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 zeroenergy 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.
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Introduction
Recent discovery of topological insulators (TIs) reveals a large class of new materials that, 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 harbour conducting surface states protected by global symmetries such as timereversal 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^{10,11,12}. 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^{13}, a large class of topological metals are classified^{14}, which harbours surface flat bands protected by global symmetries such as charge conservation. However, an experimental confirmation of these phases is still lacking^{15}.
In this work, we propose that orthorhombic perovskite iridates AIrO_{3}, where A is an alkalineearth metal, with strong spin–orbit 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 zeroenergy surface states protected by the mirrorreflection 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 fields, which breaks the mirror symmetry. All these results will be supported by topological classification in the framework of Ktheory, 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 SOC in 5dIridium (Ir) and a variety of crystal structures ranging from layered perovskites to pyrochlore lattices^{16,17}. Despite structural differences, a common ingredient of these iridates with Ir^{4+} is J_{eff}=1/2 states governing lowenergy physics, resulted from a combination of strong SOC and crystal field splitting. Among them, orthorhombic perovskite iridates AIrO_{3} (where A is an alkalineearth metal) belongs to Pbnm space group and can be tuned into a TI^{18}.
A unit cell of AIrO_{3} contains four Ir atoms as shown in Fig. 1a, and there are three types of symmetry plane: bglide, nglide and mirror plane perpendicular to axis. Each of them can be assigned to the symmetry operators as follows: Π_{b}, Π_{n} and Π_{m}, as listed in the Methods section. Introducing three Pauli matrices corresponding to the inplane sublattice (τ), layer (ν) and pseudospin J_{eff}=1/2 (σ), the tightbinding Hamiltonian H(k)^{18} has a relatively simple form shown in equation (6).
The band structure of tightbinding Hamiltonian H(k) exhibits a ringshaped onedimensional (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 confirmed by firstprinciple ab initio calculations^{18,19}, and it remains intact in the presence of Hubbard U up to 2.5 eV. Therefore, the semimetallic character of SrIrO_{3} with U around 2 eV is consistent with the previous experimental results^{20,21}. 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 SrIrO_{3}, which has a minimal distortion of octahedra, the nodal ring is centred around the point, and extends in twodimensional (2D) U–R–S–X plane (perpendicular to axis) in threedimensional (3D) Brillouin zone (BZ) as shown in Fig. 1b.
Zeroenergy surface states and dislocation helical modes
Here we show that the nodal ring FS exhibits nontrivial topology that leads to localized surface zero modes protected by the mirror symmetry, thus coined TCM. To demonstrate the existence of zeroenergy surface states, the band structure calculation for the side plane was carried out with open boundary, that is, surface perpendicular to axis. The energy dispersion at , as displayed in Fig. 1c, reveals a dispersionless zeroenergy flat band marked by red colour for all k_{a} on surface of the sample. On the other hand, the slab spectrum at k_{a}=0 shows that the surface states are gapped except at , as shown in Fig. 1d. Similar calculations for side plane show that [110] surface (perpendicular to axis) also supports localized surface zero modes at , while surface (perpendicular to axis) does not harbour any zeroenergy states. As elaborated in the Methods section, these surface states manifest a mirrorsymmetryprotected weak index labelled by a vector^{22} , where and are Bravais lattice primitive vectors. A direct consequence of this weak index is the existence of zeroenergy 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 counterpropagating zero modes (‘helical modes’) localized in a dislocation line, which respects mirror symmetry Π_{m}. The number of zeromode pairs N_{0} in each dislocation line is determined by its Burgers vector B by N_{0}=B·M/2π (ref. 23). We have performed numerical calculations that demonstrate a pair of gapless helical modes in a dislocation line along axis with Burgers vector . Detailed results are presented in the Methods section.
Classification and topological invariants
To understand the topological nature of the zeroenergy surface states, let us first clarify the symmetry of tightbinding model H(k). It turns out in the mirrorreflectionsymmetric plane, H(k) has an emergent chiral symmetry, that is, , where . Here σ, ν and τ represent pesudospin space, interplane and inplane sublattice, respectively. A chiral symmetry can be understood as the combination of timereversal symmetry and certain particlehole symmetry^{24}, hence switching the sign of Hamiltonian. The presence of chiral symmetry enforces the energy spectrum of H(k) to be symmetrical with respect to the zero energy. It is known that chiral symmetry can protect zeroenergy surface modes^{25,26}. On the other hand, various crystal symmetries such as mirror reflection Π_{m} bring extra nontrivial topological properties into the system we studied. Starting from the plane with mirror reflection Π_{m}=σ_{z}ν_{x} and chiral symmetry , we can classify possible surface flat bands in the mathematical framework of Ktheory^{27}, as summarized in Table 1 (see Methods section for details).
In particular with both mirror and chiral symmetries, the classification is 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^{14}
where k is the crystal momentum along [lmn] direction and is the 1D Hamiltonian parametrized by [lmn] surface momentum k_{} in Π_{m}=±1 subspace. For both plane ([110] surface) and plane ( surface), we have (W^{+}, W^{−})=(1, −1). These quantized winding numbers correspond to a pair of zero modes for each surface momentum with , and they cannot hybridize due to opposite mirror eigenvalues. Meanwhile, for [010] surface both W^{±} vanish, indicating a weak index^{22} . Intuitively, the system can be considered as a stacked array of planes, each plane (perpendicular to axis) with a pair of mirrorprotected zeroenergy edge modes at .
Once the mirror symmetry Π_{m} is broken, the classification becomes characterized by one integer topological invariant: the total winding number W≡W^{+}+W^{−} associated with 1D TI in symmetry class AIII^{24,27}. This total winding number vanishes for all surface momenta at though.
Topological metal/semimetal induced by magnetic fields
Timereversal (TR) breaking perturbations such as 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 is still preserved as long as the field is in plane (). Magnetic field parallels to 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 along direction. Thus, our focus below will be magnetic field in the plane. The FS topology and associated surface states with a magnetic field 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 direction magnetic field , the doubly degenerate nodal ring in Fig. 1b splits into two rings in Fig. 2a shifted along axis on U–R–S–X plane. Although the mirror symmetry is broken by the magnetic field, chiral symmetry 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^{14}. Consider surface for instance, depending on the surface momentum k_{}, the winding number 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 surfaces are displayed in Fig. 2c,d, plotted as a function of k_{a} with and , respectively. There is no zero modes at , corresponding to trivial winding number. Meanwhile, zeroenergy flat bands highlighted by red colour in Fig. 2d exist inside the nodal rings. It confirms the nontrivial topology of the bulk nodal rings with quantized winding number shown in Fig. 2b. It turns out this chiral topological metal supports localized flat band protected by chiral symmetry on any surface, as long as its normal vector is not perpendicular to 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^{14}.
Weyl semimetal
Once we apply a magnetic field along [110] direction (or axis), both mirror (Π_{m}) and nglide (Π_{n}) symmetries are broken. Consequently, the nodal ring is replaced by a pair of 3D Dirac nodes, appearing at momenta 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]} along axis.
In the presence of both magnetic field h_{[110]} and layeralternating potential mν_{z}, the system still preserves chiral symmetry , bglide Π_{b} and inversion symmetry. Two pairs of Weyl nodes emerge at and along R→U→ R line in Fig. 3a. The lowenergy Hamiltonian around k_{1} has linear dispersion along all k directions
with δ k≡k–k_{1}. Various coefficients can be expressed in terms of the tightbinding 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 that the Weyl fermions at ±k_{1} and ±k_{2} have opposite topological charge +1 (blue) and −1 (red), respectively, as shown in Fig. 3a.
The surface states on surface at , between the two Weyl nodes with opposite chirality, are plotted along k_{c} 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 onewaydispersing zero modes for all surface momenta between k_{1} and k_{2} form a ‘chiral Fermi arc’^{13,28} on surface, as shown by the green lines in Fig. 3a.
Discussion
The existence of surface zero modes in our TCM phase originates from the chiral and mirrorreflection symmetry of AIrO_{3} with Pbnm structure. Any side surface other than [010] plane should exhibit robust zeroenergy surface states independent of the details. 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 firstprinciple calculation), and a holelike pocket FS occurs around Γ point in Fig. 1b^{19}. In other words, the nodal ring occurs around , while the small bulk Fermi pocket is located near k_{c}=0. Therefore, the zeroenergy surface modes are well separated from the bulk FS pockets in momentum, and a momentumresolved probe is required to detect the surface states. Angleresolved photoemission spectroscopy (ARPES) would be the best tool to observe the momentumresolved surface states shown in Fig. 1c,d on a side plane of AIrO_{3}. Notice that, ARPES has successfully detected the topological surface states in Dirac semimetal material^{29}. 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 nontrivial surface states detectable by ARPES, despite the ‘weak breaking’ of chiral symmetry.
These surface states also contribute finite 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 (finite SDOSs) in the dI/dV curve of scanning tunnelling microscopy. However, in real materials, it will be difficult 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 lines^{23} 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, 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. 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 AIrO_{3} such as SrIrO_{3} requires high pressure to achieve the Pbnm crystal structure^{30,31}, it is desirable to grow a film of AIrO_{3}. Recently, superlattices of atomically thin slices of SrIrO_{3} was made by pulsed laser deposition along [001] plane^{32}. This work is a first step towards possible topological phases in iridates. Furthermore, a successful growth of film along [111] plane was also reported^{33}. Thus, growing a film of AIrO3 along [110] (or ) is plausible. To confirm the proposed TCM, an ARPES study should be performed on a film of AIrO3 grown along [110] (or ), where the mirror symmetry of Pbnm structure is kept. This analysis should reveal a flat surface band near below E_{F}.
Methods
Symmetry operators and tightbinding Hamiltonian
The tightbinding Hamiltonian is defined in the basis of eight component spinor , organized as^{18}
where B, R, R, G correspond to four sublattices. We define Pauli matrices τ and ν in terms of following sublattice rotations
The full space group Pbnm is generated by (see Fig. 2 in ref. 18) translations T_{x,y,z} (three Bravais primitive vectors correspond to and ) and the following three generators {Π_{b}, Π_{n}, Π_{m}}
where Π_{b} (glide plane ) and Π_{n} (glide plane ) represent two glide symmetries, while Π_{m} is the mirror reflection and σ denotes the pseudospin subspace. I=Π_{b}Π_{n}Π_{m} represents the inversion symmetry.
The tightbinding Hamiltonian of SrIrO_{3} has the following form^{18}
Here various coefficient functions have been defined in ref. 18 with additional term
where t_{1D} is the interlayer next nearestneighbour hopping due to nonvanishing rotation and tilting in local oxygen octahedra. The crystal momentum (k_{a}, k_{b}, k_{c}) relates to (k_{x}, k_{y}, k_{z}) simply by
The basis we use here is different with the basis in ref. 18 by a unitary rotation U_{k}
In this basis , the tightbinding Hamiltonian is related to H_{k} by the unitary transformation U_{k} in equation 6:
Clearly in (or ) plane the nonvanishing terms in H_{k}, as in the basis , 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
that is, they all have chiral symmetry . However, in representation, the chiral symmetry written as
Ktheory classification procedure and topological invariants
To understand surface states on surface for instance, let us focus on a 1D system H_{k} in momentum space parametrized by fixed momentum and . Such a 1D system will have mirror reflection Π_{m}=iσ_{z}ν_{x} in (5), as well as chiral symmetry in (12). We will classify such a gapped 1D system (since the bulk gap only closes at two points in 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 symmetryallowed mass matrices for a Dirac Hamiltonian^{27,34}. The mathematical framework of Ktheory applies to both global symmetry and certain spatial (crystal) symmetry^{35,36}. In 1D such a Dirac Hamiltonian can be written as
with chiral symmetry
mirror reflection Π_{m}
and U(1) charge conservation Q
Any two symmetry generators among commute with each other. Mathematically, the classification problem corresponds to the following question: given Dirac matrix γ_{1} and symmetry matrices , what is the classifying space of mass matrix γ_{0}? In particular, since each disconnected piece in classifying space corresponds to one gapped 1D phase, how many disconnected pieces does contain? Hence, the classification of gapped 1D phases with these symmetries is given by the zeroth homotopy .
In the Ktheory classification, if there are generators that 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/c plane on a generic [xy0] side surface parallel to axis, the full symmetry group is generated by as mentioned earlier. Note that these three 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}:
where the generators inside the parenthesis anticommute with each other and commute with everything outside the parenthesis. The reason we have Cl_{2} × Cl_{2} is because we can blockdiagonalize the k_{c}=π tightbinding Hamiltonian with respect to 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 . The classification of gapped 1D phases with symmetries is hence given by
There are two integervalued topological invariants (W^{+}, W^{−}) that are 1D winding numbers^{26} obtained in blockdiagonalized subspace with Π_{m}=σ_{z}ν_{y}=±1.
Now let us start to break Π_{m} and symmetries. When we break mirror reflection Π_{m} (but keep ), the Clifford algebra extension problem becomes Cl_{2}→Cl_{3}
and hence the classification is . The topological invariant is total winding number W=W^{+}+W^{−}.
If we break but keep Π_{m}, the extension problem is again (Cl_{1})^{2}→(Cl_{2})^{2}
and classification is trivial since
If we break both 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 1.
Dislocation spectrum
One implication of weak index M is the existence of pairs of counterpropagating zero modes localized in a dislocation line respecting mirror symmetry Π_{m}. The number of zeromode pairs in each dislocation line is determined by its Burgers vector B: number of helical modes=B·M/2π. (ref. 23)
To check such zero modes due to dislocation line, we consider pairedge dislocations along axis, with a pair of Burgers vectors: 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 directions (a 3D torus) so that the system does not have an open surface. Consider every unit cell (with four sublattices, two pseudospin species per site) as a lattice ‘site’ in Fig. 4a, then the system has hopping terms between nearestneighbour ‘sites’ following the tightbinding Hamiltonian in equation (6). direction is still translational invariant and k_{z} remains a good quantum number, but in plane translation symmetry is broken by the dislocation. The dislocation spectrum is obtained when there are 39 sites along axis and 15 sites along axis. The location of dislocation with Burgers vector is at (4,6), which means 4th site along axis and 6th site along axis. And the position of dislocation with Burgers vector 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 twoband 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 equation (6). The followings are the coefficients for the effective Hamiltonian describes the Weyl fermion at k_{1} of equation (2) presented in the main text:
where are the coefficients in tightbinding 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. 4c.
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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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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 Office of BES, Materials Sciences Division of the U.S. DOE under contract No. DEAC0205CH11231 (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. PHYS1066293 where a part of this work was carried out.
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Chen, Y., Lu, YM. & Kee, HY. Topological crystalline metal in orthorhombic perovskite iridates. Nat Commun 6, 6593 (2015). https://doi.org/10.1038/ncomms7593
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DOI: https://doi.org/10.1038/ncomms7593
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