
Then, TQm is also a traceless tensor.
Proof. It is straightforward to see that the new tensor TQm is real-valued and symmetric. Now,
we consider its slice matrices. As the sum of all matrix eigenvalues, the trace of a symmetric
matrix is invariant under an orthogonal transformation. Hence, we get
n m m n
t q q = t = 0 (1)
j1j2j3 jm ij1 ij2 iij3 jm
··· ···
Xi=1jX1=1jX2=1 Xi=1
3
for all j ,...,j = 1,2,...,n. By some calculations,
3 m
n n n n n n
[TQm] = t q q q q
iii3···im ··· j1j2j3···jm ij1 ij2 i3j3··· imjm
Xi=1 Xi=1jX1=1jX2=1jX3=1 jXm=1
n n n n n
= t q q q q
···  j1j2j3···jm ij1 ij2 i3j3··· imjm
jX3=1 jXm=1 Xi=1jX1=1jX2=1
= 0,  
where the last equality is valid because of (1). Hence, the new tensor TQm is also traceless.
By exploting the rotational invariance of traceless tensors, we now represent the octupolar
tensor as
a a a a a a a a a
111 112 113 112 122 123 113 123 133
A= a a a a a a a a a R[3,3] (2)
 112 122 123 122 222 223 123 223 233 
∈
a a a a a a a a a
 113 123 133 123 223 233 133 233 333 
 
by choosing a proper Cartesian coordinate system. The traceless property of A means that
a +a +a = 0,
111 122 133
a +a +a = 0, (3)
112 222 233


a +a +a = 0.
113 223 333

Hence, there are seven independentelements in A. Let
α = a , α = a , α = a , α = a , β = a , β = a , β = a .
0 123 1 111 2 222 3 333 1 122 2 233 3 113
Using the traceless property (3), we convert (2) into
A=
α1 −α2−β2 β3 −α2−β2 β1 α0 β3 α0 −α1−β1
−α2−β2 β1 α0 β1 α2 −α3−β3 α0 −α3−β3 β2 .
 
β3 α0 −α1−β1 α0 −α3−β3 β2 −α1−β1 β2 α3
 
The associated octupolar potential as defined in [7] is
3 3 3
Φ(x) Ax3 = a x x x (4)
ijk i j k
≡
i=1 j=1k=1
XXX
= α x3+α x3+α x3+6α x x x +3β x x2+3β x x2+3β x2x
1 1 2 2 3 3 0 1 2 3 1 1 2 2 2 3 3 1 3
3(α +β )x x2 3(α +β )x2x 3(α +β )x2x .
− 1 1 1 3− 2 2 1 2− 3 3 2 3
On the unit sphere S2 x= (x ,x ,x ) : x2+x2+x2 = 1 , the polynomial Φ(x) has at least
≡ { 1 2 3 ⊤ 1 2 3 }
one maximum point. Without loss of generality, we assume such a maximum point being the
North pole (0,0,1) and
⊤
α = Φ(0,0,1) 0.
3
≥
4
From the spectral theory of tensors [18], we know that α is a Z-eigenvalue of A with an
3
associatedZ-eigenvector (0,0,1) . AllZ-eigenvectors andZ-eigenvalues mustsatisfythefollowing
⊤
system:
α x2+2α x x +β x2+2β x x (α +β )x2 2(α +β )x x = λx ,
1 1 0 2 3 1 2 3 1 3− 1 1 3− 2 2 1 2 1
α x2+2α x x +2β x x +β x2 (α +β )x2 2(α +β )x x = λx ,
 2 2 0 1 3 1 1 2 2 3− 2 2 1− 3 3 2 3 2

α x2+2α x x +2β x x +β x2 2(α +β )x x (α +β )x2 = λx ,
 3 3 0 1 2 2 2 3 3 1− 1 1 1 3− 3 3 2 3
x2+x2+x2 = 1.
 1 2 3




Hence, requiring (0,0,1) to be a solution, we obtain
⊤
α +β = 0 and β =0.
1 1 2
Moreover, because Φ( x ,0,0) = Φ(x ,0,0), we could rotate the Cartesian coordinate system
1 1
− −
so that Φ(1,0,0) = 0 and we get
α = 0.
1
Now, the octupolar tensor in (2) reduces to
0 α β α 0 α β α 0
2 3 2 0 3 0
− −
A= α 0 α 0 α α β α α β 0 ,
 2 0 2 3 3 0 3 3 
− − − − −
β α 0 α α β 0 0 0 α
 3 0 0 − 3− 3 3 
 
which features four independent elements, namely, α , α , α , and β . Correspondingly, the
0 2 3 3
octupolar potential (4) is
Φ(x;α ,α ,α ,β )= α x3+α x3+6α x x x +3β x2x 3α x2x 3(α +β )x2x
0 2 3 3 2 2 3 3 0 1 2 3 3 1 3− 2 1 2− 3 3 2 3
for all x S2. Without loss of generality, we can assume
∈
α 0
2
≥
as a consequence of the following proposition.
Proposition 2.3. For the octupolar potential (12), we have
Φ(x ,x ,x ;α ,α ,α ,β )= Φ(x , x ,x ; α , α ,α ,β ).
1 2 3 0 2 3 3 1 2 3 0 2 3 3
− − −
Next, we turn to the assumption that the North pole (0,0,1) is the maximum point of the
⊤
octupolar potential with value α .
3
Theorem 2.4. Suppose that the North pole (0,0,1) is the maximum point of the octupolar
⊤
potential Φ(x) on S2. Then, we have that
3α2 4α β 4β2 4α2 0. (5)
3− 3 3− 3 − 0 ≥
If the strict inequality holds in (5), then (0,0,1) is a (local) maximum point of Φ(x) on S2.
⊤
5
Proof. We consider the spherical optimization problem:
max Φ(x) =Ax3
(6)
( s.t. x⊤x= 1.
Its Lagrangian is
1 λ
L(x,λ) = Ax3+ (x x 1).
⊤
−3 2 −
The Hessian of the Lagrangian is
2xxL(x,λ) = λI 2Ax, (7)
∇ −
which is positive semidefinite on the tangent space x y R3 : x y = 0 if x is a (local)
⊥ ⊤
≡ { ∈ }
maximum point of Φ(x) on S2 [15]. Let P I xx R3 3 be the projection matrix onto x .
⊤ × ⊥
≡ − ∈
Then, the matrix P⊤ 2xxL(x,λ)P is positive semidefinite. By use of the first-order necessary
∇
condition,
Ax2 = λx and x x = 1, (8)
⊤
we have that
P⊤ 2xxL(x,λ)P = λ(I +xx⊤) 2Ax. (9)
∇ −
Because the North pole (0,0,1) is a maximum point with the associated multiplier λ = α ,
⊤ 3
we arrive at
α 2β 2α 0
3 3 0
− −
[P⊤∇2xxL(x,λ)P]λ=α3,x=(0,0,1)⊤ =  −2α0 3α3 +2β3 0 .
0 0 0
 
 
As easily seen, this projected Hessian has eigenvalues
µ = 0, µ = 2α (α +2β )2+4α2,
1 2,3 3± 3 3 0
q
which are all required to be non-negative [15]. Hence, we obtain the following inequality
3α2 4α β 4β2 4α2 0.
3− 3 3− 3 − 0 ≥
If the strict inequality holds, i.e., if 3α2 4α β 4β2 4α2 > 0, then the North pole (0,0,1)
3 − 3 3− 3 − 0 ⊤
is a (local) maximum point of Φ(x) on S2 [15].
We first consider the case that α = 0. From Theorem 2.4, we know that α = β = 0. If
3 0 3
α > 0, then Φ(0,1,0) = α > α = Φ(0,0,1). This contradicts that (0,0,1) is a maximum
2 2 3 ⊤
point. Hence α = 0 and the octupolar tensor A is the trivial zero tensor.
2
In the remainder of this paper, we shall consider the case that α is positive. Without loss of
3
generality, by Proposition 2.3 and Theorem 2.4, we can choose
α = 1, α 0, and α2+(β + 1)2 1. (10)
3 2 ≥ 0 3 2 ≤
6
Then, the octupolar tensor
0 α β α 0 α β α 0
2 3 2 0 3 0
− −
A(α ,β ,α ) = α 0 α 0 α 1 β α 1 β 0 (11)
0 3 2  2 0 2 3 0 3 
− − − − −
β α 0 α 1 β 0 0 0 1
 3 0 0 − − 3 
 
has only three independent elements and the associated octupolar potential is given by
Φ(x) =α x3+x3+6α x x x +3β x2x 3α x2x 3(1+β )x2x (12)
2 2 3 0 1 2 3 3 1 3− 2 1 2− 3 2 3
for all x S2.
∈
3 The E-characteristic polynomial
Qi [18] introduced E-eigenvalues and E-eigenvectors for a symmetric tensor and showed that
they are invariant under orthonormal coordinate changes. E-eigenvalues and E-eigenvectors were
further studied in [19, 14, 4]. Furthermore, the coefficients of the E-characteristic polynomial of
a tensor are orthonormal invariants of that tensor [10]. If the E-eigenvalues and the associated
E-eigenvectors of a real-valued symmetric tensor are real, we call them the Z-eigenvalues and the
Z-eigenvectors of the tensor, respectively.
Using the notion of resultant from algebraic geometry [5], we write now compute explicitly
the resultant Res(Ax2), where
F (x ,x ,x )
2α x x +2β x x +2α x x 3 1 2 3
2 1 2 3 1 3 0 2 3
−
Ax2 = α x2+α x2+2α x x 2(1+β )x x F (x ,x ,x ).
 − 2 1 2 2 0 1 3− 3 2 3  ≡ 1 1 2 3
β x2 (1+β )x2+x2+2α x x
 3 1− 3 2 3 0 1 2  F2(x1,x2,x3)
 
 
Since F , F , and F are homogeneous polynomials with degree two in the variables x , x , and
1 2 3 1 2
x , the multi-polynomial system
3
F (x ,x ,x )= F (x ,x ,x ) = F (x ,x ,x ) = 0 (13)
1 1 2 3 2 1 2 3 3 1 2 3
has a trivial solution x = x = x = 0. However, we are interested in its non-trivial solutions
1 2 3
and we assume that the multi-polynomial system (13) has a non-zero common (complex) root.
According to Theorem 2.3 in Chapter 3 of [5], there is a unique irreducible polynomial
Res(F ,F ,F ) such that Res(x2,x2,x2) = 1 and the system F = F = F = 0 has a non-
1 2 3 1 2 3 1 2 3
trivial solution over C if, and only if, Res(F ,F ,F ) = 0. We follow the approach in Chapter 3,
1 2 3
4 of [5]. Since the degree of each of the F is d = 2, we set
i i
§
3
d= (d 1)+1 = 4.
i
−
i=1
X
We divide monomials xυ = xυ1xυ2xυ3 of total degree υ υ +υ +υ = d into three sets:
1 2 3 | | ≡ 1 2 3
S = xυ : υ = d,x2 divides xυ = x4,x3x ,x3x ,x2x2,x2x x ,x2x2 ,
1 { | | 1 } { 1 1 2 1 3 1 2 1 2 3 1 3}
S = xυ : υ = d,x2 doesn’t divides xυ but x2 does = x x3,x x2x ,x4,x3x ,x2x2 ,
2 { | | 1 2 } { 1 2 1 2 3 2 2 3 2 3}
S = xυ : υ = d,x2,x2 don’t divide xυ but x2 does = x x x2,x x3,x x3,x4 .
3 { | | 1 2 3 } { 1 2 3 1 3 2 3 3}
7
Clearly, there are d+2 = 15 monomials xυ with υ = d and each lies in one of sets S , S , and
2 | | 1 2
S , which are mutually disjoint. We next write the system of equations
3 (cid:0) (cid:1)
xυ/x2 F = 0 for all xυ S ,
1· 1 ∈ 1
xυ/x2 F = 0 for all xυ S ,
 2· 2 ∈ 2
xυ/x2 F = 0 for all xυ S .
3· 3 ∈ 3


Its coefficient matrix, 
α 2α α 2(1+β )
2 0 2 3
− −
α 2α α 2(1+β )
 2 0 2 3
− −
α 2α α
 − 2 0 2

α 2α
 2 0
 −
 α
 − 2

α
 2
 −
 β 2α 1 β
 3 0 − − 3
D =  β 2α 1 β
 3 0 3
 − −
 β 2α
 3 0
 β 2α
 3 0

 β
3

 2α 2β 2α
 2 3 0
 −
 2α 2β
2 3
 −
 2α
 2
 −




2(1+β )
− 3 

α 2(1+β )
2 3 
− 
α 2(1+β ) 2α 
2 − 3 0 

α 2α 2(1+β )
2 0 3 
− 
1 


1


1 β 1 
− − 3 
1 β 1 
3 
− − 
1 β 2α 1 
3 0
− − 



2α 
0

2α 2β 
0 3 

2α 2β 2α 
2 3 0
− 

in the unknowns xυ with υ = d is important in the sense that
| |
det(D) = Res(Ax2) extraneous factor.
·
8
υ
Now, we consider to the extraneous factor. A monomial x of total degree d = 4 is reduced
if x2 divides xυ for exactly one i. Let D be the determinant of the submatrix of D obtained by
i ′
deleting all rows and columns corresponding to reduced monomials, i.e.,
α
2
−
D′ =  α2 α2 .
−
β 1 β
 3 − − 3 
 
By Theorem 4.9 in Chapter 3 of [5], to within a sign, the resultant reads as
detD
Res(Ax2) = (14)
detD
′
= 16α2(48α8β +4α6(α2+β (32β2 +24β 9))+3α4(α2(52β2 +28β 1)
2 0 3 0 2 3 3 3 − 0 2 3 3 −
+4β2(8β3+8β2 9β 9))+6α2(α4(4β +1) α2β (14β3 +36β2 +35β
3 3 3 − 3 − 0 2 3 − 2 3 3 3 3
+10) 2β3(β +1)2(8β +9))+(α2 4(β +1)3)(α2 β2(2β +3))2).
− 3 3 3 2− 3 2 − 3 3
Theorem 3.1 ([5]). There exists a vector x= 0 such that Ax2 = 0 if, and only if, Res(Ax2)= 0,
6
where the formula for Res(Ax2) is given by (14).
By the same approach, we compute the E-characteristic polynomial φA(λ) of the octupolar
tensor (11), which is a resultant of the following system of homogeneous polynomial equations
x2+x2+x2 x2 = 0,
1 2 3− 0
 −α2x21+α2x22+2α0x1x3−2(1+β3)x2x3−λx0x2 = 0, (15)

 β3x21−(1+β3)x22+x23+2α0x1x2−λx0x3 = 0,
2α x x +2β x x +2α x x λx x = 0.
2 1 2 3 1 3 0 2 3 0 1
− −


Using the software Mathematica, we obtain the E-characteristic polynomial φA(λ).
Theorem 3.2. The E-characteristic polynomial of the octupolar tensor (11) is
6
φA(λ) = (λ2 1) c2iλ2i,
−
i=0
X
where
c = 256α4(48α8β +4α6(α2+β (32β2 +24β 9))+3α4(α2(52β2 +28β 1)+4β2(8β3
0 2 0 3 0 2 3 3 3 − 0 2 3 3 − 3 3
+8β2 9β 9))+6α2(α4(4β +1) α2β (14β3 +36β2 +35β +10) 2β3(β
3 − 3 − 0 2 3 − 2 3 3 3 3 − 3 3
+1)2(8β +9))+(α2 4(β +1)3)(α2 β2(2β +3))2)2,
3 2 − 3 2 − 3 3
and
c = 82944α10 11520α8(α2 36β2 36β +1) 320α6(2α2(72β2 1053β 577)+73α4
12 0 − 0 2− 3 − 3 − 0 2 3 − 3 − 2
2592β4 5184β3 2448β2 +144β +73) 240α4(α6 α4(1583β2 +1208β +922)
− 3 − 3 − 3 3 − 0 2− 2 3 3
+α2(288β4 4424β3 7328β2 116β +1203) (2β +1)2(864β4 +1728β3 +576β2
2 3 − 3 − 3 − 3 − 3 3 3 3
288β 1))+60α2(32α8 +α6( 8β2+1992β +678) α4(6168β4 +13336β3
− 3 − 0 2 2 − 3 3 − 2 3 3
+5042β2 4376β +1083) 2α2(384β6 848β5 4080β4 80β3 +4580β2 +437β
3 − 3 − 2 3 − 3 − 3 − 3 3 3
714)+8(2β +1)4(54β4 +108β3 +21β2 33β +4))+(α2 +(3β +4)2)(16α4
− 3 3 3 3 − 3 2 3 2
+α2( 12β2 132β +37)+4(2β +1)3(3β 1))2.
2 − 3 − 3 3 3 −
9
TheE-characteristic polynomialφA(λ)isapolynomialofdegree14,withnoodd-degreeterms.
By comparing the expression of c and (14), we have the following corollary:
12
Corollary 3.3. The constant term of the E-characteristic polynomial φA(λ) is
c = Res(Ax2) 2.
0
(cid:2) (cid:3)
This corollary is in agreement with Theorem 3.5 of [10] in this case.
Theorem 3.4. Suppose that Ax2 = 0 has only the trivial solution x= 0. Then, the system (15)
has a non-trivial solution (x0,...,x3) over C if, and only if, φA(λ) = 0.
Proof. Because Ax2 = 0 has only the trivial solution x= 0, we know that the system of Ax2 = 0
and x x = 0 has only a zero solution. By Proposition 2.6 in Chapter 3 of [5], we obtain the
⊤
desired conclusion.
Corollary 3.5. If Res(Ax2) = 0, then all E-eigenvalues of the octupolar tensor A are non-zero.
6
Proof. Since Res(Ax2) = 0, the system Ax2 = 0 has only the trivial solution by Theorem 3.1.
6
Let us compute φA(0) = [Res(Ax2)]2 > 0. By Theorem 3.4, λ = 0 is not an E-eigenvalue of
A.
4 Dome: the reduced admissible region
Assume that the North pole (0,0,1) is the global maximum point of the octupolar potential
⊤
Φ(x) on the unit sphere S2. That is, we assume that the octupolar tensor A has the largest
Z-eigenvalue λ = 1 with an associated Z-eigenvector (0,0,1) . In the admissible region (10),
⊤
there is a reduced region such that the maximal Z-eigenvalue of A is 1. The boundary of this
reduced admissible region is called the dome [7]: its apex is at α = 0,β = 1,α = √2, and it
0 3 −2 2 2
meets the plane α = 0 along the circle α2+β2+β = 0.
2 0 3 3
Now, we are in a position to give an explicit formula for the dome. We consider the E-
characteristic polynomialφA(λ)inTheorem3.2. Clearly, λ = 1isarootofφA(λ). Sincethedome
is thelocus wherethe maximal Z-eigenvalue is λ = 1, wesubstitute λ = 1into φA(λ)/(λ2 1)= 0
−
and we obtain the following equation
c (α ,β ,α )3 c (α ,β ,α ) c (α ,β ,α )= 0, (16)
1 0 3 2 2 0 3 2 3 0 3 2
· ·
where
c (α ,β ,α ) = 3 4α2 4β2 4β ,
1 0 3 2 − 0 − 3 − 3
c (α ,β ,α ) = 64α4 16α2(1+2β )( 12α2 +(1+2β )2)+(4α2+(1+2β )2)3,
2 0 3 2 2 − 2 3 − 0 3 0 3
and
c (α ,β ,α ) = α6(2β 1)((2β +5)2 12α2)+α4( 48α4(3β2 1)+12α2(8β4
3 0 3 2 2 3 − 3 − 0 2 − 0 3 − 0 3
+24β3 +26β2 4β 11) 16β6 96β5 168β4 72β3 21β2 24β +40)
3 3 − 3− − 3 − 3 − 3 − 3 − 3 − 3
+8α2(8α6 +6α4(4β2 2β 5)+3α2(8β4 +8β3 12β2 3β +6)+8β6
2 0 0 3 − 3− 0 3 3 − 3 − 3 3
+36β5 +42β4 +3β3 9β2 2) 16(α2 +β2+β )2(4α2 +4β2+4β 3). (17)
3 3 3 − 3 − − 0 3 3 0 3 3−
10