Topological Crystalline Metal in Orthorhombic Perovskite Iridates

Since topological insulators were theoretically predicted and experimentally observed in semiconductors with strong spin-orbit coupling, more and more attention has been drawn to topological materials which host exotic surface states. These surface excitations are stable against perturbations since they are protected by global or spatial/lattice symmetries. Succeeded 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.


INTRODUCTION
Recent discovery of topological insulators reveals a large class of new materials which, despite an insulating bulk, host robust metallic surface states [1][2][3][4][5][6][7][8][9]. Unlike conventional ordered phases characterized by their symmetries, these quantum phases are featured by nontrivial topology of their band structures, and remarkably they harbor 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 insulators [10]. 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 semimetal [11], a large class of topological metals are classified [12] which harbors surface flat bands protected by global symmetries such as charge conservation. However an experimental confirmation of these phases is still lacking [13].
In this work we propose that orthorhombic perovskite Iridates AIrO 3 where A is an alkaline earth metal, with strong spin-orbit coupling 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 reflection symmetry, and gapless helical modes located at the core of lattice dislocations. Photoemission and tunneling spectroscopy are natural experimental probes for these topological surface state. We further show how this TCM phase can be driven to metallic phases with different topology by applying magnetic fields, which breaks the mirror symmetry. All these results will be supported by topological classification in the framework of K-theory, as well as numerical calculations of topological invariants and surface/dislocation spectra, as presented in the Methods section.

RESULTS
Crystal structure and lattice symmetry in SrIrO 3 Iridates have attracted much attention due to the strong spin-orbit coupling (SOC) in 5d-Iridium (Ir) and a variety of crystal structures ranging from layered perovskites to pyrochlore lattices. Despite structural differences, a common ingredient of these iridates with Ir 4+ is J eff =1/2 states governing low energy physics, resulted from a combination of strong SOC and crystal field splitting. Among them, orthorhombic perovskite iridates AIrO 3 (where A is an alkaline earth metal) belongs to Pbnm space group and can be tuned into a topological insulator (TI) [14]. A unit cell of AIrO 3 contains four Ir atoms as shown in Fig. 1(a), 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: Π b , Π n and Π m as listed in the Methods section. Introducing three Pauli matrices corresponding to the in-plane sublattice ( τ ), layer ( ν), and pseudospin J eff =1/2 ( σ), the tight-binding Hamiltonian H(k) [14] has a relatively simple form shown in Eq. (4).
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. 1(b), which we call the nodal ring, as the energy dispersion is linear in two perpendicular directions. This nodal ring FS was confirmed by first-principle ab-initio calculations. [14,15] 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 mirror Πm chiral C Classification Topological invariants Surface zero modes Yes Yes  of SrIrO 3 which has a minimal distortion of octahedra, the nodal ring is centered around the U≡ ( π 2 , − π 2 , π 2 ) point, and extends in two-dimensional (2D) U-R-S-X plane (perpendicular tob-axis) in three-dimensional (3D) Brillouin zone (BZ) as shown in Fig. 1(b).
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 topological crystalline metal (TCM). To demonstrate the existence of zero-energy surface states, the band structure calculation for the ac-side plane was carried out with open boundary i.e.
[110] surface perpendicular tob-axis. The energy dispersion at k z = π 2 , as displayed in Fig. 1(c), reveals a dispersionless zero-energy flat band marked by red color for all k a on [110] surface of the sample. On the other hand, the slab spectrum at k a = π 2 shows the surface states are gapped except at k z = π 2 as shown in Fig. 1(d). Similar calculations for bc-side plane show that [110] surface (perpendicular toâ-axis) also supports localized surface zero-modes at k z = π 2 , while [010] surface (perpendicular toŷ-axis) does not harbor any zero-energy states. As elaborated in the Methods section, these surface states manifest a mirrorsymmetry-protected weak index labeled by a vector [16] M =â +b //ŷ, whereâ andb are Bravais lattice primitive vectors. A direct consequence of this weak index is the existence of k z = π 2 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 Π m .
The number of zero mode pairs N 0 in each dislocation line is determined by its Burgers vector B by N 0 = B · M /2π. [17] We've performed numerical calculations which demonstrate a pair of gapless helical modes in a dislocation line alonĝ c-axis with Burgers vector B = ±â, as shown in Figure 4.
Classification and topological invariants To understand the topological nature of the zero-energy surface states, let us first clarify the symmetry of tight-binding model H(k). It turns out in the mirror-reflectionsymmetric k z = π 2 plane, H(k) has the chiral symmetry, i.e. {H(k), C} = 0 where C = σ z ν y τ z . Starting from the k z = π 2 plane with mirror reflection Π m = σ z ν y and chiral symmetry C, we can classify possible surface flat bands in the mathematical framework of K-theory [18], as summarized in Table I (see Methods section for details).
In particular with both mirror and chiral symmetries, the classification 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 Π m = ±1, in each subspace we can obtain a 1D winding number [12] where k ⊥ is the crystal momentum along [lmn] direction, and h ± (k) = H ± k (k ⊥ ) is the 1D Hamiltonian parametrized by [lmn] surface momentum k in Π m = ±1 subspace. For both bc-plane ([110] surface) and ac-plane ([110] surface) we have (W + , W − ) = (1, −1). These quantized winding numbers correspond to a pair of zero modes for each surface momentum with k z = π 2 , and they cannot hybridize due to opposite mirror eigenvalues. Meanwhile for [010] surface both W ± vanish, indicating a weak index [16] M =â +b //ŷ. Intuitively the system can be considered as a stacked array of xzplanes, each plane (perpendicular toŷ-axis) with a pair of mirror-protected zero-energy edge modes at k z = π 2 . Once the mirror symmetry Π m is broken, the classification becomes Z characterized by one integer topological invariant: the total winding number W ≡ W + + W − belonging to 1D TI in symmetry class AIII [18,19]. This total winding number vanishes for all surface momenta at k z = π 2 though.
Topological metal/semimetal induced by magnetic fields Time-reversal (TR) breaking perturbations like a magnetic field can drive the system from the TCM phase to other metallic phases with different topological properties. In the presence of Zeeman coupling µ B h · σ introduced by magnetic field h, clearly the chiral symmetry C is still preserved as long as the field is in xy-plane ( h ⊥ẑ). Magnetic field parallels toẑ axis breaks chiral symmetry, which gaps the nodal ring, making the system trivial. Meanwhile mirror symmetry Π m will be broken unless the magnetic field is alongẑdirection. Thus, our focus below will be magnetic field in the xy-plane. The FS topology and associated surface states with a magnetic field along different directions are listed in Table II.
(i) Chiral topological metal protected by chiral symmetry Due to TR and inversion symmetry, the nodal ring in TCM always has 2-fold Kramers degeneracy. After applying [110] direction magnetic field h//b, the doublydegenerate nodal ring in Fig. 1(a) splits into two rings in Fig. 2(a) shifted alongĉ(ẑ)-axis on U-R-S-X plane. Though the mirror symmetry is broken by the [110] magnetic field, 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 AIII [12]. Consider [110] surface for instance, depending on the surface momentum k , the winding number W k is plotted in Fig. 2(b). It vanishes in the region where two nodal rings (blue and red) overlap, but becomes ±1 in other regions within two nodal rings.
The energy spectra on a slab geometry with open [110] surfaces are displayed in Fig. 2(c) and 2(d), plotted as a function of k a with k z = π 2 and k z = π 2 − δ, respectively. There is no zero modes at k z = π 2 , corresponding to trivial winding number. Meanwhile zero-energy flat bands highlighted by red color in Fig. 2(c) exist inside the nodal rings. It confirms the non-trivial topology of the bulk nodal rings with quantized winding number W [110] = ±1 shown in Fig. 2(b). It turns out this chiral topological metal supports localized flat band protected by chiral symmetry on any surface, as long as its normal vectorn is not perpendicular tob-axis. Meanwhile the two nodal rings are stable against any perturbations preserving chiral symmetry, since the winding number changes when we cross each nodal ring [12].
(ii) Weyl semimetal Once we apply a magnetic field along [110] direction (orâ-axis), both mirror (Π m ) and n-glide (Π n ) symmetries are broken. Consequently the nodal ring is replaced by a pair of 3D Dirac nodes, appearing at momenta (±k 0 , π ± k 0 , π 2 ) along the path R→U→R BZ line. However these Dirac nodes are not symmetry protected, since a sublattice potential mν z alternating by layers would further split each Dirac point into a pair of Weyl nodes. And this mν z term has the same symmetry as a Zeeman field h [110] alongâ-axis.
In the presence of both magnetic field h [110] and layeralternating potential mν z , the system still preserves chiral symmetry C, b-glide Π b and inversion symmetry. Two pairs of Weyl nodes emerge at ± k 1 = (±k 1 , π±k 1 , π 2 ) and ± k 2 = (±k 2 , π ± k 2 , π 2 ) along R→U→ R line in Fig. 3. The low-energy Hamiltonian around k 1 has linear disper-   sion along all k directions p(δk) ≡ (p x , p y , p z ) = with δk ≡ k − k 1 . Various coefficients 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 k a crosses k 1 , and similarly an opposite "jump" from C = 1 to C = 0 after k a passes k 2 . The different signs of jumps in Chern number indicate the Weyl fermions at ± k 1 and ± k 2 has opposite topological charge +1 (blue) and −1 (red) respectively, as shown in Fig. 3(a).
The surface states on [110] surface at k a = 0.7, between the two Weyl nodes with opposite chirality, are plotted along k z in Fig. 3(b). There is a single dispersing zero mode colored with red in Fig. 3(b) which is localized on each surface of the sample. A series of one-waydispersing zero modes for all surface momenta between k 1 and k 2 form a "chiral Fermi arc" [11,20] on [110] surface, as shown by the green lines in Fig. 3(a).

DISCUSSION
The existence of surface zero modes in our TCM phase originates from the chiral and mirror reflection symmetry of AIrO 3 with Pbnm structure. Any side surface other than [010]-plane should exhibit robust zero-energy surface states independent of the details. These localized surface states contribute to surface density of states (SDOS) near zero-energy, and can be detected by tunneling measurements. Furthermore, there exists protected propagating fermion modes in dislocation lines [17] that preserves mirror symmetry. In particular counterpropagating gapless fermions show up in pairs in the dislocation core, and the number of gapless fermion pairs is given by M · B/2π where B is the Burgers vector of dislocation as shown in the Methods section. Experimentally, these zero modes can also be detected in local SDOS near a dislocation core on [001] surface by scanning tunneling microscopy (STM). However, in a generic band structure of SrIrO 3 , this nodal ring does not sit exactly at the Fermi level E F , but slightly below E F (unless SOC is stronger than an atomic SOC used in the first-principle calculation), and a hole-like pocket FS occurs in the Γ-X-S-Y plane in Fig. 1(b) [15]. While the protected surface states are not affected by this, it will be difficult to separate the contribution of the surface states to SDOS from the bulk part. Note that the nodal ring occurs around k z = π 2 , while the bulk Fermi pocket is located near k z = 0. Thus, a momentum-resolved spectroscopy such as angle-resolved photoemission spectroscopy (ARPES) would be the best tool to observe the momentum-dependent surface states shown in Fig. 1 (c) and (d) on a side-plane of AIrO 3 .

Symmetry operators and tight-binding
Hamiltonian. The tight-binding Hamiltonian is defined in the basis of 8-component spinor ψ, organized as [14] ψ where B, R, Y, G correspond to 4 sublattices. We define Pauli matrices τ and ν in terms of following sublattice rotations The full space group P bnm is generated by (see FIG.  2 in Ref. 14) translations T x,y,z (three Bravais primitive vectors correspond to T a ≡ T x T y , T b ≡ T −1 x T y and T c = T 2 z ) and the following three generators where Π b (glide plane ⊥â ≡x+ŷ) and Π n (glide plane ⊥ b ≡ŷ−x) represents two glide symmetries while Π m is the mirror reflection. I = Π b Π n Π m represents the inversion symmetry. One subtlety is that in mirror-invariant k z = ±π/2 (or k c = π) plane, mirror symmetry is implemented by Π m = iσ z ν y instead of iσ z ν x . The tight-binding Hamiltonian of SrIrO 3 has the following form [14] Here various coefficient functions have been defined in Ref. 14 with additional term d1 where t 1d is the inter-layer next nearest neighbor hopping due to non-vanishing rotation and tilting in local oxygen octahedra. Clearly in k z = ±π/2 plane the nonvanishing terms in H(k) contain only Pauli matrices (τ x , σ z τ y ), ν y τ y and σ x,y ν x,z τ y . All these Pauli matrices anticommute with i.e. they all have chiral symmetry C.
K-theory classification procedure and topological invariants. To understand surface states on [110] surface for instance, let's focus on a 1d system H k in momentum space parametrized by fixed momentum k a ∈ [−π, π) and k c = π. Such a 1d system will have mirror reflection Π m = iσ z ν y in (3) as well as chiral symmetry C = σ z ν y τ z in (6). We'll classify such a gapped 1d system (since the bulk gap only closes at two points in k c = π plane) to see whether it has nontrivial topology, which may protect gapless surface states.
The classification of a gapped system can be understood from classifying possible symmetry-allowed mass matrices for a Dirac Hamiltonian [18,25]. The mathematical framework of K-theory applies to both global symmetry and certain spatial (crystal) symmetry [26,27]. In 1d such a Dirac Hamiltonian can be written as with chiral symmetry C and U (1) charge conservation Q Any two symmetry generators among (C, Π m , Q) commute with each other. Mathematically the classification problem correspond to the following question: given Dirac matrix γ 1 and symmetry matrices (C, Π m , Q), what is the classifying space S of mass matrix γ 0 ? In particular since each disconnected piece in classifying space S correspond to one gapped 1d phase, how many disconnected pieces does S contain? Hence the classification of gapped 1d phases with these symmetries is given by the zeroth homotopy π 0 (S).
In the K-theory classification, if there are generators which commute with all Dirac matrices and other symmetry generators, such as U (1) charge symmetry generator Q here satisfying Q 2 = (−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 k c = π plane on a generic [xy0] side surface parallel toĉ-axis, the full symmetry group is generated by (C, Π m , Q) as mentioned earlier. Note that these 3 symmetry generators all commute with each other while Q 2 = (−1) F . Together with Dirac matrix γ 1 they form a complex Clifford algebra Cl 2 × Cl 2 : parenthesis. The reason we have Cl 2 × Cl 2 is because we can block-diagonalize the k c = π tight-binding Hamiltonian w.r.t. their Π m (mirror reflection) eigenvalue ±1, and in each subspace the Clifford algebra is Cl 2 (two generator in the parenthesis). Now when we add the mass matrix γ 0 the complex Clifford algebra is extended to Cl 3 × Cl 3 generated by Therefore the classifying space of mass matrix γ 0 is determined by the extension problem of Clifford algebra Cl 2 × Cl 2 → Cl 3 × Cl 3 and we label such a classifying space as S = C 2 × C 2 . The classification of gapped 1d phases with (C, Π m , Q) symmetries is hence given by There are two integer-valued topological invariants (W + , W − ) which are 1d winding numbers [28] obtained in block-diagonalized subspace with Π m = σ z ν y = ±1. Now let's start to break Π m and C symmetries. When we break mirror reflection Π m (but keep C) the Clifford algebra extension problem becomes Cl 2 → Cl 3 and hence the classification is π 0 (C 2 ) = Z. The topological invariant is total winding number W = W + + W − .
If we break both C and Π m symmetries, our extension problem is Cl 1 → Cl 1 which leads to a trivial classification π 0 (C 1 ) = 0. Therefore we obtain the results in Table I.
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 Π m . The number of zero mode pairs in each dislocation line is determined by its Burgers vector B: number of helical modes= B · M /2π. [17] To check such zero modes due to dislocation line, we consider a pair edge dislocations alongĉ-axis, with a pair of Burgers vectors: ( B =â, B = −â) perpendicular to the dislocation line. The type of dislocation core is illustrated in Fig. 4(a). We consider now a 3D box with periodic boundary condition in all (â,b,ĉ) directions (a 3Dtorus) so that the system does not have an open surface. Consider every unit cell (with 4 sublattices, 2 pseudospin species per site) as a lattice 'site' in Fig. 4(a), then the system has hopping terms between nearest neighbor 'sites' following the tight-binding Hamiltonian in Eq. 4.ĉ direction is still translational invariant and k z remains a good quantum number, but in ab-plane translation symmetry is broken by the dislocation. The dislocation spectrum is obtained when there are 39 sites alongâ-axis and 15 sites alongb-axis. The location of dislocation with Burgers vector B =â is at (4, 6) which means 4th site alongâ-axis and 6th site alongb-axis. And the position of dislocation with Burgers vector B = −â is at (24,12). The dislocation spectrum has displayed in Fig. 4(b) highlighted by red color. 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 mν z which breaks mirror symmetry and magnetic field h [110] (σ x + σ y ) which breaks TR symmetry to H(k) in Eq. 4. The followings are the coefficients for the effective Hamiltonian describes the Weyl fermion at k 1 of Eq. 2 presented in the maintext: where t p , t o 2p , t o 1p , t o d , m, t d1 , h ≡ h [110] are the coefficients in tight-binding Hamiltonian. The Chern number C for all occupied bands as a function of crystal momentum along U → R (or k a ) is shown in Fig. 4(c).