Local Topological Markers in Odd Spatial Dimensions and Their Application to Amorphous Topological Matter

Local topological markers, topological invariants evaluated by local expectation values, are valuable for characterizing topological phases in materials lacking translation invariance. The Chern marker -- the Chern number expressed in terms of the Fourier transformed Chern character -- is an easily applicable local marker in even dimensions, but there are no analogous expressions for odd dimensions. We provide general analytic expressions for local markers for free-fermion topological states in odd dimensions protected by local symmetries: a Chiral marker, a local $\mathbb Z$ marker which in case of translation invariance is equivalent to the chiral winding number, and a Chern-Simons marker, a local $\mathbb Z_2$ marker characterizing all nonchiral phases in odd dimensions. We achieve this by introducing a one-parameter family $P_{\vartheta}$ of single-particle density matrices interpolating between a trivial state and the state of interest. By interpreting the parameter $\vartheta$ as an additional dimension, we calculate the Chern marker for the family $P_{\vartheta}$. We demonstrate the practical use of these markers by characterizing the topological phases of two amorphous Hamiltonians in three dimensions: a topological superconductor ($\mathbb Z$ classification) and a topological insulator ($\mathbb Z_2$ classification).

Topological markers in three dimensions are scarce. Examples include a marker for the odd-dimensional topological insulators with a Z invariant in the complex topological classes [29], and an extension of the spin Bott index to three dimensions [32], but there exists no general local marker for odd-dimensional topological insulators and superconductors. In fact, characterizing the three-dimensional free-fermion topological phases for sys-tems lacking translational invariance remains a hard task with few numerical options. One can either check for gapless boundary modes, by exploring if the existence of the ground-state energy gap depends on whether the boundary conditions are open or closed [5], or calculate the magnetoelectric response [37][38][39][40][41]. Another option is to produce a topological phase diagram by measuring the Witten effect [42,43] through the amount of charge bound by a magnetic monopole as a function of the parameters of the model [44]. Apart from being practically challenging, neither of these diagnostics directly predict the phase of a given state since they require comparing states in different physical settings.
In this work we present a solution to this problem by providing an analytic expression for a local topological marker in odd spatial dimensions. Specifically, we extend the formulation of the Chern marker to odd dimensions by introducing a one-parameter family of projectors traversing between the trivial and topological state, and where the parameter acts as an additional dimension. The general expression for the marker can be used as a real space expression for both the chiral winding number, a Z invariant of topological phases in odd dimensions, and the Z 2 Chern-Simons invariant. We use our real-space marker to characterize the topological phases of two different three-dimensional amorphous Hamiltonians: a topological insulator with time-reversal invariance, characterized by the Z 2 invariant θ-term (class AII), and a topological superconductor with both chiral symmetry and time-reversal invariance, characterized by a chiral winding number ν ∈ Z (class DIII).
The local Chern marker as a bundle invariant.-Two localized Slater determinant states belong to the same symmetry-protected topological phase if they can be transformed into one another through smooth trans-formations while preserving localization [45] and the given symmetry, or by adding or removing atomic-limit bands [46]. The single-particle density matrix ρ obtained from a Slater determinant-or in the presence of a local symmetry, each symmetry-resolved block of ρ-is a projector onto the space of occupied states. Given translation invariance, ρ is block-diagonal in momentum space and is a smooth function of momentum. One can therefore associate a complex vector space, the image of ρ(k), to each momentum k in a smooth way, thereby obtaining an equivalence relation to a complex vector bundle. The topological classification of Slater determinant states therefore translates into a classification of vector bundles [46]. The blocks of ρ fulfill one of ten inequivalent constraints (AZ classes) [47][48][49], leaving ten different cases to classify.
The Chern numbers (one for each even dimension) are quantized bundle invariants that in even dimensions characterize all topological phases in three out of the five nontrivial AZ classes [50]. In real space and in terms of ρ, the Chern numbers [51,52], now referred to as local Chern markers [53], take the form where ε is the Levi-Civita symbol, D is the even dimension, and the repeated indices i are summed over. X i is the ith position operator: (X i ) (r,α),(r ,β) = x i δ α,β δ r,r , where x i is the i'th component of the position r, and α, β denote local quantum numbers. Without translation invariance, the relation to vector bundles is lost (since ρ is no longer block-diagonal in momentum space) and the Chern numbers are no longer defined. The average of C(r) over large regions is, however, still a topological characteristic. The reason for this is that the states under consideration always have a translation-invariant long-wavelength limit, and one can therefore define Chern numbers characterizing the topological phase by the coarse-grained single-particle density matrix defined by the asymptotic behavior of ρ in this limit. When averaging C(r) over larger and larger regions, it will therefore approach a well-defined translation-invariant limit. These averaged Chern markers only depend on the long-wave-length properties of ρ, so they are evaluated by replacing ρ in Eq. (1) by its translation-invariant coarse-grained version-the averaged markers therefore are the coarse-grained Chern numbers characterizing the phases. This amounts to calculating the Chern marker locally for each point in the lattice and averaging over a large enough region, such that the coarse-grained ρ is effectively translation invariant.
The Chern marker is only defined in even dimensions, so constructing a local marker in odd dimensions is nontrivial. There does exist a bundle invariant in odd dimensions called the Chern-Simons invariant [52], but it is the modulo 1 part of a basis dependent expression and cannot be expressed as a function of the single particle density matrix alone. This means that the Chern-Simons invariant cannot be expressed as a sum of local expectation values, so it is not a local marker.
The local Chiral marker.-In odd dimensions three out of five nontrivial AZ classes are characterized by the chiral winding number ν ∈ Z [50]. For these classes the single-particle density matrix ρ obeys a chiral constraint; there exists a real Hermitian matrix S squaring to identity such that {ρ, S} = S. We define a local chiral marker by adopting the expression of the local Chern marker, Eq. (1), and introducing a one-parameter family of projectors P ϑ of the form where the parameter ϑ acts as the extra dimension, resulting in an even dimensional integral over real space.
Here P π/2 = ρ is the projector of interest and P 0 = (1 − S)/2, which is a trivial projector without the same chiral constraint as ρ. Expressed in terms of P ϑ , the local chiral marker is [53] where X 0 = i∂ ϑ , and D is now odd. Provided translation invariance, ν(r) mod 2 is equal to two times the difference in the Chern-Simons invariant [52] between the bundles ρ and P 0 . In the presence of the chiral constraint, the Chern-Simons invariant is a half integer valued Z 2 invariant [50], which implies that ν(r) is quantized as an integer. However, by using the P ϑ given in Eq. (2) in terms of S and ρ, not only ν(r) mod 2, but also ν(r) is a topological invariant, which in the translation invariant limit equals the chiral winding number [54]. Expanding and integrating Eq. (3) the chiral marker becomes [53] where γ D is a dimension-dependent constant: γ 1 = −2, and γ 3 = −8πi/3. By averaging ν(r) over a large enough region, it assumes the role of a coarse-grained invariant for non-translationally-invariant systems, in analog to that of the Chern marker.
The local Chern-Simons marker.-The topological phases of odd-dimensional systems that break chiral symmetry are defined by the Chern-Simons invariant, a Z 2 invariant which characterizes one of the AZ classes in each odd dimension [50,55,56]. The corresponding real space Chern-Simons marker ν cs is defined almost analogously to the chiral marker in Eq. (3), but with two crucial differences: since the Chern-Simons marker is a Z 2 invariant, it is only defined modulo 2, and since the chiral symmetry is broken, the path in parameter space must be redefined. We consider the sum of two paths in parameter space: the first between a trivial state and a chiral state with density matrix Q, and the second traversing from the same chiral state to the final state (without a chiral constraint) with density matrix ρ. Constructing the projector for the second path such that it has the same symmetries as the chiral and final states combined, results in a vanishing (mod 2) difference in the Chern-Simons marker between the initial and final points in the second path. This means that the total Chern-Simons marker comes from the path between the trivial and chiral states alone, where again, due to the enforced symmetries, the chiral endpoint gives the same contribution to the marker as ρ.
We use the topological insulator in three dimensions (class AII) as a concrete example. The single particle density matrix of this class obeys the time reversal constraint, T ρ * T † = ρ, where T = T K is the time reversal operator in real space and K is complex conjugation. We define the single particle density matrix for the chiral state as: where The operator R can be chosen to be any trivial projector for which i[ρ, S R ] ∝ i[ρ, R] has no zero modes, and thus renders Q to be a localized operator. In practice this means that R is model dependent, but as it can be constructed to be any tensor product of local operators we expect that one can always construct R such that the spectrum of i[ρ, R] is gapped [53]. Retaining an R such that i[ρ, R] has zero modes would require finetuning. Provided that both Q and ρ are local operators, any one-parameter family of projectors interpolating between them will be localized as well. To ensure that the path between the chiral and the final state is confined within the AII class, we demand that the corresponding projector, is invariant under time-reversal symmetry, which amounts to enforcing that T Q * T † = Q. Using the definition in Eq. (5), this translates to the condition T S * R T † = −S R , hence restricting the choice of R. Since the path is contained within the same symmetry class and is always characterized by a localized projector, the corresponding Chern-Simons marker is zero modulo two under this constraint. The only contribution to the total marker comes from the path between the trivial and chiral state given by the projector Since Q obeys a chiral constraint, the total Chern-Simons marker is evaluated by using the local chiral marker in Eq. (4) such that ν cs = ν mod 2. The approach in this example translates to any odd-dimensional system that breaks chiral symmetry and is characterized by the Chern-Simons invariant; for the one-dimensional superconductor (class D) which obeys the charge-conjugation constraint, one would enforce the projector in Eq. (6) to be invariant under charge conjugation. Application to topological amorphous solids.-In this section we provide examples that demonstrate how the chiral and Chern-Simons markers can be used to numerically characterize topological phases in amorphous systems. We consider two three-dimensional models, a topological insulator (class AII) and a topological superconductor (class DIII), both with the same first quantized Hamiltonian: Here φ and θ are the azimutal and polar angles between lattice sites i and j, t and t ij = 1/4 exp(1 − |r i − r j |/a) are hopping amplitudes [5] (with a the average bond length), M is a mass parameter, σ α and τ α with α ∈ (x, y, z) are Pauli matrices where the σ α acts on the spin degrees of freedom. For the superconductor λ represents a pairing potential, and τ α acts in particle-hole space, while for the insulator, λ is a spinorbit coupling parameter, and τ α acts in orbital space. For θ ∈ {0, π} and φ ∈ {0, ±π/2, π}, H ij restricts to a cubic lattice Hamiltonian, known to host different topological phases depending on the parameters [58]. To make the lattice amorphous we choose the position of each lattice site from a Gaussian distribution with a standard deviation w, centered on the lattice site positions of the crystalline limit. One can therefore continuously tune the lattice from a crystalline structure when w = 0 to an increasingly amorphous one as w increases. The model has a fixed number of six nearest neighbors which are not necessarily the same as in the crystalline limit, allowing for defects to enter as w increases.
The time-reversal invariant superconductor in class DIII obeys a chiral constraint and its phases are characterized using the chiral marker Eq. (4). In particular, setting t = 0, the chiral constraint for the Hamiltonian in Eq. (8) is imposed by S = −τ y . The averaged chiral marker for the amorphous lattice with width w = 0.1 is depicted in Figure 1(a) as a function of M for three different system sizes, L = 8, L = 10, and L = 12. As the system size increases, the chiral marker approaches an integer-quantized value, as expected in the absence of finite size effects. In the crystalline limit the chiral marker can be evaluated analytically and takes three different values depending on the parameter M , ν = −2 for |M | < 1, ν = 1 for 1 < |M | < 3 and ν = 0 for all other values of M [50]. The inset in Figure 1(a) shows the chiral marker of twelve randomly selected sites of a single realization of the lattice for M = 0, 2, 4, to highlight how the the marker fluctuates around its mean value. The effect of the amorphicity is apparent in the topological transition from ν = 1 to ν = 0 at M 2.3, which is slightly displaced compared to the crystalline limit. Figure 1(b) shows the chiral marker as a function of w for fixed M . For M = 2 the gap closes only at a single point and there is a conventional topological phase transition to a trivial state, but driven by amorphicity. This type of transition has also been reported in [33]. For M = 0, on the other hand, the energy gap remains closed for increasing w after the transition.  one-parameter family of projectors between an atomic limit state and a non-trivial topological state, which allowed us to reformulate the chiral and Chern-Simons invariants in the form of a Chern marker. We have confirmed the validity of these local topological markers by using them to characterize the topological phases of amorphous superconductors and insulators in three dimensions. Although we have only considered Slater determinants, our expressions for the markers apply more generally to all states where the spectrum of ρ is gapped; this is typically the case for interacting localized states, see, e.g., Ref.
We are grateful to A. Tiwari for fruitful discussions and input in the early stages of this work. We also thank D. Aceituno for the help with various numerical aspects. [53] See the Supplemental Material for a derivation of the Chern, and chiral markers, a detailed description of the numerical implementation of the chiral marker in three spatial dimensions, and a discussion of the chiral marker in a system with open boundary conditions. Also included is a discussion on the the closed form expression of the family of projectors P ϑ , and how to construct the projector R.
[54] In principle, ν(r) and ν could provide different integers for the same phase. However, by explicitly verify-ing that they are equal for examples with ν = 1, they are guaranteed to be equal for all phases, which follows from their additive property with respect to direct sums: νρ⊗ (r) = νρ(r) + ν (r).

SUPPLEMENTAL MATERIAL Appendix A: A derivation of the Chiral marker
We present the Chern number as the integral over the Brillouin zone of the Chern character, expressed in terms of the single particle density matrix and describe how to express the Chern marker by Fourier transforming the Chern character. The same prescription applies to the Chern-Simons invariant, from which we derive the chiral marker.

The Chern marker: Fourier-transforming the Chern character
The Bogoliubov-de Gennes single-particle density matrix for a many-body state |φ is a matrix acting in singleparticle space, with indices where {ψ † α } are Bogoliubov-de Gennes creation operators, where, s is the particle hole index (s = ±1) and a labels the single-particle space, ψ † a,s = δ s,1 c † a + δ s,−1 c a , where c † and c are the usual creation and annihilation operators. If |φ has a unitary symmetry then ρ has a block-diagonal structure. For example, for states with a U (1) fermion number conservation symmetry (states with a definite fermion number) ρ takes the block-diagonal form, It is these blocks that enter the topological classification, and in the following discussion we refer to them as ρ, or the single particle density matrix.
In terms of the single-particle density matrix, the n'th Chern character [52] takes the form where the lattice spacing is set to unity, ε is the Levi-Civita symbol, repeated indices are summed over, andρ(k) are matrices in the local Hilbert space, they are the Fourier components of the single particle density matrix ρ, Here and in the rest of these notes kets, |· , refer to single-particle states. Integrating the Chern character over an even dimensional (D = 2n) surface results in a Chern number characterizing the Brillouin zone vector bundle. Taking the surface to be the first Brillouin zone, BZ, yields the Chern number usually referred to when discussing free fermion topological states: Acting with ρ on a state whereψ(k) is a vector in the local Hilbert space, yields since k |k = (2π) −D δ D (k − k ). By defining the eigenvalue of the position operator X i through X i |r = x i |r and using the Fourier decomposition of the momentum vector |k = r e −ik·r |r , the action of the position operator on the state |ψ evaluates as A localized wave function is up to a phase a constant vector α in the local Hilbert space,ψ(k) = αe ik·r , and by denoting the corresponding localized state vector as: |ψ = |r; α , the expectation value r; α|ρX i ρX j · · · ρ|r; α becomes: Using Eq. (A9), the Chern number is given by The last expression is the local Chern marker, which we in the main article present as where the expectation value is expressed as r; α|ρX i1 ρX i2 · · · X i D ρ|r; α = [ρX i1 ρX i2 · · · X i D ρ] (r,α),(r,α) .

The Chern-Simons invariant and the local chiral marker
The Chern number is a bundle invariant in even dimensions alone. The corresponding bundle invariant for odd dimensions is the Chern-Simons invariant CS, which is the integral of the Chern-Simons form cs n over an odd dimensional surface [52]. The Chern-Simons invariant is not a local marker as it cannot be expressed as a sum of local expectation values-it is the modulo one part of a basis dependent expression and cannot be expressed in terms of the single particle density matrix alone. However, the Chern character and the Chern-Simons form are connected in that the integrals of the Chern-Simons forms over closed D − 1-dimensional surfaces ∂Ω ⊂ Ω equals integrals over Chern characters over the even D-dimensional surface Ω: By introducing a one dimensional parameter family of projectors P ϑ such that P 0 is a trivial projecor and P π/2 = ρ is the projector of interest, the difference in the Chern-Simons invariant, ∆CS between these two states becomes where the Brillouin zone is odd dimensional and the Chern character is: where P denotes the Fourier component of P . So, introducing the extra dimension through the parameter ϑ allows us to express the Chern-Simons invariant as the integral of the Chern character over an even dimensional surface, which in turn is a well defined expression in terms of the projector. Using Eq. (A9), The difference in Chern-Simons invariants is therefore ..,i D r; α|P i∂ ϑ P X i1 · · · P X i D P |r; α . (A16) A projector ρ with a chiral constraint defines the local chiral marker as ν(r) mod 2 = 2∆CS, which is integer valued since the Chern-Simons invariant in the presence of a chiral constraint is a half integer valued Z 2 invariant.
The chiral marker as presented in the main text, is thus, where X 0 = i∂ ϑ . In Eq. (A17), one may use any localized interpolation between a trivial state and ρ to get an integer related to the Chern-Simons invariant. However, different choices for P have different finite-size corrections to exact quantized integer values. For a finite system of linear size L these corrections scale as ∼ e −ξ/L , where ξ is the localization length of P . Since P equals the single-particle density matrix in question, ρ, for one value of ϑ, it cannot be more localized than ρ. The choice where S is the chiral constraint {S, ρ} = S, has the same localization length as ρ and is, therefore, the best one can do. Expanding Eq. (A17) with this expressions and using that products of the position operators X i X j vanishes when contracted with the Levi-Civita symbol, and integrating over ϑ leaves only terms with either a ρ, or a S, or their product, in between two X i 's: etc.
By using the relations {S, ρ} = S and [S, X i ] = 0 we can move the S's around and place them all at the same place, which by using S 2 = 1 leaves only two non-equivalent terms: the terms in Eq. (A19) and Eq. (A20) above. However, the term in Eq. (A19) vanishes in the translation invariant limit. Using the anti-symmetry of the Levi-Civita symbol to insert commutators into Eq. (A19) gives which vanishes, since for any string of operators where Q is a projection operator and the {A i } i are arbitrary operators and D is an odd integer. This statement can be shown by induction. The final form of the chiral marker is therefore where γ D is a dimension dependent constant.
Appendix B: Explicitly expanding and simplifying the expression for the chiral marker in three spatial dimensions We expand the general expression of the chiral marker in three spatial dimensions and describe how to implement it in practice when considering the time reversal invariant superconductor on an amorphous lattice.

Simplifying the expression for the local marker
The local chiral marker, Eq. (A17), in three spatial dimensions is where Greek letters, µ ∈ (0, 1, 2, 3). X 0 = i∂ ϑ , and X i , i ∈ (1, 2, 3) is the i'th position operator: (X i ) (r,α),(r ,β) = x i δ α,β δ r,r , where x i is the i'th component of the position r, and α, β denote local quantum numbers. P ϑ is the family of projectors, where ρ is the single particle density matrix of interest, the and the operator S obeys the chiral constraint, {ρ, S} = S. For notational convenience we will from now on refer to P ϑ as P . Expanding the operator in Eq. (B1), and using the anti-symmetry of the Levi-Civita symbol, yields the expression ε µνρσ P X µ P X ν P X ρ P X σ P = iε ijk (P ∂ ϑ P X i P X j P X k P −P X i P ∂ ϑ P X j P X k P + P X i P X j P ∂ ϑ P X k P − P X i P X j P X k P ∂ ϑ P ) , which is simplified by evaluating the derivatives and using that (1 − 2ρ) = S 2 = 1, that the chiral and position operators commute, [S, X i ] = 0, and the chiral constraint, {ρ, S} = S to move the S's around. The translational invariance of the operator in space allows us to throw away any terms with a position operator to the far right or left, as these vanish when taking the expectation value. The ϑ dependence of each term factorizes, and after integrating over ϑ we are left with: The second term is zero which can be seen by using the translational invariance of the operator, and the identity in Eq. (A24) to express this term in terms of commutators: where in the second step the projector to the far left is written as ρ = ρ 2 , where one of these two projectors is moved to the far right by using the cyclic property of the trace. Therefore the chiral marker in three dimensions takes the form: ν(r) = − 8πi 3 α ε ijk [ρSX i ρX j ρX k ρ] (r,α),(r,α) .
markers presented in this paper. The reason is that the summation of the marker over the entire open system is proportional to the trace of an operator, which for the chiral and Chern-Simons markers takes the form: Tr(ρSε i1,...,in X i1 ρX i2 ρ · · · ρX in ). (C1) Implementing the anti-commutation relation {ρ, S} = S to change the order of the first ρ and S in Eq. (C1), results in ρS being replaced by S − Sρ. The first term yields Tr(Sε i1,...,in X i1 ρX i2 ρ · · · ρX in ) = Tr(Sε i1,...,in ρX i2 ρ · · · ρX in X i1 ) = 0, where the first equality follows from the cyclic property of the trace together with the commutation relation [S, X i ] = 0, and the second follows from the fact that the product X in X i1 vanishes under anti-symmetrization. The remaining term is − Tr(Sρε i1,...,in X i1 ρX i2 ρ · · · ρX in ) = −Tr(ρε i1,...,in X i1 ρX i2 ρ · · · ρX in S), again using the cyclic property. Commuting S through all the ρ's gives the equality Tr(ρSε i1,...,in X i1 ρX i2 ρ · · · ρX in ) = −Tr(ρSε i1,...,in X i1 ρX i2 ρ · · · ρX in ), so that summing the marker over the entire open system indeed results in a vanishing result. The values at the boundary will therefore necessarily be large and grow with system size to cancel the contribution in the bulk. Figure SM1 exemplifies this effect showing the projection into the xy-plane of the chiral marker at all sites contained in a slice of width 1.5 sites in the z direction, cut from the center of an amorphous superconductor in three dimensions with open boundary conditions. Calculating the total marker numerically does not necessarily yield a vanishing result when ρ is generated from the ground state of a Hamiltonian, even though this is the case theoretically. Since the Hamiltonian for a topologically non-trivial phase is gapless on an open system, the ground state is not well defined. Choosing a state randomly in the ground state manifold will in general not result in a state which obeys the constraint {ρ, S} = S at the boundary, which therefore invalidates the argument above, as Eqs. (C1-C4) no longer hold. In this case there is no reason for the total marker to vanish. This problem does not arise when calculating the Chern marker since there are no symmetry constraints to adhere to in this case.