Higher-order topological insulators, topological pumps and the quantum Hall effect in high dimensions

Topological insulators are materials with spectral bands associated with an integer-valued index, manifesting through quantized bulk phenomena and robust boundary effects. In this Rapid Communication, we demonstrate that higher-order topological insulators are descendants from a high-dimensional chiral semimetal. Specifically, we apply dimensional reduction to an ancestor four-dimensional Chern insulator, and obtain two-dimensional (2D) second-order topological insulators when the former becomes chiral. Correspondingly, we derive the quantized charge accumulation at the corners of the 2D descendants and relate it to the topological index—the second Chern number—of the ancestor model. Our results provide a clear connection between the boundary states of higher-order topological insulators and topological pumps—the latter being dynamical realizations of the quantum Hall effect in high dimensions.

A paradigmatic example of a TI is the Chern insulator (CI), appearing in even dimensions. CIs exhibit quantized bulk transport responses [18,19] proportional to the topological indices (Chern numbers) characterizing their spectra. Interestingly, using dimensional reduction, a CI in d dimensions is mapped to a family of models in d − m dimensions, dubbed the "descendant pump family". A periodic and adiabatic scan over the descendant pump family similarly results in quantized bulk transport that is proportional to the Chern numbers of the ancestor CI. Archetypical examples of such mapping are the [two-dimensional to one-dimensional (2D → 1D)] reduction of a 2D CI to Thouless's 1D topological pump [20][21][22][23][24] and the [four-dimensional (4D) → 2D] reduction of a 4D CI to 2D topological pumps [18,25,26], see Table I co-dimension h cross the gap, i.e., states localized in h dimensions but extended in the remaining. Crucially, under dimensional reduction, the boundary states of the CI are mapped to the boundaries of the descendant pump family. For example, the Chern numbers of a 4D CI imply boundary states with codimensions h = 1 and h = 2 that, after dimensional reduction, are mapped to the edges and corners of the descendant 2D pump family [26].
Although CIs and topological pumps share equivalent Chern numbers, a relationship between topological indices in different classes and dimensions is obtained by imposing symmetry constraints when performing dimensional reduction. For example, the [4D → three-dimensional (3D)] reduction of a 4D CI yields 3D Z 2 TIs when time-reversal (T.R.) symmetry is imposed [27], see Table I. This allows for the derivation of a Z 2 index, characterizing the descendant 3D family, directly from the second Chern number of the ancestor 4D model. Correspondingly, the surface states of the 3D Z 2 TI descend from the h = 1 boundary states of the 4D ancestor model, i.e., the boundary states related to the second Chern number.
In this Rapid Communication, we reveal a connection among higher-order TIs, descendant pump families, and highdimensional CIs. Specifically, we show that 2D second-order TIs are the 2D descendants of a 4D chiral semimetal. We do so by first defining an ancestor 4D CI with well-defined first and second Chern numbers and then applying (4D → 2D) reduction to obtain the descendant 2D pump family [18,25,26]. We, then, show that the topological index, associated bulk responses, and the corresponding boundary phenomena of the 4D CI directly imply the properties of second-order TIs in the limit where the former becomes chiral. Respectively, we show that the descendant 2D family is divided into regions in parameter space separated by (bulk-or edge-) gap closures; these regions are distinguished by the appearance of midgap 0D states, localized at the corners. We derive the quantization of corner charge by extending the formula of Jackiw-Rebbi [42] to the 2D descendants and connecting it to the second Chern flux of the ancestor 4D Hamiltonian. Finally, using dimensional reduction, we generate various higher-order TIs solely via flux insertions through different planes of the ancestor model, leading to a simple procedure for finding new models for material realizations.
We consider a tight-binding model describing spinless charged particles moving on a 4D hypercubic lattice in the presence of a magnetic field [see Fig. 1

(a)],
where e μ is a lattice unit vector in direction μ, t μ is the nearest-neighbor hopping amplitude, t μν is the next-nearestneighbor hopping amplitude, and the magnetic flux is incorporated using Peierls' substitution [45]. The third term in Eq. (1) denotes the threading of b flux through each xy plaquette, Finally, a potentialV (m) = (−1) m x +m y V 0 c † m c m with V 0 as a constant (similarly when t z = 0 = t w ) gaps the spectrum, and the 4D model is a CI with well-defined first and second Chern numbers.
The chosen gauge in Eq. (2) leaves the HamiltonianĤ 4D invariant under lattice translations in the z and w directions. We can, therefore, write Eq. (1) in terms of the quasimomenta  [43]. The inset shows the spectrum for t z /t x = 0. (e) The area S enclosing the interface between a nontrivial 1D TI and the vacuum (top) and the domain wall configuration in the k z -parameter space that describes it (bottom). In addition, the charge density ρ at half-filling is sketched. and where k ν is treated as an external parameter [46]. The descendant Hamiltonianĥ μ (k ν ) defines a 1D chain with a unit cell with two degrees of freedom, identical to the Su-Schrieffer-Heeger (SSH) chain [47]. The topological index pertaining to the 1D SSH model is the bulk dipole moment P μ (k ν ) (also known as polarization), which can be calculated using the Wilson loop formalism [32,48]. As a function of k ν , the bulk dipole P μ (k ν ) mod 1 takes two values of 0 and 1/2 with the quantization imposed by chiral symmetry; this ensures the existence of gap closing points in the (k μ , k ν )-parameter space, i.e., at the 2D Dirac cones [cf. Fig. 1(b)]. An on-site potential (−1) m μ V 0 c † m c m (or similarly a nonzero hopping t ν [43] in the Creutz model) breaks chiral symmetry and gaps the spectrum. As a consequence, the (k μ , k ν )-parameter space acquires a well-defined first Chern number c 1 = 2π 0 dk ν ∂ k ν P μ (k ν ), given by the integral of the change in dipole over the entire descendant 1D familyĥ μ (k ν ), see Fig. 1(c) [21,24,49]. Subsequently, the adiabatic evolution ofĥ μ [k ν (t )] along k ν dynamically realizes the 2D quantum Hall effect, dubbed topological pumping [10,21,24], see Table I.
A nonvanishing bulk dipole P μ (k ν ) = 0 has corresponding boundary effects where 0D subgap states appear at the material's boundary, see Fig. 1(d) [49]. Using the formula of Jackiw-Rebbi [42], we calculate the charge q S accumulated in a region S enclosing the boundary [see Fig. 1(e)] by linearizing the dynamics around zero energy to obtain the low-energy Hamiltonianĥ(k ν ) = |k μ | d · σ where d = {v(k ν )k μ , μ 1 (k ν ), μ 0 (k ν )} is a real-valued vector [50], σ = {σ y , σ x , σ z } are three anticommuting matrices {σ i , σ j } = 2δ i j , and is a cutoff energy scale. The accumulated charge q S is whered = d |d| andS = S[− , ] is the integration domain. The interface between two insulators in real space is equivalent to a domain wall in the k ν -parameter space [49], see Fig. 1(e). Thus, Eq. (7) is the first Chern (Berry) flux attached to the region defined byS in the 2D Brillouin zone (BZ) of the ancestor model. Importantly,S does not cover the entire 2D BZ of the ancestor model. Hence, in the limit where chiral symmetry is restored, |q S | becomes quantized to two values of 1/2 or 0, that correspond to encircling or not encircling a singularity with ±1/2 Berry flux. As a consequence, the 1D familyĥ μ (k ν ) is divided into a trivial and a nontrivial region with q S = 0 or 1/2, respectively, in accord with the value of the bulk dipole q S = S ∂ r μ P μ dr μ , cf. Fig. 1(c). This is known as the bulk-boundary correspondence of 1D Z TIs, i.e., the relation between the quantized bulk dipole and the charge at the boundary [50]. Note that Eq. (7) also describes the accumulation of nontopological charge at the boundary between two insulators (Tamm states), arising from surface polarizability [51].
In similitude to the (2D → 1D) reduction above, the main goal of this Rapid Communication is to demonstrate that the 4D Hamiltonian (1) leads to 2D second-order TIs. A (4D → 2D) reduction of Eq. (1) yields the 2D , describing a square lattice on the xy plane made out of SSH chains [cf. Eq. (5)] in both the x and the y directions and where each xy plaquette is threaded by a magnetic flux b, see Fig. 2(a). Assuming b = π , the topological invariant of the resulting 2D descendantĥ π xy (k) is associated with a quantized bulk quadrupole moment Q xy (either 0 or ± 1 2 ), which can be calculated using nested Wilson loops [32,38,52], see Fig. 2(b). Breaking chiral symmetry with an on-site potentialV (m) [similarly with hopping t z = 0 = t w in Eq. (1)] results in a 4D CI with a well-defined second Chern number [53] Fig. 2(b). Hence, an adiabatic evolution ofĥ π xy (k) over a closed surface in thek-parameter space dynamically realizes The charge density ofĥ π xy (k) at half-filling andk = (0, 0) has ±1/2 charge deviation at the corners. In (b)-(d), we used t x = t, t x /t y = 1, and t xz /t x = t yw /t y = 0.45. In (b), we used t z = t w = 0.001t to minimally open the gap. the 4D quantum Hall effect, dubbed 2D topological pumping [18,25,26], see Table I.
The bulk responses of the descendant 2D pump family have associated boundary phenomena [26] where: (i) 1D edge states, i.e., states of co-dimension h = 1, appear in the spectrum, and (ii) 0D corner states, i.e., states of co-dimension h = 2, disperse as a function ofk, see Figs. 2(c) and 2(d). We generalize Eq. (7) and derive [53] the charge accumulation q C in a region C enclosing the corner of the 2D material [cf. Fig. 2 4 } are five anticommuting matrices { μ , ν } = 2δ μν , and is a cutoff energy scale. We obtain

(a)] with low-energy Hamiltonianĥ
whered = d |d| andC = C × [− , ] 2 is the integration domain. Since the corner of the material can be expressed as the intersection of two edges, i.e., two domain walls in thekparameter space [cf. Fig. 2(a)], q C is equivalent to the second Chern flux attached to the region defined byC in the 4D BZ of the ancestor Hamiltonian. Note thatC does not cover the entire 4D BZ of the ancestor CI. Hence, in the limit where chiral symmetry is restored, |q C | becomes quantized to 1/2 or 0, corresponding to encircling or not encircling a 4D singularity with ± 1 2 second Chern flux. For the 2D descendantĥ π xy (k), we find that thek-parameter space is divided into trivial regions with |q C | = 0 and nontrivial regions with |q C | = 1/2; the latter exhibiting zero-energy states localized at the corners, in accord with the value of the bulk quadrupole q C = C ∂ r x ∂ r y Q xy d 2 r, cf. Figs. 2(b)-2(d).
The connection between charge accumulation at the corner and the second Chern flux is a key outcome of this Rapid Communication. In general, such charges can arise due to bulk properties as well as due to boundary effects. The charge accumulation q C in a finite macroscopic 2D material interfaced with another material, e.g, the vacuum, can be calculated using the electric multipole moments [32], where ρ bulk = −∇ · P + 1 2 ∂ r μ ∂ r ν Q μν are the contributions due to the bulk dipole P and quadrupole Q xy densities, ρ ∂ =ˆ n · P| ∂ −n μ ∂ r ν Q μν | ∂ are the contributions due to a "free" edge dipoleˆ n · P| ∂ and quadrupolen μ ∂ r ν Q μν | ∂ densities, and ρ ∂∂ = 1 2n α μn β ν Q μν is the contribution due to a point charge created by a free quadrupole density at the intersection of two edges with normal vectorsˆ n α andˆ n β . Hence, a nontrivial value of q C can originate from a combination of bulk and surface terms [53].
For the 2D Hamiltonianĥ π xy (k), the only nonvanishing contributions to the corner charge q C arise from a quantized bulk quadrupole Q xy . On the other hand, starting from the 4D ancestor Hamiltonian (1) with b = 0 we obtain, using (4D → 2D) reduction, a 2D descendantĥ 0 xy (k) composed of coupled SSH chains along the x and y directions, see Fig. 3(a). This modelĥ 0 xy (k) has zero bulk quadrupole Q xy but nonzero edge dipoles P| ∂ that result in two distinct phases with q C = 0 or 1/2; the latter having zero-energy states localized at the corners that merge into the bulk or edge spectrum at gap closing points [see Figs. 3(b) and 3(c)]. As a third example, we start from Eq. (1) with b = 0 and thread a π flux through each xw plaquette. This leads, using (4D → 2D) reduction, to a 2D descendant family denoted byĥ 0,π xy (k), describing SSH chainsĥ x (k z ) along the x direction, coupled to alternating SSH chainsĥ y (k w + π x) in the y direction, see Fig. 3(d). The charge q C is now a combination of bulk and surface terms that sum to quantized values 0 or 1/2, see Fig. 3(e). The spectrum is separated by bulk-or edge-gap closing points into regions characterized by the appearance of zero-energy states localized at the corners, see Fig. 3(f). In all three cases, the corner charge accumulation is associated with a nontrivial value of the second Chern flux [53].
The demonstrated relationship between the 4D chiral semimetal (1) and the 2D second-order TIs offers a plethora of generalizations. Namely, a wide variety of 4D ancestor models can be constructed where various planes are threaded with 2π/q fluxes (where q is an even integer), and different directions are dimensionally reduced. An even q is crucial for obtaining a low-energy theory corresponding to decoupled Dirac cones [20], thus, defining regions in the BZ that are separated by gap closures. Moreover, our results readily extend to 3D, explaining the appearance of hinge modes and relating corner states to a six-dimensional CI and its third Chern number [19,30]. These charges arise from a combination of  Fig. 2(c). In (b), (c), (e), and (f), we have used t x = t, t y /t x = 1/10, t xz /t x = t yw /t y = 0.45, and an on-site staggered mass V 0 = 0.001t. octapole, quadrupole, and dipole moments. Equivalently, our procedure generates multiple topological pump realizations where charge transport is proportional to the modulation of the bulk dipole, quadrupole, and octapole moments. This naturally explains the appearance of surface, hinge, and corner modes (cf. Fig. 2(b) and Ref. [18]).
In this Rapid Communication, we reveal a connection among the physics of high-order TIs, topological pumps, and Chern insulators using dimensional reduction. This allows us to define a topological index associated with the charge accumulation at the corners, leading to a simple unifying understanding of standard TIs and higher-order TIs. It engenders a single ancestor high-dimensional insulator and uses dimensional reduction as a tool to find new higher-order TIs, each with its own low-dimensional description. Comparing our invariant with the electric multipole expansion, we establish that corner charges arise from the combination of bulk and surface multipole moments.