Type-II quadrupole topological insulators

Modern theory of electric polarization is formulated by the Berry phase, which, when quantized, leads to topological phases of matter. Such a formulation has recently been extended to higher electric multipole moments, through the discovery of the so-called quadupole topological insulator. It has been established by a classical electromagnetic theory that in a two-dimensional material the quantized properties for the quadupole topological insulator should satisfy a basic relation. Here we discover a new type of quadupole topological insulator (dubbed as type-II) that violates this relation due to the breakdown of a previously established theory that a Wannier band and an edge energy spectrum are topologically equivalent in a closed quantum system. We find that, similar to the previously discovered (referred to as type-I) quadrupole topological insulator, the type-II hosts topologically protected corner states carrying fractional corner charges. However, the edge polarizations only occur at a pair of boundaries in the type-II insulating phase, leading to the violation of the classical constraint. We propose an experimental scheme to realize such a new topological phase of matter. The existence of the new topological insulating phase means that new multipole topological insulators with distinct properties can exist in broader contexts beyond classical constraints.

Recently, the formulation of electric polarization based on the Berry phase has been extended to higher electric multipole moments, such as quadrupole moments and octupole moments [1,2].Similar to electric dipole moments, these multipole moments can be quantized due to crystalline symmetries, such as reflection symmetries, giving rise to multipole topological insulators.For a quadrupole topological insulator, besides the quantized quadrupole moment, the quantized edge polarization and fractional corner charge arise.Such fractional charges are associated with the appearance of the topologically protected corner states.The quadrupole topological insulator has ignited an intensive study of higher-order topological insulators with (n − m)-dimensional edge states with m > 1 for a n-dimensional system [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], in stark contrast to the conventional first order topological insulators with m = 1.Recently, the quadrupole topological insulator has been experimentally observed [18][19][20].
For a two-dimensional (2D) square classical system with bulk quadrupole moments q xy , a classical electromagnetic theory shows that equal amplitude edge polarizations p edge α ) describes the edge polarization per unit length on a square along the x (y) direction at the y-normal (x-normal) boundaries.The edges perpendicular to y (x) are labelled by the Greek letters β = ±y (α = ±x) with the sign denoting their relative positions.The currently discovered quantum quadrupole topological insulators indeed respect these relations [1,2] [see Fig. 1(a)].Provided that a system has bulk-independent boundary dipole moments besides the bulk quadrupole moments, the relation is summarized as Q corner +x,+y = p edge +x y +p edge +y x −q xy .While quadrupole insulating phases have been mainly studied in a system with short-range tunnelling, we con-sider a model with long-range hopping.Similar to the model in Ref. [1], this system has reflection symmetries: x → −x and y → −y, maintaining the vanishing of bulk dipole moments, while C 4 symmetry is broken.We find a novel type of quadrupole topological insulating phase where q xy = |Q corner ±x,±y | = |p edge ±y x | = e/2, but p edge ±x y = 0, which is not equal to p edge ±y x , violating the classical relation [see Fig. 1(b)].The type-II insulating phase arises from the breakdown of a previously believed theory that a Wannier band can be continuously deformed into an edge energy spectrum in a quantum system, leading to edge polarizations of equal amplitude along x and y directions [1,2,21,22].It is worthwhile to note that the type-II quadrupole insulating phase is fundamentally different from the insulator with pure edge polarizations along one direction but without bulk quadrupole moments, where the classical relation is also respected.
To characterize the edge polarization (for example, the polarization along x), we consider the Wilson loop W x = F x,kx+(Nx−1)δkx • • • F x,kx and similarly for W y .
Here [F x,kx ] mn = u m kx+δkx |u n kx , where |u n kx is the occupied eigenstate of a system Hamiltonian with n being the band index, δk x = 2π/N x with N x being the number of unit cells along x, and k x is the quasimomentum along x due to the imposed periodic boundary condition along that direction [1,2].The Wannier Hamiltonian H Wx is defined by W x ≡ e iH Wx with its eigenvalues 2πν x referred to as the Wannier spectrum, where ν x is the Wannier center that determines the polarization that each state contributes.Here, the inversion symmetry maintains the vanishing of the total polarization in the bulk [1,2].Let us first consider a cylinder geometry with open boundaries along y.In this case, when the Wannier spectrum exhibits isolated eigenvalues at ν x = ±1/2, with the corresponding eigenstates being localized at two oppo- site y-normal boundaries, the system has the boundary polarization p edge x .The appearance of the edge polarization stems from the topological property of the bulk.There are two routes to the emergence of the edge states in the Wannier spectrum.One is through the change of the topological property of the Wannier bands, the eigenvalues ν x (k y ) of H Wx (k y ), under periodic boundary conditions along y, by closing the Wannier band gap at ν x = ±1/2.An alternative route is provided by closing either the bulk energy gap or edge energy gap, resulting in an abrupt change of the quadrupole moment.
It has been believed that the Wannier band ν x (k y ) can be continuously morphed into the edge energy spectrum localized at the x-normal boundaries [21,22].This means that the Wannier band and the edge energy spectrum should close their gaps simultaneously, giving rise to edge polarizations of equal amplitude at the x-normal and y-normal boundaries.However, we find that this is not necessarily true: the vanishing of the gap of the Wannier band is not necessarily associated with the vanishing of the gap of the edge energy spectrum.Hence, the edge polarization along one direction changes while that along the other direction remains, leading to the type-II quadrupole topological insulators, as detailed in Supplementary Information.
Furthermore, we find anomalous quadrupole topological phases which have the zero Berry phase of the Wannier bands (referred to as Wannier-sector polarization) but the nonzero edge polarization.This tells us that the previously introduced nested Wilson loop formalism [1,2] cannot be used to characterize these insulating phases.Such phases arise because the Wannier Hamiltonian is fundamentally different from a static system Hamiltonian, given that the energy spectrum of the former is periodic, reminiscent of that of the effective Hamiltonian in a periodically driven system [13].This allows the Wannier bands to close their gaps at either ν x = 0 or ν x = ±1/2.When the Wannier spectrum under open boundary conditions exhibit both edge states at ν x = 0, ±1/2, the Berry phase vanishes; this resembles a periodically driven system, where although the traditional topological invariant of a Hamiltonian vanishes, the edge state persists [23].In the article, we introduce a topological invariant for a Wilson line to characterize the edge polarization.
To generate the type-II quadrupole topological insulating phase, we consider a 2D crystal with four sites in each unit cell and long-range hopping between unit cells.We enforce two reflection symmetries M x : x → −x and M y : y → −y, in order to maintain the quantization of the quadrupole moment, corner charges and edge polarizations.Specifically, the system is described by the following Hamiltonian where ) creating (annihilating) an electron at a sublattice denoted by the index α = 1, 2, 3, 4 within a unit cell denoted by a lattice vector R = R x e x + R y e y with R x and R y being integers.The sum over d x and d y depicts the on-site potential and the electron tunneling between distinct sublattices within a unit cell when d x = d y = 0 and the tunneling between neighboring unit cells, otherwise.For simplicity, we choose the lattice constant a x,y = 1.
Here, we consider the case with long-range hopping including up to the hopping with (d x = ±1, d y = ±2) and (d x = ±2, d y = ±1).To be specific, we choose ) and h (12) = it 2 τ 2 σ − , where σ, τ denote Pauli matrices for the degrees of freedom within a unit cell and σ ± = σ 1 ± iσ 2 .The other tunnelling matrices for d x = 0 and d y = 0 can be obtained by h (−dx−dy) = (h (dxdy) ) † , mx h (dxdy) m † x = h (−dxdy) and my h (dxdy) m † y = h (dx−dy) required by the Hermicity and reflection symmetries of the system with mx = τ 1 σ 3 and my = τ 1 σ 1 for zero δ.Specifically, we set ∆ = t 1 = 0.3, t 1 = 0.2, t 2 = 0.15, and t 2 = 0.1.The Hamiltonian in momentum space can be found in Supplementary Information.Compared with the model in Ref. [1], our model does not preserve the time-reversal symmetry κ (κ is the complex conjugation operator) and thus the system does not have the P T = mx my κ symmetry, which guarantees the double degeneracy of the energy bands.Our model breaks this symmetry, lifting the energy degeneracy.
Our numerical computation shows the presence of an energy gap in momentum space energy spectra with respect to γ unless γ = −0.69 and γ = 0.61, where the energy gap vanishes.This implies that the bulk of the system exhibits the insulating property in the gapped regions at half filling.The insulating feature can also be seen in the energy spectra under open boundary condi- (2,2,0,2) (2,2,0,0) (0,0,0,0)  ), the number of edge states of the Wannier Hamiltonian Nν ≡ (N 0 νx , N π νx , N 0 νy , N π νy ) are also shown.The vertical dashed lines represent the critical points where the bulk energy gap, the edge energy gap, or the Wannier spectrum gap vanish.The light red and blue regions denote the type-I AQTI(xy) (anomalousness exists in both Wannier band νx and νy) and type-I AQTI(x).Richer phase diagram for the topologically trivial phase can be found in Supplementary Information.c, The energy spectrum as a function of γ for open boundary conditions along both x and y directions with zero-energy corner modes being highlighted by a red line.d, The electron density distribution in a typical type-II QTI phase with the zero-energy corner modes marked by the green square in c.Here, a very small δ is imposed so that two corner states are occupied.e,f, The edge polarization in a type-I and a type-II QTI, respectively.tions along both x and y directions [see Fig. 2(c)].However, for −0.69 < γ < 0.34 and 0.61 < γ < 1.03, imposing open boundaries render the appearance of four zero-energy states localized at the corners corresponding to a second-order topological insulator, where corner states exist at the boundaries of boundaries as shown in Fig. 2(d).This corner states give rise to fractional charges ±e/2 localized at the corners, as detailed in the Methods section.In other parameter regions, we do not find zero-energy corner states.
To show that the insulator with zero-energy corner states is a quadrupole topological insulator, we calculate their quadrupole moments as detailed in the Methods section.Our numerical results show that the system has a quantized quadrupole moment q xy = e/2 (protected by the reflection symmetry) in the region where the zero-energy corner modes exist, as shown in Fig. 2(b).
The change of the quadrupole moment is associated with the vanishing of either a bulk energy gap or an edge energy gap, reflecting the topological properties of the quadrupole insulating phase.
To distinguish between the type-I and type-II quadrupole topological insulators, it is necessary to characterize their edge polarization by the Wilson loop.In a torus geometry, the eigenvalues of the Wilson loop W x (k y ) [similarly for W y (k x )] takes the form of e i2πν j x (ky) repeats over intervals of 1 for ν j x (k y ), we restrict ν j x (k y ) to (−0.5, 0.5].Because of the reflection symmetry M x : x → −x, ±2πν j x (k y ) are both eigenvalues of H Wx (k y ), so that the Wannier centers appear in pairs [−ν j x (k y ), ν j x (k y )], x,y over a full cycle.a, The cycle refers to the evolution of a system from a topologically trivial phase to the type-II QTI and finally return.Note that the green line is hidden behind the red one.b, The cycle refers to the evolution of a system from the type-II phase to the type-I AQTI and then return.Note that the green and black lines are hidden behind the red one.The units of all the quantities are e.
maintaining the vanishing of the bulk dipole moments in our model.The Wannier bands can be gapped with one band ν − x (k y ) ∈ (−0.5, 0) and the other ν + x (k y ) ∈ (0, 0.5) similar to a conventional band.However, it turns out that there are two gaps for the Wannier bands: one is around ν x = 0 and the other around ν x = ±1/2; the gaps can close at either ν x = 0 or ν x = ±1/2.
Fig. 2 illustrates that in the type-I phase, both Wannier spectra ν x and ν y under corresponding open boundary conditions (open along y and x, respectively) exhibit isolated eigenvalues at ν x = ±1/2, which disappear under periodic boundary conditions, implying that they are contributed by the boundary states.Their emergence indicates the presence of the dipole moments at all the four boundaries.Remarkably, in the type-II phase, only ν x = ±1/2 occurs but not for ν y , implying that the dipole moments only exist at the y-normal edge but not at the x-normal one, as shown in Fig. 1(b).
Our further numerical calculation of the polarization distribution shows that the polarization, if exists, is indeed exponentially localized at the boundaries and opposite boundaries have opposite polarizations [see Fig. 2(e,f)].While the polarization has a distribution along the direction perpendicular to a boundary, their total value for an edge is quantized, i.e., p edge x,y = ±e/2 (see the Methods section).Despite the presence of the edge dipole moments, the total polarization vanishes as opposite boundaries have opposite edge polarizations.In the type-II phase, the polarization along y remains zero, in stark contrast to the corresponding nonzero edge polarization in the type-I phase.
Although the Wannier Hamiltonian can have the edge states at ν x = ±1/2 or ν x = 0, only the former contributes to the edge polarization.In fact, both of these states at ν x = ±1/2 or ν x = 0 can appear simultaneously.In that case, we will show that the Wannier-sector polarization is zero and thus cannot be used to characterize these edge states.We refer to such a topological insulating phase as a type-I anomalous quadrupole topological insulator.While the type-II insulating phase is also anomalous from this perspective, we skip the "anomalous" as we do not find a type-II normal one.
The number of the edge states of the Wannier Hamiltonian H Wx (H Wy ) changes when the gap of the bulk energy spectrum E(k x , k y ), edge energy spectrum E edge,y (k x ) [E edge,x (k y ) ] at the y-normal (x-normal) edges or Wannier band ν x (k y ) [ν y (k x )] closes, reflecting the topological feature of the edge polarization.Associated with the vanishing of the bulk energy gap or edge energy gap is the change of the number of the edge states of the Wannier Hamiltonian corresponding to both ν x = 0 and ν x = ±1/2.However, when the Wannier bands close their gap at ν x = 0 or ν x = ±1/2, only the number of the edge states with the same eigenvalue as that where the gap vanishes changes.Specifically, the bulk energy gap closes at γ = −0.69 and γ = 0.61, leading to the phase transition between the topologically trivial phase with N ν = (2, 0, 2, 0) and type-I quadrupole insulating phase with N ν = (0, 2, 0, 2), and the transition between a trivial phase with N ν = (0, 0, 0, 0) and the type-I anomalous phase with N ν = (2, 2, 2, 2), respectively.Here, ) denoting the number of the edge sates of the Wannier Hamiltonian W λ corresponding to the eigenvalue = 0, π.The vanishing gap of the Wannier bands divides the quadrupole insulating phase for −0.69 < γ < 0.34 into three regions: type-I QTI, type-I AQTI and type-II QTI; each gap closure gives rise to the change of the number of the corresponding edge states, as shown in Fig. 2(b).The edge energy spectrum at the y-normal boundary closes its gap at γ = 0.34, resulting in the phase transition between a trivial phase with N ν = (0, 0, 0, 0) and the type-II QTI with N ν = (2, 2, 0, 0).
Our results show that the Wannier bands ν x (k y ) [ν y (k x )] do not necessarily close their gap at ν = ±1/2 at the same time as the edge energy spectrum localized at the x-normal (y-normal) boundaries (see Supplementary Information for details), which have been believed to be true [21,22].If their gaps always vanish simultaneously, there should be equal number of the edge states of the Wannier Hamiltonian W x and W y with eigenvalues ν = ±1/2, giving rise to the same amplitude edge polarization at the x-normal and y-normal boundaries [see the case for γ = 1.03 in Fig. 2(b)].With this violation in our model, we find the type-II phase where the dipole moments only exist at the boundaries vertical to y.
The Wannier-sector polarization were previously introduced to characterize the edge polarization of the type-I QTI.However, when it becomes anomalous, we find that a corresponding Wannier-sector polarization van To characterize the edge polarization along x (similarly along y), we will introduce a topological invariant based on the Wilson line with respect to defined as where W kx←0 (k y ) = F x,(kx,ky) F x,(kx−δkx,ky) • • • F x,(0,ky) is the Wilson line and is the Wannier Hamiltonian with respect to with log (e iφ ) = iφ with ≤ φ < + 2π.The reflection symmetry leads to (the details are presented in Supplementary Information): where S = σ z .In the basis consisting of eigenvectors of S, . Hence, we can define a winding number at = 0, π as As presented in Supplementary Information, the winding number can change under a gauge transformation for the occupied eigenstates, since the Wilson line is defined by these eigenstates, in sharp contrast to the Floquet case, where the winding number is defined for an evolution operator [24].It suggests that a definite physical quantity is the change of the winding number as a system parameter γ varies.During this change, the Berry phase of the occupied bands should vary continuously, as discussed in Supplementary Information.Fortunately, the quantized dipole moment is a Z 2 quantity, so that the change of the winding number is sufficient to characterize the edge polarization.
The winding number changes as the bulk energy gap and Wannier band gap close, but does not response to the closure of the edge energy gap, which is reasonable as the Wannier bands are constructed from the wave functions without any edge.However, the closure of the edge energy gap is associated with the change of the quadrupole moment.Taking into account the edge energy gap closure, we define where ∆N q,x represents the times that the quadrupole moment changes due to the gap closure of the edge energy spectrum at the boundaries perpendicular to y, when we vary γ from γ 0 to γ 1 .This shows that the topology of the bulk spectrum dictates the edge polarization as the right sides are determined by the bulk property.Provided that we start from a topologically trivial phase, i.e., p edge x (γ 0 ) = 0, the formula can be reduced to  a gauge such that W =π νx (γ 0 ) = 0.As presented in Supplementary Information, this formula correctly describes the edge polarization with respect to γ.
We now discuss the pumping phenomena as a system parameter is slowly tuned.We expect the existence of anisotropic edge currents during a pumping process.To induce the change of the edge polarization, we need to break the reflection symmetry by adding a δτ z σ 0 term so that the edge polarization is not locked to be quantized.However, to maintain the vanishing of the bulk polarization, we still preserve the inversion symmetry.We also maintain the bulk and edge energy gaps during the entire cycle for adiabaticity.
Specifically, we consider the pumping process across the topologically trivial phase and type-II QTI.To achieve the pumping, we choose δ = 0.1 sin(t) and γ = 0.35 + 0.1 cos(t).At t = 0, 2π, the system is in the topologically trivial phase without any edge polarization, quadrupole moments and corner charges, while at t = π, it is in the type-II QTI with q xy = |Q corner | = |p edge x | = 1/2 and p edge y = 0.As time progresses with changes of δ and γ, the system evolves from the topologically trivial phase to the type-II quadrupole insulating phase and then return to the original trivial phase.At each time, we evaluate the edge polarization, corner charge and quadrupole moment (see the Methods section).We find that during an entire cycle, the edge polarization at the top boundary increases by one and at the bottom boundary it decreases by one, as shown in Fig. 3(a).This also happens for corner charges and the quadrupole moment.However, the left and right edges do not exhibit a net transport for the polarization.
We also consider the pumping process across the type-I AQTI and type-II QTI described by δ = 0.1 sin(t) and γ = 0.1 + 0.1 cos(t).At t = 0, 2π, the system is in the type-II QTI phase while at t = π, the system in the type-I AQTI phase.As time evolves over a full cycle, it turns out that only the left and right boundaries show a net change of the polarization but not for the other quantities such as the edge polarization at the top and bottom boundaries, corner charges and quadrupole moments, as shown in Fig. 3(b).Both of the pumping phenomena contrast with previous works where all boundaries, corner charges and quadrupole moments exhibit a net change during a cycle [1,2,25,26].These novel pumping phenomena indicates the peculiar features of the type-II QTI.
If we regard the adiabatic parameter t as momentum k z in the third direction, we obtain two novel threedimensional higher-order topological insulators characterized by a set of topological invariants consisting of the winding of the quadrupole moment and edge polarizations along two directions: (N qxy , N p edge and (0, 0, 1), respectively, which are fundamentally different from the previously found insulator with (1, 1, 1).Although the phase with the winding number being (0, 0, 1) has a winding for the edge polarization, it does not lead to the chiral hinge modes beyond the conventional wisdom that the presence of the winding of the polarization corresponds to a Chern insulator with chiral edge modes.In fact, in our case, it is associated with the presence of chiral edge modes in the Wannier bands, as presented in Supplementary Information.
To realize the type-II QTI, we propose a scheme to simulate the Hamiltonian in electric circuits, as illustrated in Fig. 4. We note that the type-I QTI has already been observed in electric circuits [20].In the circuits, a Hamiltonian is simulated by a Laplacian, a matrix connecting electric potentials with currents, i.e., I = JV, where I and V are column vectors, each entry of which denotes the corresponding current flowing into the corresponding node and the electric potential there, respectively [27].Appropriate electric devices, such as capacitors, inductors and INICs [28,29], are applied to connect different nodes to mimic the hopping between different sites within or outside of a unit cell.The edge polarization and quadrupole moment can be obtained by measuring the single-point impedances of the circuit, and the existence of corner modes can be shown by measuring the resonance of two-point impedances near the corners, as detailed in Supplementary Information.Type-II QTIs may also be realized in other systems, such as solid-state materials, cold atoms and photonic crystals.
In summary, we discover a novel type of quadrupole topological insulator violating an established classical relation.This relation has been believed to be maintained in a quantum system by a previously proved theory that the Wannier band and edge energy spectrum have the same topology.However, the appearance of the type-II QTI indicates the breakdown of the theory.We also find the anomalous quadrupole insulating phases that cannot be characterized by the previously introduced Wanniersector polarizations.We introduce a new topological invariant to characterize them.Based on the type-II insulating phase, we find new pumping phenomena, leading to novel 3D higher-order topological insulators.Our results demonstrate that new multipole topological insulators with exotic properties can exist beyond classical constraints, opening a new direction for exploring multipole topological insulators.

METHODS
The quadrupole moments.The quadrupole moment is numerically calculated based on the following formula [25,26] where Û2 = e 2πi r qxy(r) with qxy (r) = xyn(r)/(L x L y ) being the quadrupole moment per unit cell measured with respect to x = y = 0 at the site r, L x and L y are the length of the system along x and y directions, respectively, the sum is over (x, y) ∈ (0, L x ] × (0, L y ], n(r) is the number of electrons at the site r and |Ψ G is the many-body ground state of a system.Our calculation is performed under periodic boundary conditions.Note that the atomic positive charge contribution has been deducted.
The corner charges.The corner charges are numerically calculated by performing the integration of the charge density over a quadrant of the system, where ρ(R) = 2e − e To calculate the corner charge, we include a small δ so that the fourfold degeneracy of the zero-energy states is lifted, leading to two corner states with positive energy and the other two with negative energy.At half filling, only two corner states are occupied.Suppose the atoms contribute +2e charge in each unit cell.This gives us the corner-localized fractional charges ±e/2 in the limit δ → 0.
The Wannier-sector polarizations.The Wanniersector polarization for the Wannier band ν ± x (similarly for ν ± y ) is defined as [1, 2] where being the nth occupied eigenstate of our Hamiltonian in momentum space and [ν ± x,k ] n being the nth entry of the eigenstate |ν ± x,k of the Wannier Hamiltonian in a torus geometry.
The edge polarizations.To show that the dipole moments are localized at boundaries, we calculate the polarization distribution by where is the probability density of the hybrid Wannier functions [1,2], [ν j kx ] n is the nth entry of the jth eigenvector |ν j kx of the Wannier Hamiltonian corresponding to the Wannier center ν j x in a cylinder geometry with open boundaries along y, and [u n kx ] Ry,α describes the collection of entries of the nth occupied eigenstate of our Hamiltonian in the same boundary configuration.The edge polarization is defined as the sum of p x (R y ) over a half along y, i.e., p edge −y For a tight-binding model, x is the coordinate of discrete lattice sites along the direction perpendicular to the edge and takes the value of integer numbers from −N x /2 to (N x /2 − 1) with N x being the size along the x direction.Here, V 0 (x) introduces two boundaries at x = 0 and x = N x /2 so that for −N x /2 ≤ x < 0, the system is topologically trivial and, otherwise, topologically equivalent to H(k).Indeed, we have found that the eigenvalues E V0 (k y ) of H 0 e (k y ) can be adiabatically mapped to the energy spectrum of the original Hamiltonian under open boundary conditions along x.where B mn g,k = u m Dgk |g|u n k is a unitary sewing matrix that connects states at k with states at D g k.Since we are interested in the occupied bands, the superscript only enumerates the occupied states from 1 to the total number of the occupied states N occ .
We now define the Wilson line following a path C in momentum space from k i to k f as where with m and n being the indices for the occupied bands and ∆k j dividing the trajectory into N segments and j = 0, 1, • • • , N − 1 and k j = k i + j j=1 ∆k j .In the limit N → ∞, we can write the Wilson line in the following compact form, where • dk is a Hermitian matrix, the Wilson loop is unitary, reminiscent of a time evolution operator. Since leading to which can be equivalently expressed as where D g C denotes a new path obtained by applying the symmetry operation on the original path C (we will skip this notation for simplicity).

C. Gauge transformation
Since the Wilson line is determined by the occupied eigenstates of a Hamiltonian, the winding number of the Wilson line introduced in the preceding section may be dependent of the gauge transformation of the occupied eigenstates.Specifically, if we multiply a global phase to an occupied eigenstate, i.e., |u mx k * x ,ky → e iθm x (k * x ,ky) |u mx k * x ,ky with k * x = 0, π and a reflection eigenvalue m x = ±1, then where This gives us the transformation for the Wilson line for = 0, π, Thus, the winding number changes to where . This shows that the global phase at k x = 0, π can produce an unphysical change of the winding number once the global phase exhibits a winding, implying that an isolated value of the winding number is not physical.However, if we maintain this global phase unchanged as we vary a system parameter and observe the change of the winding number, this change is physical and tells us that the edge polarization appears or disappears.This works as the polarization is a Z 2 quantity.
Numerically, we need to maintain the continuity of the global phase of the occupied energy states with respect to k y along the reflection symmetric lines k x = 0 and k x = π for a fixed system parameter γ and to maintain the continuity of the Berry phase of these states about k y with respect to γ.
To maintain the continuity of the global phase with respect to k y , we first make the wave function |u mx k * x ,ky continuous between two neighboring points (k * x , k y ) and (k * x , k y +∆k y ), where k y = n∆k y with ∆k y = 2π Ny and n = 0, 1, • • • , (N y − 1).To achieve this, we choose the maximum component of u mx k * x ,ky+∆ky |u mx k * x ,ky , that is, and eliminate the numerical phase difference by the transformation, where x ,ky ] nmax ).We repeat this process from k y = 0 to k y = 2π − ∆k y .After that, we make the wave functions continuous between k y = 2π − ∆k y and k y = 0 by performing the following phase transformations where To make sure that the Berry phase of each occupied band about k y along the reflection symmetric lines k * x = 0, π is continuous as γ varies, we numerically compute the Berry phase based on the following formula x ,ky=n∆ky with p being an integer provided that the Berry phases between two neighboring γ change by −2pπ.The procedure is similar for calculation of the winding number W νy .
In Fig. S3, we plot the edge polarizations p edge x,y as a function of γ, which is calculated based on the formula (6) in the main text.The results are consistent with the Wannier spectrum in a cylinder geometry and the edge polarization calculated using the hybrid Wannier functions.

ENERGY AND WANNIER SPECTRA DURING A PUMPING PROCESS
In the main text, we have shown the transport of the corner charge, the quadrupole moment and the edge polarizations as system parameters vary over an entire cycle.In this section, we present the energy spectrum in an open boundary geometry (see the first column in Fig. S4) and the Wannier spectrum in a cylinder geometry (see the last two columns in Fig. S4) during this full process.The first row in the figure corresponds to the pump from a topologically trivial phase to a type-II QTI and then back to the trivial phase, while the second row to the pump from a type-II QTI to a type-I AQTI and then back to the original type-II phase.It is clear that we can divide the energy spectrum and Wannier spectrum into two bands with gaps between them.In the former scenario, we see that the corner states connect the lower energy band to the higher one as time evolves, which shows the chiral property of the corner states if time is regarded as a third momentum; this agrees well with the quantized transport of corner charges shown in the main text.For the Wannier spectrum ν x , the edge states exist inside both the gaps around 0 and 1/2 and these states connect two bands of the Wannier spectrum as time progresses, similar to the energy spectrum.However, for the Wannier spectrum ν y , we do not see the presence of the edge states connecting two Wannier bands, consistent with the zero net transport for the edge polarization p edge y .In the second pumping scenario, while the corner states exist during the full cycle, they do not connect the two energy bands, and thus are not "chiral", which is consistent with the zero transport of the corner charges.Similar band patterns occur in the Wannier spectrum ν x .However, for the Wannier spectrum ν y , we find the "chiral"-like edge states, which explains the presence of the net transport of the edge polarization p edge y shown in the main text.

EXPERIMENTAL REALIZATION
In this section, we discuss how to realize our Hamiltonian in electric circuits, in which the SSH model [S3], Weyl semimetal [S4] and type-I quadrupole topological insulator [S5] have been experimentally achieved.Let us consider an electrical network consisting of many nodes simulating sites in a tight-binding model.For each node m in the circuit, suppose that I m is the external current flowing into this node and V m is the voltage at this node with respect x,y and corner charges Q corner α,β can be induced so that |q xy | = |p edge ±y x | = |p edge ±x y | = |Q corner ±x,±y | [1, 2].Here, p edge β x (p edge α y

FIG. 1 .
FIG. 1. (Color online) Schematics of edge polarizations and corner charges.a, Edge dipole moments exist at all boundaries in type-I quadrupole topological insulators (QTI) and b, they exist only at the boundaries perpendicular to y in type-II QTIs.Corner charges Q corner = ±e/2 (marked by different colors) appear in both phases.

FIG. 2 .
FIG. 2. (Color online) Schematics of our model, phase diagram and topological properties.a. Schematics of the tunnelling in our tight-binding model.b.Phase diagram with respect to a system parameter γ, where the quadrupole moment is plotted as a blue line.It includes topologically trivial insulator, type-I QTI, type-I anomalous quadrupole topological insulator (AQTI) (anomalousness exists in the Wannier band νx) and type-II QTI.The subsets display the Wannier spectrum νx (νy) in a cylinder geometry with periodic boundaries along x (y) and open ones along y (x) with the isolated Wannier centers highlighted by red circles.The edge polarization (p edge x , p edge y ), the Wannier-sector polarization (p νx y , p νy x ), the number of edge states of the Wannier Hamiltonian Nν ≡ (N 0νx , N π νx , N 0 νy , N π νy ) are also shown.The vertical dashed lines represent the critical points where the bulk energy gap, the edge energy gap, or the Wannier spectrum gap vanish.The light red and blue regions denote the type-I AQTI(xy) (anomalousness exists in both Wannier band νx and νy) and type-I AQTI(x).Richer phase diagram for the topologically trivial phase can be found in Supplementary Information.c, The energy spectrum as a function of γ for open boundary conditions along both x and y directions with zero-energy corner modes being highlighted by a red line.d, The electron density distribution in a typical type-II QTI phase with the zero-energy corner modes marked by the green square in c.Here, a very small δ is imposed so that two corner states are occupied.e,f, The edge polarization in a type-I and a type-II QTI, respectively.

FIG. 3 .
FIG. 3. (Color online) The transport of the quadrupole moment qxy, corner charge Q corner +x,+y and edge polarization p edge α ishes [see their values (p νx y , p νy x ) in Fig. 2(b)], suggesting that the Wannier-sector polarization cannot uniquely identify the edge dipole moments.

FIG. 4 .
FIG. 4. (Color online) A scheme to realize our Hamiltonian in electric circuits.a, The electric device configuration to simulate the term h (00) within a unit cell.Here, the box labelled by Za or Z b can be either a capacitor or a inductor depending on the sign of γ with their impedances being Za = −1/(iγ) and Z b = 1/(iγ), respectively.The device in the red box denotes a negative impedance converter with current inversion (INIC), the impedance of which depends on the direction of the current.The resistance of the INIC should be taken as R = 1/∆.The boxes labelled by Z a=1,2,3,4 represent device configurations to eliminate the unnecessary onsite terms (see Supplementary Information).b, A suitable electric device is applied to connect two nodes: (R, α) and (R + d, β) for distinct unit cells.The device can be either a capacitor, or an inductor or an INIC with the impedance being Z d,αβ = i/h αβ (dxdy ) , as shown in the lower right corner figure.Here, d = dxex + dyey.
with O = q xy , p edge x,y .The new insulating phases correspond to (N qxy , N p edge x , N p edge y ) = (1, 1, 0)

Nocc n=1 4 α=1
|[u n ] R,α | 2 is the charge density with the first term contributed by the atomic positive charges and the second term by the electron distribution described by the the nth occupied eigenstate |u n of our Hamiltonian under open boundary conditions with [u n ] R,α being the component at the site R with orbital index α.
FIG. S1. (Color online) The connection between the edge energy spectrum and Wannier spectrum.a, Visualization of the edge potential for V0 in the left column as well as VL with M = 1 and M = 5 in the middle and right columns, respectively.b, The energy bands of H 0 e (H L e ) with V0(y) (VL(y)) versus kx in the left column (in the middle and right columns) for γ = 0.34.The edge energy spectrum encoded in H 0 e has a vanishing gap, while the Wannier spectrum encoded in H L e for a moderate or large M (e.g., M = 5) has a finite gap.c, The same spectrum as b for γ = 0.12.The edge energy spectrum has a finite gap, which gradually vanishes as M is increased for VL(y), leading to a gapless Wannier spectrum.d, The energy spectrum of H 0 e (H L e ) with V0(x) (VL(x)) versus ky in the left column (in the middle and right columns).In this scenario, both the edge energy spectrum and Wannier spectrum are gapless.In the middle and right columns for b-d, the Wannier spectra are plotted as the red lines in comparison with the energy spectrum of H L e , showing that the former agrees very well with the latter for M = 5.For b-d, the middle and right columns correspond to the results for M = 1 and M = 5, respectively.
FIG. S2. (Color online) Distinct regions in topologically trivial insulators.The subsets display the Wannier spectrum νx (νy) for periodic boundary conditions along x (y) and open ones along y (x) with the isolated Wannier centers highlighted by red circles.The edge polarization (p edge x , p edge y ), the Wannier-sector polarization (p νx y , p νy x ), the number of edge states of the Wannier Hamiltonian Nν ≡ (N 0 νx , N π νx , N 0 νy , N π νy ) are also shown.The vertical dashed lines represent the critical points where the Wannier spectrum gap closes.
FIG. S3. (Color online)The edge polarizations calculated using the formula(6) in the main text.The edge polarization p edge x suitable transformation |u mx k * x ,ky=n∆ky → exp(−i2πp n Ny )|u mx k * FIG. S4. (Color online) The energy spectrum and Wannier spectrum during the pumping process.a, The cycle from a topologically trivial phase to a type-II QTI and back to the trivial phase.b, The cycle from a type-II QTI to a type-I AQTI and back to the type-II QTI.For both a and b, the first column corresponds to the energy spectrum with open boundary along x and y and the second (third) column correspond to the Wannier spectrum νx (νy) with open boundary along y (x) and periodic boundary along x (y).