Broken symmetry and tilting effects are ubiquitous in Weyl semimetals (WSMs). Therefore, it is crucial to understand their impacts on the materials’ electronic and optical properties. Here, using a realistic four-band model for WSMs, which incorporates both the symmetry breaking and tilting effects, we study its Landau quantization and the associated magneto-absorption spectrum. We show that the Landau bands in tilted WSMs can be obtained by considering a non-tilt Hamiltonian through Lorentz boost. However, broken symmetry effects can generate an additional term in the Hamiltonian, which equivalently lead to band reconstruction. Our work provides a more realistic view of the magnetic field response of WSMs that shall be taken into account in relevant future device applications.

1.
S.-M.
Huang
,
S.-Y.
Xu
,
I.
Belopolski
,
C.-C.
Lee
,
G.
Chang
,
B.
Wang
,
N.
Alidoust
,
G.
Bian
,
M.
Neupane
,
C.-L.
Zhang
,
S.
Jia
,
A.
Bansil
,
H.
Lin
, and
M. Z.
Hasan
, “
A Weyl fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class
,”
Nat. Commun.
6
,
7373
(
2015
).
2.
H.
Weng
,
C.
Fang
,
Z.
Fang
,
B. A.
Bernevig
, and
X.
Dai
, “
Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides
,”
Phys. Rev. X
5
,
011029
(
2015
).
3.
S.-Y.
Xu
,
I.
Belopolski
,
N.
Alidoust
,
M.
Neupane
,
G.
Bian
,
C.-L.
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 semimetal and topological Fermi arcs
,”
Science
349
,
613
(
2015
).
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.
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
).
6.
S.
Jia
,
S.-Y.
Xu
, and
M. Z.
Hasan
, “
Weyl semimetals, Fermi arcs and chiral anomalies
,”
Nat. Mater.
15
,
1140
(
2016
).
7.
B.
Yan
and
C.
Felser
, “
Topological materials: Weyl semimetals
,”
Annu. Rev. Condens. Matter Phys.
8
,
337
(
2017
).
8.
N. P.
Armitage
,
E. J.
Mele
, and
A.
Vishwanath
, “
Weyl and Dirac semimetals in three-dimensional solids
,”
Rev. Mod. Phys.
90
,
015001
(
2018
).
9.
B. Q.
Lv
,
N.
Xu
,
H. M.
Weng
,
J. Z.
Ma
,
P.
Richard
,
X. C.
Huang
,
L. X.
Zhao
,
G. F.
Chen
,
C. E.
Matt
,
F.
Bisti
,
V. N.
Strocov
,
J.
Mesot
,
Z.
Fang
,
X.
Dai
,
T.
Qian
,
M.
Shi
, and
H.
Ding
, “
Observation of Weyl nodes in TaAs
,”
Nat. Phys.
11
,
724
(
2015
).
10.
L. X.
Yang
,
Z. K.
Liu
,
Y.
Sun
,
H.
Peng
,
H. F.
Yang
,
T.
Zhang
,
B.
Zhou
,
Y.
Zhang
,
Y. F.
Guo
,
M.
Rahn
,
D.
Prabhakaran
,
Z.
Hussain
,
S.-K.
Mo
,
C.
Felser
,
B.
Yan
, and
Y. L.
Chen
, “
Weyl semimetal phase in the non-centrosymmetric compound TaAs
,”
Nat. Phys.
11
,
728
(
2015
).
11.
Z. K.
Liu
,
L. X.
Yang
,
Y.
Sun
,
T.
Zhang
,
H.
Peng
,
H. F.
Yang
,
C.
Chen
,
Y.
Zhang
,
Y. F.
Guo
,
D.
Prabhakaran
,
M.
Schmidt
,
Z.
Hussain
,
S.-K.
Mo
,
C.
Felser
,
B.
Yan
, and
Y. L.
Chen
, “
Evolution of the Fermi surface of Weyl semimetals in the transition metal pnictide family
,”
Nat. Mater.
15
,
27
(
2016
).
12.
I.
Belopolski
,
S.-Y.
Xu
,
D. S.
Sanchez
,
G.
Chang
,
C.
Guo
,
M.
Neupane
,
H.
Zheng
,
C.-C.
Lee
,
S.-M.
Huang
,
G.
Bian
,
N.
Alidoust
,
T.-R.
Chang
,
B.
Wang
,
X.
Zhang
,
A.
Bansil
,
H.-T.
Jeng
,
H.
Lin
,
S.
Jia
, and
M. Z.
Hasan
, “
Criteria for directly detecting topological Fermi arcs in Weyl semimetals
,”
Phys. Rev. Lett.
116
,
066802
(
2016
).
13.
C.
Shekhar
,
A. K.
Nayak
,
Y.
Sun
,
M.
Schmidt
,
M.
Nicklas
,
I.
Leermakers
,
U.
Zeitler
,
Y.
Skourski
,
J.
Wosnitza
,
Z. K.
Liu
,
Y. L.
Chen
,
W.
Schnelle
,
H.
Borrmann
,
Y.
Grin
,
C.
Felser
, and
B.
Yan
, “
Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbP
,”
Nat. Phys.
11
,
645
(
2015
).
14.
S. A.
Parameswaran
,
T.
Grover
,
D. A.
Abanin
,
D. A.
Pesin
, and
A.
Vishwanath
, “
Probing the chiral anomaly with nonlocal transport in three-dimensional topological semimetals
,”
Phys. Rev. X
4
,
031035
(
2014
).
15.
J.
Xiong
,
S. K.
Kushwaha
,
T.
Liang
,
J. W.
Krizan
,
M.
Hirschberger
,
W.
Wang
,
R. J.
Cava
, and
N. P.
Ong
, “
Evidence for the chiral anomaly in the Dirac semimetal Na3Bi
,”
Science
350
,
413
(
2015
).
16.
J.
Behrends
,
A. G.
Grushin
,
T.
Ojanen
, and
J. H.
Bardarson
, “
Visualizing the chiral anomaly in Dirac and Weyl semimetals with photoemission spectroscopy
,”
Phys. Rev. B
93
,
075114
(
2016
).
17.
A. C.
Niemann
,
J.
Gooth
,
S.-C.
Wu
,
S.
Bäßler
,
P.
Sergelius
,
R.
Hühne
,
B.
Rellinghaus
,
C.
Shekhar
,
V.
Süß
,
M.
Schmidt
,
C.
Felser
,
B.
Yan
, and
K.
Nielsch
, “
Chiral magnetoresistance in the Weyl semimetal NbP
,”
Sci. Rep.
7
,
43394
(
2017
).
18.
L.
Chen
and
K.
Chang
, “
Chiral-anomaly-driven Casimir-Lifshitz torque between Weyl semimetals
,”
Phys. Rev. Lett.
125
,
047402
(
2020
).
19.
J.
Gooth
,
A. C.
Niemann
,
T.
Meng
,
A. G.
Grushin
,
K.
Landsteiner
,
B.
Gotsmann
,
F.
Menges
,
M.
Schmidt
,
C.
Shekhar
,
V.
Süß
,
R.
Hühne
,
B.
Rellinghaus
,
C.
Felser
,
B.
Yan
, and
K.
Nielsch
, “
Experimental signatures of the mixed axial–gravitational anomaly in the Weyl semimetal NbP
,”
Nature
547
,
324
(
2017
).
20.
Q.
Ma
,
S.-Y.
Xu
,
C.-K.
Chan
,
C.-L.
Zhang
,
G.
Chang
,
Y.
Lin
,
W.
Xie
,
T.
Palacios
,
H.
Lin
,
S.
Jia
,
P. A.
Lee
,
P.
Jarillo-Herrero
, and
N.
Gedik
, “
Direct optical detection of Weyl fermion chirality in a topological semimetal
,”
Nat. Phys.
13
,
842
(
2017
).
21.
X.
Yang
,
K. S.
Burch
, and
Y.
Ran
, “Divergent bulk photovoltaic effect in Weyl semimetals,” arXiv:1712.09363 (2018).
22.
Y.
Zhang
,
H.
Ishizuka
,
J.
van den Brink
,
C.
Felser
,
B.
Yan
, and
N.
Nagaosa
, “
Photogalvanic effect in Weyl semimetals from first principles
,”
Phys. Rev. B
97
,
241118(R)
(
2018
).
23.
G. B.
Osterhoudt
,
L. K.
Diebel
,
M. J.
Gray
,
X.
Yang
,
J.
Stanco
,
X.
Huang
,
B.
Shen
,
N.
Ni
,
P. J. W.
Moll
,
Y.
Ran
, and
K. S.
Burch
, “
Colossal mid-infrared bulk photovoltaic effect in a type-I Weyl semimetal
,”
Nat. Mater.
18
,
471
(
2019
).
24.
M.
Koshino
and
I. F.
Hizbullah
, “
Magnetic susceptibility in three-dimensional nodal semimetals
,”
Phys. Rev. B
93
,
045201
(
2016
).
25.
C.-L.
Zhang
,
S.-Y.
Xu
,
C. M.
Wang
,
Z.
Lin
,
Z. Z.
Du
,
C.
Guo
,
C.-C.
Lee
,
H.
Lu
,
Y.
Feng
,
S.-M.
Huang
,
G.
Chang
,
C.-H.
Hsu
,
H.
Liu
,
H.
Lin
,
L.
Li
,
C.
Zhang
,
J.
Zhang
,
X.-C.
Xie
,
T.
Neupert
,
M. Z.
Hasan
,
H.-Z.
Lu
,
J.
Wang
, and
S.
Jia
, “
Magnetic-tunnelling-induced Weyl node annihilation in TaP
,”
Nat. Phys.
13
,
979
(
2017
).
26.
Y.
Jiang
,
Z. L.
Dun
,
S.
Moon
,
H. D.
Zhou
,
M.
Koshino
,
D.
Smirnov
, and
Z.
Jiang
, “
Landau quantization in coupled Weyl points: A case study of semimetal NbP
,”
Nano Lett.
18
,
7726
(
2018
).
27.
A. A.
Soluyanov
,
D.
Gresch
,
Z.
Wang
,
Q.
Wu
,
M.
Troyer
,
X.
Dai
, and
B. A.
Bernevig
, “
Type-II Weyl semimetals
,”
Nature
527
,
495
(
2015
).
28.
T. M.
McCormick
,
I.
Kimchi
, and
N.
Trivedi
, “
Minimal models for topological Weyl semimetals
,”
Phys. Rev. B
95
,
075133
(
2017
).
29.
Z.-M.
Yu
,
Y.
Yao
, and
S. A.
Yang
, “
Predicted unusual magnetoresponse in type-II Weyl semimetals
,”
Phys. Rev. Lett.
117
,
077202
(
2016
).
30.
M.
Udagawa
and
E. J.
Bergholtz
, “
Field-selective anomaly and chiral mode reversal in type-II Weyl materials
,”
Phys. Rev. Lett.
117
,
086401
(
2016
).
31.
S.
Tchoumakov
,
M.
Civelli
, and
M. O.
Goerbig
, “
Magnetic-field-induced relativistic properties in type-I and type-II Weyl semimetals
,”
Phys. Rev. Lett.
117
,
086402
(
2016
).
32.
M.
Koshino
, “
Cyclotron resonance of figure-of-eight orbits in a type-II Weyl semimetal
,”
Phys. Rev. B
94
,
035202
(
2016
).
33.
M.
Yan
,
H.
Huang
,
K.
Zhang
,
E.
Wang
,
W.
Yao
,
K.
Deng
,
G.
Wan
,
H.
Zhang
,
M.
Arita
,
H.
Yang
,
Z.
Sun
,
H.
Yao
,
Y.
Wu
,
S.
Fan
,
W.
Duan
, and
S.
Zhou
, “
Lorentz-violating type-II Dirac fermions in transition metal dichalcogenide PtTe2
,”
Nat. Commun.
8
,
257
(
2017
).
34.
S.-Y.
Xu
,
N.
Alidoust
,
G.
Chang
,
H.
Lu
,
B.
Singh
,
I.
Belopolski
,
D. S.
Sanchez
,
X.
Zhang
,
G.
Bian
,
H.
Zheng
,
M.-A.
Husanu
,
Y.
Bian
,
S.-M.
Huang
,
C.-H.
Hsu
,
T.-R.
Chang
,
H.-T.
Jeng
,
A.
Bansil
,
T.
Neupert
,
V. N.
Strocov
,
H.
Lin
,
S.
Jia
, and
M. Z.
Hasan
, “
Discovery of Lorentz-violating type II Weyl fermions in LaAlGe
,”
Sci. Adv.
3
,
e1603266
(
2017
).
35.
X.
Yuan
,
C.
Zhang
,
Y.
Zhang
,
Z.
Yan
,
T.
Lyu
,
M.
Zhang
,
Z.
Li
,
C.
Song
,
M.
Zhao
,
P.
Leng
,
M.
Ozerov
,
X.
Chen
,
N. L.
Wang
,
Y.
Shi
,
H.
Yan
, and
F.
Xiu
, “
The discovery of dynamic chiral anomaly in a Weyl semimetal NbAs
,”
Nat. Commun.
11
,
1259
(
2020
).
36.
S.
Polatkan
,
M. O.
Goerbig
,
J.
Wyzula
,
R.
Kemmler
,
L. Z.
Maulana
,
B. A.
Piot
,
I.
Crassee
,
A.
Akrap
,
C.
Shekhar
,
C.
Felser
,
M.
Dressel
,
A. V.
Pronin
, and
M.
Orlita
, “
Magneto-optics of a Weyl semimetal beyond the conical band approximation: Case study of TaP
,”
Phys. Rev. Lett.
124
,
176402
(
2020
).
37.
A. L.
Levy
,
A. B.
Sushkov
,
F.
Liu
,
B.
Shen
,
N.
Ni
,
H. D.
Drew
, and
G. S.
Jenkins
, “
Optical evidence of the chiral magnetic anomaly in the Weyl semimetal TaAs
,”
Phys. Rev. B
101
,
125102
(
2020
).
38.
S.
Bertrand
,
J.-M.
Parent
,
R.
Côté
, and
I.
Garate
, “
Complete optical valley polarization in Weyl semimetals in strong magnetic fields
,”
Phys. Rev. B
100
,
075107
(
2019
).
39.
D.
Grassano
,
O.
Pulci
,
E.
Cannuccia
, and
F.
Bechstedt
, “
Influence of anisotropy, tilt and pairing of Weyl nodes: The Weyl semimetals TaAs, TaP, NbAs and NbP
,”
Eur. Phys. J. B
93
,
157
(
2020
).
40.
We can eliminate the ty component by rotating in the y-z plane to align the tilt with an axis.
41.
A. G.
Grushin
, in
Common and Not-so-common High-energy Theory Methods for Condensed Matter Physics
, Topological Matter. Springer Series in Solid-State Sciences Vol. 190, edited by D. Bercioux, J. Cayssol, M. Vergniory, and M. Reyes Calvo (Springer, Cham, 2018).
42.
S.
Souma
,
Z.
Wang
,
H.
Kotaka
,
T.
Sato
,
K.
Nakayama
,
Y.
Tanaka
,
H.
Kimizuka
,
T.
Takahashi
,
K.
Yamauchi
,
T.
Oguchi
,
K.
Segawa
, and
Y.
Ando
, “
Direct observation of nonequivalent Fermi-arc states of opposite surfaces in the noncentrosymmetric Weyl semimetal NbP
,”
Phys. Rev. B
93
,
161112(R)
(
2016
).
43.
N.
Xu
,
G.
Autès
,
C. E.
Matt
,
B. Q.
Lv
,
M. Y.
Yao
,
F.
Bisti
,
V. N.
Strocov
,
D.
Gawryluk
,
E.
Pomjakushina
,
K.
Conder
,
N. C.
Plumb
,
M.
Radovic
,
T.
Qian
,
O. V.
Yazyev
,
J.
Mesot
,
H.
Ding
, and
M.
Shi
, “
Distinct evolutions of Weyl fermion quasiparticles and Fermi arcs with bulk band topology in Weyl semimetals
,”
Phys. Rev. Lett.
118
,
106406
(
2017
).
44.
C.-C.
Lee
,
S.-Y.
Xu
,
S.-M.
Huang
,
D. S.
Sanchez
,
I.
Belopolski
,
G.
Chang
,
G.
Bian
,
N.
Alidoust
,
H.
Zheng
,
M.
Neupane
,
B.
Wang
,
A.
Bansil
,
M. Z.
Hasan
, and
H.
Lin
, “
Fermi surface interconnectivity and topology in Weyl fermion semimetals TaAs, TaP, NbAs, and NbP
,”
Phys. Rev. B
92
,
235104
(
2015
).
45.
D.
Grassano
,
O.
Pulci
,
A. M.
Conte
, and
F.
Bechstedt
, “
Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies
,”
Sci. Rep.
8
,
3534
(
2018
).
46.
P. E. C.
Ashby
and
J. P.
Carbotte
, “
Magneto-optical conductivity of Weyl semimetals
,”
Phys. Rev. B
87
,
245131
(
2013
).
47.
Y.
Jiang
,
S.
Thapa
,
G. D.
Sanders
,
C. J.
Stanton
,
Q.
Zhang
,
J.
Kono
,
W. K.
Lou
,
K.
Chang
,
S. D.
Hawkins
,
J. F.
Klem
,
W.
Pan
,
D.
Smirnov
, and
Z.
Jiang
, “
Probing the semiconductor to semimetal transition in InAs/GaSb double quantum wells by magneto-infrared spectroscopy
,”
Phys. Rev. B
95
,
045116
(
2017
).
48.
In non-magnetic transition-metal monopnictides WSMs, the e–h asymmetry results in a much stronger electron-like transition than the corresponding hole-like transition.
49.
J. P.
Carbotte
, “
Dirac cone tilt on interband optical background of type-I and type-II Weyl semimetals
,”
Phys. Rev. B
94
,
165111
(
2016
).
50.
Y.
Jiang
,
J.
Wang
,
T.
Zhao
,
Z. L.
Dun
,
Q.
Huang
,
X. S.
Wu
,
M.
Mourigal
,
H. D.
Zhou
,
W.
Pan
,
M.
Ozerov
,
D.
Smirnov
, and
Z.
Jiang
, “
Unraveling the topological phase of ZrTe5 via magnetoinfrared spectroscopy
,”
Phys. Rev. Lett.
125
,
046403
(
2020
).
51.
Y.
Jiang
,
M. M.
Asmar
,
X.
Han
,
M.
Ozerov
,
D.
Smirnov
,
M.
Salehi
,
S.
Oh
,
Z.
Jiang
,
W.-K.
Tse
, and
L.
Wu
, “
Electron-hole asymmetry of surface states in topological insulator Sb2Te3 thin films revealed by magneto-infrared spectroscopy
,”
Nano Lett.
20
,
4588
(
2020
).
52.
C.-X.
Liu
,
X.-L.
Qi
,
H.
Zhang
,
X.
Dai
,
Z.
Fang
, and
S.-C.
Zhang
, “
Model Hamiltonian for topological insulators
,”
Phys. Rev. B
82
,
045122
(
2010
).
You do not currently have access to this content.