Robust topological invariants of topological crystalline phases in the presence of impurities
Ian Mondragon-Shem Taylor L. Hughes





^1 Department of Physics, Yale University, New Haven, Connecticut 06520, USA
^2 Department of Physics, University of Illinois-Urbana Champaign, Urbana, IL 61801, USA


Topological crystalline phases (TCPs) are topological states protected by spatial symmetries. A broad range of TCPs have been conventionally studied by formulating topological invariants (symmetry indicators) at invariant momenta in the Brillouin zone, which leaves open the question of their stability in the absence of translational invariance. In this work, we show that robust basis-independent topological invariants can be generically constructed for TCPs using projected symmetry operators.  Remarkably we show that the real-space topological markers of these invariants are exponentially localized to the fixed points of the spatial symmetry. As a result, this real-space structure  protects them against the presence of impurities that are located away from the fixed points. By considering all possible symmetry centers in a crystalline system we can generate a mesh of real-space topological markers that can provide a local topological distinction for TCPs. We illustrate the  robustness of this mesh of invariants with 1D and 2D TCPs protected by inversion, rotational and mirror symmetries. Finally, we find that the boundary modes of these TCPs can also exhibit robust topological invariants with localized markers on the edges. We illustrate this with the gapless Majorana boundary modes of mirror-symmetric topological superconductors, and relate their integer topological edge invariant with a quantized effective edge polarization.



Robust topological invariants of topological crystalline phases in the presence of impurities
    Ian Mondragon-Shem^1 and Taylor L. Hughes^2
    December 30, 2023
=============================================================================================





Introduction. Over the past decade, dramatic progress has been made in understanding the interplay between symmetry and topology in  quantum phases of matter <cit.>.
A particular focus has been the so-called topological crystalline phases (TCP) <cit.>, which are topological states protected by spatial symmetries. A number of experiments have already revealed their existence in nature <cit.>, and over the course of just a few years, significant theoretical progress has been made <cit.>.

It has been argued that the surface states of TCPs are robust to the presence of disorder as long as the disorder is symmetric on average<cit.>. 
Indeed experimental signatures of TCPs appear to be qualitatively consistent with the notion of an average symmetry, even in crystals that are expected to have moderate amounts of disorder from alloying, e.g., Pb_xSn_1-xTe<cit.>. Bulk studies have also argued the existence of statistical topological insulators which have protected boundary modes as long as the spatial symmetry is preserved on average <cit.>.  Notwithstanding these studies, a bulk understanding of the mechanism behind the robustness of TCPs against the presence of impurities is still lacking, especially in cases in which a notion of average symmetry cannot be clearly defined.  

In this work, we find that a large class of TCPs can be characterized by topological invariants that are robustly quantized in the presence of impurities. To this end, we construct basis-independent topological invariants of TCPs with point-group symmetry. We argue that, since these invariants are expressed in terms of projected symmetry operators, their topological markers are spatially localized to the fixed points of the spatial symmetry. As a result, such invariants are robust againts the presence of impurities. We exemplify this for 1D inversion-symmetric insulators and 2D rotation and mirror-symmetric topological superconductors. In the latter case, we also find phases in which the edge states are characterized by topological invariants that are themselves robustly quantized in the presence of impurities.




Case studies of localized topological markers. We begin by analyzing 1D topological insulators protected by inversion symmetry with an even number of sites L_x.  On a system with periodic boundary conditions and lattice translation symmetry, there is an extensive set of possible inversion symmetry operations P_𝒮 that each leave two positions on the lattice invariant: 𝒮={R_1+r_0, R_2+r_0},  where R_1,2 label unit cells, and r_0 takes one of two values 0,1/2. We thus consider a gapped Hamiltonian Ĥ such that [p̂_𝒮,Ĥ]=0, where p̂_𝒮 is the Hilbert space representation of P_𝒮. 

The conventional topological classification of inversion symmetric insulators proceeds by formulating the problem in momentum space Ĥ=∑_k_x c^†_k_x h(k_x) c^†_k_x. The Bloch Hamiltonian must satisfy p_𝒮(k_x) h(k_x) p^-1_𝒮(k_x)=h(P_𝒮k_x), where  p_𝒮(k_x) is the momentum representation of the inversion operator. At the inversion-invariant momenta k^inv_k_x=0,π, Bloch states are labeled by inversion eigenvalues. Let  n_𝒮^(±)(k_x^inv) denote the number of occupied states with inversion eigenvalues ± 1 at momentum k_x^inv. Let us define the topological invariant Δ^p_𝒮=∑_k_x^inv[n_𝒮^(+)(k_x^inv)-n_𝒮^(-)(k_x^inv)]. In  <cit.>,  it was shown that Δ^p_𝒮|_r_0=1/2 is the integer invariant that distinguishes inversion-symmetric topological states from the trivial atomic limit. A simple model with nontrivial Δ^p_𝒮|_r_0=1/2 is h(k_x)=sink_xσ_2+(m-cosk_x) σ_1, where the σ_a are Pauli matrices acting on two orbitals {A,B} per cell. The inversion operator is p_𝒮(k_x^inv)=e^-2i k_x^inv r_0σ_1, which leads to Δ^p_𝒮|_r_0=1/2=2 (0) in the topological (trivial) phase.

As it stands, the computation of Δ^p_𝒮 relies on momentum being a good quantum number. It is thus unclear to what extent Δ^p_𝒮 can remain quantized when translational symmetry is lost due to the presence of impurites in the system. Even with translation symmetry, using momentum states is just a choice of basis. There must necessarily be a way to compute the topological invariant in a basis-independent manner. To see that this is indeed possible, we define the projected inversion operator p̅_𝒮= 𝒫_Gp̂_𝒮𝒫_G, where 𝒫_G=∑_n∈occ.|u_n⟩⟨u_n| projects into the occupied states. We find that 

    Δ^p_𝒮=Tr[p̅_𝒮].

This basis-independent expression liberates us from the momentum-space description. In particular, instead of analyzing the contributions to Δ^p_𝒮 in momentum space, we can now determine its real-space behavior. It is revealing in this regard to examine the spatially-resolved topological marker 𝒯^p_𝒮(x)= ⟨x|Tr_∘[p̅_𝒮]|x⟩, where Tr_∘ traces over the local degrees of freedom in each unit cell, and Δ^p_𝒮=∑_x=1^L_x𝒯^p_𝒮(x). Topological markers have been used recently in other contexts to study the local properties of topological phases <cit.>. Since  ⟨x,α|p̂_𝒮|x',β⟩∝δ_x,P_𝒮x', and |⟨x',α|𝒫_G |x, β⟩| < 𝒪( e^-| x-x' |/ζ) when  | x-x' |≫ζ, we find that 𝒯^p_𝒮(x) satisfies

    |𝒯^p_𝒮(x)| < 𝒪( e^-2| x-𝒮|/ζ),   | x-𝒮|≫ζ,

where | x-𝒮| denotes the smallest distance of x to any of the points in 𝒮, and the length scale ζ is roughly of the order of the inverse gap/localization length. By taking into account the full set of operators P_𝒮 for a translationally invariant lattice, the topology of the inversion-symmetric insulator is then comprised of a real-space mesh of localized topological markers. We illustrate one of these localized markers in Fig.<ref>a; the inset shows it on a logarithmic scale to confirm its exponential localization.



The localized nature of the marker 𝒯^p_𝒮(x) protects the invariant Δ^p_𝒮 against the presence of impurities. To see this, suppose we add an impurity at a position X_0. The projection operator of the new ground state can be written as 𝒫'_G=𝒫_G+δ𝒫_G. Since the system is gapped, δ𝒫_G only has support in an exponentially small neighborhood of size ζ centered at the impurity. For an impurity placed far away from 𝒮, and using ⟨x|δ𝒫_G|X_0⟩< 𝒪(e^-| X_0-x|/ζ), we find that

    |δΔ^p_𝒮|≡|Δ^p'_𝒮(X_0)-Δ^p_𝒮|<𝒪(e^-2| X_0 -𝒮|/ζ).

Thus, the effect of the impurity on the invariant is exponentially suppressed in the distance between the impurity center and the inversion center(s). To confirm the robustness of Δ^p_𝒮, in Fig.<ref>b we show |δΔ^p_𝒮| as a function of X_0 (the inset shows its behavior on a logarithmic scale). It is clear that the change in the invariant is exponentially suppressed as the impurity moves away from 𝒮 because correlations in the insulator are short-ranged. As a result, Δ^p_𝒮 remains quantized when X_0 is further than ζ from 𝒮. We show the effect of progressively adding more impurities in Fig.<ref>c. Here we calculate the topological marker integrated over a neighborhood of one inversion center: Δ^p_𝒮_i=∑_| x-𝒮_i|  < ζ𝒯^p_𝒮(x),  such that  Δ^p_𝒮≈∑_i=1,2Δ^p_𝒮_i, where 𝒮_i=R_i+r_0 runs over the two invariant points in 𝒮. For our example model in the topological phase, and without impurities, Δ^p_𝒮_i|_r_0=1/2=1, hence Δ^p_𝒮|_r_0=1/2=2. Fig.<ref>c shows Δ^p_𝒮_i for the mesh of all possible 𝒮 as impurities (blue squares) are progressively added to the system. Without impurities we find a uniformly quantized mesh of topological invariants. The impurities destroy the quantization of the mesh at some points (gray dots), while regions subsist in which Δ^p_𝒮_i is robustly quantized (red dots). Thus, we find that even in the presence of impurities the system can retain a local distinction between trivial and non-trivial topological phases (atomic limits).




Next, we consider 2D topological superconductors protected by rotation symmetries and without time-reversal symmetry. With periodic boundary conditions, a given n-fold rotation operation C^(n)_𝒮 leaves invariant a set 𝒮={R_i+𝐫_0} of positions on the lattice (see Fig. <ref>a).  In particular, with primitive vectors 2a_1  and 2a_2, then C^(4)_𝒮 allows 𝐫_0∈{0, a_1+a_2}, whereas C^(2)_𝒮 allows r_0∈{0,a_1,a_2, a_1+a_2}. We consider models that satisfy [ĉ^(n=2,4)_𝒮,Ĥ]=0, where ĉ^(n)_𝒮 is the Hilbert space representation of C^(n)_𝒮. The conventional classification consists of studying Bloch states at the rotation-invariant momenta shown in Fig.<ref>b: ĉ^(n=4)_𝒮 (ĉ^(n=2)_𝒮) leaves invariant the momenta Γ, M (Γ, M, X, X'). At these momenta, the Bloch states can be labeled by rotation eigenvalues. One of the topological invariants of this TCP is a Chern number Ch, which exists regardless of the spatial symmetries. Additionally, there are three more integers ([X],[M_1],[M_2]) <cit.>. These invariants count the number of occupied states at the X,M points that have rotation eigenvalues (e^i π/2, e^iπ/4, e^i 3 π/4) respectively,  relative to those at Γ. 

Rather than writing out momentum-space expressions for these  invariants, we can again express them in a basis independent manner. By defining the projected rotation operators c^(n)_𝒮=𝒫_Gĉ^(n)_𝒮𝒫_G, we obtain

    [X]   =   i/4Tr[(c^(2)_𝒮|_r_0=𝐚_1)+(c̅^(2)_𝒮|_r_0=a_1+a_2)],
    [M_1,2]   =   i/4Tr[√(2)(c̅^(4)_𝒮|_r_0=𝐚_1+a_2)±(c̅^(2)_𝒮|_r_0=𝐚_1)].

The structure of c^(n)_𝒮 again leads to exponentially localized topological markers.  We exemplify this in Fig.<ref>d,e,f for a model having  (Ch, [X],[M_1],[M_2])=(1,1,1,0)  (model details in SM). Furthermore, similar to the inversion-symmetric case, this localization grants these TCPs with robustness against the addition of impurities.

Both the inversion and rotation symmetric examples suggest that other TCPs could also encode topological invariants that are robust to impurities. Naively, one might have attempted a systematic study of such robust invariants in terms of exponentially localized Wannier functions.  However, the rotational case shows that such topological markers are localized even in the presence of a nonzero Chern number, which precludes the construction of localized Wannier functions. The localized nature of topological markers is more general than a Wannier function description might allow. It is fundamentally due to the action of projected symmetry operators.




Robust topological invariants of TCPs. We now show that robust invariants can indeed be constructed using projected symmetry operators for a large class of TCPs invariant under a spatial symmetry ĝ_𝒮.  Consider a Bloch Hamiltonian h(k_G) of a d-dimensional system obtained by Fourier transforming a real-space Hamiltonian where only the spatial components r_G that change under G_𝒮 are transformed; we denote by d_G (≤ d) the number of components of r_G. At the invariant momenta, the states {|u^n_jk_G^inv⟩} are labeled by the eigenvalues {e^iϕ_j(k_G^inv)} of g_𝒮(k_G^inv) where j runs over the set of symmetry representations. 

The topology of a large class of TCPs is encoded by the projection operator at the invariant momenta  𝒫_jk_G^inv=∑_n∈occ.|u^n_jk_G^inv⟩⟨u^n_jk_G^inv|. Let us generically denote the topological invariant encoded by  𝒫_jk_G^inv as τ_j(𝐤_G^inv). As examples, τ_j(𝐤_G^inv) could be the number of occupied states with a particular rotation eigenvalue, or it could be a mirror-Chern number <cit.>, or a mirror-winding  number <cit.>. In all such examples, the τ_j(𝐤_G^inv) can be written in the generic form τ_j(𝐤_G^inv)=Tr[ℱ(𝒫_j𝐤_G^inv)], where ℱ is a function of 𝒫_j𝐤_G^inv, with the constraint that it ultimately projects into a sum over occupied states.  For example, to count the occupied rotation and inversion representations we just have  ℱ(𝒫_jk^inv_G)∝𝒫_j𝐤_G^inv where  (d_G=d in 2D). More generally, it could also involve commutators of 𝒫_jk_G^inv with position operators that are not components of r_G. Examples include mirror (y→ -y) winding numbers for which  ℱ(𝒫_jk_y^inv)∝𝒫_jk_y^inv[X̂,𝒫_jk_y^inv] where (d=2,d_G=1); and mirror (z→-z) Chern invariants for which ℱ(𝒫_jk_z^inv)∝𝒫_j k_z^inv[[X̂,𝒫_jk_z^inv], [Ŷ,𝒫_jk_z^inv]] where  (d=3,d_G=1).

Using these types of topological invariants, we can construct new robust invariants through a remarkably simple linear combination:

    T^g_𝒮≡∑_j,k_G^inv  e^iϕ_j(k_G^inv)τ_j(k_G^inv).
 
By rewriting this expression in terms of a trace over the full Hilbert space, we find that T^g_𝒮=Tr[g_𝒮ℱ(𝒫_G)],  where g_𝒮=𝒫_G ĝ_𝒮𝒫_G, with  𝒫_G=∑_j𝐤_G𝒫_j𝐤_G (see SM for details).  The presence of g_𝒮 leads to spatial topological markers that are localized in the vicinity of 𝒮, i.e.,  |𝒯^g_𝒮(r_G)| <𝒪(e^-| G_𝒮r_G-r_G|/ζ)   when  ζ≪| G_𝒮r_G-r_G|. Furthermore, if we place an impurity at R_0 far from 𝒮, then | T^g'_𝒮(R_0)-T^g_𝒮|<𝒪( e^-| G_𝒮R_0- R_0|/ζ). We thus see that the topology of these TCPs is generically encoded in localized meshes of topological markers that are robust against the presence of impurities. These localized markers are consistent with other real-space layered approaches <cit.>, as well as with approaches using path-integral formulations that use partial symmetry operators <cit.>, though the focus of these works was not on the effects of disorder.






Robust topological markers at the boundary. The boundary modes of TCPs can also exhibit robust topological markers connected with the localized bulk invariants T^g_𝒮. To exemplify this, consider 2D mirror-symmetric superconductors in class DIII <cit.>.  This symmetry class is described by a Bogoliubov-de Gennes Hamiltonian h(k_y) that commutes with a mirror operator m̂_𝒮 that sends y→ -y,  and where 𝒮=(Y_1+y_0,Y_2+y_0) denotes the mirror-invariant lines (see Fig.<ref>a); we assume m̂^2_𝒮=-1. A chiral symmetry χ̂=TC arises from having both time-reversal T and particle-hole C symmetries, and satisfies {T,χ̂}=0. This implies that, in the basis in which m̂_𝒮 is diagonal, each mirror block of h(k^inv_y) belongs to class AIII <cit.>. Using the basis-independent topological winding invariants of class AIII  <cit.>, each mirror block has a topological winding invariant  ν^±_𝒮(k^inv_y)= Tr[ℱ(𝒫_±,k^inv_y)], where  ℱ(𝒫_±,k^inv_y)= 2 L_x^-1𝒫_±,k^inv_y[X̂,𝒫_±,k^inv_y]χ̂, where X̂ is the position operator. By implementing Eq.<ref>, we obtain the robust (bulk) invariant T^m_𝒮=Tr[m_𝒮ℱ(𝒫_G) ]. We define 𝒱^m_𝒮= -i/2T^m_𝒮, with a factor to simplify later expressions. 





The presence of m_𝒮 in 𝒱^m_𝒮 localizes its real space topological markers to mirror lines. Consider three representative cases. (i) If 𝒱^m_𝒮|_y_0=0=𝒱^m_𝒮|_y_0=1/2=1, then the markers have support on every possible mirror line. This implies a non-trivial (strong) DIII ℤ_2 index, and a single Kramers' pair of propagating helical Majorana modes at the boundary. (ii) If 𝒱^m_𝒮|_y_0=0≠ 0 and 𝒱^m_𝒮|_y_0=1/2=0, then non-vanishing markers are centered at y_0=0. Such a phase has a trivial DIII ℤ_2 index, and is adiabatically connected to the limit of decoupled DIII mirror-symmetric wires parallel to the x-axis and stacked in the y-direction; as such it is a weak topological superconductor. (iii) If 𝒱^m_𝒮|_y_0=0=0 and 𝒱^m_𝒮|_y_0=1/2≠ 0, then its DIII strong invariant is trivial, but it has non-trivial markers that are localized on y-lines centered halfway between cells. Thus, while this is still adiabatically connected to a decoupled wire limit, the wires are shifted by half a unit cell in the y-direction compared to case (ii). Indeed this shift is not arbitrary and is the result of a quantized edge topological invariant discussed below.  We illustrate the markers for the latter two cases in Fig. <ref>c,d using models provided in the SM. We see that the marker for case (ii) (Fig. <ref>c) is localized around the center of the unit cell, while the marker for case (iii) (Fig. <ref>d) is localized halfway between unit cells. 

Heuristically, in cases (ii, iii) each mirror line corresponds to a non-trivial 1D topological superconductor that must have Kramers' pairs of Majorana end modes on boundaries with normal vector ±x̂. In case (ii) such Majorana modes should arise along the edge at integer sites in the y-direction, while in (iii) we expect them to be displaced to the mid-point between integers. This suggests that both phases could be distinguished by the location of Majorana modes along the edge.

To quantify this intuition, we examine the distribution of Wannier centers inside a unit cell ℛ at the edge where a mirror line terminates. First, note that with a trivial strong index, the edge spectrum can always be detached from the bulk bands while preserving the DIII and mirror symmetries (see Fig. <ref>b); we denote by 𝒫_edge the projector into this spectrally detached Majorana band (DMB). Since 𝒫_edge includes positive and negative energies related by chiral symmetry, we can write it in terms of projectors into chiral subspaces 𝒫_edge=∑_χ=±𝒫^χ_edge. The Wannier centers {e^-2π i (R_y+ν_R_yj)/L_y} and eigenstates {|Ψ^χ_R_yj⟩} of the DMB are then obtained by diagonalizing the periodic projected position operator 𝒲^χ_edge=𝒫^χ_edge e^-2π i Ŷ/L_y𝒫^χ_edge <cit.>; here, R_y labels unit cells along the edge, and j runs over the N_b states per unit cell included in the DMB. Using these states, we can locally characterize the edge modes at ℛ by the edge shift δ𝒴^χ_edge(ℛ)=(∑_jν_ℛj)mod1, where δ𝒴^+_edge(ℛ)=δ𝒴^-_edge(ℛ) due to time-reversal symmetry. Furthermore, due to mirror symmetry δ𝒴^χ_edge(ℛ)=-δ𝒴^χ_edge(ℛ)  1. Thus, the edge shift must be quantized to either δ𝒴^χ_edge(ℛ)=0 or 1/2. 

Now, the value of δ𝒴^χ_edge(ℛ) can be determined by the (parity of the) number of Wannier centers located at the symmetry centers in the unit cell at ℛ, and the mirror related cell ℛ'. Hence, consider the pair of invariant points S|_y_0=1/2={ℛ+1/2,ℛ'+1/2}. The Wannier states centered at these two symmetry centers are eigenstates of m̂_𝒮|_y_0=1/2, whereas the remaining Wannier states come in mirror-related pairs. It follows that we can write the equality (see SM): δ𝒴^χ_edge(ℛ)=[-i/4∑_R_y j⟨Ψ^χ_R_y j|(m̂_𝒮|_y_0=1/2)|Ψ^χ_R_y j⟩] mod  1. The sum can be replaced by a trace over the Hilbert space of the entire system to obtain  δ𝒴^χ_edge(ℛ)=[1/4Δ^m_𝒮|_y_0=1/2]mod  1, where Δ^m _𝒮≡ -i/2Tr[(𝒫_edgem̂_𝒮𝒫_edge)χ].  If we evaluate the edge invariant Δ^m_𝒮 and use the connection between the bulk AIII winding invariant and the number of zero-energy boundary modes, then we find (see SM for detail) Δ^m_𝒮=𝒱^m_𝒮. The edge shift is then determined by the bulk invariant

    δ𝒴^χ_edge(ℛ)=(1/4𝒱^m_𝒮|_y_0=1/2)mod  1.

We thus conclude that the two possible quantized edge shifts (0, 1/2) correspond to the two phases (ii, iii) we discussed earlier. 

A remarkable by-product of this analysis reveals that the DMB is characterized by the integer topological invariant Δ^m _𝒮 which must be equal to the bulk invariant 𝒱^m_𝒮. Its topological marker is also exponentially localized at S along the edge and is robust to the presence of impurities at the edge. In Fig. <ref>c,d we illustrate the markers of 𝒱^m_𝒮 (labeled bulk marker) and Δ^m _𝒮 (labeled edge marker) as well as representative Wannier functions |Ψ^χ_n|^2 at the edge for two cases: (𝒱^m_𝒮|_y_0=0,𝒱^m_𝒮|_y_0=1/2)=(2,0) and (𝒱^m_𝒮|_y_0=0,𝒱^m_𝒮|_y_0=1/2)=(0,2). There is a clear correspondence between the bulk and edge topological markers that are centered at the mirror-invariant lines. The Wannier functions at the edge are consistently centered at these mirror-invariant points as well. 




Conclusions. In this work we have found that a large class of TCPs are characterized by robustly quantized bulk and edge topological invariants that are constructed using projected symmetry operators. Our results constitute a starting point to further explore the real-space structure of other kinds of TCPs, for example those on non-symmorphic lattices. Furthermore, it suggests a pathway to study disordered systems with interactions by replacing single-particle projectors with Green's functions <cit.>. Finally, our findings point to the possibility of experimentally studying TCPs through local probes; for example, in ultra-cold atomic systems, where the single-particle projection operator can be extracted from time-of-flight measurements <cit.>.


Acknowledgements
We thank M. Hermele and S. Velury for useful conversations. IMS gratefully acknowledges support by the Yale Prize Postdoctoral Fellowship.  TLH thanks the US National Science Foundation (NSF) grants EFMA-1627184 (EFRI), DMR-1351895 (CAREER) for support.

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Supplementary Material
   





§ SECTION I: LOCALIZATION AND ROBUSTNESS OF TOPOLOGICAL MARKERS OF TCPS




 §.§ I.1 Conventional momentum-space classification


Consider a translationally invariant d-dimensional system described by a Hamiltonian Ĥ. Suppose the system is invariant under a spatial symmetry G_𝒮, where 𝒮 denotes collectively the set of points that are left invariant by G_𝒮 in a system with periodic boundary conditions. If we denote by ĝ_𝒮 the Hilbert space representation of G_𝒮, then

    [ĝ_𝒮,Ĥ]=0.

Depending on the dimensionality of the system, when the spatial symmetry acts on a general position vector r, it can leave invariant a subset of its components. We denote by r_I (r_G) the components of r that are invariant (changed) by the operation G_𝒮. The number of components of r_I (r_G) is d_I (d_G) such that d_I+d_G=d.

We study the Bloch Hamiltonian h(k) obtained by Fourier transforming only the spatial coordinates r_G. Thus, k is a wave vector with d_G components. The momentum-space expression of ĝ_𝒮 is

    ĝ_𝒮=∑_kg_𝒮(k)⊗|G_𝒮k⟩⟨k|,

so the action of the spatial symmetry on the Bloch Hamiltonian is then

    g̃_𝒮 (𝐤)h(𝐤) g̃^-1_𝒮(𝐤)= h(G_𝒮𝐤).

At the momenta k^inv that are invariant under G_𝒮, we have

    [g̃_𝒮(𝐤^inv),h(𝐤^inv)]=0.

Thus, the occupied states {|u^n_jk^inv⟩} can be classified according to the eigenvalues {e^iϕ_j(k^inv)} of g̃_𝒮(k^inv). Let us denote the projector into the occupied eigenstates of a given subspace by 𝒫_j𝐤^inv=∑_n∈occ|u^n_jk^inv⟩⟨u^n_jk^inv|. The conventional classification of a large class of TCPs is based on determining the topology encoded by 𝒫_j𝐤^inv. 

The set of topological invariants {τ_j(k^inv)} that characterize the invariant subspaces depend on their effective dimensionality d_s and the other symmetries of the system. Examples of possible invariants that could be relevant include occupation numbers (e.g. <cit.>), winding numbers (e.g. <cit.>) and Chern numbers (e.g. <cit.>). Such topological invariants {τ_j(k^inv)} can generally be written as the average over occupied states of an operator ℱ that is a function of 𝒫_j𝐤^inv:

    τ_j(𝐤_inv)=Tr_∘[ℱ(𝒫_j𝐤_inv)],
 
where Tr_∘ refers to the trace over the non-spatial degrees of freedom as well as the d_I spatial degrees of freedom that do not get transformed by G_𝒮. For a broad range of TCPs, the function ℱ is written as a sum of products of projection operators and position operators that are components of r_I, such that it projects the trace into a sum over occupied states. Examples of the form of ℱ are

    ℱ(𝒫)∝{[            𝒫(e.g. 0D occupation number <cit.> for 1D TCP),;         𝒫[X̂,𝒫](e.g. 1D winding number <cit.> for 2D TCP),; 𝒫[[X̂,𝒫], [Ŷ,𝒫]] (e.g. 2D Chern number <cit.> for 3D TCP), ].

where X̂,Ŷ are the x,y components of the position operator.  These commutators are the basis-independent generalizations of momentum-space derivatives that naturally arise en the calculations of Berry connections. 



 §.§ I.2 Basis-independent topological invariants of TCPs


We now set out to obtain topological invariants with localized topological markers using the momentum-space invariants {τ_j(k^inv)}. Let us now define the quantities

    T^g_𝒮≡∑_j,k^inv  e^iϕ_j(k^inv)τ_j(k^inv).
 
Since the {T^g_𝒮} are linear combinations of the momentum-space topological invariants, they are themselves topological invariants of the TCP. We will now show that these topological invariants have topological markers that are that are exponentially localized and are robust to the presence of impurities in the system. This will be done by expressing them in terms of the projector in the full ground state given by

    𝒫_G=∑_jk𝒫_kj.

We can rewrite them in a basis-independent form:

    T^g_𝒮   =   ∑_j,k^inv  e^iϕ_j(k^inv)τ_j(k^inv)
       =   ∑_j,k^invTr_∘[e^iϕ_j(k^inv)ℱ(𝒫_j,𝐤^inv)]
       =   ∑_j,k^invTr_∘[e^iϕ_j(k^inv)𝒫_j𝐤_invℱ(𝒫_j,𝐤^inv)]
       =   ∑_j,k^invTr_∘[𝒫_j𝐤^inve^iϕ_j(k^inv)𝒫_j𝐤^invℱ(𝒫_j,𝐤^inv)].

Now, since r̂_I and 𝒫_jk^inv commute with g_𝒮(k^inv), then [g_𝒮(k^inv),ℱ(𝒫_jk^inv)]=0.  This implies that ℱ(𝒫_jk^inv) is block-diagonal in the basis of g_𝒮(k^inv), and we can thus write ∑_jℱ(𝒫_j𝐤^inv)=ℱ(∑_j𝒫_j𝐤^inv). Furthermore, by assumption, the terms that make up ℱ(𝒫_G) are invariant under translations along the components of r̂_G. This means that ℱ is block-diagonal with respect to 𝐤. As a consequence

    ∑_kℱ(𝒫_𝐤)⊗|k⟩⟨k|   =   ℱ(∑_k𝒫_𝐤⊗|k⟩⟨k|)=ℱ(𝒫_G).
 
Using these properties of ℱ, we then obtain

    T^g_𝒮   =   ∑_k^invTr_∘[(∑_j_1𝒫_j_1𝐤^inv)g_𝒮(k^inv)(∑_j_2𝒫_j_2𝐤^inv)ℱ(∑_j_3𝒫_j_3𝐤^inv)]
       =   ∑_k^invTr_∘[(𝒫_𝐤^inv)g_𝒮(k^inv)(𝒫_𝐤^inv)(ℱ(𝒫_𝐤^inv))]
       =   ∑_k^inv⟨k^inv|Tr_∘[(𝒫_k^inv⊗|k^inv⟩⟨k^inv|)(g̃_𝒮(k^inv)⊗|k^inv⟩⟨k^inv|). 
          ×. (𝒫_k^inv⊗|k^inv⟩⟨k^inv|)(ℱ(𝒫_k^inv)⊗|k^inv⟩⟨k^inv|)]|k^inv⟩
       =   ∑_k^inv⟨k^inv|Tr_∘[(∑_q_1𝒫_𝐪_1⊗|q_1⟩⟨q_1|)(∑_q_2g̃_𝒮(q_2)⊗|G_𝒮q_2⟩⟨q_2|). 
          ×. (∑_q_3𝒫_𝐪_3⊗|q_3⟩⟨q_3|)(∑_q_4ℱ(𝒫_𝐪_4)⊗|q_4⟩⟨q_4|)]|k^inv⟩
       =   ∑_k⟨k|Tr_∘[(∑_q_1𝒫_𝐪_1⊗|q_1⟩⟨q_1|)(∑_q_2g̃_𝒮(q_2)⊗|G_𝒮q_2⟩⟨q_2|). 
          ×.(∑_q_3𝒫_𝐪_3⊗|q_3⟩⟨q_3|)ℱ(∑_q_4𝒫_𝐪_4⊗|q_4⟩⟨q_4|)]|k⟩
       =   ∑_k⟨k|Tr_∘[𝒫_Gĝ_𝒮𝒫_Gℱ(𝒫_G)]|k⟩
       =   Tr[g_𝒮ℱ(𝒫_G)].

which is the basis-independent expression quoted in the main text for these topological invariants.



 §.§ I.3 Localized topological markers


We now show that the presence of the projected symmetry operator g_𝒮 in our expression for T^g_𝒮 implies that the topological markers are spatially localized. Consider a position 𝐫_G far away from the fixed points 𝒮 (a condition we denote by |𝒮-𝐫_G|≫ζ), which implies that | G_𝒮𝐫_G-𝐫_G|≫ζ. We know that for gapped systems, the projection operator has the long-distance behavior

    |⟨r_G_1 ,α|𝒫_G|r_G_2,β⟩|<𝒪(e^-|r_G_1-r_G_2|/ζ)  when  ζ≪|r_G_1-r_G_2|,

where α,β denote all internal and remaining spatial degrees of freedom.
Furthermore, since ℱ is only dependent on 𝒫_G and the components of r̂_̂Î, then

    |⟨r_G_1 ,α|ℱ(𝒫_G)|r_G_2,β⟩|<𝒪(e^-|r_G_1-r_G_2|/ζ)  when  ζ≪|r_G_1-r_G_2|.

Using this, we can then write the topological marker 𝒯^g_𝒮(r_G)=⟨r_G|Tr_∘[g_𝒮ℱ(𝒫_G)]|r_G⟩ as

    𝒯^g_𝒮(r_G)   =   Tr_∘[⟨r'_G|ĝ_𝒮𝒫_Gℱ(𝒫_G)|r_G⟩]
       =   ∑_r_G1r_G2Tr_∘[⟨r_G|ĝ_𝒮|r_G1⟩⟨r_G1|𝒫_G|r_G2⟩⟨r_G2|ℱ(𝒫_G)|r_G⟩]
       =   ∑_r_G1r_G2Tr_∘[δ_r_G,G_𝒮r_G1⟨r_G1|𝒫_G|r_G2⟩⟨r_G2|ℱ(𝒫_G)|r_G⟩]
       =   ∑_r_G2Tr_∘[⟨G^-1_𝒮r_G|𝒫_G|r_G2⟩⟨r_G2|ℱ(𝒫_G)|r_G⟩]
       =   ∑_|r_G2-r_G|<ζTr_∘[⟨G^-1_𝒮r_G|𝒫_G|r_G2⟩⟨r_G2|ℱ(𝒫_G)|r_G⟩]
          +∑_|r_G2-G^-1_𝒮r_G|<ζTr_∘[⟨G^-1_𝒮r_G|𝒫_G|r_G2⟩⟨r_G2|ℱ(𝒫_G)|r_G⟩]
          +∑_|r_G2-G^-1_𝒮r_G|>ζ
    |r_G2-r_G|>ζTr_∘[⟨G^-1_𝒮r_G|𝒫_G|r_G2⟩⟨r_G2|ℱ(𝒫_G)|r_G⟩].

Now, we have that 

    A_1   =   |∑_|r_G2-r_G|<ζTr[⟨G^-1_𝒮r_G|𝒫_G|r_G2⟩⟨r_G2|ℱ(𝒫_G)|r_G⟩]|<𝒪(e^-| G^-1_𝒮r_G-r_G|/ζ)
    
        A_2   =   |∑_|r_G2-G^-1_𝒮r_G|<ζTr[⟨G^-1_𝒮r_G|𝒫_G|r_G2⟩⟨r_G2|ℱ(𝒫_G)|r_G⟩]|<𝒪(e^-| G^-1_𝒮r_G-r_G|/ζ)
    
        A_3   =   |∑_|r_G2-G^-1_𝒮r_G|>ζ
    |r_G2-r_G|>ζTr[⟨G^-1_𝒮r_G|𝒫_G|r_G2⟩⟨r_G2|ℱ(𝒫_G)|r_G⟩]|≪ A_1,2.

We thus conclude that

    |𝒯^g_𝒮(r_G)|   <   𝒪(e^-| G_𝒮r_G-r_G|/ζ)   when  ζ≪| G_𝒮r_G-r_G|.

which implies that the topological marker is concentrated around the fixed points of the spatial symmetry.



 §.§ I.4 Robustness to impurities placed far from fixed points.

We now show that although the conventional momentum-space invariants cannot be used in the presence of impurities, the invariants T^g_𝒮 can remain robustly quantized. Suppose we add an impurity V_imp(r)=V_0δ_r,R_0 at a position R_0. 
The projection operator of the new ground state can be written as 𝒫'_G=𝒫_G+δ𝒫_G. Since the system is gapped, δ𝒫_G  only has support in an exponentially small neighborhood of size ζ centered at the impurity:

    |⟨r_G,α|δ𝒫_G|R_0,β⟩|< 𝒪(e^-(|r_G-R_0|)/ζ).

Let us now define a function Γ(𝒫'_G) by

    δΓ(𝒫'_G)≡𝒫'_Gℱ(𝒫'_G)-𝒫_Gℱ(𝒫_G).

The function Γ(𝒫'_G) must vanish when δ𝒫_G=0, so it must be a sum of terms all of which possess at least one factor of δ𝒫_G. The long-distanceIt follows that 

    |⟨r_G,α|Γ(𝒫'_G) |R_0,β⟩|<𝒪(e^-|r_G-R_0|/ζ)   for  |r_G-R_0|≫ζ

For an impurity placed far away from the points in 𝒮, the topological invariant is modified as

    T^g'_𝒮(R_0)-T^g_𝒮   =   Tr[ĝ_𝒮Γ(𝒫'_G)]
       =   ∑_r_Gr_G1Tr_∘[⟨r_G|ĝ_𝒮|r_G1⟩⟨r_G1|Γ(𝒫'_G)|r_G⟩]
       =   ∑_r_Gr_G1Tr_∘[δ_r_G,G_𝒮r_G1⟨r_G1|Γ(𝒫'_G)|r_G⟩]
       =   ∑_r_GTr_∘[⟨G^-1_𝒮r_G|Γ(𝒫'_G)|r_G⟩]
       =   ∑_| G^-1_𝒮r_G-R_0|<ζTr_∘[⟨G^-1_𝒮r_G|Γ(𝒫'_G)|r_G⟩]
          +∑_|r_G-R_0|<ζTr_∘[⟨G^-1_𝒮r_G|Γ(𝒫'_G)|r_G⟩]
          +∑_| G^-1_𝒮r_G-R_0|>ζ
    |r_G-R_0|>ζTr_∘[⟨G^-1_𝒮r_G|Γ(𝒫'_G)|r_G⟩].

Now, we have that 

    B_1   =   |∑_| G^-1_𝒮r_G-R_0|<ζTr_∘[⟨G^-1_𝒮r_G|Γ(𝒫'_G)|r_G⟩]|<𝒪(e^-|R_0-G_𝒮R_0|/ζ)
    
        B_2   =   |∑_|r_G-R_0|<ζTr_∘[⟨G^-1_𝒮r_G|Γ(𝒫'_G)|r_G⟩]|<𝒪(e^-| G^-1_𝒮R_0-R_0|/ζ)
    
        B_3   =   |∑_| G^-1_𝒮r_G-R_0|>ζ
    |r_G-R_0|>ζTr_∘[⟨G^-1_𝒮r_G|Γ(𝒫'_G)|r_G⟩]|≪ B_1,2.

We thus conclude that

    | T^g'_𝒮(R_0)-T^g_𝒮|<𝒪( e^-| G_𝒮R_0- R_0|/ζ),

which implies that impurities in the system will not affect the quantization of the invariants T^g_𝒮 as long their distance to 𝒮 is many times bigger than a correlation length ζ.







§ SECTION II:  INVERSION-SYMMETRIC TOPOLOGICAL INSULATORS IN ONE DIMENSION




 §.§ II.1 Conventional classification


Consider topological insulators in one-dimensional protected by inversion symmetry. With periodic boundary conditions, an inversion symmetry operation P_𝒮 leaves invariant two positions 𝒮={R+r_0,R+1+L_x/2+r_0} on the lattice with R≤ L_x/2. The number r_0 is a fixed position within the unit cell that can take the values 0 or 1/2, where we set the unit cell length to be a=1. Consider the momentum-space expansion of the inversion operator

    p̂_𝒮=∑_k_xp_𝒮(k_x)⊗|P_𝒮k_x⟩⟨k_x|.

The conventional momentum-space classification is based on counting the number of occupied states n^±_𝒮(k^inv_x) with eigenvalues of the operator p_𝒮(k_x) at the invariant momenta k^inv_x=0,π.  The invariant is given by <cit.>

    Δ^p_𝒮   =   ∑_k_x=0,π[n_𝒮^(+)(k_x)-n_𝒮^(-)(k_x)].

which is an integer invariant.



 §.§ II.2 Basis-independent classification


We now discuss the topological invariants T^p_𝒮 for this TCP. The topological invariants at the invariant momenta are 

    τ_±(k^inv_x)=n^±_𝒮(k^inv_x)=Tr_∘[𝒫_±,k^inv_x].
 
It is clear then that, in this case, ℱ(𝒫_±,k^inv_x)=𝒫_±,k^inv_x. By using these invariants in Eq.<ref>, we find

    T^p_𝒮   =   ∑_k_x=0,π[(+1)n_𝒮^(+)(k_x)+(-1)n_𝒮^(-)(k_x)]=Tr[p_𝒮·𝒫_±,k^inv_x]=Tr[p_𝒮].

In this particular example, the linear combination T^p_𝒮 of momentum-space topological invariants is precisely the topological invariant Δ^p_𝒮 that was proposed for these TCPs <cit.>.
The topological marker for this invariant satisfies

    |𝒯^p_𝒮(x)|   <   𝒪(e^-2| x-𝒮|/ζ)   when  ζ≪| x-𝒮|,

and an impurity placed at x=X_0 will change an invariant centered at 𝒮 by an exponentially small amount

    | T^p'_𝒮(X_0)-T^p_𝒮|<𝒪( e^-2| X_0- 𝒮|/ζ)   when  ζ≪| X_0-𝒮|,

where the notation | X_0 -𝒮| means the shortest distance to any of the invariant points in the set 𝒮.



 §.§ II.3 Model example


The model presented in the main text is

    h(k_x)=sin k_x σ_2+(m-cos k_x)σ_1,

where the Pauli matrices σ_a act on two orbitals {A,B}. The momentum-space inversion operator is p_𝒮(k_x^inv)=e^-2i k_x^inv r_0σ_1. The topological phase occurs for | m|<1. For the figures in the main text we chose m=0.3.









§ SECTION III: ROTATIONALLY-SYMMETRIC TOPOLOGICAL SUPERCONDUCTORS IN TWO DIMENSIONS




 §.§ III.1 Conventional classification


Consider topological superconductors in the absence of time-reversal symmetry and protected by rotational symmetries. With periodic boundary conditions, a given n-fold rotation operation C^(n)_𝒮 leaves invariant a set 𝒮={R_i+𝐫_0} of positions on the lattice, as illustrated in Fig. <ref>.  In particular, we will consider models that satisfy [ĉ^(n=2,4)_𝒮,Ĥ]=0, where ĉ^(n)_𝒮 is the Hilbert space representation of C^(n)_𝒮. By denoting the primitive vectors as 2a_1  and 2a_2, then four-fold rotations allow the positions 𝐫_0∈{0, a_1+a_2}, whereas two-fold rotations allow r_0∈{0,a_1,a_2, a_1+a_2}. Consider the momentum-space expansion of the rotation operators

    ĉ^(n)_𝒮=∑_kc^(n)_𝒮(k)⊗|C^(n)_𝒮k⟩⟨k|.

The conventional momentum-space classification is based on counting eigenvalues of the specific operator c^(n)_𝒮(k)|_r_0=0. For a given invariant momentum Π^(n) (=Γ,X,M_i), the possible eigenvalues of c^(n)_𝒮(k)|_r_0=0 are Π^(n)_p=e^iπ(2p-1)/n, for p=1,2,…, n. Let us denote by #Π^(n)_p the number of occupied states with eigenvalue Π^(n)_p at the momentum Π^(n). The topological invariants are then obtained by defining a relative occupation number

    [Π^(n)_p]=#Π^(n)_p-#Γ^(n)_p.

In particular, as we mentioned in the main text, the topological invariants for systems that are symmetric under four-fold rotations are ([X],[M_1],[M_2]).



 §.§ III.2 Basis-independent classification


We now proceed to adapt Eq.<ref> to these TCPs. We define the topological invariants of the invariant subspaces as the occupation numbers n^(n,p)_𝒮(Π^(n)) of the eigenvalues of the operator c^(n)_𝒮(k): 

    τ_np(Π^(n))=n^(n,p)_𝒮(Π^(n))=Tr_∘[𝒫_jk^inv],

where we again have used ℱ(𝒫_jk^inv)=𝒫_jk^inv.  From Eq.<ref>, we know that this implies that

    T^c(n)_𝒮=Tr[c^(n)_𝒮ℱ(𝒫_G)]=Tr[c^(n)_𝒮].

In contrast with the inversion-symmetric case, the conventional momentum-space topological invariants are not exactly  the invariants T^c(n)_𝒮. We can, nevertheless, find a relation between both. The rotation operator at an invariant momentum for a given set of fixed points 𝒮 can be written in the form 

    c^(n)_𝒮(k)=[c^(n)_𝒮(k)|_r_0=0] e^i(C^(n)_𝒮Π^(𝐧)-Π^(𝐧))·𝐫_0,

which implies that their eigenvalues are

    e^iϕ_np(Π^(n))=e^i(π(2p-1)/n+(C^(n)_𝒮Π^(n)-Π^(n))·𝐫_0).

Since the Π^(n) are invariant momenta under rotations by definition, (C^(n)_𝒮Π^(n)-Π^(n))·𝐫_0=∑_i m_i b_i where the m_1,2 are integers and the b_1,2 are primitive vectors in reciprocal space that satisfy b_i· (2a_j)=2πδ_ij. With these definitions, the traces of the projected rotation operators for n=4 can be written as

    T^c(4)_𝒮   =   ∑^4_p=1(e^iϕ_4p(Γ)n^(4,p)_𝒮(Γ)+e^iϕ_4p(𝐌) n^(4,p)_𝒮(𝐌))
       =   ∑^4_p=1( e^i π(2p-1)/4#Γ_p+e^i π(2p-1)/4-i b_1·r_0#M_p)
       =   ∑^4_p=1e^i π(2p-1)/4( (1+e^-i b_1·r_0)#Γ_p+e^-i b_1·r_0[M_p])
       =   {(1+e^-i b_1·r_0) (∑_pe^i π(2p-1)/4#Γ_p)+i√(2)e^-i b_1·r_0([M_1]+[M_2]) },

where we have used that [M_1]=-[M_4] and [M_2]=-[M_3] due to particle-hole symmetry. For the n=2 case, the topological invariants are

    T^c(2)_𝒮   =   ∑_p(e^iϕ_2p(Γ)n^(2,p)_𝒮(Γ)+e^iϕ_2p(𝐌) n^(2,p)_𝒮(𝐌))
       =   i( (#Γ_1+#Γ_3)+e^-i (b_1+b_2)·r_0 (# M_1+# M_3)+(e^-i b_1·r_0+e^-i b_2·r_0) # X_1)
          -i( (#Γ_2+#Γ_4)+e^-i (b_1+b_2)·r_0 (# M_2+# M_4)+(e^-i b_1·r_0+e^-i b_2·r_0) # X_2)
       =   i (1+e^-i b_1·r_0+e^-i b_2·r_0+e^-i (b_1+b_2)·r_0)(#Γ_1+#Γ_3)
          +i e^-i (b_1+b_2)·r_0 ([M_1]+[M_3])+i (e^-i b_1·r_0+e^-i b_2·r_0) [ X_1]
          -i (1+e^-i b_1·r_0+e^-i b_2·r_0+e^-i (b_1+b_2)·r_0)(#Γ_2+#Γ_4)
          -i e^-i (b_1+b_2)·r_0 ([M_2]+[M_4])-i(e^-i b_1·r_0+e^-i b_2·r_0) [ X_2]
       =   2i{(1+e^-i b_1·r_0+e^-i b_2·r_0+e^-i (b_1+b_2)·r_0/2)(#Γ_1+#Γ_3-#Γ_2-#Γ_4).
          .+e^-i (b_1+b_2)·r_0 ([M_1]-[M_2])+(e^-i b_1·r_0+e^-i b_2·r_0) [ X]},

where we used [X]≡[X_1]=-[X_2] due to particle-hole symmetry as well. By evaluating these expressions at the rotation centers r_0=a_1,a_1+a_2, we obtain

    T^c(2)_𝒮|_r_0=a_1   =   -2i([M_1]-[M_2])
    
    T^c(2)_𝒮|_r_0=a_1+a_2   =   2i{  ([M_1]-[M_2])-2 [ X]}
    
    T^c(4)_𝒮|_r_0=a_1+a_2   =   -i√(2)([M_1]+[M_2]),

which finally implies, using Eq.<ref>, that

    [X]   =   i/4(T^c(2)_𝒮|_r_0=a_1+T^c(2)_𝒮|_r_0=a_1+a_2)=i/4Tr[(c^(2)_𝒮|_r_0=𝐚_1)+(c̅^(2)_𝒮|_r_0=a_1+a_2)],
    [M_1,2]   =   i/4(√(2)T^c(4)_𝒮|_r_0=a_1+a_2± T^c(2)_𝒮|_r_0=a_1)=i/4Tr[√(2)(c̅^(4)_𝒮|_r_0=𝐚_1+a_2)±(c̅^(2)_𝒮|_r_0=𝐚_1)],

which are the expressions discussed in the main text. The topological marker for this invariant satisfies

    |𝒯^c(n)_𝒮(r)|   <   𝒪(e^-2sin[π/n]|r-𝒮|/ζ)   when  ζ≪| x-𝒮|,

and an impurity placed at r=R_0 will change an invariant centered at 𝒮 by an exponentially small amount

    | T^c(n)'_𝒮(R_0)-T^c(n)_𝒮|<𝒪( e^-2sin[π/n]|R_0- 𝒮|/ζ)   when  ζ≪|R_0-𝒮|,

where the notation |𝐑_0 -𝒮| means the shortest distance to any of the invariant points in the set 𝒮.



 §.§ III.3 Model example

To exemplify the topological markers associated with these invariants, we used the model

    h(𝐤)=sin(𝐤.𝐚_1)τ_1+sin(𝐤.𝐚_2)τ_2+u_1[cos(𝐤.𝐚_1)+cos(𝐤.𝐚_2)]τ_3 
    +u_2[cos(𝐤.𝐚'_1)+cos(𝐤.𝐚'_2)]τ_3,

where a_1=a(1,0), a_2=a(0,1), a'_1=a_1+a_2, and a'_1=-a_1+a_2. The Pauli matrices τ_a act on the Nambu degree of freedom. The four-fold rotation operator is given by c^(4)_𝒮(k)|_r_0=0=1/√(2)(τ_0+iτ_3). The specific parameters chosen were u_1=1.0 and u_2=0.5.





§ SECTION IV: MIRROR-SYMMETRIC DIII TOPOLOGICAL SUPERCONDUCTORS IN TWO DIMENSIONS




 §.§ IV.1 Conventional classification


In the main text, we discussed the basis-independent approach to the classification of mirror-symmetric DIII topological supercondctors in two dimensions. Here, we will make contact with the momentum-space classification in order to connect with Eq.<ref>. 

As we mentioned in the main text, a 2D mirror-symmetric superconductor in class DIII is described by a Bogoliubov-de Gennes Hamiltonian Ĥ such that [m̂_𝒮,Ĥ]=0. The mirror-invariant lines are located at 𝒮=(Y_1+y_0,Y_2+y_0), with y_0=0,1/2. We focus on the case when m̂^2_𝒮=-1. If we Fourier transform only the coordinate that changes under mirror symmetry, the mirror constraint on the Bloch Hamiltonian is

    m_𝒮(k_y) h(k_y) m^-1_𝒮(k_y)=h(-k_y),

where m_𝒮(k_y)  is the momentum representation of the mirror operator.  At the mirror invariant momentum lines k^inv_y=0,π, the Bloch states can be labeled with mirror eigenvalues. By using the  chiral symmetry χ=TC that arises from having both time-reversal T and particle-hole C symmetries, one can define topologically invariant mirror-winding numbers given by <cit.>

    ν_𝒮^(±)(k^inv_y)= -1/N_xTr[(S_-𝒬_±,k^inv_yS_+) [X̂, (S_+𝒬_±,k^inv_yS_-)]],

where 𝒬_±,k^inv_y=2𝒫_±,k^inv_y-𝕀, S_±=(𝕀±χ̂)/2 projects into chiral subspaces with eigenvalue χ=±1, and 𝒫_±,k^inv_y is the projection operator into the occupied states with mirror eigenvalue ± i. Due to time-reversal symmetry, ν_𝒮^(+)(k^inv_y)=-ν_𝒮^(-)(k^inv_y). The integers at the two invariant momenta leads to the known ℤ×ℤ classification <cit.>. To express the winding in terms explicitly of projection operators, we write

    ν_𝒮^(±)(k^inv_y)   =   -1/N_xTr[(S_-𝒬_±,k^inv_yS_+) [X̂, (S_+𝒬_±,k^inv_yS_-)]]
       =   -1/N_xTr[(2𝒫_±,k^inv_y-𝕀) S_+[X̂,(2𝒫_±,k^inv_y-𝕀)]S_-]
       =   -4/N_xTr[𝒫_±,k^inv_y[X̂,S_+𝒫_±,k^inv_yS_-]]
       =   -4/N_xTr[𝒫_±,k^inv_y[X̂,((𝕀+ χ̂)/2)𝒫_±,k^inv_y((𝕀- χ̂)/2)]]
       =   -1/N_xTr[𝒫_±,k^inv_y[X̂,𝒫_±,k^inv_y+χ̂𝒫_±,k^inv_y-𝒫_±,k^inv_yχ̂-χ̂𝒫_±,k^inv_yχ̂]]
       =   -1/N_xTr[𝒫_±,k^inv_y[X̂,𝒫_±,k^inv_y+(χ̂-𝒫_±,k^inv_yχ̂)-𝒫_±,k^inv_yχ̂-(𝕀-𝒫_±,k^inv_y)]]
       =   -2/N_xTr[𝒫_±,k^inv_y[X̂,𝒫_±,k^inv_y]]+2/N_xTr[𝒫_±,k^inv_y[X̂,𝒫_±,k^inv_yχ̂]]
       =   2/N_xTr[𝒫_±,k^inv_y[X̂, 𝒫_±,k^inv_y] χ̂],

which is the expression we presented in the main text.



 §.§ IV.2 Basis-independent classification

We now implement Eq.<ref> for this TCP. As is clear from the previous section, the topological invariants of the invariant subspaces are

    τ^±_𝒮,b(k^inv_y)=ν^±_𝒮(k^inv_y),

where we added the index b to be reminded that it refers to the bulk invariant, as opposed to the edge invariant that we introduce in the next subsection. Furthermore, we have that

    ℱ(𝒫)=2N^-1_x𝒫[X̂,𝒫]χ.
 By applying Eq.<ref>, we then have the topological invariants

    T^m_𝒮=∑_k^inv_y[(+i)ν^+_𝒮(k^inv_y)+(-i)ν^-_𝒮(k^inv_y)]= Tr[m_𝒮ℱ(𝒫_G)].

The relation with the invariant we defined in the main text is then

    𝒱^m_𝒮=-i/2 T^m_𝒮

The topological marker for this invariant satisfies

    |𝒯^m_𝒮(y)|   <   𝒪(e^-2| y-𝒮|/ζ)   when  ζ≪| y-𝒮|,

and an impurity placed at y=Y_0 will change an invariant centered at 𝒮 by an exponentially small amount

    | T^m'_𝒮(Y_0)-T^m_𝒮|<𝒪( e^-2| Y_0- 𝒮|/ζ)   when  ζ≪| Y_0-𝒮|.




 §.§ IV.3  Integer topological invariants for the edge


In the main text, we used the bulk-boundary relation Δ^m_𝒮=-i/2Tr[m^edge_𝒮χ]=𝒱^m_𝒮, which we will now prove in this section. To do this, we begin by noting that we can write Δ^m_𝒮 in the suggestive form

    Δ^m_𝒮   =   -i/2Tr[m_𝒮ℱ(𝒫_edge)],

which has the form of the invariants in Eq.<ref>, in this case with the function ℱ(𝒫)=𝒫χ. Thus, in momentum space we must obtain that Δ^m_𝒮 receives contribution exclusively from the invariant momenta in the form

    Δ^m_𝒮   =   (-i/2)∑_k^inv_y[(+i)τ^edge_+(k^inv_y)+(-i)τ^edge_-(k^inv_y)],

where we defined new edge topological invariants

    τ^edge_m(k^inv_y)≡Tr[𝒫_edge(k^inv_y)ℱ(𝒫_edge(k^inv_y))]=Tr[𝒫_edge(k^inv_y)χ],
 
with 𝒫_edge(k^inv_y) the projector onto the boundary states with momentum k^inv_y and mirror eigenvalue m · i. Because of the form of the function ℱ in this case, the quantities τ^edge_m(k^inv_y) count the difference

    Tr[𝒫_edge(k^inv_y)χ]=∑_n ∈edge⟨u^n_k^inv_y|χ|u^n_k^inv_y⟩=𝒩^m_+(k^inv_y)-𝒩^m_-(k^inv_y)

at the invariant momentum k^inv_y in a given mirror subspace m, with 𝒩^m_χ(k^inv_y) the number of modes with chiral eigenvalue χ=± 1. Due to the bulk-boundary correspondence of AIII systems, this difference must be equal to the bulk topological winding Eq.<ref>, and thus

    τ^edge_m(k^inv_y)=ν^(m)_𝒮(k^inv_y).

Finally, putting together Eq.<ref>, Eq.<ref> and Eq.<ref>, we obtain the relation

    Δ^m_𝒮=1/2∑_k^inv_y[τ^edge_+(k^inv_y)-τ^edge_-(k^inv_y)]=1/2∑_k^inv_y[ν^(+)_𝒮(k^inv_y)-ν^(-)_𝒮(k^inv_y)]=𝒱^m_𝒮.

as we had set out to prove.



 §.§ IV.4 Quantized edge shift of gapless Majorana boundary modes


In this section, we derive Eq.7 from the main text, which relates the local edge shift δ𝒴^χ_edge(R_1) with the bulk invariant 𝒱^m_𝒮.  Consider the projected position operator 𝒲^χ_edge=𝒫^χ_edge e^-2π i Ŷ/L_y𝒫^χ_edge. We denote its eigenvalues and eigenstates by {e^-2π i λ_R_yj/L_y} and {|Ψ^χ_R_yj⟩}, respectively. Here, R_y labels unit cells along the edge, and j=1,…, N_b, where N_b is the number of subbands included in the detached Majorana band (DMB). Thus, the number of states in the edge projector 𝒫_edge is N_b L_y. Let us write λ_R_yj=R_y+ν_R_yj. The edge shift is then defined by

    δ𝒴^χ_edge(R_1)=(∑^N_b_j=1ν_R_1j)mod1.

Now, mirror symmetry imposes the constraint

    m̂_𝒮𝒲^χ_edgem̂^-1_𝒮=𝒫^χ_edgem̂_𝒮 e^-2π i Ŷ/L_ym̂^-1_𝒮𝒫^χ_edge=e^-4π iS/L_y(𝒲^χ_edge)^†.

Thus, we have that

    𝒲^χ_edge|Ψ^χ_R_yj⟩=e^-2π iλ_R_yj/L_y|Ψ^χ_R_yj⟩  ⟶  𝒲^χ_edgem̂_𝒮|Ψ^χ_R_yj⟩=e^-2π i (2S-λ_R_yj)/L_ym̂_𝒮|Ψ^χ_R_yj⟩.

Because of this constraint, for a given fixed unit cell R_y, there are three types of Wannier centers ν_R_yj:

    
  * Those that come in symmetry-related pairs {ν_R_yjmod1,-ν_R_yjmod1}.
    
  * Those that coincide with an integer mirror-invariant point ν_R_yjmod1=0.
    
  * Those that coincide with a half-integer mirror-invariant point ν_R_yjmod1=1/2.

In view of these possibilities, if we denote by N_χ,1/2(R_1) the number of Wannier states centered at R_1+1/2, then

    δ𝒴^χ_edge(R_1)=(∑^N_b_j=1λ_R_1 j)mod1=(∑^N_χ,1/2_i=11/2)mod1=[1/2N_χ,1/2(R_1)]mod1.

Thus, the only way to obtain a nonzero shift δ𝒴^χ_edge(R_1)=1/2 is for there to be an odd number of Wannier centers at the mid-point of the unit cell R_1. Next, let us write N_χ,1/2(R_1)=N^(+)_χ,1/2(R_1)+N^(-)_χ,1/2(R_1), where N^(±)_χ,1/2(R_1) is the number of Wannier states centered at R_1+1/2 with ± i mirror eigenvalues. Then

    δ𝒴^χ_edge(R_1)=[1/2(N^(+)_χ,1/2(R_1)+N^(-)_χ,1/2(R_1))]mod1=[1/2(N^(+)_χ,1/2(R_1)-N^(-)_χ,1/2(R_1))]mod1.

Consider the set of mirror-invariant points S|_y_0=1/2={R_1+1/2,R_2+1/2}. Due to the constraint of mirror symmetry, for λ_R_yj≠ R_1+1/2, the state |Ψ^χ_R_yj⟩ is orthogonal to the state m̂_𝒮|Ψ^χ_R_yj⟩.  By contrast, if λ_R_yj=R_1+1/2, then |Ψ^χ_R_yj⟩ is an eigenstate of m̂_𝒮|_y_0=1/2. It follows that

    N^(+)_χ,1/2(R_1)-N^(-)_χ,1/2(R_1)=-i/2∑_R_yj⟨Ψ^χ_R_yj|(m̂_𝒮|_y_0=1/2)|Ψ^χ_R_yj⟩.

Note that the dependence of the right-hand side on R_1 comes from the index S|_y_0=1/2 of the mirror operator that is chosen. The factor of 1/2 is due to the fact that the sum counts the two points in S|_y_0=1/2, and we defined N_χ,1/2 with respect to only one of them. Using Eq.<ref> in Eq.<ref>, we then obain

    δ𝒴^χ_edge(R_1)   =   [-i/4∑_R_yj⟨Ψ^χ_R_yj|(m̂_𝒮|_y_0=1/2)|Ψ^χ_R_yj⟩] mod  1.

Now, note that since χ̂T|Ψ^+_R_yj⟩=-χ T|Ψ^+_R_yj⟩ then T|Ψ^+_R_yj⟩∝|Ψ^-_R_yj⟩. Thus if m̂_𝒮|_y_0=1/2|Ψ^+_R_yj⟩=e^iϕ|Ψ^+_R_yj⟩, then by acting with T on both sides we obtain m̂_𝒮|_y_0=1/2|Ψ^-_R_yj⟩=e^-iϕ|Ψ^-_R_yj⟩ i.e. the two states |Ψ^+_R_yj⟩ and |Ψ^-_R_yj⟩ have opposite mirror eigenvalues. Because of this, we can write

    ⟨Ψ^χ_R_yj|(m̂_𝒮|_y_0=1/2)|Ψ^χ_R_yj⟩=1/2(⟨Ψ^+_R_yj|(m̂_𝒮|_y_0=1/2)|Ψ^+_R_yj⟩-⟨Ψ^-_R_yj|(m̂_𝒮|_y_0=1/2)|Ψ^-_R_yj⟩).

We then obtain

    δ𝒴^χ_edge(R_1)   =   [-i/8∑_R_yj{⟨Ψ^+_R_yj|(m̂_𝒮|_y_0=1/2)|Ψ^+_R_yj⟩-⟨Ψ^-_R_yj|(m̂_𝒮|_y_0=1/2)|Ψ^-_R_yj⟩}] mod  1,

Finally, we replace the sum over Wannier functions by a trace over the full Hilbert space of the system by introducing the projection operator 𝒫_edge:

    δ𝒴^χ_edge(R_1)   =   [-i/8∑_R_yjχ⟨Ψ^χ_R_yj|(m̂_𝒮|_y_0=1/2)χ̂|Ψ^χ_R_yj⟩] mod  1,
       =   [-i/8Tr[𝒫_edge(m̂_𝒮|_y_0=1/2)χ̂]] mod  1,
       =   [1/4Δ^m_𝒮|_y_0=1/2] mod  1,
       =   [1/4𝒱^m_𝒮|_y_0=1/2] mod  1,

which is the expression we discussed in the main text.



 §.§ IV.5 Model example

We here provide the details of the particular model we used to illustrate the bulk and edge topology of the two DIII-trivial topological phases that were discussed in the main text. The model we used was designed to expose clearly the features that arise in these phases, particularly at the edge. The two models we discussed had the topological invariants

    Weak TCP:      𝒱^m_𝒮|_y_0=0=2,   𝒱^m_𝒮|_y_0=1/2=0 
    Mirror-topological TCP:      𝒱^m_𝒮|_y_0=0=0,   𝒱^m_𝒮|_y_0=1/2=2.


This is achieved by the Hamiltonian that is effectively two copies of the model from <cit.>:

    h(k)=sin k_x Γ_1 + t_ysin k_y (Γ_2+m_3 Λ_3)+
    (μ+cos k_x + t_y cos k_y) Γ_3+ ∑^2_i=1 m_i Λ_i,

where

    Γ_1=f_3 τ_3 σ_3 , Γ_2=f_0 τ_3 σ_1 ,  Γ_3= f_0  τ_1 σ_0 , Λ_1=f_3 τ_1 σ_0  , Λ_2=f_2 τ_1 σ_3  Λ_3=f_1τ_3σ_2 .

The Pauli operators τ_a, σ_a act on Nambu and spin degrees of freedom, and f_a acts on the two copies. The time-reversal, charge conjugation, chiral, and mirror operators are T=σ_2 K, C=i τ_2σ_2 K, S=τ_2, and m_𝒮(k_y)=ie^2ik_y y_0σ_3 respectively. The parameters we used to achieve the two phases are:


 
     t_y     μ     m_1     m_2     m_3
 
 Weak TCP     0.25     -1.0     1.0     0.0     0.0
 
 Mirror-topological TCP     1.0     0.0     -1.0     0.5     0.5 

 


To check that the invariants yield the correct values, we can simply evaluate the Bloch Hamiltonian at the invariant momenta k^inv_y. When m_2=0, the Bloch Hamiltonian is block-diagonal in the eigenstates of m_𝒮(k_y). By denoting h^(±)(k_x,k^inv_y) the diagonal blocks, then the positive mirror block is 

    h^(+)(k_x,k^inv_y)   =   sin k_x f_3 τ_3+
    (μ+cos k_x+ t_y e^i k^inv_y) f_0τ_1+ m_1 f_3 τ_1,
       =   ( [  sin k_xτ_3+(M_+(k^inv_y)+cos k_x) τ_1                                      0;                                      0 -sin k_xτ_3+(M_-(k^inv_y)+cos k_x) τ_1 ])

where M_±(k^inv_y)=μ+ t_y e^i k^inv_y± m_1. Each block of h^(+)(k_x,k^inv_y) thus realizes a minimal two-band model of AIII topological wires. It is then straightforward to check that the momentum-space invariants for the weak TCP are ν^+_𝒮(0)=ν^+_𝒮(π)=1, which leads to Eq.<ref>. Similarly, one obtains for the mirror-topological TCP ν^+_𝒮(0)=-ν^+_𝒮(π)=1, which leads to Eq.<ref>. The added terms proportional to m_2,3 do not close the bulk gap and are chosen so as to detach the boundary modes from the bulk bands.



Finally, in order to numerically obtain the edge projector 𝒫_edge of the boundary bands of only one of the edges, it is useful to gap out the boundary modes on the opposite edge to avoid any hybridization between both edge modes. To achieve this we added the following terms to the boundaries of the weak TCP

    V_1   =   (-1.0)(f_1τ_3σ_0),
    
    V_L_x   =   (0.5)(f_1τ_3σ_1-cos k_y f_0τ_2σ_0),

and for the mirror-topological TCP we added

    V_1   =   (0.1)(f_1τ_3σ_0),
    
    V_L_x   =   (-0.8)(f_0τ_3σ_1),

where ⟨x,y|V_1|x,y⟩=δ_x,1 and ⟨x,y|V_L_x|x,y⟩=δ_x,L_x. This leaves gapless the boundary modes at the x=1 edge. These are the boundary modes for which the topological markers are shown in Fig.3 of the main text for both phases.



