Splitting the eigenvectors space for Kildal’s Hamiltonian
The rational canonical form of Kildal’s Hamiltonian has been obtained as a matrix with two identical diagonal blocks. It allowed to formulate and strictly prove few common assertions. Each of the eigenvalues of Kildal’s Hamiltonian is twice degenerated everywhere, and it is well-known Kramers’ de...
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| Опубліковано в: : | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Дата: | 2010 |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2010
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| Цитувати: | Splitting the eigenvectors space for Kildal’s Hamiltonian / G.P. Chuiko, N.L. Don // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 4. — С. 366-368. — Бібліогр.: 6 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859624488029650944 |
|---|---|
| author | Chuiko, G.P. Don, N.L. |
| author_facet | Chuiko, G.P. Don, N.L. |
| citation_txt | Splitting the eigenvectors space for Kildal’s Hamiltonian / G.P. Chuiko, N.L. Don // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 4. — С. 366-368. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| container_title | Semiconductor Physics Quantum Electronics & Optoelectronics |
| description | The rational canonical form of Kildal’s Hamiltonian has been obtained as a
matrix with two identical diagonal blocks. It allowed to formulate and strictly prove few
common assertions. Each of the eigenvalues of Kildal’s Hamiltonian is twice
degenerated everywhere, and it is well-known Kramers’ degeneration, firstly. However,
there is neither degeneration with except for Kramers’, secondly. The symmetry of
Kildal’s Hamiltonian forcedly includes the operation of inversion (i.e. the center of
symmetry), thirdly. Consequently this form of Hamiltonian is evidently not able to
describe the specific properties of crystals without the center of symmetry. The
Frobenius form (alias “the rational canonical form”) of Hamiltonian should consist of
two non-identical diagonal blocks to remove Kramers’ degeneration.
|
| first_indexed | 2025-11-29T09:11:47Z |
| format | Article |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 366-368.
PACS 71.18.+y, 71.20.-b
Splitting the eigenvectors space for Kildal’s Hamiltonian
G.P. Chuiko1, N.L. Don2
1Petro Mohyla Black Sea State University, Department of Medical Devices and Systems,
10, 68 Desantnikov str., 54003 Mykolaiv, Ukraine; e-mail: gp47@mail.ru
2Kherson National Technical University, Department of General and Applied Physics,
24, Beryslawskoe Shosse, 73008 Kherson, Ukraine; e-mail: n_don@mail.ru
Abstract. The rational canonical form of Kildal’s Hamiltonian has been obtained as a
matrix with two identical diagonal blocks. It allowed to formulate and strictly prove few
common assertions. Each of the eigenvalues of Kildal’s Hamiltonian is twice
degenerated everywhere, and it is well-known Kramers’ degeneration, firstly. However,
there is neither degeneration with except for Kramers’, secondly. The symmetry of
Kildal’s Hamiltonian forcedly includes the operation of inversion (i.e. the center of
symmetry), thirdly. Consequently this form of Hamiltonian is evidently not able to
describe the specific properties of crystals without the center of symmetry. The
Frobenius form (alias “the rational canonical form”) of Hamiltonian should consist of
two non-identical diagonal blocks to remove Kramers’ degeneration.
Keywords: Kildal’s Hamiltonian, Kramers’ degeneration, splitting the space of
eigenvectors, rational canonical form.
Manuscript received 08.12.10; accepted for publication 02.12.10; published online 30.12.10.
1. Introduction
The problem of the splitting the eigenvector space is
solved here for Hamiltonians of Kildal’s type [1].
Mathematical and physical conclusions following from
the analysis both of rational canonical form of Kildal’s
Hamiltonian as well as of characteristic polynomials
associated with it will be proved and presented. We are
going to get the strong evidences of everywhere
presented Kramer’s degeneration of each energy level
for this kind of Hamiltonian. Conversely, there is no
other degeneration except for that as follows.
The original Hamiltonian of Kildal was written
down with such a basis [1]:
( ) ( )
( ) ( ) ,,,,
,,,,
βαβα
αβαβ
iYXmZiYXmiS
iYXmZiYXmiSVb
−+−
+−=
(1)
where are basis functions of the
corresponding irreducible representations (scalar and
vector): α, β are two spin indicators whereas
ZYXS ,,,
2
1
=m
and . 12 −=i
The large-blocked structure of this Hamiltonian has
the form
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
= ⊕
12
21
HH
HH
H , (2)
where is submatrix Hermitian transpose to and
submatrices and have the following forms:
⊕
2H 2H
1H 2H
,
3/)(000
03/23/2
03/23/)(0
00
1
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
Δ++
−Δ
ΔΔ−+
=
δ
δ
δ
p
pz
p
zs
E
EPk
E
PkE
H
(3)
( ) ( )
( )
( )
,
000
0000
000
00
2
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
+
−+−
=
yx
yx
yxyx
ikkmP
ikkmP
ikkmPikkmP
H
(4)
where the symbols Es, Ep, Δ, δ, P mean the same as in
[1]: energies of s- and p-states, parameter of spin
splitting, Kildal’s parameter of crystalline field and
matrix element of the quasi-pulse. Symbols kx, ky, kz
denote three components of the wave vector.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
366
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 366-368.
We would like to transform the Hamiltonian (2)
with blocks (3) and (4) to the block-diagonal form to
split its eigenvectors space by a couple of independent
and orthogonal subspaces. This will be helpful to
formulate several strong and quite general statements
about eigenvectors and eigenvalues of such a kind of
Hamiltonians.
2. Solving the problem
Let us denote:
( )
3
Δ−δ
−−= psg EEE . (5)
Let us use thereto the spherical coordinates, where
ϕ±θ=± i
yx ekikk )sin( (6)
and 222
zyx kkkk ++= .
The submatrices (3) and (4) shall get the following
forms after these substitutions and the relocate of the
energy zero-point ⎟
⎠
⎞
⎜
⎝
⎛ =
Δ+δ
+ 0
3pE :
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−Δ−Δ
ΔΔ−
=
0000
03/3/2)cos(
03/23/20
0)ѓЖcos(0
1 δθkP
kPE
H
g
, (7)
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛ −
θ=
ϕ−
ϕ
ϕ−ϕ
000
0000
000
00
)sin(2
i
i
ii
e
e
ee
mkPH . (8)
The rational canonical form (or the so-called
“Frobenius’ form”) of this Hamiltonian can be found
using such a system of computer mathematics as
Maple11, where this possibility is a part of the package
of programmes called “Linear Algebra”. It is possible to
obtain not only the partly diagonalized Frobenius’ form
of Hamiltonian, but the matrix of the basis
transformation ( Q ) also by this way. However, this
matrix is very bulky and therefore is absent here in
paper.
The rational canonical form of Kildal’s
Hamiltonian may be written down in a kind:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
1
1
H0
0H
HFr . (9)
Thus, the Frobenius form of Hamiltonian has the
block-diagonal kind with identical (4×4) submatrices on
the main diagonal:
( )
( )
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
Δ−δ−
+Δ−Δ+δ
Δ+θδ+Δδ
θΔδ
=
g
g
g
E
PkE
PkE
kP
H
100
3/2010
)3/2)sin((2001
3/)sin(000
22
222
2
1 . (10)
Now we can get the dispersion law as the
submatrix (10) characteristic polynomial with its
dependence on energy ( E ):
( ) ( )( )
( )
( ) .03/)sin(
)3/2)sin((2
3/2
2
222
22234
=Δ−
−Δ++Δ−
−+Δ−Δ+−Δ−−−
θδ
θδδ
δδ
kP
EPkE
EPkEEEE
g
gg
(11)
It is worthy to note that coefficients of the
dispersion law (11) are presented in the latter column of
(10) as elements of this matrix. This equation may be
resolved directly and even in radicals because it is
algebraic equation of the fourth degree. However, its
four roots are somewhat bulky and thus are inconvenient
for analysis. The indirect solutions (anything like
),( θEk or even ) are looking much better from
this point of view. By the way, using these indirect
solutions is old good tradition still from Kane [2]. So, we
have from (11):
),(2 θEk
( ) ( )( )222
2
2
32)sin(3
)33)23)(((
,
EEEP
EEEEEE
Ek g
+Δ+θδΔ+
+Δ+δΔ+−
=θ . (12)
Our formula for the dispersion law is slightly more
compact in comparison with the original one [1]:
( ) ( )[
]2
222
2
2
)cos()3/2(
)sin(9/)3/)(3/(
)9/2)3/)(3/2)(((
,
θ
θδ
δ
θ
Δ++
+Δ−Δ++Δ+
Δ−Δ++Δ+−
=
EE
EEP
EEEEE
Ek g .
(13)
Nevertheless, if we will find a difference between
both expressions, we shall get zero. What means that
expressions (12) and (13) are equivalent and tantamount.
3. Conclusions
3A. Mathematical point of view
The block-diagonal structure of the Frobenius
matrix (9) determines that [3-5]:
1. Characteristic polynomials for both identical
diagonal blocks (of submatrices) are also identical.
2. At the same time, the characteristic polynomials
are the minimum polynomials of diagonal submatrices
(i.e. of blocks).
3. The sequence where is
the characteristic polynomial of the submatrix (left
side of Eq. (11)) is the sequence of all invariant
polynomials of the submatrix , where
))(,1,1,1( EPol )(EPol
1H
1HEI − I is the
identity matrix and is the submatrix determined
by (10).
1H
4. The characteristic polynomial of the
Hamiltonian matrix is a product of all its invariant
polynomials. Consequently, each of two identical
H
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
367
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 366-368.
invariant polynomials divides the characteristic
polynomial of the matrix without a rest.
)(EPol
H
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
)
)
5. The identity of characteristic and minimum
polynomials is the guaranty that all eigenvalues of the
submatrix are different in pairs. 1H
6. If is one of four ( roots of the
submatrix characteristic polynomial ( , then one
and only one eigenvector corresponds to this root, and
this eigenvector has the following coordinates with the
basis functions of Frobenius’ form (9):
jE 4,3,2,1=j
)(EPol
32 ,,,1 jjj EEE .
7. Eigenvectors mentioned above belongs to one
of these two orthogonal 4D-subspaces with the basis
functions of Frobenius’ form (9), and the complete 8D-
space of eigenvectors is the direct sum of both
subspaces.
3B. Physical point of view
1. The points 1, 3, 4 and 7 of the above list mean
that each of eigenvalues of Kildal’s Hamiltonian is twice
degenerated, and it is well-known Kramers’
degeneration evidently.
2. The points 2, 5 and 6 testify that there is neither
degeneration with except for Kramers’.
3. The symmetry of Kildal’s Hamiltonian forcedly
includes the operation of inversion (i.e. the center of
symmetry).
Consequently this form of Hamiltonian is evidently
not able to describe the specific properties of crystals
without the center of symmetry. To remove Kramers’
degeneration, the Frobenius form of Hamiltonian should
consist of two non-identical diagonal blocks, as it has
been explained in [6]. By the way, basis functions of
more general Hamiltonian [6] are “one to one” as those
for Hamiltonian (9) and can be found with the matrix Q
mentioned above and got by us.
References
1. H. Kildal, Band structure of CdGeAs2 near k = 0 //
Phys. Rev. 10(12), p. 5082-5087 (1974).
2. E.O. Kane, Band structure of indium antimonide //
J. Phys. Chem. Solids 1, p. 249-261 (1957).
3. V.V. Voievodin, Yu.A. Kuznetsov, Matrices and
Calculations. Nauka, Moscow, 1984 (in Russian).
4. F.R. Gantmacher, The Theory of Matrices. Nauka,
Moscow, 1967 (in Russian).
5. K.G. Valeiev, Splitting the Matrix Spectrum.
Vyshcha shkola, Kiev, 1986 (in Russian).
6. G.P. Chuiko, V.V. Martyniuk and V.K. Bazhenov,
The basic peculiarities of the band spectra within
the generalized model by Kildal for one-axis
semiconductors // Semiconductor Physics,
Quantum Electronics & Optoelectronics, 8(2),
p. 28-31 (2005).
368
3. Conclusions
3A. Mathematical point of view
3B. Physical point of view
|
| id | nasplib_isofts_kiev_ua-123456789-118570 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2025-11-29T09:11:47Z |
| publishDate | 2010 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Chuiko, G.P. Don, N.L. 2017-05-30T16:25:10Z 2017-05-30T16:25:10Z 2010 Splitting the eigenvectors space for Kildal’s Hamiltonian / G.P. Chuiko, N.L. Don // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 4. — С. 366-368. — Бібліогр.: 6 назв. — англ. 1560-8034 PACS 71.18.+y, 71.20.-b https://nasplib.isofts.kiev.ua/handle/123456789/118570 The rational canonical form of Kildal’s Hamiltonian has been obtained as a matrix with two identical diagonal blocks. It allowed to formulate and strictly prove few common assertions. Each of the eigenvalues of Kildal’s Hamiltonian is twice degenerated everywhere, and it is well-known Kramers’ degeneration, firstly. However, there is neither degeneration with except for Kramers’, secondly. The symmetry of Kildal’s Hamiltonian forcedly includes the operation of inversion (i.e. the center of symmetry), thirdly. Consequently this form of Hamiltonian is evidently not able to describe the specific properties of crystals without the center of symmetry. The Frobenius form (alias “the rational canonical form”) of Hamiltonian should consist of two non-identical diagonal blocks to remove Kramers’ degeneration. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Splitting the eigenvectors space for Kildal’s Hamiltonian Article published earlier |
| spellingShingle | Splitting the eigenvectors space for Kildal’s Hamiltonian Chuiko, G.P. Don, N.L. |
| title | Splitting the eigenvectors space for Kildal’s Hamiltonian |
| title_full | Splitting the eigenvectors space for Kildal’s Hamiltonian |
| title_fullStr | Splitting the eigenvectors space for Kildal’s Hamiltonian |
| title_full_unstemmed | Splitting the eigenvectors space for Kildal’s Hamiltonian |
| title_short | Splitting the eigenvectors space for Kildal’s Hamiltonian |
| title_sort | splitting the eigenvectors space for kildal’s hamiltonian |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/118570 |
| work_keys_str_mv | AT chuikogp splittingtheeigenvectorsspaceforkildalshamiltonian AT donnl splittingtheeigenvectorsspaceforkildalshamiltonian |