Higher-order topological phases: A general principle of construction

We propose a general principle for constructing higher-order topological (HOT) phases. We argue that if a $D$-dimensional first-order or regular topological phase involves $m$ Hermitian matrices that anticommute with additional $p-1$ mutually anticommuting matrices, it is conceivable to realize an $n\mathrm{th}$-order HOT phase, where $n=1,...,p$, with appropriate combinations of discrete symmetry-breaking Wilsonian masses. An $n\mathrm{th}$-order HOT phase accommodates zero modes on a surface with codimension $n$. We exemplify these scenarios for prototypical three-dimensional gapless systems, such as a nodal-loop semimetal possessing SU(2) spin-rotational symmetry, and Dirac semimetals, transforming under (pseudo)spin-$\frac{1}{2}$ or 1 representations. The former system permits an unprecedented realization of a fourth-order phase, without any surface zero modes. Our construction can be generalized to HOT insulators and superconductors in any dimension and symmetry class.

Figure 1

Boundary modes of (a) first-order (occupying six surfaces) and (b) second-order (occupying four hinges) topological Kondo insulators, obtained from an appropriate tight-binding model for its prototypical representative ${\mathrm{SmB}}_6$ [46].

Figure 2

(a) Two-dimensional Fermi arc surface states of a first-order, (b) four one-dimensional hinge states of a second-order, and (c) eight zero-dimensional corner states of a third-order spin-$\frac{1}{2}$ topological Dirac semimetal. We compute the square of the amplitude of the lowest-energy $[O\left({10}^{-3}t\right)$ roughly] state $\rho (x,y,z)$ (normalized within the interval [0,1]) on a cubic lattice with open boundaries in all three directions. For numerical diagonalization we set $t=1$, ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=0.3$ [46].

Figure 3

Same as Fig. 2, but for spin-1 topological Dirac semimetals. For numerical diagonalization we set ${\mathrm{\Delta }}_1=0.2$ for (b), and ${\mathrm{\Delta }}_1=0.1$, ${\mathrm{\Delta }}_2=0.2$ for (c). In (a) and (b) [(c)] we show the low-energy states maximally localized on the surface [at the corners] [46].

Figure 4

Topologically protected surface modes for higher-order nodal-loop semimetals (NLSMs) in the reciprocal (top panel) and real (lower panel) space. Top panel: Square of the amplitude of near zero-energy states localized on the surface. Lower panel: Corresponding LDoS in real space [(b1) and (b2)] or amplitude of one of the near zero-energy states [for (b3)], with $E\sim {10}^{-2}t$ (roughly). Here, ($\mathrm{a}n$), ($\mathrm{b}n$) show the surface states for $n\mathrm{th}$-order NLSM, for $n=1,2,3$, and (a4) depicts absence of any localized surface states for a fourth-order NLSM. The faint peripheral states in (a2)–(a4) are the shadow of the delocalized states, connecting the top and bottom surfaces through the topologically protected bulk nodal loop. For numerical diagonalization we set $t=1$, $b=1$, ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2={\mathrm{\Delta }}_3=0.3$, and $b^{\prime }=1.5$. The linear dimension in the $z$ direction, along which we always implement an open boundary, is $L_z=400$ (top panel), 10 [for (b1) and (b2)], and 25 [for (b3)] [46].