Wannier Obstruction

To understand the reason that leads to the Wannier obstruction, we first study the Chern insulator which is a paradigmatic example. It is well known that the Bloch bands with nontrivial Chern number cannot have smooth and periodic gauge in the Brillouin zone (BZ) [93, 94]. It can be understood by examining the Stokes theorem:

1 2π

I

Tr A· dk = 1 2π

Z

BZ

d²k Tr F_xy =C, (2.24)

where the loop integral integrates around the BZ torus, and C is the Chern number. If the Berry connection is periodic and smooth in the BZ, the integrals along k_x = 0, 2π and ky = 0, 2π should be cancelled out, respectively. The nonvanishing values of the integrals thus imply that there exist singularities of the Berry connection inside the BZ. If we do the inverse Fourier transform to obtain the Wannier functions (Eq. 2.16), the singularities will lead to the power­law decay of the Wannier functions. The reason is exactly the same as the correlation function: the gapless ground state has the power­law decay behavior rather than the exponential decay due to the singularities (gapless points) [97].

It is proved that all the systems with trivial Chern class can be constructed with ex­

ponentially localized Wannier functions [96, 98]. This reflects the fact that the Chern in­

sulator can exist without any symmetry protection. By going into the symmetry­protected

topological insulators, we need to require that the Wannier functions are also symmetric respect to the original Hamiltonian. In other words, the gauge we choose to construct the Wannier functions must also respect the symmetry of the Hamiltonian. The first identified example with such kind of Wannier obstruction is theZ2 TI protected byT symmetry. If one requires that the Wannier functions must form Kramers’ pairs, then there is no way to construct the exponentially localized Wannier function due to the singularities [99].

The Wannier obstruction is then found to be extremely powerful with the considera­

tion of crystalline symmetries. The topological insulators, in the strict definition, can be defined as the systems with Wannier obstruction. However, one still needs a theory to quickly and systematically identify whether the occupied space encounters Wannier ob­

struction. There are two main theories which develop approximately in the same time:

topological quantum chemistry (TQC) [44–46] and symmetry indicators (SIs) [17, 47].

Although they use different approaches to identify the topological insulators, the central idea behind them is still the Wannier obstruction.

In terms of TQC, it considers the so­called elementary band representations (EBRs).

The band representations (BRs) are basically the Bloch states that are induced by the basis orbitals. The EBRs are defined as the BRs then cannot be written as a direct sum of other BRs. It is found that the EBRs are induced from the orbitals with irreducible representa­

tions at all of the maximal Wyckoff positions for most of the space groups [44, 45]. By using the graph­theoretical method which helps to construct all the compatibility relations in the band structures, one can identify the topological insulators which are not BRs. An important theorem of TQC is that if an EBR is disconnected in the band structures, one of the isolated group of bands must be topological since an EBR cannot be written as a sum of two BRs. For example, the topology of the Kane­Mele model can be easily obtained without any knowledge of the details of the Bloch states [44, 45]. Since the complete the­

ory contains a lot of mathematical language, we will not go into the full details of TQC.

Rather, we will focus on the theory of SIs which are more elegant and have a more phys­

ical picture. Nevertheless, compared to TQC, it has some limits and deficiencies as we will discuss in the final remarks.

The idea first comes from the fact that the Z2 index in 3D withT and I symmetry

can be simplified into a product formula [7]:

(−1)^ν =Y

i N /2Y

m=1

ξ_2m(Γ_i), (2.25)

where ν is the Z2 index, ξ_2m(Γ_i) are the parity eigenvalues of the occupied bands 2m (do not count for the time­reversal pairs) at time­reversal invariant momentum Γ_i. The index i counts for all the time­reversal invariant momenta. A similar formula also applies for theZ2 index in 2D withT and I symmetry. More interestingly, there also exists a similar formula for the Chern insulator with Cnsymmetry [100], although in this case the formula can only detect the Chern numberC modulo n. The motivation is to generalize this formula to all 230 space groups with or without spin­orbit coupling and time­reversal symmetry.

The theory of SIs uses the information of the little group irreducible representations (irreps) at all the high­symmetry momenta to detect the topology. The little groups are the nontrivial subgroups of the space group that leave the high­symmetry momenta invariant, up to a translation. The SI is defined by the quotient group,

X_BS≡ {BS}

{AI}. (2.26)

Here,{BS} stands for all the possible band structures that satisfy the compatibility con­

ditions at all the high­symmetry momenta. One can write the sets of{BS} as an Abelian groupZ^dBS. In contrast,{AI} represents all the possible sets of atomic insulators induced from different maximal Wyckoff positions, which is also an Abelian group Z^dAI. The maximal Wyckoff positions are the points in real space that are left invariant (up to a translation) under the nontrivial subgroups (which is called the site­symmetry groups) of the space group. Surprisingly, it is found that d_BS = d_AIand X_BSis always a finite group for 230 space groups [17]. From the definition, it is clear that the nontrivial indices of SIs indicate the Wannier obstruction of which the occupied bands cannot be written as a sum of atomic insulators.

Let’s illustrate the theory of SIs in a simple example in 2D with I symmetry. First, the high­symmetry momenta are Γ = (0, 0), X = (π, 0), Y = (0, π) and M = (π, π).

These high­symmetry momenta are invariant under the the C₂ rotation. In this case, the little group is the same as the point subgroup C₂of the space group P 2. There are only two irreps: plus (+) or minus (−). Now we can construct the set {BS}. Since at each high­

symmetry momentum, the irreps can be (+) or (−), there are total 8 degrees of freedom, which are non­negative integers. But remember that the total band number should be the same for all these four high­symmetry momenta, therefore we have finally 8 + 1− 4 = 5 independent degrees of freedom. Here, following the same procedure in K theory, we generalize the non­negative integers toZ with total band number N → ∞. It follows that {BS} = Z⁵ and d_BS= 5.

Now, we construct the set {AI}. Similar to the high­symmetry momenta, the four maximal Wyckoff positions are a = (0, 0), b = (1/2, 0), c = (0, 1/2) and d = (1/2, 1/2) in the unit of a lattice constant. These are the points that are left invariant under the site­

symmetry group C₂, up to a translation. For each Wyckoff position, we can induce the band representations from two irreps of C2: plus (+) or minus (−), which correspond to s or p orbitals, respectively. The irreps at the four high­symmetry momenta induced from the four Wyckoff positions can be obtained by doing the Fourier transform:

as: (+, +, +, +), b_s: (+,−, +, −), c_s: (+, +,−, −), d_s: (+,−, −, +),

(2.27)

where the order is (Γ, X, Y, M ). These correspond to the BRs induced from the s or­

bitals, and the irreps of the associated p orbitals are the opposite signs of them. Although there seems to be 8 degrees of freedom totally, one should note that there are not linearly­

independent by observing the sum:

w_s⊕ wp = (+−, +−, +−, +−) (2.28)

are all the same for w = a, b, c, d. It is evident that there are only 8 + 1− 4 = 5 degrees of freedom left over. Therefore,{AI} = Z⁵ and d_AI = 5. Note that this result confirms our previous discussions that dBS= dAIin general.

Now we are going to compute the SI for the space group P 2. We can simply observe that the band structures with irreps:

t₁ = (+, +, +,−), t2 = (+, +,−, +), t₃ = (+,−, +, +), t₄ = (−, +, +, +),

(2.29)

and all the opposite signs of them cannot be induced from the atomic orbitals. These band structures cannot be obtained by summing over the atomic insulators (2.27) with integer coefficients. It must involve coefficients of±1/2. Therefore, these band structures cor­

respond to the systems with Wannier obstruction. In this case, the topology is manifested through the band­inversion which exchanges the irreps at one of the high­symmetry mo­

mentum. Therefore, they correspond to the Chern insulators with Chern numberC = 1.

What happens if we arbitrarily choose two of the band structures t_i, t_j and add them together? It is clear that if i ̸= j, the sum ti ⊕ tj can be written as a sum of two atomic insulators. If i = j, for example, i = 1, the sum t₁ ⊕ t1 = b_s ⊕ cs ⊕ ds⊖ as. Notice that although the coefficients are all integers, there exists a coefficient with negative sign.

Therefore, the band structure t₁⊕ t1 is also topological and encounters Wannier obstruc­

tion. However, contrary to the previous case, by adding the atomic band a_s, the band structure t1⊕ t1⊕ as = b_s⊕ cs⊕ dsbecomes an atomic insulator. One can conclude that the system is actually fragile topological rather than stably topological.

From these discussions, it is clear that only when the coefficients involve ±1/2 do the system has a nontrivial symmetry indicator. Since adding two topological bands leads to the integer coefficients, we conclude that XBS ≡ {BS}/{AI} = Z2 for the P 2 space group. The symmetry indicator can be further written into the simple formula in Eq. (2.25), where ξ_2mshould be replaced with ξ_msince there is no time­reversal symmetry in general and we should multiply it over all the occupied bands. The index ν =C modulo 2 is called the parity of the Chern number. In addition, there might exist band structures that can be written as a sum that includes negative integers. These cases correspond to the fragile topological bands. By generalizing this discussion into all the 230 space groups, one can

construct the full symmetry indicators with or without significant spin­orbit coupling or time­reversal symmetry, see Tab. III, IV, X, XI in Ref. [17].

There are some remarks and caveats regarding the theory of SIs. First, the SIs only provide the sufficient condition for the Wannier obstruction, but not necessary. Therefore, not all the topological insulators with Wannier obstruction can be detected using SIs. For example, for systems without any crystalline symmetries, the SIs become useless. For the Chern insulators with C_n symmetry, the SIs can only detect the Chern number modulo n. The topology or the Wannier obstruction in these cases is then hidden in the Wilson­

loop spectrum which cannot be reflected by the irreps at the high­symmetry momenta.

Nevertheless, the theory of TQC provides some additional information on the topology as an EBR cannot be written as a sum of two BRs. Therefore, even if the irreps are identical to the trivial ones, one can still detect the topology if the disconnected group of bands together form an EBR [44, 45].

Second, the theory doesn’t assume that the systems are insulators. It is possible that the bands are connected at some points outside the high­symmetry momenta and the systems become semimetals. For example, in symmetry class AI, all the nontrivial SIs are incompatible with insulators but rather the symmetry­protected topological semimet­

als [101]. The TCIs in this class that used to be considered topological have been proved to be only fragile topological [14, 58, 62, 64, 65].

Third, the theory of SIs does not provide any information on the bulk­boundary cor­

respondence. Nevertheless, for systems in symmetry class AII, the relation between the anomalous gapless surface states (and also gapless hinge states) and SIs has been well studied [48, 102]. Contrary to the symmetry class AI, all the nontrivial SIs in class AII are compatible with the topological insulators.

Chapter 3 Trivialized Topological