Energy- and Space-Resolved Topological Marker

• Energy- and space-resolved topological markers are real-space diagnostics that assign local topological information based on energy or quasienergy, applicable in varied settings.
• They employ diverse constructions—such as Green’s-function markers, spectral localizers, and Středa markers—to effectively identify bulk, edge, and interface phenomena.
• These markers enable precise mapping of topological transitions and gap closures in static, driven, and disordered systems, advancing both theoretical and computational studies.

An energy- and space-resolved topological marker is a real-space diagnostic that assigns topological information to a position and to an energy or quasienergy, thereby identifying both where topological structure is localized and which spectral window carries it. In the current literature, the term denotes not a single invariant but a family of constructions: Green’s-function markers for interacting Chern insulators, spectral-localizer signatures for static and Floquet systems, Středa-response markers based on the magnetic response of the local density of states, energy-resolved local spin Chern markers, and crystalline-symmetry markers built from energy filters and symmetry operators. Their common purpose is to replace momentum-space bulk indices by local quantities that remain meaningful in finite, disordered, amorphous, driven, interacting, bosonic, and non-Hermitian settings (Markov et al., 2020, Ghosh et al., 2024, Ghosh et al., 13 Aug 2025, Defossez et al., 8 May 2026).

1. Common structure and defining features

Across these constructions, the marker depends on a real-space label such as a site $i$, a point $(x,y)$, or a cell $r$, together with an energy variable $E$, quasienergy $E\in(-\pi,\pi]$, or Matsubara frequency $\omega$. The local quantity is built from one of several objects: a spectral projector $P(E)$, a full interacting Green’s function $G(i\omega)$, a spectral localizer $L_{x,y,E}$, or a response function such as $\partial_B \rho(x,E)$. In all cases, bulk quantization or bulk constancy is tied to the persistence of a gap or localizer gap, whereas edges, interfaces, disorder-induced puddles, or phase boundaries appear as spatial jumps, suppression, or gap closings.

A basic distinction separates integrated markers from energy-resolved ones. Integrated markers recover familiar bulk invariants after summing over space or energy. Energy-resolved markers instead retain the spectral decomposition and therefore indicate which energies contribute to the topological response. This feature is explicit in the local Green marker $(x,y)$0, the spectral-localizer signature $(x,y)$1, the Středa marker $(x,y)$2, and the energy-dependent local spin Chern marker $(x,y)$3 (Markov et al., 2020, Ghosh et al., 13 Aug 2025, Defossez et al., 8 May 2026, Júnior et al., 2024).

The literature also makes clear that locality is not merely a visualization device. In the spectral-localizer framework, the localizer gap $(x,y)$4 or the smallest singular value $(x,y)$5 provides a local criterion for whether a point in space and energy is topologically trivial or hosts a boundary or defect mode. In interacting Green’s-function constructions, locality is quasi-local: in a gapped system with exponentially decaying correlations, loops leaving a single supercell vanish in the thermodynamic limit. In Středa-based and spin-Chern constructions, locality enters through the local density of states, position operators, or site-resolved traces (Ghosh et al., 2024, Cerjan et al., 2023, Markov et al., 2020, Júnior et al., 2024).

2. Principal constructions

Several representative formulas define the main families of energy- and space-resolved markers.

Construction	Defining local quantity	Representative setting
Local Green Marker	$(x,y)$6	interacting Chern insulators
Spectral localizer	$(x,y)$7 or $(x,y)$8	static, Floquet, amorphous systems
Středa marker	$(x,y)$9	disordered Chern systems, driven-dissipative bosonic lattices
Local spin Chern marker	$r$0	Kane–Mele–Rashba-type models
Crystalline marker	$r$1 from $r$2, $r$3, $r$4, or $r$5	crystalline topology
Biorthogonal winding marker	$r$6 from spectral projectors $r$7, $r$8	interacting non-Hermitian SSH chain

For interacting Chern insulators, Markov and Rubtsov define the frequency-resolved local Green marker by keeping the Matsubara frequency unintegrated in the real-space version of the Ishikawa–Matsuyama formula: $r$9 Integrating over $E$0 and averaging over a bulk region reproduces the Hall conductance. This is a direct many-body local marker based on the full interacting single-particle Green’s function (Markov et al., 2020).

In spectral-localizer formulations, the central object in $E$1 is

$E$2

with local signature

$E$3

Closely related formulations use $E$4 in place of $E$5 and define the local Chern marker at energy $E$6 as $E$7. These constructions are used for Floquet systems, amorphous materials, and disordered driven systems (Ghosh et al., 13 Aug 2025, Ghosh et al., 2024).

The Středa construction begins from the local density of states

$E$8

and defines the energy-resolved marker as

$E$9

Its integrated counterpart $E\in(-\pi,\pi]$0 recovers the total Chern number after bulk averaging in a fermionic insulator at zero temperature. In translationally invariant Bloch bands, the energy-resolved response contains both Berry-curvature and orbital-moment contributions (Defossez et al., 8 May 2026).

For spinful time-reversal-invariant systems with non-conserved spin, the energy-resolved local spin Chern marker is obtained by replacing the valence projector $E\in(-\pi,\pi]$1 with an energy-cutoff projector $E\in(-\pi,\pi]$2, building the projected spin matrix $E\in(-\pi,\pi]$3, splitting its spectrum into positive and negative branches, and then evaluating

$E\in(-\pi,\pi]$4

An analogous construction applies with $E\in(-\pi,\pi]$5 for in-plane polarization (Júnior et al., 2024).

Crystalline and non-Hermitian extensions preserve the same logic. In crystalline topology, one may define

$E\in(-\pi,\pi]$6

or, in a spectral-localizer language, $E\in(-\pi,\pi]$7. In the interacting non-Hermitian SSH model, a biorthogonal real-space winding marker $E\in(-\pi,\pi]$8 is further decomposed spectrally into $E\in(-\pi,\pi]$9 and $\omega$0 (Cerjan et al., 2023, Sousa-Júnior et al., 4 Jun 2026).

3. Spatial locality, spectral meaning, and protection

The energy variable acquires a clear physical interpretation in each framework. In the Green’s-function construction, peaks in the bulk-averaged marker $\omega$1 appear near the edges of the single-particle gap and reflect the topological “winding” contributed by states at those energies. Inside the gap, $\omega$2 is nearly flat and positive in the topological phase and drops toward zero in the trivial phase. Spatially resolved $\omega$3 shows a stronger frequency dependence on edge sites, where the marker is enhanced at frequencies corresponding to edge-mode energies in the spectral function $\omega$4 (Markov et al., 2020).

In spectral-localizer formulations, locality is controlled by the centered position operators. For amorphous Floquet systems, the localizer probes the local physics in a ball of radius $\omega$5, where $\omega$6 is the local energy gap of $\omega$7 near $\omega$8. In the bulk of a topological region, $\omega$9 is constant and integer-valued in the interior and drops to zero outside. Sweeping $P(E)$0 across a gap detects the gap’s topology, and in a nontrivial region $P(E)$1 equals the number of boundary modes crossing energy $P(E)$2 (Ghosh et al., 13 Aug 2025).

The Středa marker has a different interpretation. It measures the magnetic-field response of the local density of states, so the energy-resolved quantity accesses quantum geometry beyond the integrated Hall response. In Bloch-band language,

$P(E)$3

This directly associates $P(E)$4 with Berry curvature and orbital moments (Defossez et al., 8 May 2026).

In symmetry-based markers, protection is made explicit by a local gap. For crystalline topology, the smallest singular value

$P(E)$5

is the minimal norm of a local perturbation needed to close the local gap at $P(E)$6. Boundary-localized or corner-localized states appear as jumps of $P(E)$7 at a fixed interface point and as localized peaks in the spatial map, while $P(E)$8 at the corresponding energy (Cerjan et al., 2023).

4. Floquet and non-equilibrium formulations

Periodic driving is one of the principal settings in which energy and space resolution become indispensable. In Floquet systems one constructs the Floquet operator

$P(E)$9

defines the effective Floquet Hamiltonian $G(i\omega)$0, and treats quasienergy $G(i\omega)$1 as the spectral parameter. The spectral-localizer construction then applies directly by replacing $G(i\omega)$2 with $G(i\omega)$3 (Ghosh et al., 13 Aug 2025).

A central result of the Floquet spectral-localizer literature is that the local markers can characterize $G(i\omega)$4- and $G(i\omega)$5-boundary modes individually. In one-dimensional chiral chains, two chiral-symmetric effective Hamiltonians $G(i\omega)$6 are built from half-period evolution operators, and the corresponding local winding markers satisfy

$G(i\omega)$7

In two-dimensional driven Chern systems, the localizer yields $G(i\omega)$8 and $G(i\omega)$9, allowing phases labeled by $L_{x,y,E}$0. This is important because the Floquet operator and effective Floquet Hamiltonian can fail to provide complete topological characterization when $L_{x,y,E}$1- and $L_{x,y,E}$2-modes coexist (Ghosh et al., 2024).

In monolayer amorphous carbon driven by circularly polarized laser light, the same formalism yields a complete topological characterization at quasienergies $L_{x,y,E}$3 and $L_{x,y,E}$4. For appropriate drive amplitude $L_{x,y,E}$5 and frequency $L_{x,y,E}$6, the local signature is $L_{x,y,E}$7 uniformly in the interior and zero outside, signaling a regular Chern $L_{x,y,E}$8 phase at quasienergy $L_{x,y,E}$9 with chiral edge modes. A smaller region of the $\partial_B \rho(x,E)$0–$\partial_B \rho(x,E)$1 plane exhibits $\partial_B \rho(x,E)$2, signaling an anomalous Floquet phase with edge modes crossing at $\partial_B \rho(x,E)$3. Spatial phase maps delimit bulk regions with constant $\partial_B \rho(x,E)$4 from edges where the local gap closes and $\partial_B \rho(x,E)$5 jumps (Ghosh et al., 13 Aug 2025).

The Green’s-function framework also admits a non-equilibrium extension. The Matsubara Green’s function is replaced by the contour-ordered Green’s function on the Keldysh contour, followed by a Wigner transform,

$\partial_B \rho(x,E)$6

and the time-resolved marker becomes

$\partial_B \rho(x,E)$7

Choosing the contour to be purely imaginary recovers equilibrium. For fast quenches or periodic driving, the same structure holds with central-time and Wigner-frequency derivatives (Markov et al., 2020).

5. Interacting, bosonic, spin, and non-Hermitian extensions

The interacting case is historically significant because local markers were first formulated for non-interacting systems, and the interacting Green’s-function construction was proposed precisely to address that gap. Markov and Rubtsov show that the local Green marker identifies topological transitions in finite lattices of a Chern insulator with Anderson disorder and Hubbard interactions, and they emphasize that the proposal directly addresses the interacting system rather than an effective non-interacting surrogate (Markov et al., 2020).

Driven-dissipative bosonic lattices provide an experimentally oriented implementation of an energy-resolved marker through the Středa response. In the bosonic protocol, uniform loss with rate $\partial_B \rho(x,E)$8 and coherent pumping with site-dependent amplitudes $\partial_B \rho(x,E)$9, where the $(x,y)$00 are independent random phases, yield steady-state occupations

$(x,y)$01

Each eigenmode is therefore populated incoherently with a Lorentzian weight centered at $(x,y)$02 and width $(x,y)$03, and scanning $(x,y)$04 reconstructs a coarse-grained $(x,y)$05 convolved with the Lorentzian kernel. The same work uses this marker to analyze disordered Haldane flakes and topological Anderson insulators (Defossez et al., 8 May 2026).

Magnonic systems admit a distinct route through local circular dichroism. In the single-magnon approximation, the ideal Bianco–Resta local Chern marker $(x,y)$06 is related to an experimentally accessible differential rate $(x,y)$07 induced by weak global circular drives of opposite chirality. The measurable marker is

$(x,y)$08

and in the bulk $(x,y)$09 is quantized to the band’s Chern number. Because the response is frequency dependent, the construction is explicitly energy-resolved through the drive frequency $(x,y)$10 (Bermond et al., 24 Apr 2025).

For strong Rashba spin-orbit coupling, where $(x,y)$11, the local spin Chern marker remains applicable after projecting the spin operator into the occupied subspace or, in the energy-resolved version, into the subspace below the cutoff $(x,y)$12. The topological protection of $(x,y)$13 requires both the single-particle gap $(x,y)$14 and the valence-projected spin gap $(x,y)$15 or $(x,y)$16 to remain open. The finite honeycomb-flake calculations show that both the energy and valence-projected spin matrix eigenvalues exhibit a gap that protects the marker (Júnior et al., 2024).

The non-Hermitian interacting SSH chain requires biorthogonal projectors. The global winding marker $(x,y)$17 is built from the chiral operator $(x,y)$18, the unit-cell position operator $(x,y)$19, and the biorthogonal projectors $(x,y)$20 and $(x,y)$21, while the local marker

$(x,y)$22

is quantized in the bulk under both periodic and open boundary conditions. Its spectral decomposition

$(x,y)$23

provides an energy-resolved form. In the interacting calculations, the marker remains a robust diagnostic of the topological phases and vanishes at the onset of a charge density wave (Sousa-Júnior et al., 4 Jun 2026).

6. Numerical implementation and computational scaling

Despite their conceptual differences, these markers are designed for finite real-space calculations. For the local Green marker, one computes $(x,y)$24 on a finite cluster via exact diagonalization or many-body solvers such as DMFT or QMC, discretizes Matsubara frequencies $(x,y)$25 up to a cutoff $(x,y)$26, forms the loop sum for each site and frequency, averages over a bulk region excluding a boundary strip, averages over $(x,y)$27 disorder realizations when required, and checks convergence in $(x,y)$28, $(x,y)$29, $(x,y)$30, and $(x,y)$31 (Markov et al., 2020).

For spectral localizers, direct diagonalization of the $(x,y)$32 localizer can be replaced by an $(x,y)$33 or $(x,y)$34 factorization. In the amorphous Floquet construction, Sylvester’s law of inertia reduces the signature evaluation to two $(x,y)$35 matrices,

$(x,y)$36

which cuts the cost roughly by a factor of $(x,y)$37. The localizer requires the system to be larger than the localization radius $(x,y)$38, so $(x,y)$39 is chosen small enough that bulk sites remain well away from boundaries; in practice $(x,y)$40. For the Floquet operator $(x,y)$41, time evolution is Trotterized on a sufficiently fine time grid, for example $(x,y)$42 steps per period, and the marker is averaged over $(x,y)$43 independent amorphous samples to obtain smooth phase diagrams (Ghosh et al., 13 Aug 2025).

The driven-dissipative Středa protocol can be implemented either from the Lindblad steady state or by diagonalizing the Hamiltonian and evaluating the finite difference $(x,y)$44. Scanning $(x,y)$45 in steps $(x,y)$46 reconstructs the coarse-grained marker. For larger systems, the same work explicitly invokes the Kernel-Polynomial Method (Defossez et al., 8 May 2026).

A full KPM treatment of an energy- and space-resolved marker in three dimensions is given for class AII topological insulators. There the projector $(x,y)$47 is expanded in Chebyshev polynomials of the rescaled Hamiltonian $(x,y)$48, with Jackson kernels $(x,y)$49, and the local marker is assembled as

$(x,y)$50

The method is described as an $(x,y)$51 memory method, with $(x,y)$52 equal to the number of orbitals times sites, and is intended to make higher-dimensional markers numerically efficient (Roy et al., 2 Dec 2025).

7. Representative phenomena, diagnostics, and recurring misunderstandings

The principal physical use of these markers is the diagnosis of topological transitions in inhomogeneous spectra. In the interacting Chern-insulator study, bulk-averaged $(x,y)$53 decreases from $(x,y)$54 at $(x,y)$55 to $(x,y)$56 near the transition at $(x,y)$57 and to $(x,y)$58 in the Anderson insulator at $(x,y)$59. For the Hubbard-driven transition on a $(x,y)$60 cluster, $(x,y)$61 at $(x,y)$62, is suppressed to $(x,y)$63 at $(x,y)$64, and tends to zero deep in the Mott regime for $(x,y)$65. In both cases, the disappearance of the peaks in $(x,y)$66 accompanies the collapse of the topological structure of the Green’s function (Markov et al., 2020).

In disordered Chern systems, the Středa marker makes the disorder evolution visible in energy space. In the non-trivial regime of the disordered Haldane model, $(x,y)$67 shows two hot-spot lines at the lower and upper band edges that ride in energy as $(x,y)$68 increases, until they merge at $(x,y)$69 and the gap closes. In the trivial regime with broken time-reversal symmetry, disorder drives one valley through zero energy and inverts its sign, producing a window $(x,y)$70 in which the integrated Středa response becomes non-zero and signals a Topological Anderson Insulator. The bulk-averaged local Chern marker $(x,y)$71 reproduces the integrated Středa response, and finite-size scaling shows convergence toward quantized plateaus in the TAI regime (Defossez et al., 8 May 2026).

In bosonic and magnonic settings, the markers remain local but the measured quantity changes. For topological magnons, $(x,y)$72 equals $(x,y)$73 in the bulk of the $(x,y)$74 lattice and exhibits large negative deviations near the edges; finite-size scaling shows exponential convergence of both the exact $(x,y)$75 and the finite-time $(x,y)$76 to the integer. In the non-Hermitian SSH model, the bulk marker realizes the phase sequence $(x,y)$77–$(x,y)$78–$(x,y)$79–$(x,y)$80 in the non-interacting limit and vanishes when strong interactions drive a charge density wave. Under open boundary conditions, boundary noise grows near exceptional points, but the central cells still settle on the same plateaus (Bermond et al., 24 Apr 2025, Sousa-Júnior et al., 4 Jun 2026).

A recurrent simplification is to treat every energy-resolved marker as merely a local restatement of a global Chern number. The cited works show a broader picture. The spectral localizer distinguishes $(x,y)$81- and $(x,y)$82-modes individually in Floquet systems; the Středa marker resolves Berry-curvature and orbital-moment structure in energy; crystalline markers encode where symmetry-protected boundary modes occur and how robust they are through $(x,y)$83; the Green’s-function marker identifies which Matsubara frequencies carry the bulk winding; and the spin-Chern construction shows that quantization can persist even when spin is not conserved, provided the relevant projected spin gap remains open (Ghosh et al., 2024, Defossez et al., 8 May 2026, Cerjan et al., 2023, Júnior et al., 2024).

Taken together, these developments suggest that “energy- and space-resolved topological marker” is best understood as a methodological class rather than a unique invariant. What unifies the class is the replacement of translationally invariant bulk formulas by local spectral diagnostics capable of resolving bulk, edge, interface, and disorder physics in the same framework.