We classify two-by-two traceless Hamiltonians depending smoothly on a three-dimensional Bloch wavenumber and having a band crossing at the origin of the wavenumber space. Recently these Hamiltonians attract much interest among researchers in the condensed matter field since they are found to be effective Hamiltonians describing the band structure of the exotic materials such as Weyl semimetals. In this classification, we regard two such Hamiltonians as equivalent if there are appropriate special unitary transformation of degree 2 and diffeomorphism in the wavenumber space fixing the origin such that one of the Hamiltonians transforms to the other. Based on the equivalence relation, we obtain a complete list of classes up to codimension 7. For each Hamiltonian in the list, we calculate multiplicity and Chern number [D. J. Thouless et al., Phys. Rev. Lett. 49, 405 (1982); M. V. Berry, Proc. R. Soc. A 392, 45 (1983); and B. Simon, Phys. Rev. Lett. 51, 2167 (1983)], which are invariant under an arbitrary smooth deformation of the Hamiltonian. We also construct a universal unfolding for each Hamiltonian and demonstrate how they can be used for bifurcation analysis of band crossings.

1.
D. J.
Thouless
,
M.
Kohmoto
,
P.
Nightingale
, and
M.
den Nijs
, “
Quantized Hall conductance in a two-dimensional periodic potential
,”
Phys. Rev. Lett.
49
,
405
(
1982
).
2.
M. V.
Berry
, “
Quantal phase factors accompanying adiabatic changes
,”
Proc. R. Soc. A
392
,
45
(
1983
).
3.
B.
Simon
, “
Holonomy, the quantum adiabatic theorem, and Berry’s phase
,”
Phys. Rev. Lett.
51
,
2167
(
1983
).
4.
B. Q.
Lv
,
H. M.
Weng
,
B. B.
Fu
,
X. P.
Wang
,
H.
Miao
,
J.
Ma
,
P.
Richard
,
X. C.
Huang
,
L. X.
Zhao
,
G. F.
Chen
,
Z.
Fang
,
X.
Dai
,
T.
Qian
, and
H.
Ding
, “
Experimental discovery of Weyl semimetal TaAs
,”
Phys. Rev. X
5
,
031013
(
2015
).
5.
S.-Y.
Xu
,
I.
Belopolski
,
N.
Alidoust
,
M.
Neupane
,
G.
Bian
,
C.
Zhang
,
R.
Sankar
,
G.
Chang
,
Z.
Yuan
,
C.-C.
Lee
,
S.-M.
Huang
,
H.
Zheng
,
J.
Ma
,
D. S.
Sanchez
,
B.
Wang
,
A.
Bansil
,
F.
Chou
,
P. P.
Shibayev
,
H.
Lin
,
S.
Jia
, and
M. Z.
Hasan
, “
Discovery of a Weyl fermion seminetal and topological Fermi arcs
,”
Science
349
,
613
(
2015
).
6.
X.
Wan
,
A. M.
Turner
,
A.
Vishwanath
, and
S. Y.
Savrasov
, “
Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates
,”
Phys. Rev. B
83
,
205101
(
2011
).
7.
L.
Balents
, “
Weyl electrons kiss
,”
Physics
4
,
36
(
2011
).
8.
G.
Xu
,
H.
Weng
,
Z.
Wang
,
X.
Dai
, and
Z.
Fang
, “
Chern semimetal and the quantized anomalous Hall effect
,”
Phys. Rev. Lett.
107
,
186806
(
2011
).
9.
S.
Adler
, “
Axial-vector vertex in spinor electrodynamics
,”
Phys. Rev.
177
,
2426
(
1969
).
10.
J. S.
Bell
and
R.
Jackiw
, “
A PCAC puzzle: π0γγ in σ-model
,”
II Nuovo Cimento A
60
,
47
(
1969
).
11.
H. B.
Nielsen
and
M.
Ninomiya
, “
The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal
,”
Phys. Lett. B
130
,
389
(
1983
).
12.
P.
Hosur
and
X.-L.
Qi
, “
Recent developments in transport phenomena in Weyl semimetals
,”
C. R. Phys.
14
,
857
(
2013
).
13.
S. A.
Parameswaran
,
T.
Grover
,
D. A.
Abanin
,
D. A.
Pesion
, and
A.
Vishwanath
, “
Proving the chiral anomaly with nonlocal transport in three-dimensional topological semimetals
,”
Phys. Rev. X
4
,
031035
(
2014
).
14.
A. A.
Burkov
, “
Chiral anomaly and transport in Weyl metals
,”
J. Phys.: Condens. Matter
27
,
113201
(
2015
).
15.
A. P.
Schnyder
,
S.
Ryu
,
A.
Furusaki
, and
A. W. W.
Ludwig
, “
Classification of topological insulators and superconductors in three spatial dimensions
,”
Phys. Rev. B
78
,
195125
(
2008
).
16.
M.
Golubitsky
and
D. G.
Schaeffer
,
Singularities and Groups in Bifurcation Theory
, Volume I of Applied Mathematical Science (
Springer
,
1985
).
17.

In the context of band theory, it is natural to restrict a domain of the definition of the Bloch wavenumber k to the first Brillouin zone, but, here, we consider the Bloch wavenumber k in the extended Brillouin zone19 for simplicity. In this manuscript, our main interest is a local classification of Hamiltonians in a neighborhood of band crossings and bifurcations of the band crossings. There, the difference between the two representations does not make any difference.

18.
O.
Vafek
and
A.
Vishwanath
, “
Dirac fermions in solids: From high-Tc cuprates and graphene to topological insulators and Weyl semimetals
,”
Annu. Rev. Condens. Matter Phys.
5
,
83
(
2014
).
19.
N. W.
Ashcroft
and
N. D.
Mermin
,
Solid State Physics
(
Thomson Learning
,
1976
).
20.
F.
Bloch
, “
Über die quantenmechanik der elektronen in kristallgittern
,”
Z. Phys.
52
,
555
(
1928
).
21.
F.
Rellich
, “
Störungstheorie der spektralzerlegung
,”
Math. Ann.
113
,
600
(
1937
).
22.
T.
Kato
,
Perturbation Theory for Linear Operators
, Classics in Mathematics (
Springer
,
1976
).
23.
G. A.
Hagedorn
, “
Molecular propagation through electron energy level crossings
,”
Mem. Am. Math. Soc.
111
,
13
(
1994
).
24.
H. A.
Kramers
, “
General theory of the paramagnetic rotation in crystals
,”
Proc. Amsterdam Acad.
33
,
959
(
1930
).
25.

Here, we would like to consider the class of C diffeomorphisms that do not necessarily fix the origin because if we apply a generic perturbation to the Hamiltonian having a band crossing at the origin, the band crossing typically moves away from the origin but the Hamiltonian in a neighborhood of the band crossing does not necessarily change qualitatively.39 

26.
Choose
in Eq. (36).
27.
S.
Izumiya
,
M.
Takahashi
, and
H.
Teramoto
, “
Geometric equivalence among smooth section-germs of vector bundles with respect to structure groups
” (unpublished).
28.
J.
Martinet
,
Singularities of Smooth Functions and Maps
, Volume 58 of Lecture Note Series (
London Mathematical Society
,
1982
).
29.
G.-M.
Greuel
and
G.
Pfister
,
A Singular Introduction to Commutative Algebra
, 2nd ed. (
Springer
,
2008
).
30.
M. Z.
Hasan
and
C. L.
Kane
, “
Colloquium: Topological insulators
,”
Rev. Mod. Phys.
82
,
3045
(
2010
).
31.
X.-L.
Qi
and
S.-C.
Zhang
, “
Topological insulators and superconductors
,”
Rev. Mod. Phys.
83
,
1057
(
2011
).
32.
J. N.
Mather
, “
Stability of C mappings, IV: Classification of stable germs by R algebras
,”
Publ. Math., Inst. Hautes Etud. Sci.
37
,
223
(
1969
).
33.
J. N.
Damon
, The Unfolding and Determinacy Theorems for Subgroups of A and K, Memoirs of the American Mathematical Society (
American Mathematical Society
,
1984
), Vol. 306.
34.
W.
Decker
,
G.-M.
Greuel
,
G.
Pfister
, and
H.
Schönemann
, Singular 4-0-2—A computer algebra system for polynomial computations, 2015, http://www.singular.uni-kl.de.
35.
R.
Soset Sinha
and
R.
Wik Atique
, Classification of multigerms (from a modern viewpoint), http://www.worksing.icmc.usp.br/main_site/2016/minicourse3_notes.pdf.
36.
A. A.
Soluyanov
,
D.
Gresch
,
Z.
Wang
,
Q.
Qu
,
M.
Troyer
,
X.
Dai
, and
B. A.
Bernevig
, “
Type-II Weyl semimetals
,”
Nature
527
,
495
(
2015
).
37.
B.
Sturmfels
,
Algorithms in Invariant Theory
, 2nd ed. (
SpringerWien
,
NewYork
,
1993
).
38.

This result follows from Ref. 29, Lemma A.9.2. in p. 484 because formal power series (for both vector and scalar fields) are equivalent to smooth functions modulo M by the Borel-Ritt theorem.

39.
W.
Witczak-Krempa
,
G.
Chen
,
Y. B.
Kim
, and
L.
Balents
, “
Correlated quantum phenomena in the strong spin-orbit regime
,”
Annu. Rev. Condens. Matter Phys.
5
,
57
(
2014
).
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