4-fold degeneracy

•

4-fold degenerate ring protected by PT and glide reflection

CuBi

O

: Double drumhead surface states

X Γ

S Y Z

R T Z U

(100) surface

•

Double drumhead surface states bound by Dirac ring

Acknowledgments

Ching-Kai Chiu   (Uni Maryland) M. Hirschmann  

(MPI-FKF)

Yang-Hao Chan   (A. Sinica, Taiwan) Yuxin Zhao 

(MPI-FKF/Nanjing)

Xue-Yang Song  (U Peking)

Wenbin Rui  (MPI-FKF)

•

Classification of crystalline topological materials 

Rev. Mod. Phys 88, 035005 (2016)

•

Topological nodal line semimetals 

  Tabelle I Classification of topological insulators and superconductors (“TI/TSC”) as well as of stable Fermi surfaces (“FS1” and ”FS2”) and nodal points/lines in 27 symmetry classes with reflection symmetry, in terms of the spatial dimensiondof topological insulators and superconductors, and the codimensionpof Fermi surfaces (nodal lines). “FS1” denote Fermi surfaces (nodal lines) which are within mirror planes and at high-symmetry points, whereas “FS2” denote those that are away from high-symmetry points.Z2,MZ2andTZ2invariants only protect Fermi surfaces of dimension zero (dFS= 0) at high-symmetry points of the Brillouin zone. For entries labeled withZ2,MZ2,TZ2, Fermi surfaces located within the mirror plane but away from high symmetry points cannot be protected by aZ2orMZ2topological invariant. Nevertheless, the system can exhibit gapless surface states that are protected by aZ2orMZ2topological invariant. For gapless topological materials the presence of translation symmetry is always assumed. Hence, there is no distinction betweenTZ2andZ2for gapless topological materials.

Figure 3: current generated in gapped nodal loop system by electrical field in ˆey(left panel) and ˆez

(right panel) direction.

which is depicted in Fig. 3.

Electrical field can also be applied in ˆexor ˆeydirection, the corresponding currents are dj

Next we calculate the current in the gapped case, j=e²

We consider angular di↵erential current, dj

From above discussion, the integrand approximates delta function, which suggests contribution around nodal loop dominates. By using k_k = _B^R

k +_2RB^⇢

where the integration range is⇢2(0,p

µ² ²), which comes fromE2( , µ). With this expres-sion, we can calculate the currents and conductance withEin di↵erent direction.

Below, we calculate conductance when electrical field are in ˆex, ˆeyand ˆexdirections.

PRB 90, 205136 (2014)

PRB 95, 195108 (2017) PRL 116, 156402 (2016)

PRB 93, 205132 (2016) PRL 116, 156402 (2016)

•

Quantum anomalies in nodal-line semimetals 

— Parity anomaly & anomalous transport

arXiv:1703.05958

to be published

•

Dirac line nodes with non-symmorphic symmetries 

— CuBi2O4

3 momentum space the (anti)-commutation relations can be

al-tered as the centers of the two crystalline symmetry operations are di ↵ erent. We first consider the operations M _x and PT in two di ↵ erent orders

( x , y , z) ^M !

( x , y , z + 1 / 2) ^PT ! ( x + 1 / 2 , y + 1 / 2 , z 1 / 2) (11) ( x , y , z) ^PT ! ( x + 1 / 2 , y + 1 / 2 , z)

M

! ( x 1 / 2 , y + 1 / 2 , z + 1 / 2) (12)

Since in the x direction M _x and PT reflecting at di ↵ erent centers and M _x has an additional half lattice shifting, the commutation relation between these two symmetry opera-tions posses an extra momentum-dependent phase pt ^s m ^s _x = e ⁱ⁽ ^k

⁺ ^k

⁾ m ^s _x pt ^s . The relation for the entire symmetry presence of 4-fold degenerate nodal lines protected by M _x reflection-glide symmetry only in the k _x = ⇡ plane. As shown in fig. 3(a,b), the two two-fold degenerate bands cor-respond to two di ↵ erent M _x eigenvalues. The 4-fold degen-erate band crossing (nodal line) is robust under the M _x pro-tection since the band hybridization mixes the M _x eigenval-ues and breaks M _x symmetry. On the other hand, as k _x = 0, M _x PT | _± (k) i = ⌥ e ^ik

^/ ² PT | _± (k) i the two degenerate states correspond to di ↵ erent M _x eigenvalues. If a 4-band cross-ing occurs in k _x = 0 plane, two pairs of the crossings can be gapped without breaking any symmetries since each pair is in the same eigenspace of M _x . Therefore, robust nodal lines are absent in k _x = 0 plane as a trivial case. Since in most of the physical systems, time-reversal inversion operator and reflection operator usually anticommutes with each other (if ‘i’ is recovered in the reflection operator for spin-1 / 2 sys-tems, the anti-commutation relation is changed to commuta-tion relacommuta-tion); hence, symmetry-protected nodal lines is com-monly absent. As the reflection has a di ↵ erent reflection cen-ter than inversion symmetry, the emerge of the commutation relation between M _x and PT protects nodal lines in URSX plane (k _x = 0). Bi ₂ CuO ₄ is the first concrete material realiza-tion (c.f. the early theoretical proposals [11,12]).

To show additional 4-fold degeneracy in UR, we further include M _z symmetry by considering the relation between M _x and M _z en-ergy scale is weird Blue(red) indicates 2-fold degenerate energy band corresponding reflection eigenvalue M _x = e ^ik

^/ ² ( e ^ik

^/ ² ). Yang-hao check. maybe it’s other way around. (b) two double nodal rings appear at di ↵ erent energy levels. Green(purple) represents M _x = e ^ik

^/ ² ( e ^ik

^/ ² ) Check M _x eigenvalue. R point in (b) is ? (c) (100) surface spectrum shows surface states connecting the pro-jected double nodal ring. zoom in for the surface states (d) In S R and UX , any two 2-fold degenerate bands merge in UR as a 4-fold degenerate band.

the glide property of these two nonsymmorphic symme-try operators. In UR of the BZ ( ⇡, k _y , ⇡ ), the additional same energy due to the symmetries. Back to Bi ₂ CuO ₄ , first due to PT symmetry, any band in BZ is two-fold degeneracy and each band in UR is four-fold degeneracy, as shown in fig.

3(d), due to the interplay of PT , M _x , and M _z symmetries. This analysis leads to that for the (magnetic) space group having a subgroup of this magnetic space group #56 . 367, UR is always 4-degenerate. For example, BaP ₂ (HO ₂ ) ₄ preserves time rever-sal symmetry and belongs SG #56 and WO ₃ TaTe ₄ SG 130.

•

CuBi

O

: Double drumhead surface states

(100) surface

•

Acknowledgments

•

•

•

•

3 momentum space the (anti)-commutation relations can be

al-tered as the centers of the two crystalline symmetry operations are di ↵ erent. We first consider the operations M _x and PT in two di ↵ erent orders

( x , y , z) ^M !

( x , y , z + 1 / 2) ^PT ! ( x + 1 / 2 , y + 1 / 2 , z 1 / 2) (11) ( x , y , z) ^PT ! ( x + 1 / 2 , y + 1 / 2 , z)

M

! ( x 1 / 2 , y + 1 / 2 , z + 1 / 2) (12)

Since in the x direction M _x and PT reflecting at di ↵ erent centers and M _x has an additional half lattice shifting, the commutation relation between these two symmetry opera-tions posses an extra momentum-dependent phase pt ^s m ^s _x = e ⁱ⁽ ^k

⁺ ^k

To show additional 4-fold degeneracy in UR, we further include M _z symmetry by considering the relation between M _x and M _z en-ergy scale is weird Blue(red) indicates 2-fold degenerate energy band corresponding reflection eigenvalue M _x = e ^ik

^/ ² ( e ^ik

^/ ² ). Yang-hao check. maybe it’s other way around. (b) two double nodal rings appear at di ↵ erent energy levels. Green(purple) represents M _x = e ^ik

^/ ² ( e ^ik

^/ ² ) Check M _x eigenvalue. R point in (b) is ? (c) (100) surface spectrum shows surface states connecting the pro-jected double nodal ring. zoom in for the surface states (d) In S R and UX , any two 2-fold degenerate bands merge in UR as a 4-fold degenerate band.

As this double nodal line is protected by reflection glide

symmetry, it is natural to ask if the surface states should

be robust under symmetry protection. Although as shown

in fig.3(c) (100) surface spectrum the boundary of the

face states connect the projected double nodal ring, the

sur-face state cannot be protected by magnetic space group

sym-metries in the absence of spin S U (2) symmetry[13]. The

main reason is M _x operation, which is o ↵ -centered of the unit

cell, is always k-dependent in momentum space.

Momentum-dependence reflection unable to quantize Berry phase; hence,

the bulk cannot be described by well-defined topological

in-variant so the surface states are unstable. In the following, we

use 1D reflection-symmetric toy model (7) to show

4-fold degeneracy

•

CuBi

O

: Double drumhead surface states

(100) surface

•

Acknowledgments

•

•

•

•

3 momentum space the (anti)-commutation relations can be

al-tered as the centers of the two crystalline symmetry operations are di ↵ erent. We first consider the operations M x and PT in two di ↵ erent orders

( x , y , z) M !

( x , y , z + 1 / 2) PT ! ( x + 1 / 2 , y + 1 / 2 , z 1 / 2) (11) ( x , y , z) PT ! ( x + 1 / 2 , y + 1 / 2 , z)

M

! ( x 1 / 2 , y + 1 / 2 , z + 1 / 2) (12)

Since in the x direction M x and PT reflecting at di ↵ erent centers and M x has an additional half lattice shifting, the commutation relation between these two symmetry opera-tions posses an extra momentum-dependent phase pt s m s x = e i( k

+ k

To show additional 4-fold degeneracy in UR, we further include M z symmetry by considering the relation between M x and M z en-ergy scale is weird Blue(red) indicates 2-fold degenerate energy band corresponding reflection eigenvalue M x = e ik

/ 2 ( e ik

/ 2 ). Yang-hao check. maybe it’s other way around. (b) two double nodal rings appear at di ↵ erent energy levels. Green(purple) represents M x = e ik

/ 2 ( e ik

/ 2 ) Check M x eigenvalue. R point in (b) is ? (c) (100) surface spectrum shows surface states connecting the pro-jected double nodal ring. zoom in for the surface states (d) In S R and UX , any two 2-fold degenerate bands merge in UR as a 4-fold degenerate band.

the glide property of these two nonsymmorphic symme-try operators. In UR of the BZ ( ⇡, k y , ⇡ ), the additional same energy due to the symmetries. Back to Bi 2 CuO 4 , first due to PT symmetry, any band in BZ is two-fold degeneracy and each band in UR is four-fold degeneracy, as shown in fig.

As this double nodal line is protected by reflection glide

symmetry, it is natural to ask if the surface states should

be robust under symmetry protection. Although as shown

in fig.3(c) (100) surface spectrum the boundary of the

face states connect the projected double nodal ring, the

sur-face state cannot be protected by magnetic space group

sym-metries in the absence of spin S U (2) symmetry[13]. The

main reason is M x operation, which is o ↵ -centered of the unit

cell, is always k-dependent in momentum space.

Momentum-dependence reflection unable to quantize Berry phase; hence,

the bulk cannot be described by well-defined topological

in-variant so the surface states are unstable. In the following, we

use 1D reflection-symmetric toy model (7) to show

al-tered as the centers of the two crystalline symmetry operations are di ↵ erent. We first consider the operations M _x and PT in two di ↵ erent orders

( x , y , z) ^M !

( x , y , z + 1 / 2) ^PT ! ( x + 1 / 2 , y + 1 / 2 , z 1 / 2) (11) ( x , y , z) ^PT ! ( x + 1 / 2 , y + 1 / 2 , z)

Since in the x direction M _x and PT reflecting at di ↵ erent centers and M _x has an additional half lattice shifting, the commutation relation between these two symmetry opera-tions posses an extra momentum-dependent phase pt ^s m ^s _x = e ⁱ⁽ ^k

⁺ ^k

To show additional 4-fold degeneracy in UR, we further include M _z symmetry by considering the relation between M _x and M _z en-ergy scale is weird Blue(red) indicates 2-fold degenerate energy band corresponding reflection eigenvalue M _x = e ^ik

^/ ² ( e ^ik

^/ ² ). Yang-hao check. maybe it’s other way around. (b) two double nodal rings appear at di ↵ erent energy levels. Green(purple) represents M _x = e ^ik

^/ ² ( e ^ik

^/ ² ) Check M _x eigenvalue. R point in (b) is ? (c) (100) surface spectrum shows surface states connecting the pro-jected double nodal ring. zoom in for the surface states (d) In S R and UX , any two 2-fold degenerate bands merge in UR as a 4-fold degenerate band.

main reason is M _x operation, which is o ↵ -centered of the unit