--- title: Energy- and Space-Resolved Topological Marker url: https://www.emergentmind.com/topics/energy-and-space-resolved-topological-marker type: topic --- # Energy- and Space-Resolved Topological Marker 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 [2009.01801], [2408.08548], [2508.09571], [2605.07948]. ## 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 \(C(i,\omega)\), the spectral-localizer signature \(\nu(x,y;E)\), the Středa marker \(m(x,E)\equiv \partial_B\rho(x,E)\), and the energy-dependent local spin Chern marker \(\mathfrak C_s(r;E)\) [2009.01801], [2508.09571], [2605.07948], [2409.02319]. The literature also makes clear that locality is not merely a visualization device. In the spectral-localizer framework, the localizer gap \(\sigma_L^{(x,E)}\) or the smallest singular value \(\mu(r,E)\) 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 [2408.08548], [2310.01398], [2009.01801], [2409.02319]. ## 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 | \(C(i,\omega)\) | interacting Chern insulators | | Spectral localizer | \(\nu(x,y;E)=\frac12\,\mathrm{sig}[L_{x,y,E}]\) or \(C(x,y;E)=\frac12\,\mathrm{Sig}[L]\) | static, Floquet, amorphous systems | | Středa marker | \(m(x,E)=\partial_B\rho(x,E)\) | disordered Chern systems, driven-dissipative bosonic lattices | | Local spin Chern marker | \(\mathfrak C_s(r;E)\) | Kane–Mele–Rashba-type models | | Crystalline marker | \(M(r,E)\) from \(P(E)\), \(f_r\), \(O_g\), or \(\widetilde L_r(E)\) | crystalline topology | | Biorthogonal winding marker | \(M(i;E_1,E_2)\) from spectral projectors \(P(E)\), \(Q(E)\) | 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: \[ C(i,\omega)\equiv -\frac{i}{6}\,\epsilon_{\mu\nu\rho}\sum_{jkl} \mathrm{Tr}\!\Bigl[ G_{ij}(i\omega)\,r_j^\mu\, G^{-1}_{jk}(i\omega)\,r_k^\nu\, G_{kl}(i\omega)\,r_l^\rho\, G^{-1}_{li}(i\omega) \Bigr]. \] Integrating over \(\omega\) 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 [2009.01801]. In spectral-localizer formulations, the central object in \(d=2\) is \[ L_{x,y,E}=\kappa\bigl[(X-xI)\tau_x+(Y-yI)\tau_y\bigr]+(H_F-EI)\tau_z, \] with local signature \[ \nu(x,y;E)=\frac12\,\mathrm{sig}[L_{x,y,E}]. \] Closely related formulations use \(H_{\rm eff}\) in place of \(H_F\) and define the local Chern marker at energy \(E\) as \(C(x,y;E)=\frac12\,\mathrm{Sig}[L_{x,y;E}]\). These constructions are used for Floquet systems, amorphous materials, and disordered driven systems [2508.09571], [2408.08548]. The Středa construction begins from the local density of states \[ \rho(x,E)=\sum_\lambda |\psi_\lambda(x)|^2\delta(E-\epsilon_\lambda), \] and defines the energy-resolved marker as \[ m(x,E)\equiv \partial_B \rho(x,E)\quad (B\to0). \] Its integrated counterpart \(M(x)\equiv \partial_B n(x)\) 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 [2605.07948]. For spinful time-reversal-invariant systems with non-conserved spin, the energy-resolved local spin Chern marker is obtained by replacing the valence projector \(P\) with an energy-cutoff projector \(P(E)\), building the projected spin matrix \(M_v^{s_z}(E)=P(E)\hat s_z P(E)\), splitting its spectrum into positive and negative branches, and then evaluating \[ \mathfrak C_s(r;E)=\frac{\mathfrak C_+(r;E)-\mathfrak C_-(r;E)}{2}. \] An analogous construction applies with \(\hat s_n\) for in-plane polarization [2409.02319]. Crystalline and non-Hermitian extensions preserve the same logic. In crystalline topology, one may define \[ M(r,E)=\sum_{g\in G}\chi(g)\,\mathrm{Tr}\!\left\{f_r\,[P(E),O_g]^\dagger [P(E),O_g]\right\}, \] or, in a spectral-localizer language, \(M(r,E)=\frac12\,\mathrm{sig}[\widetilde L_r(E)]\). In the interacting non-Hermitian SSH model, a biorthogonal real-space winding marker \(M(i)\) is further decomposed spectrally into \(M(i;E_1,E_2)\) and \(M(i,E)=\sum_{E'}M(i;E,E')\) [2310.01398], [2606.06466]. ## 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 \(\overline C(\omega)\) appear near the edges of the single-particle gap and reflect the topological “winding” contributed by states at those energies. Inside the gap, \(\overline C(\omega)\) is nearly flat and positive in the topological phase and drops toward zero in the trivial phase. Spatially resolved \(C(i,\omega)\) shows a stronger frequency dependence on edge sites, where the marker is enhanced at frequencies corresponding to edge-mode energies in the spectral function \(A(i,\omega)\) [2009.01801]. 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 \(\sim \Delta/\kappa\), where \(\Delta\) is the local energy gap of \(H_F\) near \(E\). In the bulk of a topological region, \(\nu(x,y;E)\) is constant and integer-valued in the interior and drops to zero outside. Sweeping \(E\) across a gap detects the gap’s topology, and in a nontrivial region \(\nu\) equals the number of boundary modes crossing energy \(E\) [2508.09571]. 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, \[ \Phi_0\,\partial_B \rho(E) = \sum_\beta \int_{BZ}\frac{d^2k}{2\pi} \left[ \mathcal F^\beta_{xy}(k)\,\delta(E-\epsilon_{\beta k}) +\frac{\Phi_0}{2\pi}m^\beta_z(k)\,\partial_E\delta(E-\epsilon_{\beta k}) \right]. \] This directly associates \(m(x,E)\) with Berry curvature and orbital moments [2605.07948]. In symmetry-based markers, protection is made explicit by a local gap. For crystalline topology, the smallest singular value \[ \mu(r,E)=\min_{\psi\neq 0}\|\widetilde L_r(E)\psi\| \] is the minimal norm of a local perturbation needed to close the local gap at \((r,E)\). Boundary-localized or corner-localized states appear as jumps of \(M(r,E)\) at a fixed interface point and as localized peaks in the spatial map, while \(\mu(r,E)\to0\) at the corresponding energy [2310.01398]. ## 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 \[ U(T,0)=T\exp\!\left[-i\int_0^T dt\,H(t)\right], \] defines the effective Floquet Hamiltonian \(H_F=-i\ln U(T,0)\), and treats quasienergy \(E\in(-\pi,\pi]\) as the spectral parameter. The spectral-localizer construction then applies directly by replacing \(H\) with \(H_F\) [2508.09571]. A central result of the Floquet spectral-localizer literature is that the local markers can characterize \(0\)- and \(\pi\)-boundary modes individually. In one-dimensional chiral chains, two chiral-symmetric effective Hamiltonians \(H_{\rm eff}^{a,b}\) are built from half-period evolution operators, and the corresponding local winding markers satisfy \[ W_0(x)=\frac{\nu_a(x)+\nu_b(x)}{2},\qquad W_\pi(x)=\frac{\nu_a(x)-\nu_b(x)}{2}. \] In two-dimensional driven Chern systems, the localizer yields \(C(x,y;0)\) and \(C(x,y;\pi)\), allowing phases labeled by \((C_0,C_\pi)=(0,0),(1,0),(1,1),(0,1)\). This is important because the Floquet operator and effective Floquet Hamiltonian can fail to provide complete topological characterization when \(0\)- and \(\pi\)-modes coexist [2408.08548]. In monolayer amorphous carbon driven by circularly polarized laser light, the same formalism yields a complete topological characterization at quasienergies \(0\) and \(\pm\pi\). For appropriate drive amplitude \(A\) and frequency \(\Omega\), the local signature is \(\nu(x,y;0)=-1\) uniformly in the interior and zero outside, signaling a regular Chern \(1\) phase at quasienergy \(0\) with chiral edge modes. A smaller region of the \(A\)–\(\Omega\) plane exhibits \(\nu(x,y;\pi)=-1\), signaling an anomalous Floquet phase with edge modes crossing at \(E=\pm\pi\). Spatial phase maps delimit bulk regions with constant \(\nu\) from edges where the local gap closes and \(\nu\) jumps [2508.09571]. 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, \[ G(T,\omega;\mathbf r,\mathbf r') = \int d\tau\,e^{i\omega\tau} G\!\left(T+\frac{\tau}{2},T-\frac{\tau}{2};\mathbf r,\mathbf r'\right), \] and the time-resolved marker becomes \[ C(i,T,\omega) = -\frac{i}{6}\epsilon_{\mu\nu\rho}\sum_{jkl} \mathrm{Tr}\!\Bigl[ G_{ij}(T,\omega)\,r_j^\mu\, G^{-1}_{jk}(T,\omega)\,r_k^\nu\, G_{kl}(T,\omega)\,r_l^\rho\, G^{-1}_{li}(T,\omega)\,r_i^\rho \Bigr]. \] 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 [2009.01801]. ## 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 [2009.01801]. 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 \(\gamma\) and coherent pumping with site-dependent amplitudes \(f_i=f\,e^{i\theta_i}\), where the \(\theta_i\) are independent random phases, yield steady-state occupations \[ \langle |c_\lambda|^2\rangle=\frac{|f|^2}{(\omega_0-\epsilon_\lambda)^2+\gamma^2}. \] Each eigenmode is therefore populated incoherently with a Lorentzian weight centered at \(\omega_0\) and width \(\gamma\), and scanning \(\omega_0\) reconstructs a coarse-grained \(m_A(\omega)\) convolved with the Lorentzian kernel. The same work uses this marker to analyze disordered Haldane flakes and topological Anderson insulators [2605.07948]. Magnonic systems admit a distinct route through local circular dichroism. In the single-magnon approximation, the ideal Bianco–Resta local Chern marker \(C(r_i)\) is related to an experimentally accessible differential rate \(\Delta\Gamma_i(t)\) induced by weak global circular drives of opposite chirality. The measurable marker is \[ \widetilde C(r_i) = \lim_{t\to\infty}\frac{\hbar^2}{V_{\rm cell}\epsilon^2}\Delta\Gamma_i(t), \] and in the bulk \(\widetilde C(r_i)=C(r_i)\) is quantized to the band’s Chern number. Because the response is frequency dependent, the construction is explicitly energy-resolved through the drive frequency \(\omega\) [2504.17374]. For strong Rashba spin-orbit coupling, where \([\mathcal H,\hat s_z]\neq0\), 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 \(E\). The topological protection of \(\mathfrak C_s(r;E)\) requires both the single-particle gap \(\Delta^E\) and the valence-projected spin gap \(\Delta^{s_z}(E)\) or \(\Delta^{s_y}(E)\) 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 [2409.02319]. The non-Hermitian interacting SSH chain requires biorthogonal projectors. The global winding marker \(W\) is built from the chiral operator \(\hat S\), the unit-cell position operator \(\hat X\), and the biorthogonal projectors \(P\) and \(Q\), while the local marker \[ M(i)=\mathrm{Tr}\!\bigl[\Pi_i\,\hat S\,(P X Q+Q X P)\bigr] \] is quantized in the bulk under both periodic and open boundary conditions. Its spectral decomposition \[ M(i;E_1,E_2) = \mathrm{Tr}\!\Bigl[ \Pi_i\hat S\bigl(P(E_1)XQ(E_2)+Q(E_2)XP(E_1)\bigr) \Bigr] \] 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 [2606.06466]. ## 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 \(G_{ij}(i\omega_n)\) on a finite cluster via exact diagonalization or many-body solvers such as DMFT or QMC, discretizes Matsubara frequencies \(i\omega_n=(2n+1)\pi T\) up to a cutoff \(n_{\max}\), forms the loop sum for each site and frequency, averages over a bulk region excluding a boundary strip, averages over \(\sim 10^2\!-\!10^3\) disorder realizations when required, and checks convergence in \(\omega_{\max}\), \(n_{\max}\), \(L_x\), and \(L_y\) [2009.01801]. For spectral localizers, direct diagonalization of the \(2N\times 2N\) localizer can be replaced by an \(LDL^T\) or \(LDL^\dagger\) factorization. In the amorphous Floquet construction, Sylvester’s law of inertia reduces the signature evaluation to two \(N\times N\) matrices, \[ \tilde H,\qquad \tilde H+(\tilde X+i\tilde Y)\tilde H^{-1}(\tilde X-i\tilde Y), \] which cuts the cost roughly by a factor of \(4\). The localizer requires the system to be larger than the localization radius \(\sim \Delta/\kappa\), so \(\kappa\) is chosen small enough that bulk sites remain well away from boundaries; in practice \(\kappa\sim O(1/\mathrm{size})\). For the Floquet operator \(U(T)\), time evolution is Trotterized on a sufficiently fine time grid, for example \(N\sim100\!-\!500\) steps per period, and the marker is averaged over \(\sim 30\!-\!50\) independent amorphous samples to obtain smooth phase diagrams [2508.09571]. The driven-dissipative Středa protocol can be implemented either from the Lindblad steady state or by diagonalizing the Hamiltonian and evaluating the finite difference \(\partial_B\rho_A(\omega)\). Scanning \(\omega_0\) in steps \(\Delta\omega_0\sim\gamma\) reconstructs the coarse-grained marker. For larger systems, the same work explicitly invokes the Kernel-Polynomial Method [2605.07948]. A full KPM treatment of an energy- and space-resolved marker in three dimensions is given for class AII topological insulators. There the projector \(P(E)=\theta(E-H)\) is expanded in Chebyshev polynomials of the rescaled Hamiltonian \(\tilde H\), with Jackson kernels \(g_n\), and the local marker is assembled as \[ M(\mathbf r,E) = \sum_{n,m=0}^{N_{\rm KPM}} g_n g_m\,\mu_{nm}(\mathbf r)\,T_n(\epsilon)T_m(\epsilon), \qquad \epsilon=\frac{E-b}{a}. \] The method is described as an \(O(M\times N_{\rm KPM})\) memory method, with \(M\) equal to the number of orbitals times sites, and is intended to make higher-dimensional markers numerically efficient [2512.03022]. ## 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 \(\overline C(0)\) decreases from \(0.50\) at \(W/t=0\) to \(0.25\) near the transition at \(W/t=3\) and to \(0.03\) in the Anderson insulator at \(W/t=6\). For the Hubbard-driven transition on a \(4\times4\) cluster, \(\overline C(0)\approx0.50\) at \(U=0\), is suppressed to \(\sim0.2\!-\!0.3\) at \(U\approx1\!-\!2\), and tends to zero deep in the Mott regime for \(U\gtrsim4\). In both cases, the disappearance of the peaks in \(C(i,\omega)\) accompanies the collapse of the topological structure of the Green’s function [2009.01801]. 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, \(m_A(\omega;W)\) shows two hot-spot lines at the lower and upper band edges that ride in energy as \(W\) increases, until they merge at \(W_c\) 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 \(W_1