Unified characterization for higher-order topological phase transitions

Higher-order topological phase transitions (HOTPTs) are associated with closing either the bulk energy gap (type-I) or boundary energy gap (type-II) without changing symmetry, and conventionally the both transitions are captured in real space and characterized separately. Here we propose a momentum-space topological characterization of the HOTPTs, which unifies the both types of topological transitions and enables a precise detection by quench dynamics. Our unified characterization is based on a novel correspondence between the mass domain walls on real-space boundaries and the higher-order band-inversion surfaces (BIS) which are characteristic interfaces in the momentum subspace. The topological transitions occur when momentum-space topological nodes, dubbed higher-order topological charges, cross the higher-order BISs after proper projection. Particularly, the bulk (boundary) gap closes when all (part of) topological charges cross the BISs, characterizing the type-I (type-II) HOTPTs. These distinct dynamical behaviours of higher-order topological charges can be feasibly measured from quench dynamics driven with control in experiments. Our work opens an avenue to characterize and detect the two types of HOTPTs within a unified framework, and shall advance the research in both theory and experiment.

Since the bulk, as well as partially the boundary, is gapped for higher-order topological states, the higherorder topological phase transitions (HOTPTs) are associated with closing either the bulk (type-I) or boundary (type-II) energy gap without changing symmetry [45][46][47][48][49][50][51][52][53][54][55][56].Currently, the type-I and type-II HOTPTs are characterized by the topological invariants defined on the bulk and Wannier bands [57], respectively.For instance, the multipole moments [8] and bulk polarization [9] can only be applied to identify the type-I HOTPTs, while the nested Wilson loop [3] and Wannier band polarizations [52] are limited to the type-II HOTPTs.Nevertheless, when the topological transitions occur, these invariants defined by the bulk and boundary properties are not unified and can not fully capture all the transitions [58][59][60].Hence the independent characterization can not essentially describe the HOTPTs and is not conducive to uncover the novel higher-order topological states.
Meanwhile, the current characterizations bring difficulties for identifying the both types of topological transitions in experiments [61][62][63].Very recently, the experi-mental realizations of the higher-order topological states have been widely reported in cleaning synthetic systems in a controllable fashion [64][65][66][67][68][69][70][71][72].The bulk physics can be conveniently simulated in synthetic systems like ultracold atoms [73][74][75][76], nitrogen-vacancy center [77][78][79], and nuclear magnetic resonance [80,81], while the classical simulators (such as phononic crystals [64], photonic crystals [65], and electric circuits [66]) provide ideal grounds to play with the higher-order boundary modes.However, while having high controllability, it is still challenging for these synthetic systems to observe the two types of HOTPTs due to the lack of a full manipulation and detection of both the bulk and boundary physics.
Motivated by these considerations, in this Letter, we propose a unified characterization for the both fundamental types of HOTPTs, which goes beyond the traditional independent characterization and enables a feasible detection of HOTPTs via quench dynamics.We first show a generic duality that for a dD nth-order topological phase, the existence of (d − n)D gapless boundary states uniquely corresponds to the emergence of nthorder band-inversion surfaces (BISs) [82][83][84][85][86][87][88][89][90][91] which are (d−n)D interfaces in the momentum space characterizing where the energy bands cross and are inverted.Based on this nontrivial duality, the topological phase transitions occur when the higher-order topological charges cross the higher-order BISs after proper projection, with the type-I (or type-II) transitions being characterized by all (or part of) charges pass through the BISs, providing an elegant and unified characterization of both types of HOTPTs.We finally show that both topological charges and BISs can be well measured in quantum quench experiments.
Our work provides a new way to simulate the higherorder topological phases and detect the HOTPTs.
Duality between mass domain wall and BIS.-We start with deriving a duality between mass domain wall (MDW) and BISs for a generic dD nth-order topological insulator (TI) captured by the Hamiltonian (1) where k = (k 1 , k 2 , • • • , k d ) is the dD momentum.The Gamma matrices obey the anticommutation relation of Clifford algebra [92,93] and can be regarded as the (pseudo)spin operators.Here we use the convention that h j d denotes (pseudo)spin-orbit coupling coefficients, while h d+l represents mass terms which include the Zeeman terms.Without mass terms the Hamiltonian H k characterizes a massless Dirac semimetal.The mass term for n = 1 opens a bulk gap and gives the 1st-order topological model, such as the 1D Su-Schrieffer-Heeger (SSH) chain [94] and 2D Haldane model [95].For n > 1 the additional mass terms further open gaps on boundary and give rise to the higher-order topological phases, including the 2nd-order TIs with order-two symmetry [16] and the 3rd-order TIs with inversion and reflection symmetries [87], where the crystalline symmetries determine the configurations of (d − n)D gapless boundary modes.
The boundary states of higher-order topological phases can be characterized through the dimensional reduction approach.Namely, the boundary states of an nthorder topological phase are obtained as Jackiw-Rebbi We first start from the 1st-order TIs (n = 1).The corresponding gapless surface states can be described as bound modes at the (d − 1)D MDWs between the system and vacuum on real-space boundary.On the other hand, these surface states are uniquely determined by the bulk topology, which is known to be further characterized by the (d − 1)D 1st-order BIS B 1 in momentum space with vanishing mass term h d+1 (k) = 0 [82].This renders the MDW-BIS duality for the 1st-order TIs.Further, the 2nd-order topological phase is obtained when an additional mass term h d+2 is added to the Hamiltonian of 1st-order TIs; see Eq.  Hence the h d+2 term gaps out the (d − 1)D surface states almost everywhere but leaves the MDWs on the (d − 2)D boundary, yielding a 2nd-order topological phase.Repeating the above procedures, we obtain all of the higherorder topological phases, rendering the generic duality between the nth-order BISs and (d − n)D MDWs for the nth-order topological phases.This nontrivial duality reveals a new correspondence between the momentumspace bulk physics and the real-space boundary physics.Moreover, it is also faithful for the higher-order topological phases with stacking 1D SSH chain, such as the 2D and 3D BBH models [3,8] which can arrive at Hamiltonian (1) by rescaling the Gamma matrices [32].
Topological characterization of HOTPTs.-Wenow develop the unified characterization of HOTPTs based on the above MDW-BIS duality.Here a key idea is that an nth-order topological system can be equivalently transformed into the superposition of n effective 1st-order topological subsystems by using the dimensional reduction.Specifically, the (d − i)D MDWs are introduced to the (d − i + 1)D boundary states by adding an additional mass term h d+i to the Hamiltonian with i = 1, 2, . . ., n, turning the (i − 1)th-order topological phases into a ithorder topological phase.We treat the (d−i+1)D gapless boundary modes as a massless Dirac system, and then the MDWs of the ith-order topological phases are indeed the boundary states of an effective 1st-order (d − i + 1)D gapped topological phase given by with D (i−1) being a subset of {1, 2, . . ., d} with (d − i + 1) elements [97].For these effective 1st-order topological subsystems, the (d − i + 1)D momentum subspace k (i−1) characterizes an effective (d − i + 1)D Brillouin zone BZ (i−1) obtained by projecting B i−1 onto the subspace spanned by all k j , with the Gamma matrices γ (i−1) j being generally superpositions of the original ones [98].
With above observation, the topological index V n of the nth-order topological phase (1) is then determined by all of the invariants w i of the effective 1st-order topological Hamiltonian H k (i−1) , given by ( This can be easily understood in process of constructing an nth-order topological phase from the (n − 1)thorder TI with the topological index V n−1 .The sign function sgn(|V n−1 |) = 1 (or 0) characterizes the presence (or absence) of the (d − n + 1)D boundary states for the (n − 1)th-order topological phase.As indicated by Eq. ( 2), the topology of the nth-order TI is inherited from these boundary modes [99, 100] and is characterized by the invariant w n , while the absence of these boundary states always leads to a trivial nth-order phase.Thus the nth-order TI has the topological invariant V n = sgn(|V n−1 |)w n .Repeating the same analysis for all V i n−1 yields the topological index (3).The last step for the unified characterization is to represent w i in terms of topological charges, which are dual to the BISs [101].For above effective 1st-order topological Hamiltonian, an sth-order topological charge and quantified by Jacobian determinant J hso (k (i−1) ) ≡ det(∂h so,j /∂k j ) [97,102].Accordingly, the invariant w i equals the total monopole charges enclosed by , dubbed as the projective sth-order BISs.We then obtain where B(i−1) proj,s is the momentum region enclosed by B (i−1) proj,s with h d−i+3−s < 0. Unlike the higher-order BISs in the original bulk system, these projective higher-order BISs are defined in the effective BZ (i−1) , since proj,s is actually the projection of the ith-order BIS B i onto BZ (i−1) .The Eqs. ( 3) and (4) give the characterization for a broad class of higher-order topological phases with various lattice symmetries and described by Hamiltonian (1).We show later that while the characterization is built on the topological indices w i of the effective 1st-order topological system, it can be precisely measured in experiment by quench dynamics.
We are now ready to write down the unified characterization of the HOTPTs which must be associated with the change of w i for one or multiple effective 1st-order topological subsystems (2).Equivalently, in a HOTPT the topological charges C (i−1) s,q must cross either the projective BIS B (i−1) proj,s or the border of BZ (i−1) [see Fig. 2(a)].Namely, a topological transition of nth-order phase .Moreover, it characterizes the HOTPT between an initial n-order phase and a final pth-order phase with p n, given that w i p is nonzero according to Eq. (3).Dynamical detection and Application.-Weshow now that the unified characterization can facilitate the precise detection of the HOTPTs and propose the applications based on quantum quenches.Our scheme is based on sequentially quenching all of the (pseudo)spin axes γ α=1,2,...,d+n , while only measuring a single (pseudo)spin component γ d+1 in each quench.For this we suddenly tune the Hamiltonian , where ρ α is density matrix for initial state.The projective BISs are determined as is further detected by the dynamical field  for s = 1 with β = d + i and s > 1 with β = d − i + 2 − s, since it can be shown that g j = h so,j near the node point k Here N k (i−1) is a normalization factor.This dynamical detection scheme is highly feasible in experiment as we demonstrate below.
We exemplify the application of the unified characterization with a 3D 2nd-order TI, constructed by adding a mass term into the 3D chiral TI [79,80], with the bulk Hamiltonian Here σ and τ are both Pauli matrices and k 1,2,3 = k x,y,z .From the time-averaged spin texture shown in Fig. 3(a), we observe a ring-shaped projective BIS B (1) proj,1 , manifesting the existence of the BIS B 2 in original momentum space and identifying the emergence of hinge states according to the MDW-BIS duality.Moreover, one negative (positive) topological charge Then the two-fold degenerate zero energy states are localized at the hinges of the top and bottom surfaces along z-direction, and protected by the C z 4 -rotation symmetry and the anti-reflection symmetry R † j H kj Rj = −H −kj along the j = x, y, z axis.However, when B 2 disappears [see Fig. 3(b)], there is no B (1) proj,1 and topological charge C (1) 2,q in BZ (1) [see Fig. 3(d)].Hence we have w 2 = 0 and no zero energy state exists in the hinges.
Based on the MDW-BIS duality, the existence of B 2 gives the 2nd-order topological phase diagram 0 < |m 5 | < 2t 0 and |m 4 −m 5 | < t 0 , as shown in Fig. 3(e).One can see that the bulk energy bands become gapless at (m 4 , m 5 ) = ±(3t 0 , 2t 0 ), ±(t 0 , 2t 0 ), ±(t 0 , 0) (green squares), while the surface energy gap is only closed at m 5 = 0, ±2t 0 (solid lines) and m 4 = m 5 ± t 0 (dot-dashed lines), which are confined to the real-space interfaces along and perpendicular to z direction, respectively [97].We shall choose two different parameter paths to observe the phase transitions.In path A-B-C-D [see Fig. 3(e)], there is one negative topological charge crossing the border (purple dashed curves) of BZ (1) [see Fig. 3(g)], and the surface energy gap closes in both xz and yz planes for B [see Fig. 3(f)].This renders a type-II transition with further increased, all topological charges simultaneously move to the projective BISs for D, then the bulk energy gap closes, rendering a type-I transition.In another path C-E-F [see Fig. 3(e)], the surface energy gap closes in the xy plane for E.There is one negative topological charge crossing the projective BISs and changed into a positive topological charge.The higher-order topological transition occurs as The unified characterization has explicit advantages that the topological phase transitions of different types can be resolved in quench dynamics.
Discussion and Conclusion.-Theunified characterization also shows that the type-II transitions are further classified into different m-orders which can be precisely determined by quench detection.In Supplementary Material [97], we have presented more relevant examples for the 2D 2nd-order and 3D 3rd-order TIs, which further showcase the broad applicability of the unified characterization.Moreover, our unified theory may be applied to study the Floquet higher-order phases and phase transitions, such as clarifying which type of phase transitions dominates the emergence of Floquet corner modes [103].This shall further promote the study of topological phase transitions in Floquet higher-order systems.
In summary, we have shown a unified characterization in momentum space for the higher-order topological phase transitions and further proposed the detection by quench dynamics.The unified characterization is built on the MDW-BIS duality which relates the higherorder boundary modes in real space and the higher-order BISs with topological charges in the momentum space.The topological phase transitions of two types and various orders are generically identified by the higher-order topological charges crossing over the BISs after proper projection, which can be precisely detected by quench dynamics.This work establishes a unified and fundamental characterization of the higher-order topological phases and phase transitions, and shall advance the further broad studies in theory and experiment. This has the half size of γ (i−1) .
[99] R. S. K. Mong  Supplementary Material for "Unified characterization for higher-order topological phase transitions" In this Supplementary Material, we provide the detailed proof for the duality between mass domain wall (MDW) and band inversion surface (BIS), i.e., the MDW-BIS duality.We also provide the details of characterization for higherorder topological phase transitions (HOTPTs) and the completely numerical results for the dynamical characterization of 2D 2nd-order, 3D 2nd-order, and 3D 3rd-order topological phases.
I. MDW-BIS duality and its applications 1. 1st-order topological phases Firstly, we demonstrate the MDW-BIS duality of 1st-order topological phases.Our starting point is a 1st-order topological insulator (TI) or a 1st-order topological superconductivity (TSC) obeying the lattice Hamiltonian [S1] where dD) momentum and Gamma matrices γ satisfy the anticommutation relation of Clifford algebra.Here d is the spatial dimension of the system and k j is the momentum in the j-th direction.For convenience, we use the convention that h j (k j ) are (pseudo)spin-orbit (SO) coupling components while denotes the mass term with Zeeman coupling constant.
To capture the MDWs of this 1st-order topological system, we take k → 0 for the Hamiltonian (S1) and obtain its low-energy effective model , where h j (k j → 0) = k j and h d+1 (k → 0) = M d+1 .Now one can consider a MDW in the r d direction, i.e., md+1 = m 0 (−m 0 ) is a positive (negative) constant effective mass in the region r d > 0 (r d < 0), which is a topological (trivial) phase.Thus k d is not a good quantum number and can be replaced by −i∂/∂ r d .The low-energy effective model can be rewritten as with with On the other hand, since iγ d+1 γ d commutes with γ j =d+1,d , these (d − 1)D edge states can also be characterized by a projective Hamiltonian [S1].Namely, we use the projection operator where γ (1) j = P (0) † γ j P (0) is denoted as the Gamma matrices in (d − 1)D subspace and has the half size of γ j .
Obviously, the (d − 1)D edge energy spectrum is j=1 k 2 j and shows the gapless behaviour.Since Real space Momentum space BZ (1)   (e3) BZ (2)   FIG.S1.MDW-BIS duality for 1st, 2nd, and 3rd-order topological phases in a 3D system.), showing that all (d − 1)D real-space interfaces host the zero-energy edge states.Correspondingly, the effective mass vanishes in the all (d − 1)D interfaces, which renders md+1 = 0 and M d+1 = 0 due to k → 0. The physical consequence is that (d − 1)D MDWs are emerged on all boundaries, which is similar to the interfaces between the material and vacuum.Now we return to the lattice Hamiltonian H dD (k).After using the projection operator P (0) = (1 − iγ d+1 γ d )/2 in the eigenspace of iγ d+1 γ d = −1, we obtain a (d − 1)D projective Hamiltonian P (0) † H dD (k)P (0) , which is explicitly written as From above MDW-BIS duality of the 1st-order TIs, we also see that the two h-components h d (k d ) and h d+1 (k) vanish in the projection process of k d -OBC.This implies that the (d − 1)D edge states, perpendicular to the r d direction, are only characterized by the projection of 1st-order BISs B 1 on the surfaces of h d (k d ) = 0, defining an effective Brillouin zone BZ (1) with the momenta k (1) = (k 1 , • • • , k d−1 ).Correspondingly, BZ (1) shall be limited within the projective 2nd-order BISs B For example of the 1st-order topological phase with n = 1, B (0) proj,1 = B 1 is located at the original BZ with the momenta k (0) = k [i.e.BZ (0) ].The projective sth-order BISs B (0) proj,s ≡ {k|h d+1 = h d = • • • = 0} is defined by using s terms of h-components of H dD (k), which has no essential difference with the definition of higher-order BISs in Ref. [S2].Hence w 1 can also be further treated as an integer invariant on B (0) proj,s [S2].

2nd-order topological phases
We add a mass term S1) and obtain where the Gamma matrices still satisfy the Clifford algebra.This Hamiltonian (S6) describes the dD 2nd-order topological phases.To capture the MDWs of this 2nd-order topological system (S6), we still consider its low-energy effective model Compared with the 1st-order topological phase, H dD (k) is now rewritten as For the Eqs.(S7) and (S8), it is clear that their M d+1 -terms are same but M d+2 -terms are different.After taking the zero-energy modes Φ(r d 0) and Φ(r d 0) of Eq. (S3) into the above two Hamiltonians respectively, both M d+1 -term shall vanish.The remaining terms give Next we follow the previous projection approach and obtain all (d − 1)D edge Hamiltonians For a 2nd-order topological phase obeying the Hamiltonian (S6), the case ii and case iii are both covered.These (d − 2)D edge modes should hold md+1 = 0 and md+2 − md+1 = 0 simultaneously, i.e., M d+1 = M d+2 = 0. Similar to 1st-order topological phase, for all (d − 2)D projective Hamiltonians, M d+1 and M d+2 are the results of h d+1 (k → 0) and h d+2 (k → 0) respectively, which implies that these MDWs on the (d − 2)D real-space interfaces correspond to the (d−2)D momentum-space surfaces where both h d+1 (k) and h d+2 (k) vanishes, defined the 2nd-order BISs B 2 ≡ {k|h d+1 = h d+2 = 0}.This gives a MDW-BIS duality of 2nd-order topological phases, showing that the existence of (d − 2)D gapless boundary states uniquely corresponds to the emergence of 2nd-order BIS [see Fig. S1].It is worth mentioning that we hereby do not emphasize the crystalline symmetries of the Hamiltonian (S6), because the above proof is general.On the contrary, we focus on a lattice Hamiltonian can satisfy the above requirements which leads to the MDWs in Eq. (S9).And then, no matter what kinds of crystalline symmetries it has, we can always use the 2nd-order (or higher-order) BISs to characterize the 2nd-order (or higher-order) topological phases in momentum space.
For the simplest case iii, we only use the (d − 1)D edge Hamiltonian (S9) of k d -OBC to characterize the 2nd-order topological properties of bulk Hamiltonian, since there is no MDW in the (d − 2)D interfaces along the r d direction.We regard the (d − 1)D edge Hamiltonian of k d -OBC as a subsystem and further obtain its (d − 2)D edge Hamiltonian (S10) by using the projection operator d−1 )/2 in the eigenspace of iγ d−1 = −1.Namely, we have H edge (d−2)D = P (1) † P (0) † H dD (k)P (0) P (1) .Note that H edge (d−2)D are topologically equivalent to project the (d − 1)D projective Hamiltonian along k d−1 in momentum space, i.e., the (d − 2)D projective Hamiltonian Considering that H proj (d−1)D (k (1) ) describes a subsystem which is similar to the 1st-order topological phases, thus its topology can be treated as an integer invariant on the projective sth-order BISs B (1) proj,s ≡ {k (1) We next give the topological number V 2 to characterize the bulk Hamiltonian (S6) through the two topological indexes V 1 ≡ w 1 and w 2 .Since the construction of the 2nd-order topological phases is based on the 1st-order topological phases with the nonzero w 1 , i.e., the existence of the (d − 1)D edge states, the extra mass term h d+2 (k) can open energy gap of (d−1)D edge states and whose dispersion is also destroyed [S3, S4].If the system before adding the h d+2 (k)-term is trivial (V 1 = 0) and then there is no (d − 1)D edge states, the 2nd-order topological phases should be trivial even if adding the additional mass term.Hence we take sgn(|V 1 |) to characterize the topology of (d − 1)D edge states [or say the existence of (d − 1)D edge states], where sgn(|w 1 |) = 1 is for topological and sgn(|V 1 |) = 0 is for trivial.Finally, for the Hamiltonian (S6), these (d − 2)D edge states should inherit the topological properties of (d − 1)D edge states and characterized by On the other hand, for k d k d−1 -OBCs, the two components of h d+2 (k (2) ) and h d−1 (k d−1 ) further vanish in this projection process of P (1) .The (d − 2)D edge states on the (d − 2)D interfaces perpendicular to both r d and r d−1 directions are only characterized by BZ (2) with k , which is the projection of the projective 1storder BISs B proj,1 ≡ {k (1) |h d+2 = 0} on the surfaces of h d−1 (k d−1 ) = 0 and has the boundaries B proj,2 ≡ {k (1) |h d+2 = h d−1 = 0} [see Fig. S1(e2)].Particularly, the vanishing h d+1 (k)-term and h d+2 (k)-term gives B 2 , which is equivalent the union of B (1) proj,1 in (d − 1)D momentum subspace k (1) under the MDW-BIS duality.For the case ii, the establishment of the topological characterization is similar to the case iii, except that we need to confirm which (d − 1)D projective Hamiltonian captures these (d − 1)D gapless boundary states.Here we will not discuss it in detail.

3rd-order topological phases
We add a mass term ) in the Hamiltonian (S6) and obtain where the Gamma matrices still satisfy the Clifford algebra.This Hamiltonian (S14) can describe the dD 3rd-order topological phases.Similar to the 1st-order and 2nd-order topological phases, we can obtain the duality between the MDWs in (d − 3)D real-space interfaces and 3rd-order BISs [see  ), thus here each w i is classified by Z invariant and the nth-order topology of bulk Hamiltonian is also classified by the integer invariant.
Remarkably, if the topology of the projective Hamiltonian H proj but it can be determined by sgn(w i ) where w i characterizes a Z-classified Hamiltonian which keeps the dimensions and h-components be same as the Z 2 -classified Hamiltonian but changes the symmetries (i.e., keeping the dimensions but changing the symmetries result in Z 2 → Z), our characterization theory still applies but now H proj (d−i+1)D (k (i−1) ) is needed to be replaced by the Z-classified Hamiltonian.Accordingly, the index w i can be obtained by the topological charges of this Z-classified Hamiltonian and the topological number sgn|w i | for the Z 2 -classified Hamiltonian is naturally recovered in V n .As we show that in the below dynamical characterization, the former case can be observed in 3D 2nd-order TI, while the later special case can be observed in 2D 2nd-order and 3D 3rd-order TIs.

Applications of MDW-BIS duality in three typical models
We first focus on a 2D 2nd-order TI which can be constructed by copying the quantum anomalous hall insulator and adding a mass term to break its time-reversal symmetry [S5].The corresponding Hamiltonian is with Under the approximation of (k x , k y ) → (0, 0), we project the low-energy Hamiltonian of H 2D into the edges of x and y by using P (0) = (1 − iγ 3 γ 2 )/2 and P (0) = (1 − iγ 3 γ 1 )/2 respectively.The Hamiltonians at four edges [I(x > 0), II(y > 0), III(x < 0), IV(y < 0)] are written as One can find that all the terms of brackets ( alternately change along the counterclockwise of edges, which implies that the MDWs should be produced at four corners and each corner holds a zero-energy mode [see Fig. S2(a) and Fig. S3(g)].Accordingly, we clearly observe that the MDWs of four corners are corresponding to the 2nd-order BISs B 2 , which are four points in Fig. S3(d).
Further, we consider the 3D 2nd-order TI of main text and write the corresponding Hamiltonian again with k 1,2,3 = k x,y,z .The Gamma matrices are taken as γ 0), III(y > 0), IV(z < 0), V(x < 0), VI(y < 0)] are written as One can find that the terms of brackets (• • • ) in H surf II,III and H surf V,VI are exactly same, thus the mass terms do not change sign no matter what parameters of m 4 and m 5 .Naturally, there is no MDW in the hinges along z-axis.However, under the parameter conditions of |m 5 | < 2t 0 and |m 4 − m 5 | < t 0 (i.e. the existence B 2 ), the last terms of H surf I,IV have different signs with the remaining Hamiltonians, and the mass terms change signs on the two sides of x and y edges, which induces the MDWs [see Fig. S2(b)].These MDWs on the hinges along x and y are corresponding to the 2nd-order BISs B 2 (i.e., h 4 = h 5 = 0) which presents two rings in the inset of Fig. 3(a) of the main text.
Finally, we consider a 3D 3rd-order TI and the corresponding Hamiltonian is where s x,y,z , σ x,y,z and τ x,y,z are both Pauli matrices.s 0 , σ 0 , and τ 0 are identify matrices.This 3D Hamiltonian only has chiral symmetry with U c = s y ⊗ σ y ⊗ τ y and belongs to the class AIII.The systems hosts the inversion symmetry I = s x ⊗ σ z ⊗ τ x and reflection symmetry R x = s 0 ⊗ σ 0 ⊗ τ x , R y = s 0 ⊗ σ x ⊗ τ y , R z = s x ⊗ σ y ⊗ τ y along x, y, z-direction.We can easily obtain the parameters of the existence of B 3 are |m 6 | < t 0 , |m 4 − m 5 | < t 0 and |m 5 − m 6 | < t 0 .Then we project the low-energy model of H 3D into the hinges of x, y and z by using P (0) = (1 − iγ 4 γ 3 )/2 and P (1) = (1 − iγ BZ (1)   FIG.S4.Configurations of higher-order topological charges and the corresponding kxky-OBC energy spectrum for the 2D 2nd-order TI, where (a) and (b) are for h4 = m4 − t0 cos kx, (c) and (d) are for h4 = m4 − t0 cos (2kx).In (a) and (b), there is a 2nd-order topological charge in B(0) proj,2 , while no 2nd-order BISs B2 exists in BZ, which does not gives BZ (1) .In (c), although there are four 2nd-order BISs and gives BZ (1) , there is no 1st-order topological charge on it.All of these give a trivial 2nd-order topological phase and no corner state.In (d), there is a positive 2nd-order (1st-order) topological charge in B(0) proj,2 ( B(2) proj,1 ), which holds a 2nd-order topological phases and gives four zero-energy modes in each corner.Here the other parameter is tso = t0.
in Fig. S4(d) and there are corner states.While it is trivial in Fig. S4(c) and no corner state exists.This results once again show that the existence of corner states is completely determined by the n-order BISs B n in the original BZ, together with the non-zero invariant on a series of effective BZs.
Besides, we give the phase diagram in Fig. S5(i) again, where the existence of hinge states is determined by the emergence of 2nd-order BISs B 2 , i.e., |m 5 | < 2t 0 and |m 4 − m 5 | < t 0 .For the different parameter points B, D, E, and G, the distribution of zero energy modes in real space further confirm two types of HOTPT points.For B, the modes [96] by introducing Dirac MDWs into the (d − n + 1)D boundary states of a (n − 1)th-order topological phase [see Figs.1(a) and 1(b)].Accordingly, the (d − n)D MDWs in real space can correspond to the momentum-space (d−n)D closed surfaces with vanishing mass terms of Hamiltonian (1), defining the nth-order BISs B n ≡ {k|h d+l = 0, l = 1, 2, . . ., n} [see Fig. 1(c)].This renders a MDW-BIS duality for the nth-order topological phases.Below we briefly illustrate this duality of the Hamiltonian (1).More details of the generic proofs are provided in Supplementary Material [97].

FIG. 1 .
FIG. 1. Schematic of MDW-BIS duality.(a) Construction of dD nth-order TIs from massless semimetals by adding additional mass terms.(b) The corresponding (d − n)D MDWs in real space.(c) The corresponding nth-order BISs Bn in BZ.

FIG. 2 .
FIG. 2. Schematic of HOTPTs.(a) Behaviour of topological charges in the phase transitions.Here the charges cross either the projective BISs [black (i) and green (iii) dots] or the border [red stars (ii)] of BZ (n−1) (gray regions).The boundary gaps for i < n − 1 are assumed to be open.(b) In ii or iii, driving an (n − 1)th-order topological phase.While i together with iii gives an (n − 2)th-order topological phase.The nth-order topological phase shall remains unchanged when only i occurs.
) occurs for i taking values from m to n, yielding at the critical point (d − m + 1)D gapless boundaries characterized by V m−1 [see Fig. 2(b)].For m = 1, all topological charges cross the projective BISs, and the bulk gap closes at the critical point, manifesting the type-I transition.For m > 1, only part of the topological charges cross the projective BISs (or the border of effective BZ).Accordingly, the energy gap closes only for the (d−m+1)D boundaries parallel (or perpendicular) to the lower-dimensional BZ (m−1) , giving a type-II transition.This characterization also further precisely classifies the type-II transition into different m-orders [97]
FIG.S1.MDW-BIS duality for 1st, 2nd, and 3rd-order topological phases in a 3D system.(a1)-(a7) Sketches of the emergence of 1st, 2nd, and 3rd-order BISs in original momentum space, being denoted as a sphere (h4 = 0), two ring-shaped lines (h4 = h5 = 0), and eight points (h4 = h5 = h6 = 0) respectively.The zero-value SO coupling components h1 = 0, h2 = 0, and h3 = 0 are 2D planes of the original BZ.(b)-(c) Sketches of the duality between 2D surface states, 1D hinge states, and 0D corner states in real space and 1st-order, 2nd-order, and 3rd-order BISs in momentum space, where the orange and purple regions denotes the different sign for the effective mass in real space and for the mass term in outside and inside of Bn in momentum space, respectively.The three paths denoted by green, blue, and pink dashed line give that the signs of effective mass of real space and mass term of momentum space have the same changing.(d1)-(d3) Sketches of 2D surface states, 1D hinge states, and 0D corner states in real space, being denoted as the blue surfaces, blue thick lines, and blue points respectively.(e1)-(e3) The positive and negative topological charges "⊕" and " " live in the effective Brillouin zones BZ (i−1) (gray regions)

)
This projective Hamiltonian essentially describes a gapless subsystem in the momentum subspace perpendicular to k d .Since its low-energy effective model is exactly equivalent to Eq. (S4), this projective Hamiltonian H proj (d−1)D is topologically equivalent to H edge (d−1)D and can characterize the (d − 1)D edge states perpendicular to r d direction.Besides, the other (d − 1)D projective Hamiltonians along k d =d -OBC with d = 1, 2, • • • , d − 1 have similar forms with Eq. (S5), and they characterize the (d − 1)D edge states perpendicular to r d direction.With this observation, only the mass term h d+1 (k → 0) gives all M d+1 -terms in the low-energy model, thus these MDWs with M d+1 = 0 on all (d − 1)D real-space interfaces correspond to the (d − 1)D momentum-space surfaces where h d+1 (k)-term vanishes, defined as the 1st-order BISs B 1 ≡ {k ∈ BZ|h d+1 (k) = 0}.This gives the MDW-BIS duality for 1st-order topological phases, showing that the existence of (d − 1)D gapless boundary states uniquely corresponds to the emergence of 1st-order BIS [see Figs.S1(a3), S1(b1), S1(c1), and S1(d1)].Moreover, the numbers of (d − 1)D edge states are characterized by a topological invariant w 1 , defining as a winding on the 1st-order BIS B 1 [S6].

and ignored the k 2 d
-term due to k → 0. One can find that all (d − 1)D edge states can open gap, when the effective masses md+2 and md+2 − md−1 are both non-zero.
S9) by using the projection operator P (0) = (1 − iγ d+1 γ d(d ) )/2 for each low-energy H dD (k) along k d(d ) directions.Clearly, the emergence of (d − 2)D MDWs can be determined by observing the change of the effective mass sign for above (d − 1)D edge Hamiltonians.It should be noted that here γ (1) j[or γ(1) d+2 ] in Eq. (S9) have same forms for all edge Hamiltonians, because we can perform rotation transformation for Gamma matrices without changing the topological property of each edge Hamiltonian and finally this only makes difference for the effective mass signs.Accordingly, the topological change of these edge Hamiltonians as just the changing in the effective mass sign.That can be classified by the following three cases:(i) md+2 and all md+2 − md−1 have same signs, which implies that there is no gapless boundary state in the (d − 2)D real-space interfaces;(ii) The part of md+2 − md−1 have different signs, which naturally makes their signs are different with md+2 , leading to the (d − 2)D gapless boundary states.Yet, these (d − 2)D real-space interfaces hosted MDWs depend on the specific edge Hamiltonians with the different effective masses;(iii) All md+2 − md−1 have same signs but that have different signs with md+2 , leading to the gapless boundary states with the emergence of MDWs in the real-space (d − 2)D interfaces perpendicular to a certain direction r d .

1 . 1
This implies that B 2 shall generate B at the surface of h d = 0. Hence B (1) proj,1 reflects the existence of B 2 , which allows us to identify 2nd-order topological phases though B Fig. S1].Firstly, we regard these (d − 1)D edge Hamiltonians H edge (d−1)D of k d -OBC and k d -OBCs as the (d − 1)D subsystems in Eq. (S9) and obtain their zero-energy modes of (d − 2)D edges, which is similar to the 1st-order case.After performing the straightforward calculation, the (d − 2)D edge Hamiltonians reads ) d+3 , k d =d ,d k j − OBCs (S15) by taking OBCs along two arbitrary directions, where d = 1, 2, • • • , d − 1 and d = 1, 2, • • • , d − 2. Similarly, the signs of effective masses still have three cases as follows: (i) md+3 , all md+3 − md+2 , and all md+3 − md+2 + md+1 have same signs, leading to no gapless boundary state in the (d − 3)D real-space interfaces; (ii) Some of md+3 − md+2 or some of md+3 − md+2 + md+1 have different signs, which naturally makes them have different signs with the remaining effective masses, leading to the (d − 3)D gapless boundary states.But these (d−3)D real-space interface hosted MDWs depend on the specific edge Hamiltonians with the different effective masses; (iii) All md+3 − md+2 and all md+3 − md+2 + md+1 have same signs but have different signs with md+3 , leading to the (d − 3)D gapless boundary states due to emerging MDWs in the real-space (d − 3)D interfaces perpendicular Note that w i describes a winding or Chern number of the projective Hamiltonian H proj (d−i+1)D (k (i−1) and γ 4 = τ y ⊗ σ 0 , where σ x,y,z and τ x,y,z are both Pauli matrices.τ 0 and σ 0 are identity matrices.This 2D Hamiltonian only has chiral symmetryU † c H k U c = −H k with U c = τx ⊗ σ 0 and belongs to the class AIII.The system hosts inversion symmetryI † H k I = H −k with I = τ 0 ⊗ σ z , x-direction reflection symmetry R † x H kx,ky R x = H −kx,ky with R x = τ y ⊗ σ x and C 4 -rotation symmetry C † 4 H kx,ky C 4 = H −ky,kx with C 4 = τ 0 ⊗ e −iπ 4 σz .The existence of 2nd-order BISs B 2 can easily given by |m 4 | < t 0 and |m 3 − m 4 | < t 0 .

and γ 5 =
FIG.S2.Sketches of 0D corner states for 2D and 3D system and 1D hinge states in 3D system.Four edges and whose mass sign are I(+), II(-), III(+), and IV(-) in (a), where the signs of mass term alternately change along the counterclockwise and hold four corner states in 2D 2nd-order TI.Six surfaces and whose mass sign are I(-), II(+), III(+), IV(+), V(+), and VI(-) in (b), where the hinges along x and y hold MDWs due to the sign change of mass terms in 3D 2nd-order TI.In 3D 3rd-order TI of (c), the changing sign of mass terms only are along the hinges of x and y, which products eight corner states.
and V. Shivamoggi, Edge states and the bulk-boundary correspondence in Dirac Hamiltonians, • • • ) have the same sign under the parameter conditions of |m 4 | < t 0 and |m 3 − m 4 | < t 0 (i.e. the existence of B 2 ).The signs of last terms in H edge