Real-Space Topological Markers

• Real-space topological markers are operator-valued invariants that quantify and localize global topological features on spatial lattices.
• They effectively diagnose phases in disordered, defect-laden, and fragile or higher-order systems where traditional momentum-space methods fail.
• Their evaluation leverages projector techniques, Clifford algebra, and machine learning, providing practical insights into complex quantum materials.

Real-space topological markers are operator-valued, spatially resolved invariants or feature maps that quantify and localize global topological properties—such as Chern number, winding number, ℤ₂ index, Euler class, or crystalline symmetry-protected invariants—directly on the lattice or in real space. These markers are essential for diagnosing, visualizing, and tracking topology in systems lacking translational symmetry, exhibiting disorder or defects, or supporting fragile and higher-order phases. Their definitions often employ projectors onto occupied states, position operators, symmetry operators, or statistical/entropic signatures extracted from wavefunctions or eigenvectors, and their spatial averages yield the corresponding bulk invariant. In modern contextual applications, real-space markers extend to quantum geometric, nonlocal, or statistical (machine learning-derived) formulations, as well as boundary- or defect-sensitive settings.

1. Formal Definitions and Marker Types

Real-space topological markers span several formulations, each targeting specific symmetries and physical settings:

• Chern and Winding Markers: The local Chern marker for a noninteracting fermion system is given by

$$\mathcal{M}(\mathbf{r}) = -4\pi\,\Im \langle \mathbf{r}|P\,\hat{x}\,Q\,\hat{y}\,P|\mathbf{r}\rangle,$$

where $P$ is the projector onto occupied states and $Q = 1 - P$ (Caio et al., 2018). The spatial average over all sites yields the global Chern number.

The real-space winding marker in 1D chiral systems is

$$w(x) = \frac{1}{2\pi i} \langle x|C\,P\,[X, P]|x\rangle,$$

with $C$ the chiral symmetry operator (Jezequel et al., 31 Jul 2025).

• Universal Dirac-Type Markers: For Dirac models in any dimension and symmetry class,

$$\hat{\mathcal{C}} = N_D\, W \Bigl[ Q \hat{i}_1 P \hat{i}_2 Q \cdots \hat{i}_D P + (-1)^{D+1} P \hat{i}_1 Q \hat{i}_2 P \cdots \hat{i}_D Q \Bigr],$$

where $N_D$ normalizes the trace to the topological degree, $W$ is a Dirac matrix product, and $\hat{i}_j$ are position operators. The site-resolved marker is $$\mathcal{C}(\mathbf{r}) = \langle \mathbf{r}|\hat{\mathcal{C}}|\mathbf{r}\rangle$$, with the total sum reproducing the integer-valued invariant (Oliveira et al., 2024, Roy et al., 2 Dec 2025).

• Spectral Localizer: The spectral localizer is a real-space operator encoding topological invariants via its signature. In two dimensions:

$P$0

and the local marker is $P$1. The localizer index is robust in the presence of disorder and directly matches the Chern marker for small $P$2 (Jezequel et al., 31 Jul 2025).

• Shannon Entropy and Information-Theoretic Markers: Machine learning–based approaches extract site-resolved markers $P$3 representing the normalized entropy reduction at each site/component in a decision-tree classifier, revealing the most topologically informative regions in the system (Holanda et al., 2019).
• Symmetry-Protected and Crystalline Markers: Markers built from projected symmetry operators (e.g. inversion, rotation, mirror) are spatially localized to symmetry centers and yield symmetry-resolved invariants,

$P$4

with $P$5 and $P$6 operator-dependent. The result is an exponentially localized marker robust to perturbations away from the symmetry center (Mondragon-Shem et al., 2019).

• Euler Class and Fragile Topology: For real two-band systems, the local Euler marker is

$P$7

where $P$8 are unitary position operators, and $P$9 is the Pfaffian over the occupied band block (Li et al., 2023).

• Nonlocal Quantum Geometric Correlators: Nonlocal markers such as $Q = 1 - P$0 probe the real-space correlations and criticality near phase transitions and are sensitive to the nature (momentum channel) of topological transitions (Marsal et al., 12 Nov 2025).

2. Marker Construction and Mathematical Foundations

The computation of real-space topological markers relies on several operator-theoretic and algebraic techniques:

• Projector-based Formulations: Most markers are operations involving projectors onto occupied (or specified) bands and matrix elements of local operators (positions, symmetry operators).
• Clifford Algebra and Spectral Localizer: The spectral localizer encodes topology by combining the Hamiltonian and coordinate differences with auxiliary Clifford matrices, with the integer index given by the signature or kernel dimension (Jezequel et al., 31 Jul 2025, Franca et al., 2023).
• Symmetry Projectors and Group Cohomology: For crystalline symmetry-protected phases, invariants are extracted via traces of projectors onto irreducible representations of the site-symmetry group at fixed points or Wyckoff positions, implementing Smith normal forms and Schur multipliers for projective symmetry extensions in a magnetic field (Herzog-Arbeitman et al., 2022).
• Pfaffian and Commutator Structures: For Euler class and fragile topology, the use of Pfaffians over the real occupied-band subblock is essential, and commutator structures [e.g., $Q = 1 - P$1] are critical for gauge-invariant local evaluation (Li et al., 2023).
• Kernel Polynomial Method (KPM): Large-scale evaluation of markers in 3D or amorphous systems is enabled by approximating projectors via Chebyshev expansions, significantly reducing computational cost (Roy et al., 2 Dec 2025).
• Information-Theoretic Approaches: In the presence of complex symmetry or data-driven features, machine learning extracts marker profiles by ranking entropy reductions in nonlinear classifiers, compressing the description to maximally informative spatial regions (Holanda et al., 2019).

3. Physical Interpretation and Critical Properties

• Spatial Structure and Criticality: Markers are typically uniform within the bulk of a topological phase, show sharp peaks or distinctive profiles at critical points, and may decay or oscillate in the vicinity of symmetry centers, boundaries, or defects.
  • In Chern insulators, the local marker exhibits one-parameter scaling collapse near transitions, with a width $Q = 1 - P$2 and $Q = 1 - P$3 (Caio et al., 2018).
  • Nonlocal correlators $Q = 1 - P$4 exhibit distinctive real-space modulation patterns (isotropic, cross-shaped, checkerboard) corresponding to different critical momenta, each surviving strong disorder and indicating the precise nature of the transition (Marsal et al., 12 Nov 2025).
  • Markers in crystalline-symmetric phases are exponentially localized to fixed sets and robustly preserve their quantized values against disorder away from these centers (Mondragon-Shem et al., 2019).
• Defects, Boundary, and Dimensional Reduction:
  • Markers provide a real-space diagnostic for defect-bound states such as skin effects in non-Hermitian systems—jumps in the localizer index track the location and number of topologically trapped modes at boundaries or dislocations (Chadha et al., 2023).
  • The transition from 2D to 1D topological behavior can be quantified by dimensional crossover of marker profiles, with sharp changes in local marker gaps and invariants indicating the minimal geometry required for topological protection (Rodriguez-Vega et al., 15 May 2025).
• Disorder Robustness and Fragility:
  • Markers constructed from projectors and position operators are invariant under impurities or disorder that modify only nonzero matrix elements of the original Hamiltonian, provided the disorder strength and density remain below thresholds set by the bulk gap and correlation length (Oliveira et al., 2024).
  • For fragile topology, such as characterized by Euler markers, disorder exceeding a critical strength or density destroys the quantized spatial average, marking the collapse of global topological order (Li et al., 2023).
• Information Compression and Experiment:
  • Shannon-entropy markers reveal that topological information is concentrated at a small set of spatial locations in models such as the SSH chain, allowing for significant lattice compression with minimal loss of topological fidelity (Holanda et al., 2019).
  • Experimentally, markers can in principle be accessed via scanning probe or quantum-gas microscope measurements of the one-particle density matrix, STM imaging of local pseudospin textures, or NV-center magnetometry of orbital magnetization (Borhani et al., 2024, Caio et al., 2018).

4. Symmetry, Moiré Systems, and Fragile/Obstructed Topology

• Moiré Systems and Fictitious Magnetic Fields:
  • In moiré materials (twisted bilayers, TMDs), the real-space Chern marker for a single Bloch state is generically fragile—nonzero only if the spinor amplitude vanishes at certain points in the unit cell. This is only realized in fine-tuned chiral or adiabatic limits, otherwise the real-space Chern number of a single state vanishes (Kolář et al., 30 Jun 2025).
  • Robust, symmetry-protected markers emerge at the ensemble level via collective pseudospin textures, which lead to a well-defined, gap-protected real-space Chern number,

  $Q = 1 - P$5

  even in the absence of momentum-space band structure (Kolář et al., 30 Jun 2025). - Symmetry indicators (e.g., $Q = 1 - P$6, $Q = 1 - P$7) enforce nonzero markers across all twist angles, leading to spatial patterns measurable by STM as skyrmion- or dipole-like pseudospin textures.

• Crystalline and Higher-Order Topology:

  • Real-space symmetry-resolved markers (real-space invariants, RSIs) classify fragile, obstructed, or higher-order topological phases—not only in crystals but also under flux (Hofstadter response) or in amorphous and quasicrystalline systems (Herzog-Arbeitman et al., 2022, Li et al., 2023).
  • For Euler class and Stiefel–Whitney phases (and their corner modes), site- or bond-resolved Euler markers remain quantized in open or aperiodic lattices, provided the relevant spectral gap is open (Li et al., 2023).

5. Applications Across Platforms and Novel Regimes

• Disordered, Amorphous, and Quasicrystalline Materials: Real-space markers remain valid and computable in amorphous and quasicrystalline systems where momentum-space methods fail, enabling robust diagnosis of topology under extreme inhomogeneity (Marsal et al., 12 Nov 2025, Li et al., 2023).
• Non-Hermitian Systems and Skin Effects: The spectral localizer provides a real-space diagnostic for both boundary- and dislocation-induced skin effects in non-Hermitian lattices, including strongly disordered and translation-breaking regimes. Localizer index jumps pinpoint the locations and count of skin modes, unifying point-gap topology, and defect-induced localization (Chadha et al., 2023).
• Interacting and Many-Body Systems: Many-body generalizations of real-space markers—e.g., twist-operator ℤ₂ markers—are directly evaluable in Fock space for ground states of correlated (e.g., Hubbard) Hamiltonians, circumventing the need for Bloch-band or Wilson-loop analyses (Gilardoni et al., 2022).
• Quantum Critical Phenomena: Nonlocal real-space markers act as correlation functions, with their spatial range diverging near critical points, allowing quantitative finite-size scaling and critical exponent extraction directly from real-space observables (Roy et al., 2 Dec 2025).

6. Limitations, Computational Considerations, and Extensions

• Scalability and Computation: Direct evaluation of projector-based markers may require full diagonalization, becoming infeasible for large systems; kernel polynomial methods (KPM) and Chebyshev expansion approaches render 3D and large 2D systems tractable (Roy et al., 2 Dec 2025).
• Fragility of Certain Definitions: Real-space markers for single Bloch states can be fragile, vanishing under generic perturbations; robust quantization is only ensured at the ensemble/projector or many-body level and when a spectral gap protects the definition (Kolář et al., 30 Jun 2025).
• Boundary and Gauge Considerations: For periodic boundary conditions, position operators are ill defined; new definitions based on "single-point" or large-supercell limits circumvent this, extending Chern marker evaluations to true bulk systems (Baù et al., 2023).
• Open Questions: Current lines of inquiry include robust markers for interacting systems without Slater-determinant structure, extension to other characteristic classes (e.g., Pontryagin, higher Stiefel–Whitney), real-space markers for dynamical or Floquet systems, and further connections between statistical learning-theoretic and operator-theoretic markers (Li et al., 2023, Holanda et al., 2019).

7. Representative Examples of Real-Space Topological Markers

Marker Type	Mathematical Expression	Application Domain
Local Chern	$Q = 1 - P$8	2D insulators, Chern/topological phases
Euler Marker	$Q = 1 - P$9	2D real/fragile systems
Spectral Localizer	Signature of Hermitian $$w(x) = \frac{1}{2\pi i} \langle x|C\,P\,[X, P]|x\rangle,$$0 as in section 2	All symmetry classes, disordered/defected
Machine-Learning Entropy	Normalized site entropy reductions $$w(x) = \frac{1}{2\pi i} \langle x|C\,P\,[X, P]|x\rangle,$$1	Data-driven phase diagram identification
Symmetry-Invariant Marker	$$w(x) = \frac{1}{2\pi i} \langle x|C\,P\,[X, P]|x\rangle,$$2	Crystalline, higher-order, fragile topology
Nonlocal Chern	$$w(x) = \frac{1}{2\pi i} \langle x|C\,P\,[X, P]|x\rangle,$$3	Correlations, criticality, amorphous

The breadth and continual refinement of real-space topological markers has established them as indispensable diagnostics and classification tools—beyond the reach of momentum-space invariants—for contemporary studies of topological matter, defect engineering, amorphous systems, quantum criticality, and the machine learning–aided discovery of novel topological phases.