Real-Space Topological Invariants

• Real-space topological invariants are spatially localized markers that extend traditional band theory to systems lacking translational symmetry.
• They utilize local markers, spectral localizers, and symmetry-projected operators to diagnose edge, surface, and defect modes in disordered and inhomogeneous environments.
• These invariants enable practical computation and direct experimental measurement, advancing studies in non-Hermitian, Floquet, and many-body topologies.

Real-space topological invariants are spatially localized, physically meaningful, and computationally accessible quantities characterizing topological phases independently of translational symmetry. They generalize conventional band-theory invariants (such as Chern numbers or symmetry indicators) to inhomogeneous, disordered, interacting, non-Hermitian, and non-periodic settings, and can encode subtle crystalline, fragile, or higher-order topology not visible in momentum space. Their rigorous foundation spans operator K-theory, Clifford algebra localizers, symmetry-projected markers, and Wannier-based classifications. Real-space invariants provide direct diagnostics of protected bulk and boundary phenomena and underpin the modern bulk-boundary correspondence in topology.

1. Fundamental Constructions: Local Markers, Spectral Localizers, and Projection Operators

The principal families of real-space invariants include local Chern/winding markers, Clifford spectral localizers, symmetry-projected markers, and stable real-space invariants (SRSIs).

Local Chern and Winding Markers: For Hamiltonians with chiral, particle–hole, or time-reversal symmetry, local invariants can be constructed using flattening (spectral projectors) and position operators: $$C(\mathbf{r}) = -2\pi\,\mathrm{Im} \langle \mathbf{r} | P\hat{x}P\hat{y}P - P\hat{y}P\hat{x}P | \mathbf{r} \rangle$$ for the Chern marker, with $P$ the projector onto occupied states (Carvalho et al., 2018, Hattori et al., 23 Nov 2025, Arjas et al., 5 Aug 2025). The real-space winding number in 1D chiral-symmetric chains is

$$\nu = \frac{1}{2\pi i}\mathrm{Tr} \log(\mathcal{X}_A\mathcal{X}_B^{-1})$$

where $\mathcal{X}_{A,B}$ are projected/twisted position operators in sublattices (Lin et al., 2021).

Spectral Localizer Invariants: The spectral localizer $L_\lambda$ forms a Hermitian matrix by combining position and Hamiltonian information using Clifford algebra: $$L_\lambda (X, H) = \sum_{j} \kappa (X_j - x_j I) \otimes \Gamma_j + (H - EI) \otimes \Gamma_{d+1}$$ The signature or determinant of $L_\lambda$ yields integer or $\mathbb{Z}_2$ indices (e.g., $\operatorname{sign} \det[\kappa (X-xI) - i H]$ in 1D class D) that are robust to disorder and dimension changes (Rodriguez-Vega et al., 15 May 2025, Qi et al., 2024, Chadha et al., 2023). The spectral localizer can be used for both Hermitian and non-Hermitian systems, and local index jumps diagnose the precise spatial location and count of protected edge, boundary, or defect modes.

Symmetry-projected and Wannier-based Invariants: For crystalline and topological crystalline phases, projected symmetry operators $\overline{g}_\mathcal{S} = P \hat{g}_\mathcal{S} P$ yield real-space topological traces and local markers

$$\mathcal{T}^{\,g}_{\mathcal S}(r) = \langle r | \overline{g}_{\mathcal S}\mathcal{F}(P) | r \rangle$$

which are exponentially localized at symmetry fixed points and robust to spatial disorder/impurities (Mondragon-Shem et al., 2019). Composite and stable real-space invariants (SRSIs) are constructed as Smith-normal-form invariants of the vector of Wannier multiplicities at all Wyckoff positions in a unit cell, modulo all symmetry-allowed adiabatic moves (Hwang et al., 14 May 2025). SRSIs span $\mathbb{Z}$ and $\mathbb{Z}_n$-valued classes and are complete for atomic and fragile topology.

2. Disorder, Symmetry, and Topology without the Brillouin Zone

One of the major advances from real-space invariants is their capacity to characterize topological phases in the absence of good momentum quantum numbers.

• Disorder Robustness: Real-space Chern and $\mathbb{Z}_2$-type indices remain quantized as long as a local spectral gap is preserved, making them ideal for diagnosing transitions under disorder, both for standard (normal) and sublattice-polarized randomness (Hattori et al., 23 Nov 2025, Hattori et al., 2023, Rodriguez-Vega et al., 15 May 2025, Setescak et al., 2024).
• Topological Crystalline Phases: Real-space symmetry markers identify crystalline topology and fragile phases in the presence of impurities or defects away from symmetry centers. Each real-space marker is exponentially localized to a symmetry center and is unchanged by disorder not overlapping that center (Mondragon-Shem et al., 2019, Herzog-Arbeitman et al., 2022).
• Non-Hermitian and Floquet Contexts: Clifford spectral localizers, their signatures, and related determinants are directly applicable to non-Hermitian systems (e.g., skin effect, dislocation-skin) and time-(quasi)periodically driven systems with dense spectra. They provide a unified framework for topology in the absence of a spectral gap or in the presence of point or line gaps (Qi et al., 2024, Chadha et al., 2023, Song et al., 2019).
• Bulk-boundary Correspondence and Defect Modes: Local marker jumps and spatially resolved spectral localizer indices predict the existence and precise location of edge, surface, or defect-bound (e.g., Majorana, skin) modes, even in aperiodic, amorphous, or strongly disordered lattices (Rodriguez-Vega et al., 15 May 2025, Qi et al., 2024, Chadha et al., 2023).

3. Real-space Invariants in Topological Quantum Chemistry

Topological quantum chemistry provides a group-theoretical real-space classification of band topology in atomic and crystalline systems:

• Wannier Multiplicity Vectors and SRSIs: The vector $p$ of multiplicities (occupancies of irreps at all Wyckoff positions) generates all possible symmetric, exponentially localized atomic configurations. The space of symmetry-allowed adiabatic moves is encoded by an integer matrix $q$ whose Smith normal form classifies all deformations. SRSIs are the invariants left after quotienting by all band-movable directions (Hwang et al., 14 May 2025).
• Connecting to Symmetry Indicators: Integer-valued SRSIs correspond one-to-one to momentum-space symmetry data (symmetry indicators), but additional $\mathbb{Z}_n$-valued SRSIs (arising from, e.g., Kramers degeneracy or multiple admissible adiabatic moves) classify atomic equivalence beyond what is possible in the Brillouin zone. For split EBRs and fragile topologies invisible to symmetry indicators, SRSIs provide sufficient (though not always necessary) nontrivial real-space diagnostics (Hwang et al., 14 May 2025, Song et al., 2019, Herzog-Arbeitman et al., 2022).
• Crystalline, Higher-Order, and Fragile Topology: Real-space invariants serve as a minimal classification of crystalline topological phases, including higher-order (corner-mode) and fragile phases. In these settings, only SRSIs (not symmetry indicators alone) can diagnose all topologically distinct configurations (Mondragon-Shem et al., 2019, Song et al., 2019, Kooi et al., 2020).

4. Physical Measurement, Implementation, and Computing Techniques

Real-space invariants are suited for both theoretical and experimental characterization, as well as for scalable computation:

• Direct Measurement in Artificial Lattices: The mean chiral displacement, measured from steady-state real-space imaging at fixed momentum (e.g., in polaritonic honeycomb lattices), directly reconstructs 1D winding invariants underlying the bulk–edge correspondence in graphene-type Hamiltonians (St-Jean et al., 2020).
• Local Markers and Density-matrix Learning: Topological invariants can be mapped in real space by constructing local markers from either the full density matrix or local patches, enabling machine-learning-based assignment of topology site by site—including in large, inhomogeneous, disordered, or non-periodic samples (Carvalho et al., 2018).
• Numerical Algorithms: Spectral localizer-based indices, Chern marker calculations, and SRSI evaluation can all be implemented in a straightforward manner with efficient finite-system scaling. For systems with large Hilbert spaces or strong disorder, kernel polynomial methods, sparse diagonalization, and SVD-based routines provide scalable robust approaches (Hattori et al., 23 Nov 2025, Rodriguez-Vega et al., 15 May 2025, Hwang et al., 14 May 2025, Carvalho et al., 2018).
• Coarse-geometric Frameworks: Operator-algebraic (Roe algebra) descriptions supply a physically transparent, computationally tractable, and mathematically rigorous setting for the calculation of strong topological invariants (e.g., 3D $\mathbb{Z}_2$ index) via projection and localizer determinants in real space (Setescak et al., 2024).

5. Interacting and Many-body Real-space Topology

Interacting systems require generalizations of real-space invariants to the many-body setting:

• Many-body RSIs: Local and global many-body RSIs are defined as quantized eigenvalues of symmetry operators (e.g., $e^{i\pi/n\:\hat N C_n}$, $C_n^2$), acting on the ground state or its projected subspace. These invariants are independent of the choice of boundary and provide a classification of many-body fragile, stable, and higher-order topological phases (Herzog-Arbeitman et al., 2022).
• Bulk–corner correspondence and Wen–Zee response: Many-body RSIs appear as quantized coefficients in Wen–Zee-like topological quantum field theories and determine physical observables such as corner charges, angular-momentum densities, and response to lattice defects. Their detection requires only local operator measurements or local entanglement properties in open or periodic geometries (Kooi et al., 2020, Herzog-Arbeitman et al., 2022).

6. Extensions: Non-Hermitian Systems, Floquet/Quasiperiodic Systems, and Green’s-function Formulations

• Non-Hermitian Invariants: Real-space winding, spectral localizer, and Chern marker approaches admit generalization to non-Hermitian topology, providing complete characterization of skin effect, dislocation-induced localization, and boundary correspondence beyond Bloch-band theory (Song et al., 2019, Chadha et al., 2023).
• Quasiperiodic/Floquet Systems: Spectral localizer and determinant-type invariants allow for the identification of topological Majorana modes and other robust features in systems with dense, continuous, or incommensurately driven spectra, where traditional band-structure methods fail (Qi et al., 2024).
• Interacting Green’s-function Topology: For smooth textures and solitons, real-space invariants built from Green’s-functions (e.g., $N_{D+2}(\mathcal{G})$) classify and attribute conserved quantum numbers such as fractional soliton charge or quantized Hall conductivity to defect configurations, and are robust to interactions (Vayrynen et al., 2011).

7. Conceptual Impact and Outlook

Real-space topological invariants provide a unifying, robust, and scalable language for classifying and diagnosing topological phases beyond the paradigm of band theory and the Brillouin zone. Their physical meaning is tied to the spatial structure or localization of electronic, photonic, or atomic configurations and to the persistence of topology under spatial inhomogeneity. By enabling real-space diagnostics, they have transformed the practical identification of topological phases in computation, experiment, and materials design, and are now central to modern approaches in quantum chemistry, metamaterials, and engineered platforms (Hwang et al., 14 May 2025, Herzog-Arbeitman et al., 2022, Hattori et al., 23 Nov 2025, Qi et al., 2024, Chadha et al., 2023). Their further development will underpin advances in strongly correlated, aperiodic, non-Hermitian, and higher-order topological matter.