Real-Space Topological Invariants

• Real-space topological invariants are defined using local operators and spatial properties, offering a clear classification of topological phases even in disordered environments.
• They provide quantized observables such as fractional edge charges and Hall conductance, directly linking local measurements to global topological features.
• Computational methods like spectral localizers, bott indices, and machine learning techniques enable practical extraction of these invariants in complex, non-periodic systems.

A real-space topological invariant is a mathematical object—often an integer or discrete quantum number—defined directly in terms of spatial, rather than momentum-space, properties of a condensed matter system. Such invariants provide a way to classify and diagnose topological phases of matter, including those in systems with disorder, spatial inhomogeneity, defects, or lack of translation symmetry, where momentum-space band-structure-based invariants fail or become ill-defined. Real-space invariants arise from the properties of local operators (such as the Hamiltonian, position operators, or local symmetry operators), and frequently relate intimately to experimentally measurable quantities and robust physical phenomena such as quantized edge states, fractional charges, or topological responses at defects and interfaces.

1. Principles and Definitions

A real-space topological invariant is constructed from local or global features of the system’s wavefunctions, Green's functions, or operator algebras, as opposed to the global characteristics of the Bloch bands in momentum space. The archetypal example is the real-space winding number for 1D chiral-symmetric insulators, expressed as: $$\nu = \frac{1}{2\pi i} \mathrm{Tr}\left[ \log\left( \mathcal{X}_A \mathcal{X}_B^{-1} \right) \right]$$ where $\mathcal{X}_\sigma$ are projected position operators onto sublattice sectors (Lin et al., 2021). For higher-dimensional systems, real-space invariants can be realized as Bott indices, local Chern markers, or as spectral localizer signatures (Hattori et al., 2023, Rodriguez-Vega et al., 15 May 2025, Chadha et al., 2023).

In interacting or noninteracting fermionic systems, Green’s function-based invariants capture the winding of the single-particle Green’s function over phase-space including spatial coordinates, providing invariants such as

$$N_3 = \frac{1}{4\pi^2} \, \mathrm{tr} \int dx\,d\omega\,dk_x \; {\cal G}\partial_{[k_x} {\cal G}^{-1} {\cal G}\partial_{\omega} {\cal G}^{-1} {\cal G}\partial_{x]} {\cal G}^{-1}$$

for 1D (fermion parity) or its 3D analog for quantized Hall conductance (Vayrynen et al., 2011).

These invariants remain well defined in the presence of symmetry-breaking disorder that destroys translation symmetry, so long as the protecting symmetry (e.g., chiral, particle-hole, or time-reversal) is preserved.

2. Formalism, Construction, and Examples

The construction of real-space invariants depends on the system’s spatial structure and symmetry:

• Chiral-symmetric systems (1D): The real-space winding number, expressed with projected position operators or as a Bott index, provides a direct integer invariant diagnosing topological phases even with disorder (Lin et al., 2021). For systems lacking chiral symmetry, a global invariant Q involving all energy states is quantized so long as a spectral gap remains open (Hattori et al., 2023).
• Higher dimensions: In two and three dimensions, generalizations use projectors onto occupied local states (Chern markers) or the spectral localizer framework, which relates topological invariants to the signature of a matrix combining the Hamiltonian and position operators (Rodriguez-Vega et al., 15 May 2025, Chadha et al., 2023). For a 2D system, for example:

$$L^{(2D)}_\lambda = (\mathcal{H} - E I)\otimes \sigma_z + \kappa [(X-xI)\otimes\sigma_x + (Y-yI)\otimes\sigma_y]$$

and the local Chern number at $\lambda = (x, y, E)$ is given by $\frac{1}{2}$ of the signature of $L^{(2D)}_\lambda$.

• Defects and Inhomogeneity: Real-space invariants robustly characterize the topology in the vicinity of defects (e.g., dislocations, disclinations, domain walls) using local projectors, the spectral localizer, or additive rules between topology associated with lattice defects and bulk quantum geometry (Chadha et al., 2023, He et al., 2022).
• Fragile and higher-order topology: Invariants (often called RSIs—real-space invariants) are defined via site symmetry representations of orbitals at Wyckoff positions and remain quantized under all symmetry-allowed adiabatic processes (Song et al., 2019, Hwang et al., 14 May 2025). Higher-order topological invariants, such as quadrupole or octupole moments, are constructed from nested Wilson loops or real-space operator algebra in systems with suitable symmetry (Qiu et al., 29 Sep 2024, Liu et al., 2021).
• Disordered and incommensurate systems: Real-space formulations are essential for robustness in systems with disorder or without translational invariance, including time-quasiperiodic and non-Hermitian systems where the spectral localizer and related constructs provide practical, computable invariants (Qi et al., 19 Apr 2024, Song et al., 2019).

3. Physical Implications and Robustness

Real-space topological invariants have broad physical consequences:

• Quantized observables: Invariants such as the real-space winding number, Bott index, or quadrupole/octupole moments yield quantized values for physical observables (fractional charge, quantized Hall conductivity, or topological polarizations) even in strongly disordered or interacting environments (Vayrynen et al., 2011, Liu et al., 2021).
• Bulk-defect and bulk-boundary correspondence: The presence and number of boundary, corner, or defect modes is inferred from real-space invariants, as in the robust prediction of dislocation or boundary-localized skin effects in non-Hermitian systems from the localizer index (Chadha et al., 2023), or in the direct linkage of bound states at disclinations to additive real/reciprocal invariants (He et al., 2022).
• Resilience to symmetry-breaking and disorder: As long as the spectral gap is maintained (or the protecting symmetry is unbroken), real-space invariants retain their quantization and diagnostic power even for strong disorder, amorphous structures, or spatially inhomogeneous phases. This is evidenced by robust topological indices in disordered Rice-Mele chains (Hattori et al., 2023), topological Anderson insulators (Liu et al., 2021), and 3D time-reversal invariant insulators (Setescak et al., 23 Jul 2024).
• Higher-order and fragile phases: Real-space invariants track intrinsic higher-order topological phases (e.g., quantized octupole or quadrupole response) as well as extrinsic, boundary-obstructed, or fragile topologies invisible to momentum-space symmetry indicators (Qiu et al., 29 Sep 2024, Hwang et al., 14 May 2025).

4. Classification Schemes and Relation to Band Theory

The real-space invariant framework generalizes and refines momentum-space symmetry indicator diagnostics:

• Stable Real-Space Invariants (SRSIs): SRSIs are global invariants formed from linear combinations of Wannier state multiplicities at Wyckoff positions, remaining fixed under all allowed adiabatic processes. They may be integer- ($\mathbb{Z}$) or modulo-valued ($\mathbb{Z}_n$), and fully classify atomic insulators up to stable equivalence (Hwang et al., 14 May 2025).
• Equivalence to or extension beyond symmetry indicators: While $\mathbb{Z}$-valued SRSIs are determined by momentum-space irreps and thus determine symmetry indicators (SIs), $\mathbb{Z}_n$-SRSIs contain extra topological information not detectable in momentum space, distinguishing phases that are trivial by SIs but topologically distinct by real-space analysis.
• Diagnosing split band representations and fragile phases: SRSIs rigorously identify nontriviality in cases where elementary band representations (EBRs) split in energy, and momentum-space decompositions would incorrectly suggest triviality. The additivity and Smith normal form structure in the SRSI construction are key (Hwang et al., 14 May 2025, Song et al., 2019).
• Interacting phases and topological quantum chemistry: Real-space invariants extend to interacting systems, where global RSIs reduce to familiar band invariants in weak coupling, but remain well-defined in regimes of strong correlation, directly appearing as quantized coefficients in effective field theories (e.g., the coefficient of the Wen-Zee term) (Herzog-Arbeitman et al., 2022).

5. Computational and Experimental Approaches

The calculation and measurement of real-space topological invariants employ various techniques:

• Tensor network and renormalization group: Real-space tensor renormalization group (TRG) methods extract invariant quantities for classical and quantum lattice models (e.g., spin glasses, frustrated magnets) by exploiting coarse-grained contractions and environment tensors, enabling extraction of local order parameters and identification of multicritical points (Wang et al., 2013).
• Spectral localizer and local markers: The spectral localizer approach systematically computes local invariants and associated local spectral gaps using only the system’s finite-size Hamiltonian and position operators, without reference to crystalline periodicity. This framework is suited both for numerical paper of finite or disordered systems and for experimental mapping of topological domains via, e.g., STM or spectroscopic techniques (Chadha et al., 2023, Rodriguez-Vega et al., 15 May 2025).
• Neural network and machine learning: Supervised learning using artificial neural networks trained on local density matrices—extracted via the kernel polynomial method—enables rapid mapping of topological invariants in large systems with spatial inhomogeneity or disorder, circumventing the need to reconstruct global wavefunctions (Carvalho et al., 2018).
• Experimental realization: Electrical circuits implementing Laplacians analogous to tight-binding Hamiltonians, microwave arrays, and photonic crystals have been used to measure real-space invariants via local impedance, confirming theoretical predictions even in non-toroidal momentum-space settings (Qiu et al., 29 Sep 2024, He et al., 2022).

6. Extensions, Open Problems, and Outlook

Current real-space invariant formalism covers extensive ground, but key directions and open challenges remain:

• Generalization to arbitrary symmetry classes and dimensions: The spectral localizer and real-space K-theory methods cover arbitrary symmetry and spatial dimension, allowing for local diagnosis in complex systems, including higher-order and strongly interacting phases (Rodriguez-Vega et al., 15 May 2025, Setescak et al., 23 Jul 2024).
• Interplay with coarse geometry and operator algebras: Operator-algebraic (Roe algebra) and coarse geometry approaches formalize the linkage between real-space invariants and bulk-boundary correspondence, ensuring the appearance of robust edge states whenever the bulk invariant is nontrivial, even in strongly disordered or amorphous matter (Setescak et al., 23 Jul 2024).
• Systematic tabulation and classification: Recent enumerations of all SRSIs and $\mathbb{Z}_n$-valued invariants across non-magnetic space groups (with or without SOC) provide a comprehensive taxonomy and systematic toolkit for cataloguing topological insulators, including phases latent to SI analysis (Hwang et al., 14 May 2025).
• Fragile and Obstructed Topology: Real-space invariants are essential for distinguishing fragile and obstructed atomic insulators, for which symmetry data alone is insufficient. Twisted boundary condition protocols provide experimental access to these invariants (Song et al., 2019).
• Dynamical and time-dependent topology: The extension of real-space invariants to driven (Floquet or time-quasiperiodic) systems, using the spectral localizer in the enlarged frequency-physical space, opens new possibilities for diagnosing dynamically generated Majorana modes and non-equilibrium topological effects (Qi et al., 19 Apr 2024).

Real-space topological invariants, by leveraging locality, robustness, and operator algebraic structure, offer a unifying paradigm for topological classification and characterization well beyond the idealized momentum-space scenario, enabling the paper and realization of topological matter in a broad variety of physical, disordered, non-Hermitian, or interacting contexts.