Topological Crystalline Insulators
The recent discovery of topological insulators has revived interest in the band topology of insulators. In this work, we extend the topological classification of band structures to include certain crystal point group symmetry. We find a class of three-dimensional “topological crystalline insulators” which have metallic surface states with quadratic band degeneracy on high symmetry crystal surfaces. These topological crystalline insulators are the counterpart of topological insulators in materials without spin-orbit coupling. Their band structures are characterized by new topological invariants. We hope this work will enlarge the family of topological phases in band insulators and stimulate the search for them in real materials.
The topology of band structures is important in the study of topological phases of matter. Historically the quantization of Hall conductance in the integer quantum Hall effect was explained by the Thouless-Kohmoto-Nightingale-den Nijs integer (also known as Chern number) of occupied energy bandsthouless . Recently the topological classification of spin-orbital coupled band structures with time reversal symmetry has played a key role in the theoretical prediction of topological insulatorskanereview ; moorereview ; zhangreview . The subsequent development of topological band theoryfukane , combined with realistic band structure calculations, has proven useful in the material search for topological insulators.
Inspired by the discovery of topological insulators, the classification of band structures has been extended to other discrete symmetry classes such as particle-hole symmetryludwig ; kitaev ; ying , which leads to a rich family of topological phases such as topological superconductorszhangtsc ; ludwig . More recently, the classification of magnetic insulators with certain magnetic translation symmetry has been studiedzhangmagnetic ; mooremagnetic . Finding these phases in real materials is interesting and challenging.
The purpose of this work is to extend the classification of band structures in a different direction—to include crystal point group symmetries. We introduce the notion of “topological crystalline insulators”, which cannot be smoothly connected to a trivial atomic insulator when time reversal () symmetry and certain point group symmetry are respected. Unlike time reversal symmetry, crystal symmetries can be broken by sample surfaces. As a result, a low-symmetry surface of a topological crystalline insulator does not have robust surface states. This motivates us to study a class of three-dimensional topological crystalline insulators which have four-fold () or six-fold () rotational symmetry. As we will show, its (001) surface, which preserves the rotational symmetry, supports gapless surface states. These topological crystalline insulators are the counterpart of topological insulators in materials without spin-orbit coupling. Instead electron’s orbital degrees of freedom play a role similar to spin. Unlike the linearly dispersing Dirac surface states of topological insulators, the (001) surface states of topological crystalline insulators have quadratic band degeneracy protected by time reversal and discrete rotational symmetrywen ; sun .
The outline of this paper is as follows. First we introduce a simple tight-binding model in a tetragonal crystal with symmetry. We explicitly show that gapless surface states exist on the (001) crystal face. The topological stability of surface states suggests a nontrivial phase in this model. Next we define a new topological invariant for generic time reversal invariant band structures with or symmetry in three dimensions. This establishes the topological crystalline insulator phase. Finally we discuss the electronic properties of quadratic surface bands.
Tight-Binding Model: Consider a tetragonal lattice with a unit cell of two inequivalent atoms and along -axis, as shown in Fig.1a. The 3D crystal can be viewed as a stack of bilayer square lattices in the plane. We now introduce a tight-binding model to describe the band structure of electron’s orbitals (or orbitals, see the next paragraph). In particular, we are interested in the energy bands derived from and orbitals. We assume that these bands do not overlap with the bands, and construct a tight-binding model from Wannier functions with the same symmetry as and orbitals. The Hamiltonian consists of intra-layer hopping and , as well as inter-layer hopping :
Here each site is specified by : labels the bilayer unit cell along -axis; labels the -plane coordinate; labels the sublattice. label the orbital. Two types of -orbital hopping terms appear in . The intra-layer hopping in is of -bonding type: it depends on the relative orientation of -orbital and the hopping direction . The inter-layer hopping in is orbital-independent. Written in this form, the tight-binding Hamiltonian manifestly preserves crystal symmetries. The hopping amplitudes , and between two sites depend on their -plane distance . Throughout this work, we assume spin-orbit coupling is negligible, so that electron’s spin index is omitted.
We emphasize that the form of the Hamiltonian is entirely determined by crystal symmetry. Because orbitals transform in the same way as orbitals under , (1) also applies to materials with these -orbital. Therefore (1) is potentially revelant to a large class of materials including transition metal compounds with orbitals near Fermi energy.
To obtain a minimal model for topological crystalline insulators, we include the nearest and next-nearest neighbor intra-layer hoppings in with the amplitude and , as well as nearest and next-nearest neighbor inter-layer hoppings in with the amplitude and . The corresponding Bloch Hamiltonian is obtained by Fourier transform:
The band structure is shown in Fig.2a for the following set of parameters: . We have checked that the energy gap is finite everywhere in the Brillouin zone. As long as the energy gap does not close, the system remains in the same topological class within a finite parameter range.
To study surface states, we solve in a slab geometry. We find that the existence of gapless surface states crucially depend on the surface termination. Consider the high symmetry (001) surface, which preserves the symmetry. Here surface states exist and transverse the whole energy gap as shown in Fig.2, leading to a 2D surface metal. Note that surface states are doubly degenerate at : one in orbital and the other in orbital. Because is a fixed point under four-fold rotation, the doublet form a two-dimensional irreducible real representation of symmetry, as a result of time reversal symmetry for electrons without spin-orbit coupling (i.e., spinless fermions). Surface band dispersion near can be obtained from theory. We introduce a pseudospin to label the and orbital of the doublet. In this basis, rotation is represented by and time reversal operator is represented by complex conjugation. Up to a gauge transformation, the form of the Hamiltonian is dictated by symmetry:
Perturbations which break either or symmetry can open up an energy gap and destroy the protected surface states. This can be seen explicitly by adding a -breaking term or a -breaking term to . Similar quadratic bands have been recently studied in 2D photonic crystalswen and fermion modelssun . Our tight-binding model does not have gapless surface states for other surface terminations which break the symmetry.
When and symmetry are preserved, the surface states shown above are topologically protected. This can be understood by considering the surface band dispersion along . Within our model, surface states (if present) must be doubly degenerate at and . As a consequence, there exists two distinct types of surface band connectivity along , having an even or odd number of band crossings at Fermi energy respectively. Surface states with even crossings are fragile: they can be pushed out of the energy gap by changing the surface potential. Surface states with odd crossings have a Fermi surface which encloses an odd number of times. They cannot be removed. These two types of surface states imply the existence of two topologically distinct phases of time-reversal-invariant band insulators with four-fold symmetry. The reasoning here closely follows the previous study of topological insulatorskane .
Topological Invariant: We now show that a topological invariant characterizes the band structure of 3D time-reversal-invariant insulators with four-fold rotation symmetry (without spin-orbit coupling). corresponds to a trivial phase adiabatically connected to an atomic insulator. corresponds to a topological crystalline insulator which has gapless surface states on the surface.
We start by examining the symmetry property of Bloch wavefunctions of occupied bands: where is the cell-periodic eigenstates of the Bloch Hamiltonian . Here the unit cell is chosen to be invariant under rotation around the axis. satisfies
Here the unitary operator implements rotation within the unit cell ( is angular momentum operator). The anti-unitary operator (complex conjugation) implements time reversal transformation for spinless fermions, with the property . As a result, time reversal symmetry by itself does not guarantee a two-fold degeneracy. However, the combination of time reversal and four-fold rotational symmetry can lead to protected degeneracies at four special momenta in the 3D Brillouin zone (Fig.1b). At such a high symmetry point , commutes with , so that the energy bands are eigenstates of four-fold rotation with possible eigenvalues and . Moreover, because is real, energy bands at with eigenvalues are guaranteed to be degenerate, forming a two-dimensional irreducible real representation of . For example, such doublet bands can derive from orbitals or orbitals. From now on, we consider a set of energy bands which are doubly degenerate at and . As we will see below, only these doublet bands admit a classification.
The topological invariant is defined in terms of the electron wavefunctions:
is the Berry connection. is an antisymmetric U(2N) matrix because and . stands for the Pfaffian. We now specify the integration path in (6). For , the integral is along an arbitrary line connecting and which lies within the 2D plane in the Brillouin zone as shown in Fig.1b. Likewise, the integration path for lies in the plane .
Both and depend on the choice of basis , but we now prove that is gauge invariant. Two different basis and are related by a gauge transformation:
This leads to the corresponding gauge transformation of and :
Using the identity , it is straightforward to verify that is gauge invariant.
We now further show that is a quantity. It follows from (4) that at the two 2D planes and , has the symmetry property: This allows us to choose a real gauge along the integration path to evaluate (6):
Because is anti-unitary, everywhere along the integration path. So in this gauge reduces to
Because belong to the two-dimensional representation of group, we have , so that the gauge condition (9) at is equivalent to , i.e., the wavefunction is real. Under this reality condition, the matrix simplifies to
Now by choosing a particular kind of real basis , reduces to a standard form , which is a direct sum of Levi-Civita tensors. This means that in a generic real basis, can be written as
So we have
This proves that is a gauge invariant quantity. So both and are topological invariants which characterize the band structures in the 2D momentum space and respectively. Their product is defined for 3D time-reversal-invariant insulators with symmetry, and determines the surface state property: gapless surface states exist on (001) face only when . This is verified in the tight-binding model (1). The relation between and is analogous to that of strong and weak index in 3D topological insulatorsfkm ; moore ; roy .
Generalization to Crystals with Symmetry: The study of topological crystalline insulators with symmetry can be generalized to hexagonal crystal structures with symmetry. In that case, (001) surface states have quadratic degeneracy points either at or two equivalent . The topological invariant is similarly defined by Eq.(6) from the electron wavefunctions on the lines and in the Brillouin zone. A concrete tight-binding model can be obtained by placing (1) in a layered triangular lattice. The details will be reported elsewhere.
The fact that the above invariant is only defined for doublet bands illustrates the important interplay between symmetry representation and topology of energy bands in solids, a feature absent in other known classes of topological insulators. We are not aware of a systematic mathematical approach to classifying vector bundles (e.g., electron wavefunctions over the Brillouin zone) with given representations of group action (e.g., rotation) at fixed points (e.g. high symmetry momenta ). We hope the concrete topological invariant (6) can stimulate continuing research on topological crystalline insulators with other lattice structures.
The occupied bands of real materials usually have both doubly degenerate and nondegenerate singlet components. Provided that the doublet bands are energetically separable from the singlet ones (i.e., no band crossing between them), the invariant remains well-defined for the former. The coexistence with singlet bands can in principle weaken the stability of surface states (see supplementary material), although this senario seems unlikely to happen in reality.
Surface States with Quadratic Degeneracy: Quadratic degeneracies in 2D band structure have attracted considerable interest recently. The Hamiltonian (3) is widely used as a low energy approximation for the band structure of bilayer graphene near points. However, unlike topological crystalline insulators, the quadratic degeneracy there is not protected by symmetry. Instead, the band dispersion very close to becomes linear due to trigonal warping effects.
Electron interactions can drive 2D bands with quadratic degeneracy into a variety of interesting broken symmetry phases including quantum anomalous Hall state, nematic phase and etcsun . The competition between different ordered phases is of great interest in bilayer graphene. It remains to see what happens in the single-valley quadratic surface states of topological crystalline insulators.
It would be interesting to generalize the concept of topological crystalline insulators to interacting systems. For example, crystal symmetry is known to play an important role in the topological classification of spin liquidsspin , spin chainsAKLT , and Mott insulatorsyao . A unifying theory of symmetry protected topological phases remains to be developed. We also note that analogs of topological crystalline insulators can be potentially realized in photonic crystals.
Part of this work was done in early 2009 at University of Pennsylvania under the support of NSF grant DMR-0906175. We thank Charlie Kane and Jeffrey Teo for very helpful discussions, and acknowledge support from the Harvard Society of Fellows.
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I Supplementary Material
We study the stability of surface states with quadratic degeneracy when the doublet bands coexist with singlet band in the bulk. The latter may come from, e.g., orbitals so far neglected in our tight-binding model (1) on tetragonal lattice. Provided there is no band crossing between doublet and singlet bands, the invariant for the former remains well-defined. However, as we now demonstrate, the quadratic surface states of a nontrivial phase can in principle be damaged by hybridizing with additional singlet surface band.
We assume that the singlet band in the bulk gives rise to a branch of unprotected nondegenerate surface state with the usual dispersion . is the energy difference between the doublet and singlet surface state at . This extra surface state can hybridize with the original surface states with quadratic degeneracy. A hybridization term is symmetry allowed in the Hamiltonian:
Here is the Hamiltonian defined in Eq.(3). is the coupling strength between them responsible for hybridization. For and , singlet and doublet surface bands cross at . A finite coupling turns the band crossing into an avoided crossing and opens up an energy gap. The resulting surface states cease to be protected: they can be smoothly pushed into the bulk continuum. This shows that the intervention from singlet bands can in principle destroy the gapless surface state with quadratic degeneracy. However, the above scenario requires the coexistence of singlet surface state under particular conditions, which seems very nongeneric in real materials. Therefore we are optimistic that surface states with quadratic degeneracy on topological crystalline insulators are robust for practical purposes.