Abstract
Recently, real topological phases protected by PT symmetry have been actively investigated. In two dimensions, the corresponding topological invariant is the Stiefel-Whitney number. A recent theoretical advance is that in the presence of the sublattice symmetry, the Stiefel-Whitney number can be equivalently formulated in terms of Takagi’s factorization. The topological invariant gives rise to a novel second-order topological insulator with odd PT-related pairs of corner zero modes. In this article, we review the elements of this novel second-order topological insulator, and demonstrate the essential physics by a simple model on the honeycomb lattice. Novelly, the higher-order topological boundary modes can not only be tuned by the parameters but also the geometric shape of the sample.
Introduction
The symmetry-protected topological phases, such as topological (crystalline) insulators (TIs) and superconductors (TSCs), have been one of the most active fields of physics during the last 15 years [1–9]. Based on the topological K theory, the topological band theory has been established to classify and characterize various topological states [10–12]. Symmetry plays an fundamental role in the classification of topological phases. Considering three discrete symmetries, namely time reversal , charge conjugation and chiral symmetry , physical systems can be classified into ten symmetry classes, termed Altland-Zirnbauer (AZ) classes [10–15], among which the eight ones with at least or are called real AZ classes. The topological classifications in the framework of the eight real AZ classes correspond to the real K theory. Using the real K theory, gapped systems including topological insulators and topological superconductors were first classified [11, 12, 16], and then gapless systems were classified as well [17–20]. All the classification tables exhibit an elegant eightfold periodicity along the dimensions for the eight real AZ classes. After internal symmetries like and , more and more spatial symmetries were involved to enrich symmetry-protected topological matter. It was noticed that combined symmetries and correspond to the orthogonal K theory with the spatial inversion, since they leave every k point fixed in the reciprocal space. Hence, the topological classification table was worked out [21]. A remarkable feature is that groups , and 0 in the table appear in the reversed order in dimensionality, compared with previous tables for the real AZ classes. and are fundamental in nature, and therefore the classification table has been applied to explore topological phases in various physical systems, such as quantum materials [6, 22, 23], topological superconductors [24–27], and photonic/phononic crystals and electric-circuit arrays [27–35], and can generate unique topological structures with many novel consequences, such as non-Abelian topological charges, cross-order boundary transitions, and nodal-loop linking structures [22, 23, 36–40].
Remarkably, from the classification table, the symmetry class with corresponds to the classification for d = 1 and d = 2. As revealed in Ref. [22], leads to real band structures in contrast to conventional complex band structures. Then, the topological invariant w1 for d = 1 can be formulated as the quantized Berry phase in units of π modulo 2π. The case of d = 2 is much fascinating. The topological invariant is the Euler number, a real version of the Chern number, for two valence bands. The Euler number is valued in , but only its parity is stable if more trivial valence bands are added into consideration. The parity, namely the Euler number modulo 2, is just the Stiefel-Whitney number w2 in two dimensions, which determines whether the real vector bundle can be lifted into a spinor bundle.
The topological invariant w2 gives rise to novel topological phases with extraordinary properties. In 3D, it characterizes a real Dirac semimetal, which can be transformed into a nodal ring with symmetry-preserving perturbations. Then, the nodal ring is characterized by two topological charges (w1, w2). In 2D, it describes a topological insulator. The common topological wisdom is that the bulk topological invariant determines a unique form of the boundary modes, namely the well-known one-to-one bulk-boundary correspondence. However, a remarkable discovery in Ref. [41] is that w2 corresponds to multiple forms of boundary modes, extending the one-to-one correspondence to one-to-many. The 2D topological insulator can host various second-order phases with odd -related pairs of corner zero-modes, which are mediated by first-order phases with helical edge states. Similarly, the 3D semimetal can host second-order hinge Fermi arcs and first-order surface Dirac states as well. Recently, graphynes have been proposed as the material candidates which can realize both the 2D topological insulator and the 3D topological semimetal [37, 42, 43].
As aforementioned, the second-order phases of the 2D topological insulator feature odd -related pairs of corner zero modes. It is interesting to look for its 3D analog, which has been presented in Ref. [44]. Referring to the topological classification table for and symmetries, we notice that although the classification for is trivial, with an additional chiral symmetry with the classification is preserved as in 2D and, more importantly, becomes non-trivial as in 3D. It is found that the corresponding topological invariants can be formulated in terms of Takagi’s factorization. The topological invariant in 2D is equivalent to w2, while that in 3D is a new topological invariant. Either in 2D or in 3D the bulk topological invariant can be manifested as odd -related pairs of corner zero-modes. Now, with the chiral symmetry, the two zero-modes in each pair are eigenstates with opposite eigenvalues of the chiral symmetry.
In this article, we review the elements of 2D -protected topological insulators with or without chiral symmetry. The essential physics is demonstrated by the Honeycomb-lattice model, with only the nearest-neighbor hopping amplitudes. We show that under certain dimerization patterns the model is a topological insulator with non-trivial Stiefel-Whitney number or the Takagi topological invariant, and therefore presents all the non-trivial topological phenomena. Particularly, under various -invariant geometries, there are always odd -related pairs of corner zero-modes for the second-order topological phase. Before diving into the details, it is noteworthy that the dimerized honeycomb model can be regarded as an abstraction from the graphynes [37, 42, 43].
The Honeycomb-Lattice Model
Let us start with presenting the honeycomb-lattice model, the lattice structure is shown in Figure 1A. The Hamiltonian in momentum space is given bywhere with i = 1, 2, 3. Here, ai are the bond vectors connecting the centers of nearest-neighbor unit cells, as indicated in Figure 1A with ∑iai = 0. The Hamiltonian has inversion symmetry with , spinless time-reversal symmetry with , and therefore spacetime-inversion symmetry with , where is the complex conjugation and is the inversion of momenta. Note that σ’s are the Pauli matrices acting on the sublattice space (see Figure 1A), and I3 is the 3 × 3 identity matrix. Each inversion center is taken as the center of a hexagon in real space. The sublattice symmetry operator is . Since the inversion exchanges sublattices, both and anti-commute with , namely, .
FIGURE 1
(A) Schematic of the honeycomb-lattice model. ti represents intracell hoppings and represents intercell hoppings with i = 1, 2, 3. The six atomic sites in a unit cell can be divided into two sublattice, as marked by the gray and brown circles, so that a site in one sublattice has all its nearest neighbors from the other sublattice. (B) The winding of Wilson loop around ky for topological nontrivial case. The Wilson loop is computed along a large circle parametrized by kx for fixed ky (where kx,y represents periodic direction along a1, a2 or a3). The loop exhibits a cross at θ = π and ky = 0, which means the topological invariant ν = 1. The parameters are set as .
To obtain the non-trivial topological phases, we calculate the determinant of the Hamiltonian (Eq. 1) at Γ point1 in the Brillouin zone as
Since the bulk topological criticality generally corresponds gap-closing point, we can obtain the topological phase-transition points by letting , which gives
Interestingly, if (Eq. 3) holds, the system is generally reduced to a topologically equivalent graphene model with two Dirac points in the first Brillouin zone [45]. When , the system steps into a topological phase, while conversely the system becomes a trivial phase, which can be checked by computing Stiefel-Whitney number or Takagi’s factorization.
Topological Invariants
The topology can be determined by various formulas of the topological invariant. We now briefly review them. First, as given in Ref. [22], the topological invariant can be determined by the Wilson loop
(with P indicating the path order) along large circles parametrized by kx. is the contour at a fixed ky and is the non-Abelian Berry connection for the valence bands. The topological information is encoded in the phase factors of the N eigenvalues λm(ky) of for valence bands:
Different from the conventional TIs and Chern insulators, the Wilson loop spectral flow for real phases are mirror symmetric with respect to the θ = 0 axis see Figure 1B. This is because is equivalent to a mapping from ky ∈ S1 to O(N) up to a unitary transformation [22]. The topological information can be pictorially derived from counting how many times ζ the trajectories cross θ = π as
For honeycomb lattice with , a single crossing exsit as shown in Figure 1B, namely, w2 = 1, which indicates the model is in a topological non-trivial phase.
As aforementioned, our system is protected by spacetime inversion symmetry and sublattice (chiral) symmetry . These symmetries constraint the classifying space of to be symmeric unitary matrices. Thus the invariant from the Takagi’s factorization can be defined [44], which leads to an alternative formulation for w2. We now prove the equivalence of the two formulas. For technical simplicity, we assume the momentum space as a sphere S2, which is sufficient to present the essential ideas.
In general, requires the Hamiltonian to be block anti-diagonal and requires the upper-right block to be symmetric. Thus the flattened Hamiltonian is given bywhere is a unitary symmetric matrix for each k and M denotes the number of valence (conduction) bands. is the Takagi factor. The classifying space for this symmetric class is US(M) = U(M)/O(M) [44]. Here, corresponds to the topological invariant of our system. Consider a 2D sphere S2, which is divided into north and south hemispheres , overlapping along the equator S1. The Takagi factors over , respectively, can be transformed to each other by a gauge transformation over the equator S1, as shown in Figure 2. is given by for M > 2 leads to obstructions for a global Takagi’s factorization over S2.
FIGURE 2
The Takagi factors in 2D. and represent the south hemisphere and north hemisphere respectively, the overlapping region is equator S1. is the Takagi factors over .
The conduction and valence wavefunctions of can be given bywhere n ∈ {1, 2, … , M}. φn = (0 0 ⋯ 0 1 0 0 ⋯ 0)T is a unit vector with “1” locating at the n-th position.
Performing a unitary transformation on this system, the Hamiltonian and valence wavefunctions both become real. Meanwhile, and are transformed to and σ2, respectively. Over the intersection S1, transition function of real valence wavefunctions can be given by
Thus, we know the transition function of real valence wavefunctions is equal to the gauge transformation . As noted in Ref. [22], w2 is just the parity of the winding number of the transition function for valence bands. Thus, we see the equivalence of two 2D topological invariants.
Physical Consequence
The high-order topological phase in the honeycomb model features novel bulk-boundary correspondence that is different from the traditional one, namely, it has one-to-many bulk-boundary correspondence. Furthermore, the boundary modes can be tuned by the boundary selection. According to analytical and numerical methods, we reveal that three pairs of hopping parameters ti and (with i = 1, 2, 3) jointly determine the configuration of topological boundary modes. To facilitate understanding the relation between distinct boundary modes and parameters, we define a boundary effective mass term mi for each edge:where the subscript i denotes the hopping along the primitive vector ai direction (a3 = −a1 − a2). The above Eq. 10 can be derived from the boundary effective Hamiltonian2. Hence, if mi = 0, the corresponding edges are gapless, which is also the boundary critical condition to separate two second-order topological phases.
To demonstate the boundary modes, we consider a rhombic-shaped 2D sample with armchair termination, i.e., by opening boundary along a1 and a2 direction, as shown in Figure 3. If m1 = 0 and m2 ≠ 0, the helical edge modes along periodic a1 can be obtained, as shown in Figure 3B. However, once m1,2 ≠ 0, the helical edge modes will be gapped and the localized corner modes will emerge. More specifically, for the case with sgn(m1) = sgn(m2) (sgn(m1) = − sgn(m2)), the corner modes will locate at 120° (60°) corners, as shown in Figures 3A,C respectively. The symmetry requires that the corner zero-modes always come in pairs and the chiral symmetry sets the midgap modes exactly at zero energy 3.
FIGURE 3
(A–C) Possible topological boundary modes for the rhombic-shaped sample with 10, ×, 10 unit cells. Black circles indicate the distribution of density of zero-mode (A,C) Second-order TI phases with a single pair of zero-modes corners in diagonal and off-diagonal (or horizontal and vertical) directions, respectively. (B) Helical edge modes with the boundary effective mass mi = 0, which is a critical state separating two second-order TI phases. Parameters are set as (A), (B), (C).
To keep the completeness of honeycomb unit cell in a rhomboid sample, one only has three kinds of armchair edges, namely the edge parallel to ai direction with i = 1, 2, 3. If the edge connected by the same corner has the same mass term mi sign, the corner zero-modes will be localized at the obtuse angle of the rhomboid, otherwise at acute corners. We shall theoretically explain these numerical results in the next section. It is emphasized that in the whole process of the edge-phase transitions, the bulk gap is always open and the symmetries are preserved, therefore, the bulk invariant ν is unchanged. Thus the conventional bulk-boundary correspondence is not appliable for TTI, namely, the bulk invariant can not uniquely determine the boundary modes, but dictates an edge criticality, as the concept mentioned in previous work [41].
As promised in introduction, we now proceed to tune the boundary modes with fixed parameters. In the rhombic case, all samples terminate with armchair edges and exhibit parameter-depended boundary modes. As long as and are not violated, the finite samples can be cut with not only rhomb as shown in Figure 3, but also hexagon(see Figure 5). Beside armchair edges, zigzag edges can serve as termination too. Creatively, with fixed parameters but different boundary selections, one can also find various distinguishable boundary modes. For instance, helical edge states emerge on the zigzag edges in a rectangle sample as shown in Figure 4B, with the same hopping parameters as Figure 4A. This result further proves that the bulk topological invariant can not uniquely determine the topological boundary modes. We also study lots of other patterns with the same parameters, and abundant topological boundary modes consisting of corner zero modes and gapless edge modes can be obtained (see Supplementary Appendix SAD). They are all boundary-selection-depended. Hence, we propose that one can obtain needed topological boundary modes by choosing particular boundary geometry, without tuning parameters, which is usually difficult to perform in real systems.
FIGURE 4
(A) Topological corner modes in rhombic sample with . (B) Topological edge modes in rectangle sample with same parameters as rhombic.
Novelly, in both situations discussed above, for second-order topological phases, the number of the zero-energy corners must be odd pairs. For example, for a hexagonal sample, we can only find one or three pairs of zero-mode corners, as shown in Figures 5A,B. Similarly, the Octagonal sample also has one or three pairs of zero-mode corners as shown in Figures 5C,D.
FIGURE 5
(A,D) Orthohexagonal sample with one and three pairs of zero-mode corners respectively. The parameters are set as for (A) and for (D). (B,E) Octagonal sample with one and three pairs of zero-mode corners respectively. The parameters are set as . for (B) and for (E). (C,F) Hexagonal prisms sample with one and three pairs of corner zero-modes respectively. The parameters can be seen in Supplementary Appendix SAC.
We find that the peculiarity of odd-pair-zero-modes is universal and it can be generalized to a higher-dimensional situation, such as 3D [44]. The 3D model is constructed by stacking the 2D honeycomb TTI discussed above in a staggered manner to preserve the sublattice symmetry . The details of the construction of the model can be found in Supplementary Appendix SAC. The inversion center is chosen as the center of hexagonal in one layer. Thus the anti-commuting relation of and are preserved. We cut a finite hexagonal prisms sample that keeps the symmetries. One can find only odd pairs (one or three) of corner states related by appear. Note that the corner zero-modes can be driven to other corners by tunning the hopping parameters like in a 2D situation.
Analytic Method
We first proceed to solve the boundary criticality along the periodic a1 direction and openning boundary with a2. Replace by S in Hamiltonian Eq. 1, with S ladder operator and S|i⟩ = |i + 1⟩, S†|i⟩ = |i − 1⟩. Then can be represented by since a3 = −(a1 + a2). After a series of tedious derivation (see Supplementary Appendix SAA), we obtian the effective Hamiltonian of the bottom boundary:
Following the same argument with aforementioned bulk criticality, we can obtain the boundary criticality by letting , which leads to
The above Eq. 12 holds only at k ⋅a1 = 0. Thus, when the system is in a topological non-trivial case, Eq. 12 related edge criticality separates two different second-order topological phases with corner zero modes. Likely, we can obtain similar results for periodic a2 and a3 directions. For convenience, we can define boundary effective mass by the left of Eq. 12 for edges parallel to a1. Or, generally Eq. 10 for edges parallel to ai. Different from the armchair edges, the zigzag terminations has an additional boundary criticality, namely, the effective masses can be defined by
The derived details can be found in the Supplementary Appendix SAB.
With the effective mass orderly distributing on each edge (2), the existence of corner zero-modes is reduced to a Jackiw-Rebbi problem [46]. The corners with opposite effective masses on both sides can have zero-modes.
Discussion
In this article, we present a simple 2D realizable honeycomb-lattice model to demonstrate the essential physics of the Takagi topological insulator. It is found that with unchanged topological invariant, one can tune topological boundary modes by not only parameters, but also boundary selections. It goes beyond the common wisdom about bulk-boundary correspondence, and gives rise to much richer boundary physics.
Our model with novel physics is closely related to real systems. It is easier to realize our model by photonic/phononic crystals, electric-circuit arrays and mechanics systems, since only have nearest-neighbor hopping amplitudes are included into the model. Several special cases of our model have been recently realized in photonic/phononic crystals [47, 48], where hopefully the general form of our model can be further experimentally examined.
Statements
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
The authors acknowledge the support from the National Natural Science Foundation of China under Grants (Nos 11874201, 12174181, and 12161160315).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2022.915764/full#supplementary-material
Footnotes
1.^By the numerical calculation of Wilson loop, one can easily check that only the algebra of parameters obtained at Γ point is the real bulk criticality, while the others are not.
2.^Note that a pair of opposite edges parallel to ai have opposite mass terms since symmetry inverses the effective mass.
3.^Finite-size effects can split the degenerate zero modes and deviate them from zero to form a ingap corner modes, but the deviation is exponentially suppressed with the sample size.
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Summary
Keywords
real topology, pt symmetry, higher-order topological insulators, topological insulator (TI), chiral symmetry
Citation
Liu Q, Wang K, Dai J-X and Zhao YX (2022) Takagi Topological Insulator on the Honeycomb Lattice. Front. Phys. 10:915764. doi: 10.3389/fphy.2022.915764
Received
08 April 2022
Accepted
19 April 2022
Published
01 June 2022
Volume
10 - 2022
Edited by
Rui Wang, Chongqing University, China
Reviewed by
Wenbin Rui, The University of Hong Kong, Hong Kong SAR, China
Xian-Lei She Ng, Beihang University, China
Updates
Copyright
© 2022 Liu, Wang, Dai and Zhao.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Y. X. Zhao, zhaoyx@nju.edu.cn
†These authors have contributed equally to this work
This article was submitted to Condensed Matter Physics, a section of the journal Frontiers in Physics
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.