Antiunitary symmetry protected higher order topological phases

Higher-order topological (HOT) phases feature boundary (such as corner and hinge) modes of codimension $d_c>1$. We here identify an \emph{antiunitary} operator that ensures the spectral symmetry of a two-dimensional HOT insulator and the existence of cornered localized states ($d_c=2$) at precise zero energy. Such an antiunitary symmetry allows us to construct a generalized HOT insulator that continues to host corner modes even in the presence of a \emph{weak} anomalous Hall insulator and a spin-orbital density wave orderings, and is characterized by a quantized quadrupolar moment $Q_{xy}=0.5$. Similar conclusions can be drawn for the time-reversal symmetry breaking HOT $p+id$ superconductor and the corner localized Majorana zero modes survive even in the presence of weak Zeeman coupling and $s$-wave pairing. Such HOT insulators also serve as the building blocks of three-dimensional second-order Weyl semimetals, supporting one-dimensional hinge modes.

Introduction. The hallmark of topological phases of matter is the presence of gapless modes at the boundary, protected by the nontrivial bulk topological invariant. Traditionally, a d-dimensional bulk topological phase (insulating or gapless) hosts boundary modes that are localized on d − 1 dimensional surfaces, characterized by codimension d c = 1 [1][2][3][4][5][6][7]. Nevertheless, the family of topological phases of matter nowadays includes its higher order cousins, and an nth-order topological phase features boundary modes of codimension d c = n > 1 [8], such as the corner (with d c = d) and hinge (with d c = d − 1) states of topological insulators (electrical and thermal) and semimetals . In this language, the traditional topological phases are first order. While the bulk topological invariant assures the existence of boundary modes, often (if not always) the localized topological modes get pinned at precise zero energy due to the spectral symmetry, which we exploit here to propose the most general setup for a two-dimensional higher-order topological (HOT) insulator, characterized by a quantized quadrupolar moment Q xy = 0.5 and supports four corner localized zero-energy modes. The central results are summarized in the phase diagram, shown in Fig. 1.
The HOT phases can be constructed (at least, in principle) by systematically reducing the dimensionality of the boundary modes at the cost of some discrete crystalline and fundamental (such as time-reversal) symmetries in the bulk of the system. For example, a twodimensional HOT insulator, supporting four corner localized zero-energy modes (d = 0, d c = 2), can be realized in the presence of a four-fold (C 4 ) and time-reversal (T ) symmetry breaking perturbation that acts as a mass for two one-dimensional counter propagating helical edge modes (d = 1, d c = 1) of a first-order topological insulator. The corresponding effective single-particle Hamilto-  (2)]. For small ∆1 and ∆2, the system supports four zeroenergy corner modes (Fig. 3), protected by an antiunitary operator (A) and representing a generalized higher order topological insulator (GHOTI). For charged fermions GHOTI is characterized by a quantized quadrupolar moment Qxy = 0.5. The bulk band gap closes either at the Γ point (solid line) or along the Γ − M line (dashed line) (Fig. 2) beyond which the system becomes a trivial or normal insulator (NI), with Qxy = 0 for charged fermions. The phase diagram possesses a reflection symmetry about (∆1, ∆2) = (0, 0), where the bands recover two-fold degeneracy [see Fig. 2 (left column)], and the system describes a regular HOTI (red dot). The phase boundaries do not depend on ∆ (C4 symmetry breaking mass).
nian can be decomposed asĥ 2D HOT =ĥ 0 +ĥ 1 , witĥ sin(k j a)Γ j + m + t 0 where Γ j 's are mutually anticommuting four-component Hermitian matrices, satisfying {Γ j , Γ k } = 2δ jk for j, k = 1, · · · , 5, a is the lattice spacing (set to be unity) and k is spatial momenta. For 0 < m/t 0 < 2,ĥ 0 describes a firstorder topological insulator. But, depending on the spinor basis and the corresponding representation of Γ matrices (about which more in a moment), this phase represents a quantum spin-Hall insulator (QSHI) or a topological pwave pairing. On the other hand,ĥ 1 lacks both C 4 and T symmetries. It (1) acts as a mass for the edge modes, since {ĥ 1 ,ĥ 0 } = 0, and (2) changes sign under the C 4 rotation, thus assuming the profile of a domain wall. Then a generalized Jackiw-Rebbi index theorem [40], guarantees the existence of four corner localized zero energy modes, with d c = 2. We then realize a second-order topological insulator. Respectively for charged and neutral fermions,ĥ 1 represents either a spin-orbital density wave ordering and a d-wave pairing. In the latter case, the resulting phase stands as HOT p + id pairing [27]. We here seek to answer the following question. What is the most general form of the Hamiltonian operator that supports topologically protected corner modes at precise zero energy and describes a two-dimensional HOT insulator? We note that the corner modes are pinned at zero energy due to the spectral symmetry ofĥ 2D HOT , generated by a unitary (U ) as well as an antiunitary (A) operators, such that {ĥ 2D HOT , U } = 0 = {ĥ 2D HOT , A}. Since the maximal number of mutually anticommuting four-component Γ matrices is five and only four of them appear inĥ 2D HOT , one is always guaranteed to find U = Γ 5 . On the other hand, the existence of A can be assured in the following way. Note all representations of mutually anticommuting four-component Hermitian Γ matrices are unitarily equivalent. Hence, without any loss of generality, we commit to a representation in which Γ 1 and Γ 2 (Γ 3 and Γ 4 ) are purely real (imaginary) [41][42][43]. Then A = K, where K is the complex conjugation [44]. Identification of the antiunitary operator A allows us to construct the most general form of the Hamitonian operatorĥ gen HOT =ĥ 2D HOT +ĥ p , such that {ĥ gen HOT , A} = 0 (with real ∆ 1 and ∆ 2 ), wherê For small ∆ 1 and ∆ 2 , the system continues to support four zero-energy corner modes [see Fig. 3] and a quantized quadrupolar moment Q xy = 0.5 (modulo 1). The resulting phase then describes a two-dimensional generalized higher order topological insulator (GHOTI). However, for sufficiently large ∆ 1 or ∆ 2 , the system enters into a trivial or normal insulating phase, where Q xy = 0 (modulo 1), following a band gap closing (see Fig. 2). These findings are summarized in Fig. 1. The physical meanings of ∆ 1 and ∆ 2 are of course representation dependent [45].
Once we turn onĥ p [see Eq. (2)], the bands loose the two-fold degeneracy (see Fig. 2). Note that under T , P and T ps , the term proportional to ∆ 1 (∆ 2 ) is odd (even), even (odd) and odd (odd). Therefore, it is impossible to find an antiunitary operator that commutes witĥ h gen HOT and squares to −1. As a result, the energy spectra ofĥ gen HOT only contains non-degenerate bands. Still {ĥ gen HOT , A} = 0, assuring the spectral symmetry among the bands about the zero energy. It is worth pointing out thatĥ gen HOT is algebraically similar to the generalized Jackiw-Rossi Hamiltonian, yielding zero-energy modes bound to the core of a vortex in d = 2 [42,43,[46][47][48].
Next, we assess the stability of the HOT insulator in the presence of two perturbations, ∆ 1 and ∆ 2 . As shown in Fig. 2 (second column) that despite loosing the twofold degeneracy, the bands are still gapped for small ∆ 1 and/or ∆ 2 . But, at an intermediate ∆  Fig. 1). On the other hand, the gap closing along the Γ − M line takes place at momenta k = (±, ±)k * and the corresponding phase boundary (the dashed line in Fig. 1) . At the gap closing points, the system is described in terms of linearly dispersing massless twocomponent Weyl fermions at low energies. For stronger ∆ 1 or ∆ 2 , the system reenters into an insulating (but trivial) phase (see the fourth column of Fig. 2). Note that the phase boundaries between GHOTI and the trivial insulator do not depend on ∆, asĥ 1 vanishes at the Γ point and along the Γ − M line.
We now anchor the topological nature of these insulators, separated by a band gap closing. To this end, we numerically diagonalize the effective tight-binding model, namelyĥ gen HOT , on a square lattice of linear dimension L and with an open boundary in each direction for various choices of ∆ 1 and ∆ 2 . The results are shown in Fig. 3. For ∆ 1 = 0 = ∆ 2 , the system supports four near (due to a finite system size) zero energy states that are highly localized near the corner of the system, yielding a conventional HOT insulator [see Fig. 3(a)].
An HOT insulator can be identified from the quantized quadrupolar moment Q xy = 1/2 (modulo 1) [49][50][51]. In order to compute Q xy , we first evaluate whereq xy (r) = xyn(r)/L 2 andn(r) is the number operator at r = (x, y), and U is constructed by columnwise arranging the eigenvectors for the negative energy states. The quadrupolar moment is defined as Q xy = n − n al , where n al = (1/2) r xy/L 2 represents n in the atomic limit and at half filling. Indeed for a HOT insulator, we find Q xy = 0.5 (within numerical accuracy). While a quantized quadrupolar moment is solely supported by the C 4 symmetry breaking Dirac mass (ĥ 1 ), the antiunitary operator (A) allows us to construct GHOTI.
In the above-mentioned representation, ∆ 1 denotes the Zeeman coupling, while ∆ 2 corresponds to the amplitude of spin-singlet (real) s-wave pairing. Hence, our discussion on GHOTI suggests that a two-dimensional HOT pairing can be realized in the form of p + s + id pairing even in the presence of (sufficiently weak) Zeeman coupling, at least when the amplitude of the s-wave pairing is small enough. Therefore, a quantum phase transition between HOT and a trivial paired state can be triggered by tuning the Zeeman coupling between the quasiparticles and external magnetic field.
Note that when a d-wave pairing sets in, it also causes a lattice distortion or electronic nematicity in the system that in turn induces a (small) s-wave pairing [54]. Nonetheless, the amplitude of the s-wave pairing can be amplified and the system can also be tuned through the HOT-trivial pairing critical point by applying an external uniaxial strain along the 11 directions, for example.
Three dimensions. Using two-dimensional GHOTI as the building blocks, one can construct three-dimensional HOT phases, by stacking them along the k z direction in the momentum space. This is accomplished by replacing the term proportional to Γ 3 in Eq. (1) by For example, when ∆ 1 = ∆ 2 , the system describes a second order Weyl semimetal (since all bands are nondegenerate) with two Weyl nodes at (0, 0, ±k * z ), where k * z = cos −1 (|m|/t z ) for t z > |m| and m/t 0 < 1. It supports localized one-dimensional hinge modes for |k z | < k * z [see Fig. 4(a)]. However, the corner localization of the hinge modes decreases monotonically as one approaches the Weyl nodes from the center of the Brillouin zone (k z = 0), similar to the situation with the Fermi arcs of a first-order Weyl semimetal (WSM) [55,56]. Within this range of k z , the quadrupolar moment is quantized to 0.5, but vanishes for |k z | > k * z [see Fig. 4(b)]. By con-trast, for ∆ 2 = 0, four Weyl nodes appear at (0, 0, ±k α z ), where k α z = cos −1 ([m + α∆ 1 ]/t z ) for α = ±. Four pairs of Weyl nodes can be found at (±k 0 , ±k 0 , ±k 0 z ) when ∆ 1 = 0, where k 0 = sin −1 (∆ 2 /[ √ 2t 0 ]) and k 0 z = cos −1 ([m − 2t 0 − 2t 0 cos(k 0 )]/t z ). A complete analysis of three-dimensional second-order Weyl semimetals in the (∆ 1 , ∆ 2 ) plane is left for a future investigation. It should be noted that so far only second-order Dirac semimetals (supporting linearly touching Kramers degenerate valence and conduction bands) have been discussed in the literature [23,27,29], whereas we here demonstrate that it is conceivable to realize its Weyl counterparts (yielding linear touching between Kramers non-degenerate bands), protected by an antiunitary symmetry.
Summary and discussions. To summarize, we identify an antiunitary operator (A) that assures the spectral symmetry of a two-dimensional HOT insulator [see Eq. (1)] and pins four corner modes at precise zero energy. Such an antiunitary symmetry allows us to construct a GHOTI for charged as well as neutral fermions, in terms of two additional perturbations [see Eq. (2)], that continues to support corner localized zero-energy mode (see Figs. 1 and 3), at least when they are small. In particular, our findings suggest that the corner localized Majorana zero modes of a HOT p + id superconductor survive even in the presence of a weak Zeeman coupling and a parasitic or strain engineered s-wave pairing. Concomitantly, a transition between a HOT to trivial paired state can be triggered by tuning the strength of the external magnetic field or uniaxial strain, which can be instrumental for topological quantum computing based on Majorana fermions. The proposed anitiunitary symmetry protected corner and hinge modes can also be observed in highly tunable metamaterials, such as electrical circuits [57].