Twisted bulkboundary correspondence of fragile topology

Understanding fragile topology.Exploiting topological features in materials is being pursued as a route to build in robustness of particular properties. Stemming from crystalline symmetries, such topological protection renders the properties robust against defects and provides a platform of rich physics to be studied. Recent developments have revealed the existence of socalled fragile topological phases, where the means of classification due to symmetry is unclear. Z.D. Song et al. and Peri et al. present a combined theoretical and experimental approach to identify, classify, and measure the properties of fragile topological phases. By invoking twisted boundary conditions, they are able to describe the properties of fragile topological states and verify the expected experimental signature in an acoustic crystal. Understanding how fragile topology arises could be used to develop new materials with exotic properties.Science, this issue p. 794, p. 797.Abstract.A topological insulator reveals its nontrivial bulk through the presence of gapless edge states: This is called the bulkboundary correspondence. However, the recent discovery of “fragile” topological states with no gapless edges casts doubt on this concept. We propose a generalization of the bulkboundary correspondence: a transformation under which the gap between the fragile phase and other bands must close. We derive specific twisted boundary conditions (TBCs) that can detect all the twodimensional eigenvalue fragile phases. We develop the concept of realspace invariants, local good quantum numbers in real space, which fully characterize these phases and determine the number of gap closings under the TBCs. Realizations of the TBCs in metamaterials are proposed, thereby providing a route to their experimental verification.Topological insulators are materials that conduct no electricity in the bulk but that allow perfect passing of the current through their edges. This is the basic concept of the bulkboundary correspondence: A topological bulk is accompanied by a gapless edge. New theories (1–4) have developed systematic methods for searching topological materials (5–7). This led to the discovery of higherorder topological insulators (HOTIs) (8–11) and fragile topological states (12–16), the latter being predicted (17, 18) to exist in the newly discovered twisted bilayer graphene (19). The fragile phases generally do not exhibit gapless edges, thereby violating the bulkboundary correspondence.We show that fragile phases exhibit a new type of bulkboundary correspondence with gapless edges under “twisted” boundary conditions (TBCs). TBCs were introduced (20) to prove the quantization of Hall conductance. On a torus, a particle under U(1) TBCs gains a phase eiθx,y whenever it undergoes a period in the x/y direction. This phase was generalized to a complex number λ=reiθ(0 ≤ r ≤ 1) (21) for a trivial state with two pairs of helical edge states, with unclear results. We consider a slow deformation of the boundary condition controlled by a single parameter, λ. If the fragile phase, determined by eigenvalues, can be written as a difference of a trivial atomic insulator and an obstructed atomic insulator (with electron center away from atoms), the energy gap between the fragile bands and other bands must close as we tune λ on a particular path.We develop a realspace invariant (RSI) (22, 23), to classify eigenvalue fragile phases (EFPs) and their spectral flow under TBCs. RSIs are local quantum numbers protected by point group (PG) symmetries. With translation symmetry, they can be calculated from symmetry eigenvalues of the band structure. Under a specific evolution of the boundary condition, where the symmetry of some lattice (Wyckoff) position is preserved, the system goes through a gauge transformation, which does not commute with the symmetry operators. The symmetry eigenvalues and the RSIs at this Wyckoff position also go through a transformation: If the RSIs change, a gap closing happens during the process. We find that EFPs always have nonzero RSIs: Therefore, TBCs generally detect fragile topology. A realspace approach has also been useful for (non) interacting (24–26) strong crystalline topological states. We obtain the full classification of RSIs for all twodimensional (2D) PGs with and without spinorbit coupling (SOC) and/or timereversal symmetry (TRS), and we derive their momentum space formulae [table S5 (27)]. For each 2D PG, we introduce a set of TBCs to detect the RSIs (27). Criteria for stable and fragile phases are written in terms of RSIs [table S6 (27)] and exemplified on a spinless model.The symmetry property of bands is fully described by its decompositions into irreducible representations (irreps) of little groups at momenta in the first Brillouin zone (BZ). Topological quantum chemistry (1) and related theories (3, 4) provide a general framework to diagnose whether a band structure is topological from the irreps. If the irreps of a band structure are the same as those of a band representation (BR), which is a space group representation formed by decoupled symmetric atomic orbitals, representing atomic insulators, then the band structure is consistent with topologically trivial state; otherwise, the band structure must be topological. Generators of BRs are called elementary BRs (EBRs) (1). The EBRs in all space groups are available on the Bilbao Crystallographic Server (BCS) (1, 28). We will demonstrate this principle using a tightbinding model in the following.There are two distinct categories of topological band structures. If a topological band structure becomes a BR (in terms of irreps) after being coupled to a topologically trivial band, the band structure has at least a fragile topology. We refer to such a phase as an EFP. An EFP may also have a stable topology undiagnosed from symmetry eigenvalues (14, 29). If the band structure remains inconsistent with a BR after being coupled to any topologically trivial bands, the band structure has a stable topology. Further discussions about the classifications of topological and nontopological bands can be found in (27).We build a spinless model whose bands split into an EFP branch and an obstructed atomic insulator branch. Consider a C4 symmetric square lattice (wallpaper group p4 ). Its BZ has three maximal momenta Γ(0,0) , M (π,π) , and X (π,0) . The little group of Γ and M is PG 4, and the little group of X is PG 2, with irreps tabulated in Table 1. The irreps form coirreps when we impose TRS. We tabulate all the EBRs of p4 with TRS in Fig. 1. Here we consider the EFP 2Γ1+2M2+2X1 , a state of two bands where each band forms the irreps Γ1 , M2 , X1 at Γ,M,X . These bands are topological: They cannot decompose into a sum of EBRs. The EFP is (necessarily) a difference of EBRs as 2(A)b↑G⊕(1E2E)b↑G⊖(1E2E)a↑G .Table 1 Character tables of irreps of PGs 2 and 4. First column: BCS notations (28) of the PG irreps. Second column: notations of momentum space irreps at X, Γ, and M for wallpaper group p4 . The third column is the atomic orbitals forming the corresponding irreps. In the presence of TRS, the two irreps 1E (Γ3 , M3 ) and 2E (Γ4 , M4 ) of PG 4 form the coirrep 1E2E (Γ3Γ4 , M3M4 ).View this table: View popup.View inline. Download highres image . Open in new tab . Download Powerpoint . Fig. 1 EBRs of wallpaper group p 4 without SOC with TRS. [See BCS (1, 2, 28)]. The square represents the unit cell. a(0,0) , b (12,12) , c(0,12),(12,0) are maximal Wyckoff positions. The yellow, red and blue, and green and gray orbitals represent the s, px,y , and dx2−y2 orbitals, respectively. Consider a fourband model of two s (s1 and s2), one px, and one py orbitals at the b position (Fig. 2A). Per Table 1, s1,2 orbitals (irrep A) induce the BR 2(A)b↑G=2Γ1+2M2+2X2 ; px,y orbitals (irrep 1E2E ) induce the EBR (1E2E)b↑G=Γ3Γ4+M3M4+2X1 . Let the px,y orbitals have a higher energy than the s1,2 orbitals. We band invert at the X point such that the upper two bands’ irreps become Γ3Γ4+M3M4+2X2=(1E2E)a↑G (an EBR), and the lower two bands have the EFP irreps 2Γ1+2M2+2X1 . Because the upper band forms an EBR at the empty Wyckoff position, where no atom exists, it forms an obstructed atomic insulator band. The model, in the basis (px,py,s1,s2) , is (27) H(k)=τzσ0(E+2t1cos(kx+ky)+2t1cos(kx−ky))+τyσzt2sin(kx)+τyσxt2sin(ky) (1) E (−E ) is the onsite energy for the px,y (s1,2) orbitals, t1 band inverts at X, and t2 guarantees a full gap between the upper and lower two bands. We introduce ΔH(k) (27) to break two accidental symmetries, Mz (z→−z ) =τzσy , chiral τxσ0 . The band structure of H(k)+ΔH(k) is plotted in Fig. 2B. Download highres image . Open in new tab . Download Powerpoint . Fig. 2 Spectral flow of fragile phase under TBCs. (A) Fragile phase model (wallpaper group p4 with TRS). The yellow, green, and red and blue orbitals are the two s and the px,y orbitals. The gray parallelogram is the unit cell, and black lines are the hoppings. (B) Band structure of the fragile phase. (C) The C4 symmetric TBCs of a finite size system. Black dots are the atoms; bonds are hoppings; four yellow circles are corner states of the fragile state. Four shaded regions (μ= I, II, III, IV) transform to each other under C4 action. Hoppings from the μth part to the (μ+1) th part (red bonds), from the μth part to the (μ+2) th part (green bonds), and from the μth part to the (μ−1) th part (red bonds) are multiplied by a complex λ/real Re(λ2) /complex λ . (D) The C2 and TRS symmetric TBCs. The two shaded regions (μ= I, II) transform to each other under C2 rotation. The hoppings between I, II (red bonds) are multiplied by a real number λ. (E) Spectral flow under C4 symmetric TBC. (F) Spectral flow under C2 and TRS symmetric TBCs. We construct a finitesize (30 × 30) TRS Hamiltonian with C4 rotation symmetry preserved at the coordinate origin on the a site (Fig. 2C). The spectrum consists of 1798 occupied states, 4 degenerate partially occupied levels at the Fermi level, and 1798 empty levels; they form the representations 450A⊕450B⊕449(1E2E) , A⊕B⊕1E2E , and 449A⊕449B⊕450(1E2E) , respectively. The partially occupied states are corner states, or the “filling anomaly” of fragile topology (Fig. 2C). The gap between the four partially occupied levels and the occupied or empty levels is about 0.3/0.01 , as ΔH(k) breaks the accidental chiral symmetry. Every four states forming the irreps A⊕B⊕1E2E can be recombined as 1〉=(A〉+B〉+1E〉+2E〉)/2 , 2〉=(A〉−B〉−i1E〉+i2E〉)/2 , 3〉=(A〉+B〉−1E〉−2E〉)/2 , and 4〉=(A〉−B〉+i1E〉−i2E〉)/2 , which transform to each other under the C4 rotation and have Wannier centers away from the C4 center: We hence move the occupied and empty states, both of which form the representation 449A⊕449B⊕449(1E2E) , away from the C4 center. We are left with two occupied states, A⊕B , and two empty states, (1E2E) , at the C4 center. These four states form a level crossing under TBC evolution.We divide (Fig. 2C) the system into four parts (μ= I, II, III, IV), which transform into each other under C4 . We introduce the TBC by multiplying the hoppings between different parts by specific factors such that the twisted and original Hamiltonians are equivalent up to a gauge transformation. Specifically, the multiplication factors on hoppings from μth part to (μ+1) th part, from μth part to (μ+2) th part, from μth part to (μ−1) th part are i,−1,−i . The twisted and untwisted Hamiltonians H^(i),H^(1) satisfy〈μ,αH^(i)ν,β〉≡(i)ν−μ〈μ,αH^(1)ν,β〉 (2) μ,α〉 is the αth orbital in the μth part, and H(λ) is the Hamiltonian with multiplier λ. We introduce the twisted basis V^μ,α〉=(−i)μ−1μ,α〉 . The elements of H^(i) on the twisted basis equal those of H^(1) on the untwisted basis: 〈μ,αV^†H^(i)V^ν,β〉=〈μ,αH^(1)ν,β〉 . C4 transforms the μth part into the (μ+1) th part: The twisting phases of μ,α〉 and C^4μ,α〉 under V^ are (−i)(μ−1) and (−i)μ , implying V^C^4=−iC^4V^ . If ψ〉 is an eigenstate of H^(1) with C4 eigenvalue ξ, then V^ψ〉 will be an eigenstate of H^(i)=V^H^(1)V^† of equal energy but different C4 eigenvalue iξ . The irreps A, B, 1E , 2E become 2E , 1E , A, B under the gauge transformation (Table 1). Therefore, two of the irreps A⊕B in the occupied states interchange with two of the irreps 1E⊕2E in the empty states after the gauge transformation; all other irreps, (449A⊕449B⊕449(1E)⊕449(2E) ), remain unchanged. We generalize the C4 symmetric TBC:〈μ,αH^(λ)ν,β〉=(〈μ,αH^(1)ν,β〉,ν=μλ〈μ,αH^(1)ν,β〉,ν=μ+1mod4λ〈μ,αH^(1)ν,β〉,ν=μ−1mod4Re(λ2)〈μ,αH^(1)ν,β〉,ν=μ+2mod4 (3) Re(λ2) is the real part of the complex λ2 . The factor between the μth and (μ+2) th part is real owing to C2 (27). Equation 2 is the λ=i case of Eq. 3. Under continuous tuning of λ from 1 to i, two occupied irreps A⊕B interchange with two empty irreps 1E⊕2E . Their level crossings are protected by C4 symmetry (Fig. 2E) [see (27) for other C4 paths].Now we consider C4 breaking but C2  and TRSpreserving TBCs. Divide the system into two parts (I,II), transforming into each other under C2 (Fig. 2D), and multiply all hoppings between orbitals in part I and II by a real λ. The gauge transformation relating the twisted and untwisted Hamiltonians anticommutes with C2 : {V^,C^2}=0 . V^ transforms between eigenstates of H^(1),H^(−1) with equal energy but opposite C2 eigenvalue. Under a continuous tuning of λ from 1 to −1 , the two final occupied (empty) states must have the C2 eigenvalue −1 (1). This inconsistency implies C2 protected gap closing, as shown in Fig. 2F. The unitary transformation relating H(−1) to H(1) also maps H(−λ) to H(λ) , and the gap must close as λ changes from 1 to 0. In (27), we generalize the TBCs to all the 2D PGs. The gapless states under TBCs are the experimental consequences of the fragile states.We introduce the RSI as an exhaustive description of the local states, pinned at the C4 center, that undergo gap closing under TBCs. The Wannier centers of occupied states of a Hamiltonian can adiabatically move if their displacements preserve symmetry. Orbitals away from a symmetry center can move on it and form an induced representation of the sitesymmetry group at the center. Conversely, orbitals at a symmetry center can move away from it symmetrically if and only if they form a representation induced from the sitesymmetry groups away from the center. The RSIs are (27) linear invariant—upon such induction processes—functions of irrep multiplicities. For the PG 4 with TRS, we assume a linearform RSI of the occupied levels δ=c1m(A)+c2m(B)+c3m(1E2E) . The induced representation at the C4 center from four states at C4 related positions away from the center is A⊕B⊕1E2E (27). After the induction process, the irrep multiplicities at the C4 center change as m(A)→m(A)+1 , m(B)→m(B)+1 , m(1E2E)→m(1E2E)+1 . The two linear combinations of irrep multiplicities that remain invariant areδ1=m(1E2E)−m(A),δ2=m(B)−m(A) (4) In our model, the occupied states that can be moved away from the C4 center form the representation 449A⊕449B⊕449(1E2E) and have vanishing RSIs. The states pinned at the C4 center form A⊕B with RSIs δ1=−1 , δ=0 . If an RSI is nonzero, spectral flow exists upon a particular TBC (27). We calculate all the RSIs in all 2D PGs with and without SOC or TRS [table S4 (27)]. The groups formed by RSIs are shown in Table 2. PG 4 with TRS has two integervalued RSIs: The RSI group is ℤ2 . Most RSIs are ℤtype; some groups with SOC and TRS have ℤ2 type RSIs, the parities of the number of occupied Kramer pairs.Table 2 The RSI groups of 2D PGs. View this table: View popup.View inline.For the C4 symmetric TBC (3), the occupied irrep multiplicities m′ at λ=i are determined by the multiplicities m at λ=1 as m′(A)=m(1E) , m′(B)=m(2E) , m′(1E)=m(B) , and m′(2E)=m(A) . The changes of irreps in the evolution λ=1→i are Δm(A)=m′(A)−m(A)=m(1E)−m(A)=δ1 , Δm(B)=δ1−δ2 , Δm(1E)=δ2−δ1 , and Δm(2E)=−δ1 . Therefore, there will be δ1 crossings formed by A and 2E and δ2−δ1 crossings formed by B and 1E . This and the similar analysis for C2 and TRSsymmetric TBCs are given in Tables 3 and 4 and expanded in (27). Our model (δ1=−1 and δ2=0 ) has two level crossings protected by C2 in the process λ=1→−1 .Table 3 View this table: View popup.View inline.Table 4 View this table: View popup.View inline.The RSI can be calculated either from the momentum space irreps of the band structure or from symmetrycenter PGrespecting disordered configurations. In (27), we develop a general framework to calculate the RSIs from momentum space irreps and obtain the expressions of RSIs in all wallpaper groups [table S5 (27)]. Here we give the expressions for RSIs of wallpaper group p4 . p4 has two inequivalent C4 Wyckoff positions, a and b, and one C2 Wyckoff position, c (Fig. 1). PG 4 has two RSIs, δ1 and δ2 (Eq. 4), and PG 2 has a single RSI, δ1=m(B)−m(A) [table S2 (27)]. The band structure expressions areδa1=−m(Γ1)−m(Γ2)2−m(Γ3Γ4)+m(M2)2+m(M3M4)+m(X2)2 (5) δa2=−m(Γ1)−m(Γ3Γ4)+m(M2)+m(M3M4) (6) δb1=12m(Γ2)+m(Γ3Γ4)−12m(M2)−12m(X2) (7) δb2=m(Γ2)+m(Γ3Γ4)−m(M2)−m(M3M4) (8) δc1=m(Γ3Γ4)−m(M3M4) (9) One can immediately verify that the band structure shown in Fig. 2B has the RSIs δa1=−1 , δa2=0 at the a position, which are the same as the results calculated from the disordered configuration.We find that the RSIs fully describe eigenvalue band topology: EFP is diagnosed by inequalities or modequations of RSIs (15), and stable topology is diagnosed by fractional RSIs. We prove this in all the wallpaper groups in (27). For the wallpaper group p4 , the stable topology is diagnosed by fractional δa1 and δb1 , which imply topological semimetal phase with Dirac nodes at general momenta (30). The fragile topology, by contrast, is diagnosed by the inequalityN

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