Topological crystalline metal in orthorhombic perovskite iridates

Abstract

Since topological insulators were theoretically predicted and experimentally observed in semiconductors with strong spin-orbit coupling, increasing attention has been drawn to topological materials that host exotic surface states. These surface excitations are stable against perturbations since they are protected by global or spatial/lattice symmetries. Following the success in achieving various topological insulators, a tempting challenge now is to search for metallic materials with novel topological properties. Here we predict that orthorhombic perovskite iridates realize a new class of metals dubbed topological crystalline metals, which support zero-energy surface states protected by certain lattice symmetry. These surface states can be probed by photoemission and tunnelling experiments. Furthermore, we show that by applying magnetic fields, the topological crystalline metal can be driven into other topological metallic phases, with different topological properties and surface states.

ARTICLE

Received 23 Sep 2014 | Accepted 10 Feb 2015 | Published 16 Mar 2015

Since topological insulators were theoretically predicted and experimentally observed in semiconductors with strong spinorbit coupling, increasing attention has been drawn to topological materials that host exotic surface states. These surface excitations are stable against perturbations since they are protected by global or spatial/lattice symmetries. Following the success in achieving various topological insulators, a tempting challenge now is to search for metallic materials with novel topological properties. Here we predict that orthorhombic perovskite iridates realize a new class of metals dubbed topological crystalline metals, which support zero-energy surface states protected by certain lattice symmetry. These surface states can be probed by photoemission and tunnelling experiments. Furthermore, we show that by applying magnetic elds, the topological crystalline metal can be driven into other topological metallic phases, with different topological properties and surface states.

DOI: 10.1038/ncomms7593

Topological crystalline metal in orthorhombic perovskite iridates

Yige Chen1, Yuan-Ming Lu2,3 & Hae-Young Kee1,4

1 Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada. 2 Department of Physics, University of California, Berkeley, California 94720, USA. 3 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA. 4 Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada. Correspondence and requests for materials should be addressed to H.-Y.K. (email: mailto:[email protected]

Web End [email protected] ).

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Recent discovery of topological insulators (TIs) reveals a large class of new materials that, despite an insulating bulk, host robust metallic surface states19. Unlike conventional

ordered phases characterized by their symmetries, these quantum phases are featured by non-trivial topology of their band structures, and remarkably they harbour conducting surface states protected by global symmetries such as time-reversal and charge conservation. More recently, it was realized that certain insulators can support surface states protected by crystal symmetry, and they are named topological crystalline insulators1012. The rich topology of insulators in the presence of symmetries lead to a natural question: are there similar topological metals hosting protected surface states? After the proposal of Weyl semimetal13, a large class of topological metals are classied14, which harbours surface at bands protected by global symmetries such as charge conservation. However, an experimental conrmation of these phases is still lacking15.

In this work, we propose that orthorhombic perovskite iridates AIrO3, where A is an alkaline-earth metal, with strong spinorbit coupling (SOC) and Pbnm structure, can realize a new class of metal dubbed topological crystalline metal (TCM). Topological properties of such a TCM phase include zero-energy surface states protected by the mirror-reection symmetry, and gapless helical modes located at the core of lattice dislocations. Photoemission and tunnelling spectroscopy are natural experimental probes for these topological surface states. We further show how this TCM phase can be driven to metallic phases with different topology by applying magnetic elds, which breaks the mirror symmetry. All these results will be supported by topological classication in the framework of K-theory, as well as numerical calculations of topological invariants and surface/dislocation spectra, as presented in the Methods section.

ResultsCrystal structure and lattice symmetry in SrIrO3. Iridates have attracted much attention due to the strong SOC in 5d-Iridium (Ir) and a variety of crystal structures ranging from layered perovskites to pyrochlore lattices16,17. Despite structural differences, a common ingredient of these iridates with Ir4 is

Jeff 1/2 states governing low-energy physics, resulted from a

combination of strong SOC and crystal eld splitting. Among them, orthorhombic perovskite iridates AIrO3 (where A is an alkaline-earth metal) belongs to Pbnm space group and can be tuned into a TI18.

A unit cell of AIrO3 contains four Ir atoms as shown in Fig. 1a, and there are three types of symmetry plane: b-glide, n-glide and mirror plane perpendicular to ^c axis. Each of them can be assigned to the symmetry operators as follows: Pb, Pn and Pm, as listed in the Methods section. Introducing three Pauli matrices corresponding to the in-plane sublattice (s), layer (n) and pseudo-spin Jeff 1/2 (r), the tight-binding Hamiltonian H(k)18

has a relatively simple form shown in equation (6).

The band structure of tight-binding Hamiltonian H(k) exhibits a ring-shaped one-dimensional (1D) Fermi surface (FS) close to the Fermi level as shown in Fig. 1b, which we call the nodal ring, as the energy dispersion is linear in two perpendicular directions. This nodal ring FS was conrmed by rst-principle ab initio calculations18,19, and it remains intact in the presence of Hubbard U up to 2.5 eV. Therefore, the semimetallic character of SrIrO3 with U around 2 eV is consistent with the previous experimental results20,21. It was further shown that the size of nodal ring is determined by rotation and tilting angles of the oxygen octahedra around each Ir atom. In the case of SrIrO3, which has a minimal distortion of octahedra, the nodal ring is centred around the U

ka 0; kb pa ; kc pc

point, and extends in two-dimensional (2D) URSX plane (perpendicular to ^

b axis) in three-dimensional (3D) Brillouin zone (BZ) as shown in Fig. 1b.

Zero-energy surface states and dislocation helical modes. Here we show that the nodal ring FS exhibits non-trivial topology that leads to localized surface zero modes protected by the mirror symmetry, thus coined TCM. To demonstrate the existence of zero-energy surface states, the band structure calculation for the ^

a^c-side plane was carried out with open boundary, that is, 1 10

surface perpendicular to ^

b axis. The energy dispersion at kc pc, as

displayed in Fig. 1c, reveals a dispersionless zero-energy at band marked by red colour for all ka on 1

10 surface of the sample. On

the other hand, the slab spectrum at ka 0 shows that the surface

states are gapped except at kc pc, as shown in Fig. 1d. Similar

calculations for ^

b^c-side plane show that [110] surface (perpendicular to ^a axis) also supports localized surface zero modes at kc pc, while 010 surface (perpendicular to ^y axis) does not

harbour any zero-energy states. As elaborated in the Methods section, these surface states manifest a mirror-symmetry-protected weak index labelled by a vector22 M ^a

^

b==^y, where ^a and ^

b

are Bravais lattice primitive vectors. A direct consequence of this weak index is the existence of kc pc zero-energy states for any side

surface, except for [010] surface perpendicular to vector M.

Another consequence of weak index M is the existence of pairs of counter-propagating zero modes (helical modes) localized in a dislocation line, which respects mirror symmetry Pm. The

number of zero-mode pairs N0 in each dislocation line is determined by its Burgers vector B by N0 B M/2p (ref. 23). We

have performed numerical calculations that demonstrate a pair of gapless helical modes in a dislocation line along ^c axis with Burgers vector B ^a. Detailed results are presented in the

Methods section.

Classication and topological invariants. To understand the topological nature of the zero-energy surface states, let us rst clarify the symmetry of tight-binding model H(k). It turns out in the mirror-reection-symmetric kc pc plane, H(k) has an

emergent chiral symmetry, that is, fHk; Cg 0, where C sznxtz. Here r, n and s represent Jeff 12 pesudo-spin space,

inter-plane and in-plane sublattice, respectively. A chiral symmetry can be understood as the combination of time-reversal symmetry and certain particle-hole symmetry24, hence switching the sign of Hamiltonian. The presence of chiral symmetry C

enforces the energy spectrum of H(k) to be symmetrical with respect to the zero energy. It is known that chiral symmetry can protect zero-energy surface modes25,26. On the other hand, various crystal symmetries such as mirror reection Pm bring extra non-trivial topological properties into the system we studied. Starting from the kc pc plane with mirror reection

Pm sznx and chiral symmetry C, we can classify possible surface

at bands in the mathematical framework of K-theory27, as summarized in Table 1 (see Methods section for details).

In particular with both mirror and chiral symmetries, the classication is Z Z characterized by a pair of integer topological

invariants (W, W ). Since the Hilbert space can be decomposed into two subspaces with different mirror eigenvalues Pm 1, in

each subspace we can obtain a 1D winding number14

W lmn kk

1

2pi

; 1

where k? is the crystal momentum along [lmn] direction and

h k H kk k? is the 1D Hamiltonian parametrized by [lmn]

surface momentum k|| in Pm 1 subspace. For both

^

Z dk?Tr Ch 1 k

@ @k?

h k

b^c plane

([110] surface) and ^a^c plane (1 10 surface), we have (W ,

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kc

Y

G

X

Z

U

T

Y

(z)

R

T

z =c/4

b

Z

X

B R x

T

S

kb

ka

Y

E/t

0.3

0.15

0

0.15

0.3

E/t

0.2

0.1

0

0.1

0.2

T T

Z Z

Figure 1 | Crystal structure and surface states on 1

10 side plane. (a) Crystal structure of AIrO3 with ^

a axis along [110], ^

b axis along 1

10 and ^c axis along

^

z direction, respectively. The unit cell contains four Ir atoms: blue (B), red (R), yellow (Y) and green (G) represent different oxygen octahedra environment.

The length of Bravais lattice unit vectors ^

a=^

b and ^c axis are a/a and c, respectively. A mirror symmetry plane (coloured with light orange), mapping z to z,

locates at z c4, where four oxygen atoms (purple solid circles) are within the same plane. (b) Special k-points in the 3D bulk BZ and 2D surface BZ for 1

10

surface. The location of the nodal ring depict as red circles on URSX plane in 3D BZ. Slab-geometry surface spectrum (c) for 1

10 surface at kc pc,

plotted along the high-symmetry line (green dashed line)

T !

T, and (d) for 1

10 surface at ka 0 by varying kc to follow

G (purple dashed

line), where red lines represent surface states.

Table 1 | Classication of topological crystalline metals and possible surface states on a generic side surface parallel to ^

z-axis.

Mirror Pm Chiral C Classication Topological invariants Surface zero modes

Yes Yes Z Z (W, W ) (1, 1) Yes kc pc

No Yes Z W W W Yes kc pc d; j d j 1

Yes No 0 No No No 0 No

Note that m; C 0. (W , W ) is the pair of winding numbers obtained in P 1 subspaces of block-diagonalized Hamiltonian.

W ) (1, 1). These quantized winding numbers correspond to

a pair of zero modes for each surface momentum with kc pc, and

they cannot hybridize due to opposite mirror eigenvalues. Meanwhile, for [010] surface both W vanish, indicating a weak index22 M ^a

^

b== ^y. Intuitively, the system can be considered as a stacked array of ^x^z planes, each plane (perpendicular to ^y axis) with a pair of mirror-protected zero-energy edge modes at kc pc.

Once the mirror symmetry Pm is broken, the classication becomes Z characterized by one integer topological invariant: the total winding number W W W associated with 1D TI in

symmetry class AIII24,27. This total winding number vanishes for all surface momenta at kc pc though.

Topological metal/semimetal induced by magnetic elds. Time-reversal (TR) breaking perturbations such as a magnetic eld can drive the system from the TCM phase to other metallic phases with different topological properties. In the presence of Zeeman coupling mBh r introduced by magnetic eld h, clearly

the chiral symmetry C is still preserved as long as the eld is in ^x^y

plane (h?^z). Magnetic eld parallels to ^z axis breaks chiral

symmetry, which gaps the nodal ring, making the system trivial. Meanwhile, mirror symmetry Pm will be broken unless the

magnetic eld is along ^z direction. Thus, our focus below will be magnetic eld in the ^x^y plane. The FS topology and associated surface states with a magnetic eld along different directions are listed in Table 2.

Chiral topological metal protected by chiral symmetry. Due to TR and inversion symmetry, the nodal ring in TCM always has twofold Kramers degeneracy. After applying 1 10 direction

magnetic eld h==^

b, the doubly degenerate nodal ring in Fig. 1b splits into two rings in Fig. 2a shifted along ^c^z axis on URSX

plane. Although the mirror symmetry is broken by the 1 10

magnetic eld, chiral symmetry C is still preserved. Therefore,

the topological properties of the two nodal rings are captured by an integer winding number W in the symmetry class AIII14. Consider 1 10 surface for instance, depending on the

surface momentum k||, the winding number Wkk is plotted in

Fig. 2b. It vanishes in the region where the two nodal rings (blue and red) overlap, but becomes 1 in other regions within the two nodal rings.

The energy spectra on a slab geometry with open 1 10 surfaces

are displayed in Fig. 2c,d, plotted as a function of ka with kc pc

and kc pc d, respectively. There is no zero modes at kc pc,

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corresponding to trivial winding number. Meanwhile, zero-energy at bands highlighted by red colour in Fig. 2d exist inside the nodal rings. It conrms the non-trivial topology of the bulk nodal rings with quantized winding number W1

10 1 shown

in Fig. 2b. It turns out this chiral topological metal supports localized at band protected by chiral symmetry on any surface, as long as its normal vector ^

n is not perpendicular to ^

b axis.

Meanwhile, the two nodal rings are stable against any perturbations preserving chiral symmetry, since the winding number changes when we cross each nodal ring14.

Weyl semimetal. Once we apply a magnetic eld along [110] direction (or ^a axis), both mirror (Pm) and n-glide (Pn) symmetries are broken. Consequently, the nodal ring is replaced by a pair of 3D Dirac nodes, appearing at momenta k0; pa ; pc

and k2

k2; pa ; pc

along R-U- R line in Fig. 3a. The low-energy Hamiltonian around k1 has linear dispersion along all k directions

Heffdk pdk s; 2

pdk px; py; pz A1dkb A2dkc; B1dkc; D1dka

along

the path R-U-R BZ line. However, these Dirac nodes are not symmetry protected, since a sublattice potential mnz alternating by layers would further split each Dirac point into a pair of Weyl nodes. And this mnz term has the same symmetry as a Zeeman eld h[110] along ^a axis.

In the presence of both magnetic eld h[110] and layer-

alternating potential mnz, the system still preserves chiral symmetry C, b-glide Pb and inversion symmetry. Two pairs of

Weyl nodes emerge at k1 k1; pa ; pc

; with dk kk1. Various coefcients can be expressed in terms of

the tight-binding hopping parameters (see Methods).

There exists a jump for the Chern number C for all occupied bands from C 0 to C 1 when ka crosses k1, and similarly an

opposite jump from C 1 to C 0 after ka passes k2. The

different signs of jumps in Chern number indicate that the Weyl fermions at k1 and k2 have opposite topological charge 1

(blue) and 1 (red), respectively, as shown in Fig. 3a.

The surface states on 1 10 surface at ka 0:7a, between the two

Weyl nodes with opposite chirality, are plotted along kc in Fig. 3b. There is a single dispersing zero mode coloured with red in Fig. 3b, which is localized on each surface of the sample. A series of one-way-dispersing zero modes for all surface momenta between k1 and k2 form a chiral Fermi arc13,28 on 1 10 surface,

as shown by the green lines in Fig. 3a.

Table 2 | Various TR-breaking perturbations and consequences on the nodal ring and surface states (for example, on 110 side

surface).

B eld b-glide Pb n-glide Pn Mirror Pm Chiral C Fermi surfaces Zero-energy surface states 1 10 No Yes No Yes Nodal rings Flat bands at kc pc d

[110] Yes No No Yes Nodal points Fermi arcs between Weyl points[001] No No Yes No Gapped No

When the magnetic eld is along [110] direction, the nodal ring splits into a pair of Dirac nodes along U-R. On the other hand, the nodal ring is completely gapped if the magnetic eld has a nonzero ^z^c

direction component.

X

Z

R

U

T

T

R

T

Y

+1

0

Z

1

0

X

S

Y

E/t

0.5

0.25

0

0.25

0.5

E/t

0.5

0.25

0

0.25

0.5

[afii9843]/a [afii9843]/a

Figure 2 | Evolution of the nodal ring under various TR-breaking terms and corresponding surface modes. (a) When the magnetic eld ^

h k

^

T

Z

T

0

ka

b, a pair of

nodal rings contain one ring (blue) shifted upwards along ^c^z axis and the other one (red) shifted downwards. (b) The corresponding winding number

W1

10 distribution on

G plane. Slab spectrum (c) when kc pc for 1

10 surface plotted as a function of ka along the high-symmetry line

T, and (d) when kc pc d indicated by the green dashed line in (b).

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E/t

R

+1

k1

U X

R

E/t

0.1

0

0.1

0

1

k2

S

[afii9843]/c

2[afii9843]/c

kz

Figure 3 | Emergence of Weyl fermions on 1

10 side surface. (a) Two pairs of Weyl node located along the high-symmetry line U-R at k1 and k2,

respectively. One of the Weyl fermion (blue) has 1 chirality but the other Weyl node (red) has opposite chirality. The Fermi arc connecting those two

Weyl nodes is coloured by green. (b)The edge states spectra when ka 0:7a, which is in between two Weyl nodes plotted as a function of kc.

DiscussionThe existence of surface zero modes in our TCM phase originates from the chiral and mirror-reection symmetry of AIrO3 with

Pbnm structure. Any side surface other than [010] plane should exhibit robust zero-energy surface states independent of the details. In a generic band structure of SrIrO3, this nodal ring does not sit exactly at the Fermi level EF, but slightly below EF (unless SOC is stronger than an atomic SOC used in the rst-principle calculation), and a hole-like pocket FS occurs around G point in

Fig. 1b19. In other words, the nodal ring occurs around kc pc,

while the small bulk Fermi pocket is located near kc 0.

Therefore, the zero-energy surface modes are well separated from the bulk FS pockets in momentum, and a momentum-resolved probe is required to detect the surface states. Angle-resolved photoemission spectroscopy (ARPES) would be the best tool to observe the momentum-resolved surface states shown in Fig. 1c,d on a side plane of AIrO3. Notice that, ARPES has successfully detected the topological surface states in Dirac semimetal material29. Due to the presence of extremely small orbital overlap amplitudes between further Ir sites, the surface states acquire a slight dispersion. However, as we emphasized above, the mirror symmetry is a crucial ingredient to support such non-trivial surface states detectable by ARPES, despite the weak breaking of chiral symmetry.

These surface states also contribute nite surface density of states (SDOSs) near zero energy. In contrast, the semimetallic bulk band contribution to SDOS vanishes around zero energy due to the presence of bulk nodal ring. Therefore, protected surface modes can be detected as a zero bias hump (nite SDOSs) in the dI/dV curve of scanning tunnelling microscopy. However, in real materials, it will be difcult to separate the contribution of the surface states to SDOS from the bulk part. On the other hand, there exists protected propagating fermion modes in dislocation lines23 that preserves mirror symmetry. One advantage is that these topological helical modes are protected by mirror and chiral symmetries, and hence will not be destroyed by hybridization with bulk gapless excitations. In particular, counter-propagating gapless fermions show up in pairs in the dislocation core, and the number of gapless fermion pairs is given by M B/2p, where B is the Burgers vector of

dislocation as shown in the Methods section. Unlike other gapless and dispersive bulk excitations that are extended in space, these helical fermion zero modes are localized near the dislocation core, which in hence can be detected by scanning tunnelling microscopy.

Since a bulk sample of AIrO3 such as SrIrO3 requires high pressure to achieve the Pbnm crystal structure30,31, it is desirable to grow a lm of AIrO3. Recently, superlattices of atomically thin

slices of SrIrO3 was made by pulsed laser deposition along [001] plane32. This work is a rst step towards possible topological phases in iridates. Furthermore, a successful growth of lm along [111] plane was also reported33. Thus, growing a lm of AIrO3 along [110] (or 1 10 ) is plausible. To conrm the proposed TCM,

an ARPES study should be performed on a lm of AIrO3 grown along [110] (or 1 10 ), where the mirror symmetry of Pbnm

structure is kept. This analysis should reveal a at surface band near kc pc below EF.

Methods

Symmetry operators and tight-binding Hamiltonian. The tight-binding Hamiltonian is dened in the basis of eight component spinor c, organized as18

c cB"; cR"; cY"; cG"; cB#; cR#; cY#; cG# T

; 3 where B, R, R, G correspond to four sublattices. We dene Pauli matrices s and n in terms of following sublattice rotations

B !

t R; Y !

t G;

B !

n Y; R !

n G: 4 The full space group Pbnm is generated by (see Fig. 2 in ref. 18) translations

Tx,y,z (three Bravais primitive vectors correspond to Ta TxTy; Tb T 1xTy and

Tc T2z) and the following three generators {Pb, Pn, Pm}

ck ;k ;k !

ei is s

2

p nxtx e

i ck ; k ;k ;

ck ;k ;k !

ei is s

2

p tx ei c k ;k ;k ;

ck ;k ;k !

5

isznxck ;k ; k ; ck !

I c k:

where Pb (glide plane ?^a ^x ^y) and Pn (glide plane ?

^

b ^y ^x) represent

two glide symmetries, while Pm is the mirror reection and r denotes the pseudo-spin subspace. I PbPnPm represents the inversion symmetry.

The tight-binding Hamiltonian of SrIrO3 has the following form18

Hk Re Epk

t

x Im Epk

n

xtx Im Edk

s

zty Ezknx

Re Edk

n

yty Re Epok

s

ynzty

Im Epok

s

ynytx Im Ed1k

s

xnytz Re Edok

s

ynxty Im Edok

s

xnxty

6

Re Ed1k

s

xnytx :

Here various coefcient functions have been dened in ref. 18 with additional term Ed1k

Ed1k td1 cosky i coskx

cosk

z

; 7 where t1D is the inter-layer next nearest-neighbour hopping due to non-vanishing rotation and tilting in local oxygen octahedra. The crystal momentum (ka, kb, kc)

relates to (kx, ky, kz) simply by

ka

kx kya ; kb

ky kxa ; kc

2kzc : 8

The basis ck ;k ;k we use here is different with the basis ck ;k ;k in ref. 18 by a

unitary rotation Uk

ck ;k ;k Uy

k ck ;k ;k ; Uk ei n ei t : 9

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In this basis ck ;k ;k , the tight-binding Hamiltonian is related to Hk by the unitary

transformation Uk in equation 6:

Hk Hka; kb; kc Uy

k HkUk: 10

Clearly in kc pc(or kz p2) plane the non-vanishing terms in Hk, as in the basis

ck ;k ;k , contain only Pauli matrices (tx, szty), nyty and sx,ynx,zty. All these Pauli

matrices anti-commute with

C sznytz; 11

that is, they all have chiral symmetry C. However, in ck ;k ;k representation, the

chiral symmetry C written as

C sznxtz: 12

K-theory classication procedure and topological invariants. To understand surface states on 1

10 surface for instance, let us focus on a 1D system Hk in

momentum space parametrized by xed momentum ka 2 pa ; pa

and kc pc.

Such a 1D system will have mirror reection Pm isznx in (5), as well as chiral

symmetry C sznxtz in (12). We will classify such a gapped 1D system (since the

bulk gap only closes at two points in kc pc plane) to see whether it has non-trivial

topology, which may protect gapless surface states.

The classication of a gapped system can be understood from classifying possible symmetry-allowed mass matrices for a Dirac Hamiltonian27,34. The mathematical framework of K-theory applies to both global symmetry and certain spatial (crystal) symmetry35,36. In 1D such a Dirac Hamiltonian can be written as

H1dDirac kag1 mg0; 13 with chiral symmetry C

fC; g0g fC; g1g 0: 14 mirror reection Pm

m; g0 m; g1 0: 15 and U(1) charge conservation Q

Q; g0 Q; g1 0: 16 Any two symmetry generators among C; m; Q commute with each other.

Mathematically, the classication problem corresponds to the following question: given Dirac matrix g1 and symmetry matrices C; m; Q, what is the classifying

space S of mass matrix g0? In particular, since each disconnected piece in

classifying space S corresponds to one gapped 1D phase, how many disconnected

pieces does S contain? Hence, the classication of gapped 1D phases with these

symmetries is given by the zeroth homotopy p0S.

In the K-theory classication, if there are generators that commute with all Dirac matrices and other symmetry generators, such as U(1) charge symmetry generator Q here satisfying Q2 ( 1)F, then we say the gapped system belong to a

complex class. Otherwise, it belongs to a real class. Clearly, our case belongs to the complex class because both Q and M commute with any other matrices.

For the kc p/c plane on a generic [xy0] side surface parallel to ^c axis, the full

symmetry group is generated by C; m; Q as mentioned earlier. Note that these

three symmetry generators all commute with each other while Q2 ( 1)F.

Together with Dirac matrix g1, they form a complex Clifford algebra Cl2 Cl2:

fg1; Cg m Q; 17 where the generators inside the parenthesis anti-commute with each other and commute with everything outside the parenthesis. The reason we have Cl2 Cl2 is

because we can block-diagonalize the kc p tight-binding Hamiltonian with

respect to their Pm (mirror reection) eigenvalue 1, and in each subspace the

Clifford algebra is Cl2 (two generator in the parenthesis). Now when we add the mass matrix g0, the complex Clifford algebra is extended to Cl3 Cl3 generated by

fg1; C; g0g m Q 18 Therefore, the classifying space of mass matrix g0 is determined by the extension problem of Clifford algebra Cl2 Cl2-Cl3 Cl3 and we label such a classifying

space as S C2 C2. The classication of gapped 1D phases with C; m; Q

symmetries is hence given by

p0

C2 C2 p0C2 p0C2 Z Z 19 There are two integer-valued topological invariants (W , W ) that are 1D winding numbers26 obtained in block-diagonalized subspace with

Pm szny 1.

Now let us start to break Pm and C symmetries. When we break mirror

reection Pm (but keep C), the Clifford algebra extension problem becomes

Cl2-Cl3

fg1; Cg Q ! fg1; C; g0g Q 20 and hence the classication is p0C2 Z. The topological invariant is total

winding number W W W .

If we break C but keep Pm, the extension problem is again (Cl1)2-(Cl2)2

fg1g m Q ! fg1; g0g m Q 21 and classication is trivial since

p0

C1 C1 p0C1 p0C1 0 0 0: 22

If we break both C and Pm symmetries, our extension problem is Cl1-Cl1

fg1g Q ! fg1; g0g Q; 23 which leads to a trivial classication p0(C1) 0. Therefore, we obtain the results in

Table 1.

Dislocation spectrum. One implication of weak index M is the existence of pairs of counter-propagating zero modes localized in a dislocation line respecting mirror symmetry Pm. The number of zero-mode pairs in each dislocation line is determined by its Burgers vector B: number of helical modes B M/2p. (ref. 23)

To check such zero modes due to dislocation line, we consider pair-edge dislocations along ^c axis, with a pair of Burgers vectors: B ^a; B ^a

perpendicular to the dislocation line. The type of dislocation core is illustrated in Fig. 4a. We consider now a 3D box with periodic boundary condition in all

^

a; ^

b;^c

directions (a 3D torus) so that the system does not have an open surface. Consider every unit cell (with four sublattices, two pseudo-spin species per site) as a lattice site in Fig. 4a, then the system has hopping terms between nearest-neighbour sites following the tight-binding Hamiltonian in equation (6). ^c direction is still translational invariant and kz remains a good quantum number, but in ^a^

b plane translation symmetry is broken by the dislocation. The dislocation spectrum is obtained when there are 39 sites along ^a axis and 15 sites along ^b axis. The location of dislocation with Burgers vector B ^a is at (4,6), which means 4th

site along ^a axis and 6th site along ^

b axis. And the position of dislocation with Burgers vector B ^a is at (24,12). The dislocation spectrum has displayed in

Fig. 4b highlighted by red colour. It shows two pairs of gapless helical modes localized on each dislocation cores.

Effective Hamiltonian of Weyl fermion. By projecting the states around the Weyl node, the two-band Hamiltonian with linear Weyl fermion form can be obtained after adding mnz, which breaks mirror symmetry and magnetic eld h[110](sx sy),

which breaks TR symmetry to H(k) in equation (6). The followings are the coefcients for the effective Hamiltonian describes the Weyl fermion at k1 of

0.06

1

0

b

0.03

y

E/t

0

0.03

x

0.06

1.45

1.51 [afii9843]/2 1.65 1.7

k1 k2 ka

kz

Figure 4 | Disloction spectrum and Chern number. (a) Top view of an edge dislocation with Burgers vector ^

a (coloured by red arrow), whose dislocation line is along ^c axis. Dashed green line denotes a trajectory around the dislocation core (red circle). Clearly when we go around the dislocation once, we need an extra translation by the Burgers vector (the red arrow), equal to Bravais primitive vector ^

a in this case) to return to the starting point.(b) The dislocation spectrum for system size 39 15 contains two pairs of gapless helical modes (highlighted with red colour) localized at each dislocation

line. (c) Chern number as a function of ka for all occupied bands. Opposite Chern number jump indicates that the Weyl fermion at k1 and k2 has different chirality.

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equation (2) presented in the main text:

A1

2tptodsin2k1 to2p to1p

cosk1

;

A2 2

p h m

to

z

p to2p to1p

cosk1

;

2

24

B1

td1 m 2

p h

2

p to2p to1p

;

sink1

to2p to1p

;

tod

2

D1

to2p to1p

2

where tp; to2p; to1p; tod; m; td1; h h110 are the coefcients in tight-binding

Hamiltonian. The Chern number C for all occupied bands as a function of crystal momentum along U-R (or ka) is shown in Fig. 4c.

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Acknowledgements

This work is supported by Natural Science and Engineering Research Council of Canada (NSERC), Center for Quantum Materials at the University of Toronto (Y.C. and H.-Y.K.) and Ofce of BES, Materials Sciences Division of the U.S. DOE under contract No. DE-AC02-05CH11231 (Y.-M.L.). H.-Y.K. thanks S. Ryu for informing topology of gapless superconductors in Ref. 36. Y.-M.L. and H.-Y.K. acknowledge the hospitality of the Aspen Cetner for Physics supported by National Science Foundation Grant No. PHYS-1066293 where a part of this work was carried out.

Author contributions

All authors performed the theoretical calculations, discussed the results and wrote the manuscript. H.-Y.K. planned and supervised the project.

Additional information

Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Chen, Y. et al. Topological crystalline metal in orthorhombic perovskite iridates. Nat. Commun. 6:6593 doi: 10.1038/ncomms7593 (2015).

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