A A brief review of Wannier functions

-dimensional edge states of rotation symmetry protected topological states

Abstract

We study fourfold rotation invariant gapped topological systems with time-reversal symmetry in two and three dimensions (). We show that in both cases nontrivial topology is manifested by the presence of the -dimensional edge states, existing at a point in 2D or along a line in 3D. For fermion systems without interaction, the bulk topological invariants are given in terms of the Wannier centers of filled bands, and can be readily calculated using a Fu-Kane-like formula when inversion symmetry is also present. The theory is extended to strongly interacting systems through explicit construction of microscopic models having robust -dimensional edge states.

Introduction. A symmetry protected topological state (SPT) is a gapped quantum state that cannot be continuously deformed into a product state of local orbitals without symmetry breaking Gu and Wen (2009); Chen et al. (2012); Lu and Vishwanath (2012). SPT is known to have gapless boundary states in one lower dimension Chen et al. (2011), i. e., the -dimensional edge, such as the spin-1/2 excitations at the end of a Haldane chain Haldane (1983) or the Dirac surface states at the surface of a topological insulator Hasan and Kane (2010); Qi and Zhang (2011). The gapless states are protected by the symmetries on the -dimensional edge, and when the symmetry is a spatial symmetry, they only appear on the boundary that is invariant under the symmetry operation Fu (2011); Turner et al. (2010); Hughes et al. (2011); Hsieh et al. (2012).

Very recently, the possibility of having gapped -dimensional edge but gapless -dimensional edge has been discussed Benalcazar et al. (); Schindler et al. (2017a); Slager et al. (2015); Peng et al. (2017). In Ref. Benalcazar et al. (), it was shown that in a 2D spinless single particle (i.e., no spin-orbit coupling) system that has anti-commuting mirror planes, all four side edges can be gapped without symmetry breaking on an open square, but there are four modes localized at the four corners (0D edge) protected by mirror symmetries. Here we first extend the theory of 0D edge states to spin-1/2 fermion systems without mirror symmetries but with fourfold rotation symmetry and time-reversal symmetry. We point out that the presence of 0D-edge states can be understood as the result of a mismatch between the locations of the centers of the Wannier states and those of atoms. Then we generalize the theory to 3D, and define a new topological invariant by classifying the ‘spectral flow’ of the Wannier centers between the - and the -slices in the Brillouin zone. When this invariant is nontrivial, there are four helical edge modes on the otherwise gapped side surfaces of the 3D system. We further show that when space inversion is also present, there is a Fu-Kane-like formula Fu et al. (2007) relating this invariant to certain combinations of rotation and inversion eigenvalues of the filled bands at high-symmetry crystal momenta. Finally, we generalize the theory to strongly interacting systems, by constructing microscopic models of boson and fermion SPT states that have -dimensional edge states for using coupled wires construction. We remark that these edge states, protected by and some local symmetry such as time-reversal, are not pinned to the corners or hinges of the system, and can even appear in geometries having smooth side surfaces.

Mismatch between the atom sites and the Wannier centers. Wannier functions for the filled bands can be constructed for all 2D gapped insulators that have zero Chern number Marzari and Vanderbilt (1997). When symmetries are involved (time-reversal and/or spatial), the set of Wannier functions may or may not form a representation of the symmetry group Soluyanov and Vanderbilt (2011). If they do, then we call these Wannier functions ‘symmetric’. If a set of symmetric Wannier functions cannot be found for all filled bands, we know that the system cannot be adiabatically deformed into an atomic insulator: this is considered a generalized definition of topologically nontrivial insulators Po et al. (2017); Bradlyn et al. (2017), since atomic orbitals automatically form a set of symmetric wavefunctions. Atomic insulators are usually considered trivial. Nevertheless, we realize that even they can also be somewhat nontrivial if there is a mismatch between the Wannier centers and the atomic positions, as shown in the left panel of Fig. 1(a). A Wannier center (WC) can be understood as the middle of the Wannier function (but see Ref. Sup () for a rigorous definition), and if the Wannier functions are symmetric, their centers are also symmetric. When the mismatch happens, it means that while the insulator can be deformed into some atomic insulator, it would not be made by the atoms forming the lattice. The presence of 0D edge states of the system put on an open disk is the manifestation of the ‘mismatch’.

To be specific, let us consider a square lattice model

(1)

in which all the atomic orbitals are put on the lattice sites. Here , () are Pauli matrices representing the orbital degrees of freedom, and () representing the spin. This model can be thought as two copies of 2D topological insulator plus a mixing term with as coefficient; and it has time-reversal symmetry and a rotation symmetry . The system put on a torus is fully gapped because the four terms in Eq. (1) anti-commute with each other and their coefficients do not vanish at the same time.

As shown in Ref. Sup (), whatever value takes, the insulator is equivalent to an atomic one, and its WCs are located at the plaquette centers. We have explicitly constructed a set of symmetric Wannier functions and prove that, protected by the time-reversal and symmetries, the Wannier centers stay invariant under any gauge transformation that keeps the Wannier functions symmetric. This model hence realizes the mismatch between the WCs at plaquette centers and the atomic positions at sites.

Now we cut along the dotted lines in the left panel of Fig. 1(a) and turn the 2D torus into an open square. Since this cut preserves symmetries, the states centered at the plaquette center will be equally divided into the four quarters, so that each quarter carries one extra electron on top of some even integer filling. Due to , this means that a pair (Kramers’ pair) of zero modes are located near each of the four corners of the square. One may observe that in the absence of particle-hole symmetry (which is an accidental symmetry of the model), the modes can be moved away from zero and pushed into the bulk states, but we argue that even when this happens, the corners are still nontrivial in the following sense. The total eight modes (two near each corner) come from both the conduction and the valence bands, each having and electrons respectively, where and are the total number of bands and the filling number respectively, and the length of the square (Fig. 1(b)). No matter where the Fermi energy is, a gapped ground state must have electrons on an even-by-even lattice, so that each corner has exactly one (or minus one) extra electron on top of the filling of the bulk. This is in sharp contrast with the systems having trivial corner states, whose energy levels are plotted in Fig. 1(c). In that case, the in-gap states can be pushed into the conduction bulk and there is no extra charge at each corner. In Fig. 1(d)-(e), we plot the charge density at in real space, and plot the extra electric charge within a small area near the corner as a function of radius in the Slater-product many-body ground state.

To see how the odd parity of the corner charge is protected by , we contrast the above scenario with the one having a nematic perturbation breaking down to , so that the Wannier centers are shifted to the positions shown in the right panel of Fig.1(a). When the system is cut along the dotted lines, quarter has inside it an integer number of Kramers’ pairs, and the degeneracy at each corner is absent.

Figure 1: Nontrivial 0D edge modes of 2D fermion. In (a) we sketch the mismatch between the atom sites and the WCs in the presence (left panel) and the absence (right panel) of , where the atom sites are represented by the block circles and the WCs are represented by the colored orbitals. In (b) and (c), level counting for systems with nontrivial and with trivial 0D edge state are shown, respectively. In (d) the numerical calculated density profile of the 2D model with a finite size of is plotted, where the Fermi level is set at . The four bright regions in (d) show the additional charges located at the corners. To count the number of additional charges around a corner, we plot the integral of the density deviation from the filling () in (e).

1D helical state and Wannier center flow. A natural generalization of the 0D state in 2D is the 1D edge state in 3D, where both the 3D bulk and 2D side surfaces are insulating, as shown in Fig. 2(c). Our construction of this state is also based on the WC picture. Assume the 3D system has and symmetries, so we can take a invariant tetragonal cell and transform the Hamiltonian along the -direction to momentum space. Each slice with fixed can be thought as a 2D system, wherein the -slices are time-reversal and invariant while the others are only invariant. Consider an insulator that has four filled bands, or four WCs for each -slice. Due to , the four WCs are related to each other by fourfold rotations; and due to , at or , two WCs that form a Kramers’ pair must coincide. Therefore, at and , there are only three possible configurations for the four WC: all four at , all four at and two at each Wyckoff positions. Wyckoff positions are points in a lattice that are invariant under a subgroup of the lattice space group. For a square lattice in a Wigner-Seitz unit cell, and are the center and the corner invariant under , are the middles of the edges invariant under , and 4d are generic points invariant under identity (the trivial subgroup). If the configurations at and at are different, the evolution of the WC between the two slices forms a ’-flow’, a robust topological structure revealing that the 3D insulator is not an atomic one. Out of several different combinations of the configurations at and , there are two topologically distinct -flows, where the four WCs flow from to and from to [solid yellow and dashed green lines in Fig. 2(a)], respectively. The latter -flow can be shown equivalent to a weak topological index (Ref. Sup ()), and we from now on focus on the first -flow from to . Whether this flow is present or not gives us a new -invariant, and its edge manifestation is the existence of 1D helical edge modes on the side surface of a bulk sample. (For more rigorous definition and classification of the WC flow for arbitrary number of filled bands, see Ref. Sup ().)

To see this bulk-edge correspondence, we cut the bulk along both and directions, keeping the periodic boundary condition along . From top-down perspective, a corner of the sample takes the shape of the dotted lines shown in Fig. 2(a). One can see that at the corner, the boundary cuts through exactly one (or three) line in the WC flow, corresponding to one helical mode along the hinge between the two open surfaces. To make the picture more concrete, we consider the following 3D model, which is a simple extension of the 2D model in Eq. (1).

(2)

The -slice is equivalent with the 2D model in Eq. (1), thus having four charges locating at the plaquette center. The -slice is, however, a 2D atomic insulator with four charges locating at the lattice site. The mismatch between the WCs at - and -slices means that the -flow exists. To confirm the -flow, we also choose a smooth gauge for all the slices from to and plot the WC flow explicitly, which indeed gives the -flow, shown in Ref. Sup (). The 1D helical state is also confirmed by a numerical calculation of the band structure of a finite tetragonal cylinder, as plotted in Fig. 2(b). For this particular model, the helical edge states can be viewed from another perspective. The edge between the two open surfaces can be considered as the domain wall between them. On each surface there is a mass gap, and the rotation symmetry in this model enforces the two masses to be opposite, so that at the domain wall there is a helical mode Hsieh et al. (2012) [see Fig. 2(c) for a schematic and see Ref. Sup () for more details].

Figure 2: Nontrivial 1D helical modes of 3D insulator. In (a) we plot the two generators of nontrivial -flows from the -slice to the -slice, where the lattice site (), the plaquette center (), and the edge midpoint () are represented by black planchet, hollow circle, and grey planchet, respectively. In (b) the numerically calculated helical modes of our 3D model on a tetragonal cylinder geometry is plotted. The length along and directions is 50. In (c) we sketch the domain wall between surfaces of opposite masses, enforced by the rotation symmetry.

Symmetry indicators for the -invariant. To see if a given insulator has 1D helical edge modes on the side surface, one needs to calculate the evolution of the WCs as a function of , which in turn requires finding symmetric, smooth and periodic Bloch wave functions for all bands at each -slice as is done for our model Hamiltonian. This is practically impossible in real materials. Now we show that in the presence of additional inversion symmetry, this -invariant can be determined by the rotation and inversion eigenvalues at all high-symmetry momenta, simplifying the diagnosis. We call this method a “Fu-Kane-like formula”, likening it to the Fu-Kane formula for time-reversal topological insulators Fu et al. (2007), where inversion is not required to protect the nontrivial topology, but when present greatly simplifies the calculation.

This formula is derived based on the new theory of symmetry indicators Po et al. (2017); Bradlyn et al. (2017): given any insulator, a full set of eigenvalues of the space group symmetry operators for filled bands at all high-symmetry points generates a series of indicators. They tell us if this set is consistent with any atomic insulator, and if yes, the theory further gives where the atomic orbitals are located. Our goal is to find such an indicator that is equivalent to the -invariant for the WC flow. Following the WC flow picture, we require: (i) at and , the eigenvalues of , and are consistent with atomic insulators; (ii) there is no surface state on the side surfaces; and (iii) comparing the two slices at and , the numbers of atomic orbitals at and at change by and , respectively. For a concrete example, let us consider space group , whose indicators form a group Po et al. (2017), so that insulator according to its and eigenvalues can be denoted by (, , ), and an insulator with a nonzero indicator cannot be adiabatically deformed into an atomic insulator. Using the three criteria above, we find that the -flow is nontrivial only if . We have found the explicit formulas to calculate these indicators directly from the symmetry eigenvalues, which can be applied to all space groups having both C4 and P. (See Ref. Sup () for the results, and find a MATLAB script therein for automated diagnosis for materials in these space groups.)

Extension to strongly interacting SPT. In the above we have established the theory of -dimensional edge modes for free fermions through the WC picture. Since WC is a single particle object, the same picture does not apply for strongly interacting bosons or fermions. Here we rebuild a 3D free fermion model with robust 1D helical edge modes using coupled wires construction Teo and Kane (2014); Oreg et al. (2014); Neupert et al. (2014); Klinovaja and Tserkovnyak (2014); Klinovaja et al. (2015), a method that can be easily extended to strongly interacting SPT. These SPT can either be bosonic Thorngren and Else (2016) or fermionic, and are in general protected by spatial symmetry Huang et al. (2017) plus some internal symmetry Wang and Cheng (2017).

Figure 3: Coupled wires construction for a 3D SPT with robust 1D edge modes. Each filled circle is a wire from topdown, and each open circle including four elementary wires is a ‘physical’ wire that can be realized in 1D lattice models. The breaking of mirror symmetry in (a) causes the edge modes to move to the positions in (b).

Consider an arrangement of 1D wires shown in topdown view in Fig. 3(a), each of which represents a helical mode. Due to the fermion doubling theorem, each wire alone cannot be physically realized in 1D, but an even number of these wires can be realized as a 1D wire fine tuned to a critical point. In our model, four wires make a physical, critical 1D wire. For concreteness we assume that under -rotation the four wires inside cyclically permute. Then we couple the wires in the following way: the four wires, in topdown view, which share a plaquette are coupled diagonally, i.e., 1 coupled to 3 and 2 to 4. For a 3D torus these couplings (solid red lines) make the coupled wire system an insulator. For a cylinder geometry open in and directions, however, there are ‘dangling helical wires’ on the side surfaces, which can again be gapped by turning on a dimerizing coupling (dotted lines). But one soon discovers that, as long as is preserved, there are always four unpaired wires on the side surface (represented by green dots), which are in fact the same 1D helical edge mode protected by and studied above.

This construction can be easily extended to strongly interacting SPT. One simply replaces each helical wire with a -dimensional edge of a -dimensional SPT protected by some local symmetry. For example, each ‘wire’ can be a 0D spin-1/2, which is the edge of a 1D Haldane chain protected by SO(3) symmetry. In that case, the resultant construction in Fig. 3(a) is nothing but an AKLT-like state Affleck et al. (1987, 1988) formed by spins, but unlike previously considered AKLT states in 2D, it has gapped 1D edge but four 0D gapless spin-1/2 excitations localized at the four corners in an open square. We can also replace each wire by the edge of a Levin-Gu state Levin and Gu (2012), protected by a local symmetry, then the construction in Fig. 3(a) is a 3D bosonic SPT with 1D gapless modes at four corners. Notice that in these boson examples, time-reversal symmetry is not necessary. Similar construction can be used to obtain SPT states protected by both the local symmetries (being , SO(3) or ) and -rotation symmetry.

Discussion. It is important to note that, while in examples studied so far, the -dimensional edge modes sit at the corners or hinges in the disk or cylinder geometry, it is not always the case. In the model shown in Fig. 3(a), the edge modes are pinned to the corners by the mirror symmetries (dotted lines), and breaking these mirror planes in the bulk or on the surface causes the edge modes to move away. In the example shown in Fig. 3(b), we break the mirror symmetry of the construction on the surface, so that the dangling wires move from the corners to some generic points on the side. As long as is present, the -dimensional edge modes are stable, yet not pinned to corners or hinges in the absence of mirror symmetries. In fact, they still appear even if the whole side surface is smooth without hinges at all. We also emphasize that while these edge modes are protected by -rotation symmetry, breaking the symmetry perturbatively in the bulk or on the boundary does not in general gap out the modes, because time-reversal alone is sufficient to protect 1D helical edge modes. The only way of gapping the modes is to annihilate them in pairs, and this means large -breaking either in the bulk or on the boundary. Similar discussions may be extended to systems with twofold, threefold, sixfold rotations.

Experimentally, the four helical edge modes of a 3D electronic insulator contribute a quantized conductance of that may be measured in electric transport Qi and Zhang (2011). Also the -dimensional edge modes may be detected by local probes such as scanning tunneling microscopy, either on a bulk sample or at the step edge of a thin film.

Acknowledgement. C. F. thanks Xi Dai, Meng Cheng, Yang Qi and B. Andrei Bernevig for helpful discussion. The work was supported by the National Key Research and Development Program of China under grant No. 2016YFA0302400, and by NSFC under grant No. 11674370.

Note added. We are aware of works on related topics that have appeared on arXiv after our posting Benalcazar et al. (2017); Schindler et al. (2017b); Langbehn et al. (2017); their results have finite overlap with ours and seem consistent.

Supplemental materials for “-dimensional edge states of rotation symmetry protected topological states”


Appendix A A brief review of Wannier functions

Wannier functions (WFs) are defined as the Fourier transformations of Bloch wave functions, with a gauge freedom

(S1)
(S2)

Here is the -th Bloch state at , is a lattice vector in real space, is the number of cells, is the wannier index, and can be an arbitray unitary matrix. Generally speaking, as long as takes a smooth gauge in the whole Brillouin zone, the corresponding WF will be well localized around the lattice .

The WC (Wannier center) is defined as the position expectation on the WF. It can be calculated from the Berry connection in momentum space

(S3)

where the Berry connection is defined as

(S4)

and is the periodic part of the Bloch wavefunction . In general, WCs are gauge dependent, i.e. they depend on the choice of the gauge . An exception is the 1D insulating system where the 1D WCs can be thought as the spectrum of the Wilson loop along the 1D Brillouin zone and thus are gauge invariant quantities. Such a property leads to the modern theory of polarization and has greatly facilitated the studies of topological insulators because the spectrum of wilson loop is believed to be isomorphic with the surface dispersion.

To define the symmetric WFs, let us firstly review the concept of Wyckoff positions. General positions in the unit cell can be classified into a few types of Wyckoff positions by the their site symmetry groups (SSGs). The SSG for a given site can be defined as the collection of all the space group elements that leave invariant (module a lattice vector), i.e.

(S5)

Here with the point group operation of and the translational operation of . The equivalent positions of can be generated from a complete set of the representatives of the quotient group

(S6)

where is the identity, and is the lattice containing such that locates inp the home cell. Hereafter, for convenience some times we will split the Wannier index into a site index and an orbital index , indicating that the WF is the -th orbital locating at the site . A set of WFs is called symmetric if (i) they form representations (reps) of the SSGs

(S7)

, (ii) WFs at a general equivalent position of can be generated from the WFs at by a symmetry operation relating the two positions, and (iii) for time-reversal invariant systems, we further ask the WFs to form Kramers’ pairs, i.e.

(S8)

where is an anti-symmetric unitary matrix. For convenience in the following we take its standard form as

(S9)

The transformations of under symmetry operations are completely determined by the symmetry property of WFs. For the space groups considered in our work, we have

(S10)
(S11)

, where if module a lattice, and

(S12)

The transformation matrices and will be referred as the sewing matrices in the following.

Appendix B Gauge invariant 2D Wannier centers

The discussion about the mismatch between atom sites and WCs in the text presumes the gauge invariance of the occupied 2D WCs. Otherwise, we can choose a gauge where the WCs move away from the plaquette center to negative all the arguments. Similarly, the gauge invariance of the -flow discussed in the text also needs the 2D WCs at -slices to be gauge invariant. Here, we will set such a cornerstone by proving that in some special cases the 2D WCs indeed are gauge invariant guaranteed by the crystalline symmetry.

Since all the symmetry properties of symmetric WFs are encoded in the sewing matrices, in the proof below we follow the logic that two sets of symmetric WFs can be deformed to each other only if the sewing matrices generated from them can be transformed to each other by a smooth gauge transformation. A relevant useful concept is the band representation (BR), i.e. the set of irreducible reps (irreps) at high-symmetry momenta, which is indeed the diagonal blocks of sewing matrices in momentum space. In some cases, the information in BR is enough to demonstrate two sets of WFs are inequivalent. Therefore, before the proof, let us figure out what BR can tell us. The smallest 2D space group containing is . As shown in table SI, it has four types of wyckoff positions in real space, wherein the and positions are the site and plaquette center, respectively. To describe the BR we only need to count the irreps at and , because the irreps at and general momentum are always same. After a few derivations according to Eq. (S11), we find the mappings from symmetric WFs to BRs as

(S13)
(S14)
(S15)
(S16)
(S17)
(S18)

Here we use the symbol of an irrep decorated with a Wyckoff position to represent the WFs forming this irrep at this Wyckoff position, and use the symbol of an irrep decorated with a momentum to represent the bands forming this irrep at this momentum. We follow the notations for irreps in Ref. [Altmann and Herzig, 1994]. In the following, we will make use of these mappings.

SSG W K Coordinates irreps at W irreps at K
Table SI: The Wyckoff positions and high-symmetry momenta in 2D space group (with time-reversal symmetry). Due to the time-reversal symmetry, all the irreps at Wyckoff positions and time-reversal invariant momenta are double degenerate.

b.1 Gauge invariant Wannier centers at (site)

As will be proved latter, a set of WFs at can be moved away by a symmetric gauge transformation only if they form a rep consist of an even number of combined rep . Therefore, the conclusion is that, for a given set of WFs at , taking off all the even number of the rep , the left WFs consisted of

(S19)

, where , and one of equals to zero, will stay still under any symmetry allowed gauge transformation. We will prove this statement in three steps.

Firstly, we will show that the eight WFs forming the rep can be moved to four Kramers’ pairs at positions without breaking any symmetry. Apply an unitary transform for the bases in

(S20)
(S21)
(S22)

, where the subscript is used to distinguish the two same irreps, we find that the time-reversal rep has the standard form , while the rep is identical with the WFs

(S23)

Therefore, the WFs in can be moved to without breaking any symmetry. We denote this equivalent relation as . Readers may find that such a equivalence is consistent with the BR mappings in Eq. (S13)-(S18).

Secondly, it is obvious that the left or WFs alone can not be gauged away because the BR generated from or can not be reproduced by any combination of WFs at other sites.

Thus, to complete the proof, we only need to prove a single rep must stay at under any symmetric gauge transformation. From the BR mappings, we find that can only be moved to or . Such transformations are very unnatural from an intuitive perspective, because if we continuously move the four WFs at to or , the intermediate process will break either time-reversal or . (Enforced by the symmetry, the four WFs must move to four different directions, leading to separation of Kramers’ pairs). To prove this statement, here we show that the gauge transformation from to or must be singular. Let us first consider the transformation from to . For convenience, here we choose the WF bases with a “cyclical” gauge

(S24)
(S25)

where is the rotation operation centered at and for the and WFs, respectively. In this guage, for both and positions, and form a Kramers’ pair and the time-reversal rep matrix has the standard form . According to Eq. (S11) the sewing matrices generated from at and positions can be derived respectively as

(S26)
(S27)

Now assume there is a well defined gauge transformation that relates these two sets of WFs, then from Eq. (S10) and (S12) such a gauge transform must satisfy

(S28)
(S29)

leading to the equation . Thus we can write as an orthogonal matrix multiplied by a phase factor

(S30)

Substitute this back to Eq. (S28) and (S29), we get the constraints on as (i) and (ii) , wherein the first constraint is implied by the second one. By requiring to be periodic, we get another two constraints as (iii) and (iv) . In fact, such constraints make must be singular at some momenta. To see this, express as

(S31)

where is the determinant, and is a 4 by 4 imaginary Hermition matrix parameterized as

(S32)

Here we take the convention that , are real and positive, and , are unit vectors. It should be noticed that as the first three matrices and the last three matrices respectively form a set of generators, and these two sets are commutative with each other, such a parameterization realizes an isomorphic mapping from to . Therefore, the orthogonal matrix can be expressed as a product of two commutative matrices

(S33)

where

(S34)
(S35)

The constraints (i) and (ii) lead to

(S36)
(S37)
(S38)

And the anti-periodic property in constraints (iii) and (iv) must be realized by either or . In fact, whatever which is chosen to be anti-periodic, the anti-periodic condition together with the symmetry constraints (Eq. (S36)-(S38)) will make singular at some momenta. Here we only take as an example. With the anti-periodic condition, only two branches of solutions exist, the first is

(S39)
(S40)

and the second is

(S41)
(S42)

Obviously, the first breaks the symmetry constraint (Eq. (S36)) because . While, the second solution is singular at , since there must be due to the constraint (Eq. (S36)) and the anti-periodic constraint (Eq. (S42)). Therefore, we achieve the conclusion that there is no smooth gauge transformation can deform to . The proof for the inequivalence between and completely parallelizes the above process.

b.2 Gauge invariant Wannier centers at other positions

As and positions can be renamed to each other by re-choosing the origin point, there is no physical difference between them and all the statements about should also hold for . Therefore a set of WFs at can be moved away by a symmetric gauge transformation only if the WFs consist of an even number of rep . And, taking off all the even the number of the rep , the centers of the left WFs are gauge invariant.

As for the position, just like the WFs in section B.1, a pair of can reduce to four Kramers’ pairs at the position, i.e. . Thus, if there are an even number of at the position all of them can be gauged away to positions, however, if there is an odd number of , at least two WFs (a Kramers’ pair) will stay at under any symmetric gauge transformation.

In summary, a set of WFs at any wyckoff position can be moved by symmetric gauge transformations if and only if the rep (of the SSG at the Wyckoff position) they form is consistent with a set of WFs. In other words, all the move of WFs should pass through positions, which is very consistent with the intuitive picture.

b.3 Occupied Wannier functions for the 2D model

The occupied bands of our 2D model give the BR , which is irrelevant with the parameter since the corresponding term vanish at and . However, such a BR can not give a concrete real space information because it can be generated from , or , or . Here, by constructing the WFs explicitly, we will show that the occupied states are equivalent to WFs. We follow the projection procedure described in Ref. [Soluyanov and Vanderbilt, 2011]. Firstly, let us guess four trial local orbitals, denoted by , at the home cell and define them by the model orbitals

(S43)

Here is the -th atomic orbital in the lattice in our 2D model, (8 by 4) is the overlap matrices between atomic orbitals and the trial orbitals, and the summation is taken only for a few lattices around the home cell. Assume form the rep and limit the the summation over within , , , and , the symmetry properties satisfied by imply a set of constraints on

(S44)
(S45)

Here is the operator on the atomic orbitals, is the rep matrix of , is the time-reversal operator on the atomic orbitals, and is the time-reversal rep on . Aligning the gauge of occupied Bloch states with respect to the trial orbitals, we can define a set of projected Bloch-like states

(S46)

and the overlap matrix between them

(S47)

Then by the Lwdin orthonormalization procudure, a set of orthonormal Bloch-like states are obtained

(S48)

The WFs transformed from these Bloch-like states will be well localized and satisfy the rep as long as the overlap matrix is non-singular over the whole Brillouin zone.

In practice, we generate a random matrix and symmetrize it due to Eq. (S44)-(S45). A non-singular has been successfully obtained. The WCs are also confirmed by the wilson loop method and are found indeed to locate at the position. In the wilson loop method, the center of the -th WF is calculated by

(S49)
(S50)
(S51)
(S52)

where , and is the periodic part of the orthonormal Bloch-like state .

It should be noticed that, whatever the value of takes, the occupied WCs should locate at . This is simply because the term can not close the band gap and the four WFs can not be moved away from by any adiabatic process, as proved before.

Appendix C Wannier center flow in 3D system

c.1 Classification of the flows

For a 3D system with both time-reversal and symmetries, we can choose a tetragonal cell with its principal axis along the direction and apply a fourier transformation along . Here we focus on the case where all the -slices are equivalent to some 2D atomic insulator. The above conclusions about the gauge invariant 2D WCs applies for the -slices because both the time-reversal and symmetries present there. However, in general intermediate -slices, the above conclusions fail because of the absence of time-reversal symmetry. Then an immediate observation follows is that a bulk state must be nontrivial if there is a mismatch between its 2D WCs in - and -slices. We argue that the bulk topology is only determined by the WFs at the -slices, because, as the sewing matrices in the intermediate slices are completely determined by the two ends (from the compatibility relation) and all the 2D atomic insulator with same sewing matrices are topologically equivalent, all the possible evolutions must be equivalent with each other. Therefore, for a given set of 2D WFs at the - and -slices, a nontrivial flow exists if (i) the WCs at - and -slices are inequivalent with each other and (ii) the WCs at the two ends can be continuously deformed to each other by a time-reversal breaking process. Considering that the gauge invariant WCs at -slices can only locate at , and positions and the flows between and can be generated form the flows between and and the flows between and , to figure out the flow classification we only need to discuss the latter two cases.

Let us start with the flows between and . For both and in - or -slices, we only need to study the left immovable irreps

(S53)

as discussed in section B.1. Here and the one of equals to zero. Firstly, we will show that the () 2D WFs in the () irreps at or can not move along the flow. This can be seen by presuming an infinite small move and comparing the rep matrix before and after the move. Here we take irreps for an example. Before the move, the rep matrix is a direct sum of rep matrices, thus the trace gives . While, after the move, the WFs locating at positions must form a traceless , because the four equivalent positions transform to each other in turn under the rotation. Therefore the presumption of the infinite small move is untenable. Secondly, notice that a single rep can be separated into four WFs at positions by a time-reversal breaking process, which can be achieved in two steps. In the first step, we take the “cyclical” gauge defined in Eq. (S24)-(S25), where transforms the WFs to each other in turn and time-reversal transforms to . In the second step, we split and in the direction and split the and in the direction. Thus, through the positions, there can be a flow or , where the arrow represents a flow from to . Such flows are gauge invariant because both and at the two ends can not be gauged away. However, double of the flows must be trivial because the or the can be gauged away, as discussed in section B.1. It should also be noticed that the flow is equal to the flow module a trivial flow . Therefore, in summary, we obtain a class of the flow between and , wherein the nontrivial element manifests itself by 1D helical modes, as discussed in the text.

The discussion of the flow between and is much more simple. As discussed above, at position, a nontrivial flow must start or end with irreps. While, at the side