Local Chern Marker: Real-space Topological Diagnosis

• The local Chern marker is a spatially defined topological invariant that quantifies Berry curvature locally, even in systems lacking translational symmetry.
• Its real-space formulation via projectors enables evaluation in finite, amorphous, and fractal structures, overcoming limitations of traditional momentum-space invariants.
• Extensions to spin, bosonic, and interacting systems highlight its versatility, while its critical behavior tracks topological phase transitions and dynamic responses.

The local Chern marker is a spatially resolved, real-space quantity that assigns a local value to the topological invariant characterizing Chern insulators and related quantum phases. Unlike traditional global invariants—typically defined via integrals of Berry curvature over the Brillouin zone—the local Chern marker can be evaluated even in systems with disorder, inhomogeneities, boundaries, or lacking translational symmetry. Its real-space formulation enables the robust diagnosis of topological phases in complex material architectures, including finite, disordered, amorphous, and even fractal systems, and is generalizable to interacting settings, bosonic excitations, and higher-order topological phases.

1. Formal Definition and Variants

The canonical form of the local Chern marker for a two-dimensional non-interacting electronic system is

$$c(\mathbf{r}) = -4\pi\, \mathrm{Im}\left\langle \mathbf{r} \left| \hat{P}\, \hat{x}\, (\mathbb{I} - \hat{P})\, \hat{y}\, \hat{P} \right| \mathbf{r} \right\rangle$$

where $\hat{P}$ is the projector onto occupied states, $\hat{x}$ and $\hat{y}$ are the position operators, and $|\mathbf{r}\rangle$ denotes a local orbital basis (Caio et al., 2018, Baù et al., 2023). The marker quantifies the accumulation of Berry curvature at position $\mathbf{r}$ and, in the bulk of a Chern insulator, approaches the quantized integer value of the global Chern number. Its definition is strictly local in real space due to the exponential decay of $\hat{P}$ matrix elements in gapped phases.

Spinful and bosonic settings admit closely related constructions. The local spin Chern marker (LSCM) resolves spin-resolved topological invariants via the eigenstructure of a valence-projected spin operator: $$\mathfrak{c}_s(\mathbf{r}) = \frac{\mathfrak{c}_+(\mathbf{r}) - \mathfrak{c}_-(\mathbf{r})}{2}$$ where each partial marker is evaluated as (Júnior et al., 3 Sep 2024, Lage et al., 4 Mar 2025)

$$\mathfrak{c}_\sigma(\mathbf{r}) = 2\pi\, \mathrm{Im} \langle \mathbf{r} | Q_\sigma X P_\sigma Y Q_\sigma - P_\sigma X Q_\sigma Y P_\sigma | \mathbf{r} \rangle$$

with $P_\sigma$ the projector onto the $\sigma$-spin sector defined via the sign of the projected spin operator, and $Q_\sigma = 1 - P_\sigma$.

In bosonic (e.g., magnonic) systems, a local topological marker can be accessed via a circular dichroic response measured after a local driven-dissipative preparation, yielding markers of the form (Bermond et al., 24 Apr 2025): $$\mathcal{C}(\mathbf{r}_i) = -\frac{4\pi}{V_{\text{cell}}}\, \mathrm{Im} \sum_{g \in \text{LB}} \langle \downarrow_i | g \rangle\, \langle g | x \, Q \,y | g \rangle\, \langle g | \downarrow_i \rangle$$ where $Q$ projects onto higher bands and LB is the set of lowest-band states.

Alternative formulations, such as the spectral localizer (Spataru et al., 1 Mar 2025, Jezequel et al., 31 Jul 2025), cast the topological problem as a Clifford algebraic index: $$L_{x,y,E}(X,Y,H) = \kappa (X-xI)\otimes\sigma_x + \kappa(Y-yI)\otimes\sigma_y + (H-EI)\otimes\sigma_z$$ and the local invariant as the half-signature of $L$. Expanding the localizer's inverse square root demonstrates the exact equivalence to the local Chern (or winding) marker in the small-$\kappa$ limit (Jezequel et al., 31 Jul 2025).

2. Local Chern Marker in Heterogeneous and Disordered Systems

Traditional Brillouin-zone-based invariants are ill-defined in disordered, aperiodic, or heterogeneous systems. In contrast, the local Chern marker remains well-defined and robust against impurities, inhomogeneities, or even the absence of a bulk spectral gap (d'Ornellas et al., 2022, Spataru et al., 1 Mar 2025). This is exemplified by studies of amorphous networks, position-space generated Voronoi lattices, modulated superlattices, and systems with engineered defects.

Empirically, marker values computed in the bulk converge rapidly (exponentially in system size or with supercell size (Baù et al., 2023)) to the quantized topological invariant, even when disorder is sufficiently strong to close the global gap. The hallmark behavior is the clean distinction between topologically nontrivial (quantized marker) and trivial (zero marker) regions, with edge or defect regions displaying sharp transitions.

In experimental platforms—such as artificial graphene and 2D electron gases patterned with antidot arrays—direct comparison between the spatially averaged local Chern marker and measured Hall conductivity confirms this correspondence and resilience to disorder (Spataru et al., 1 Mar 2025). Additionally, local Chern marker maps reveal topological origins of localized states not visible in transport, e.g., antidot-confined Landau levels and interface-localized chiral edge states.

3. Extensions: Spin, Bosons, Interactions, and Higher-Dimensional Topology

Spin Chern Markers and Quantum Spin Hall Systems

For Hamiltonians with spin-orbit coupling and broken spin conservation (e.g., Kane–Mele–Rashba or sublattice-dependent models in graphene heterostructures), the LSCM provides a method to define and compute local equivalents of the $\mathbb{Z}_2$ invariant. The procedure requires diagonalizing the valence-projected spin matrix $M^{\hat{s}} = P\, \hat{s}\, P$ to define spin-resolved projectors, ensuring a finite spin gap $\Delta^{(\hat{s})}$ for quantization (Júnior et al., 3 Sep 2024, Lage et al., 4 Mar 2025).

Bosonic Topological Bands

Extensions to bosonic systems, particularly topological magnons in Heisenberg-Dzyaloshinskii-Moriya models, are achieved by combining local driven-dissipative steady-state preparation with global circular drives, culminating in a local, quantized dichroic response that directly probes the magnon bands' topology (Bermond et al., 24 Apr 2025).

Interactions and Many-Body Markers

For interacting systems, topological markers can be generalized via the single-particle Green's function (Markov et al., 2020), the one-particle density matrix (with adiabatic flattening for gapped spectra) (Hannukainen et al., 2023), or non-equilibrium generalizations within the Keldysh formalism. For instance, the local Green marker $\mathcal{G}(\mathbf{r})$ relates to the many-body Hall response via triangle-like products of Green functions and position operators, enabling identification of topological-to-trivial transitions in Chern insulators subject to both disorder and interactions.

Odd and Higher Dimensions

Local topological markers have been generalized to odd spatial dimensions, constructing chiral or Chern–Simons markers via an interpolation $P_\vartheta$ between trivial and topological density matrices, thereby encoding the $\mathbb{Z}$ winding number or $\mathbb{Z}_2$ invariant in systems such as 3D topological superconductors and insulators (Sykes et al., 2020, Hannukainen et al., 2022). In such constructions, the marker takes the form of local antisymmetric products of position operators and density matrices, often integrating over the synthetic parameter $\vartheta$.

Fractals and Nonperiodic Geometries

LSCMs remain quantized (up to edge corrections) in systems with nontrivial geometry, such as Sierpinski carpet fractal lattices. The valence-projected spin gap remains robust to increased fractal generation, ensuring applicability of the marker even as the system's self-similarity and internal boundaries proliferate (Lage et al., 4 Mar 2025).

4. Topological Phase Transitions, Dynamics, and Criticality

Real-Space Critical Behavior

Near topological quantum phase transitions, the spatial inhomogeneities of the local Chern marker encode the critical behavior: As control parameters approach the transition point, the size of domains with different marker values grows as a power-law, with critical exponent matching those extracted from Berry curvature analysis in clean systems. In disordered systems, the Kibble–Zurek scaling describes the non-equilibrium development of spatial inhomogeneities in the marker, with typical domain sizes increasing as $\tau^{1/2}$ for quench time $\tau$ (Ulčakar et al., 2020).

Out-of-Equilibrium and Time-Dependent Phenomena

In dynamics following a quantum quench or periodic drive (e.g., adiabatic ramp-up in Floquet Chern insulators), the local marker provides a real-time probe of the bulk topological transition. During Floquet protocols, as parameters are varied, the bulk marker transitions smoothly from zero to its quantized value, signifying an emergent nonequilibrium topological phase (Privitera et al., 2015). Out of equilibrium, the marker propagates via local currents that obey a continuity equation, in analogy to local order parameter dynamics (Caio et al., 2018, Golovanova et al., 2021, Markov et al., 2023).

New formulations based on the Streda response (i.e., local density response to a probe magnetic field) provide physically local, easily measurable definitions of the marker that automatically satisfy a lattice continuity equation—enabling both control and experimental accessibility (Golovanova et al., 2021, Markov et al., 2023).

5. Experimental Implementation and Physical Interpretation

Experimental Access

The local marker has been realized in cold atom experiments, optical lattice systems, photonic platforms, and solid-state artificial graphene (Spataru et al., 1 Mar 2025, Irsigler et al., 2019). In ultracold atoms, quantum gas microscopes enable the measurement of the single-particle density matrix required for marker tomography, including off-diagonal coherence via local two-level quench protocols (Irsigler et al., 2019). In photonic systems and engineered materials, spatially resolved measurements can access the local electrical response or density in the presence of small probe fields, thereby reconstructing the marker via the Streda formula (Golovanova et al., 2021, d'Ornellas et al., 2022).

Physical Meaning and Locality

The marker diagnoses bulk topology by mapping the local accumulation of Hall conductivity or Berry curvature. Its values are quasi-quantized deep in topological regions and deviate only in the vicinity of boundaries, interfaces, or sharp disorder. The spatial sum of the marker over the entire system is constrained to vanish in finite samples; its robustness derives from the “short-sightedness” of the projector in gapped systems (Caio et al., 2018). In higher-order topological insulators, the marker's profile reveals both surface and hinge contributions, distinguishing them from conventional topological phases (Pozo et al., 2019).

Spectral Properties and Protection

Quantization of local Chern and spin Chern markers requires a finite spectral gap—either in energy or in the eigenvalues of the valence-projected spin operator. This protects the marker from small perturbations and ensures correct identification of topological phases even in the presence of strong spin mixing, interaction, or complex sample geometry (Júnior et al., 3 Sep 2024, Lage et al., 4 Mar 2025).

6. Connections to Alternative Topological Diagnostics

Bott Index, Spectral Localizer, and Equivalence

Alternative real-space diagnostics—such as the Bott index or the spectral localizer—offer complementary and sometimes computationally advantageous approaches. The spectral localizer, in particular, is shown to be explicitly equivalent to the local Chern (even dimensions) or winding marker (odd dimensions) at leading order in the perturbative expansion in the localization parameter $\kappa$ (Jezequel et al., 31 Jul 2025). This bridges matrix-index-based and marker-based approaches and ensures that both are suitable for identifying and quantifying topology in systems without translational symmetry or well-defined momentum-space invariants.

Quantum Metric, Localization Markers, and Insulating Character

Generalizations to localization markers based on the commutator of position operators and projectors (Marrazzo et al., 2018) quantify local metallic versus insulating character and are able to distinguish topological boundary modes from insulating bulk. These metrics are sensitive only to the ground-state electron distribution and are thus applicable in regimes where spectral diagnostics (e.g., local density of states) become ambiguous, such as gapless Anderson insulators.

7. Generalizations and Outlook

The local Chern marker's scope encompasses topological characterization in disordered, amorphous, interacting, bosonic, higher-order, fractal, and out-of-equilibrium systems. Its various forms, mathematical equivalence with other real-space invariants, and experimental accessibility position it as a standard diagnostic tool across condensed matter, photonics, cold atoms, and quantum materials research. Future directions likely include extending these concepts to fractionalized phases (e.g., fractional Chern and spin liquids), quantum dynamics in higher-dimensional or synthetic dimension systems, material discovery pipelines for topological properties, and real-time monitoring of topological transitions in engineered quantum devices. A plausible implication is that position-space diagnostics—embedded in electronic structure and experimental workflows—will continue to expand the range of systems where topological order can be robustly classified and manipulated.