Spectral Localizer in Topological Systems

• Spectral localizer is a finite-dimensional Hermitian operator that encodes topological invariants, translating real-space data into indices like Chern or winding numbers.
• It employs a tunable parameter and finite-volume projection to achieve robust, numerical computation of topological markers even in disordered or non-translational systems.
• The framework extends to complex systems including non-Hermitian and Floquet models, enabling efficient analysis of topological phases via index-signature correspondence.

A spectral localizer is a finite-dimensional Hermitian operator, constructed from a (typically lattice or tight-binding) Hamiltonian and associated position or Dirac operators, whose signature encodes the topological index (e.g., Chern number, winding number, or strong/weak invariants) of the infinite system. This framework provides an explicit, real-space prescription for computing topological invariants that extends to non-translationally invariant, disordered, quasicrystalline, amorphous, or non-Hermitian systems, and allows for the efficient and robust extraction of topological information directly from finite samples without recourse to momentum-space analysis or abstract functional-analytic $K$-theory.

1. Mathematical Definition and Algebraic Structure

The spectral localizer is constructed from three primary ingredients: a Hamiltonian $H$ (bounded or self-adjoint on a Hilbert space $\mathcal H$, typically $\ell^2(\mathbb Z^d)\otimes\mathbb C^N$), position operators $X_j$, and a (possibly unbounded) Dirac operator $D$ formed from the $X_j$'s and a representation of the Clifford algebra $\{\Gamma_j\}_{j=1}^{d+1}$: $$L_\kappa(x) = \kappa \sum_{j=1}^d (X_j - x_j)\Gamma_j + (H - E_0)\Gamma_{d+1}$$ where $\kappa>0$ is a tuning parameter (setting relative weighting of position vs. spectral gap), $x$ is a probe center in real space, and $E_0$ is a spectral reference energy (Cerjan et al., 17 Jun 2025, Berkolaiko et al., 26 Dec 2025, Loring et al., 2018, Loring et al., 2018, Viesca et al., 2019, Jezequel et al., 31 Jul 2025).

Depending on context, one distinguishes between the odd case (pairing a unitary $A$ with a Dirac operator $D$) and even case (pairing a projection $P$ with a Dirac operator), with corresponding $2\times 2$ block matrix forms.

In practice, the localizer is usually compressed to a finite-volume (e.g., sites within $\|X-x\|\le\rho$), so $$L_{\kappa,\rho}(x) = \pi_\rho(x)L_\kappa(x)\pi_\rho(x)^*$$ is a Hermitian matrix of finite dimension.

2. Index-Signature Theorem and Stability

The central result is the index-signature theorem: under suitable spectral gap and locality hypotheses, the topological index (Fredholm index pairing) equals (half) the signature of the finite-volume localizer: $$\mathrm{Index}(T) = \frac{1}{2}\mathrm{Sig}(L_{\kappa,\rho})$$ where $\mathrm{Sig}$ is the signature—the number of positive minus negative eigenvalues of $L_{\kappa,\rho}$. This holds for both odd and even pairings and extends to systems with chiral, time-reversal, or particle-hole symmetry (Loring et al., 2018, Loring et al., 2018, Viesca et al., 2019, Berkolaiko et al., 26 Dec 2025).

For the result to hold, the tuning parameter $\kappa$ and window size $\rho$ must satisfy explicit bounds ensuring invertibility and a uniform spectral gap of $L_{\kappa,\rho}$, typically: $$0 < \kappa \leq \frac{g^3}{12C}, \quad \rho > \frac{2g}{\kappa}, \quad \text{with } g = \|A^{-1}\|^{-1}$$ where $C$ denotes a commutator norm between $D$ and $A$ (odd) or $H$ (even) (Loring et al., 2018). The signature is constant on open regions of allowed $(\kappa,\rho)$.

In the presence of a local (rather than global) spectral gap, the result remains stable, and the local index is invariant under perturbations that do not close the gap (Cerjan et al., 17 Jun 2025).

3. Connections to Spectral Flow, $K$-Theory, and Real-Space Topological Markers

The proof of the index-signature correspondence hinges on relating both the Fredholm index and the localizer signature to spectral flow. Specifically, the index can be expressed as the spectral flow between pairs of self-adjoint Fredholm operators linked by conjugations or deformations involving the unitary part of $A$ or flattenings of $H$ (Loring et al., 2018, Viesca et al., 2019, Berkolaiko et al., 26 Dec 2025). The $K$-theoretic class of the system (complex or real, even or odd) is thereby captured by the signature.

For spatially local invariants, a perturbative expansion in $\kappa$ demonstrates that the half-signature of the localizer in the small-$\kappa$ limit agrees with established real-space topological markers such as the Chern or winding marker: $$\frac{1}{2}\mathrm{Sig}(L(\kappa)) \xrightarrow{\kappa \to 0} \text{Chern or winding marker}$$ This is a direct algebraic bridge between the spectral localizer and the Bianco–Resta marker and real-space approaches (Jezequel et al., 31 Jul 2025).

Table 1: Correspondence of localizer index to bulk invariants

Dimensionality	Class	Localizer index	Conventional invariant
Even ($d$)	A	$\tfrac12 \mathrm{Sig}(L)$	Chern number
Odd ($d$)	AIII	$\tfrac12 \mathrm{Sig}(L)$	Winding number

4. Extensions: Disorder, Mobility Gap, and Non-Hermitian Systems

The spectral localizer framework applies beyond clean, periodic systems. For strongly disordered or aperiodic systems with only a mobility gap (no extended bulk states at Fermi energy), the (half-)signature of the localizer remains quantized and robust under disorder-averaged or local-gap-preserving homotopies; even in the absence of a spectral gap, the probability law of the topological invariant is continuous along continuous disorder or parameter variation (Stoiber, 2024, Viesca et al., 2019, Schulz-Baldes et al., 2021).

In non-Hermitian, line-gapped, or Floquet systems, the localizer is appropriately modified to incorporate the non-Hermitian Hamiltonian and dual representation, with the signature defined via real parts of eigenvalues (or Pfaffian sign for $\mathbb Z_2$ invariants), providing a robust, real-space characterization of topological invariants such as the Chern or strong index in these contexts (Cerjan et al., 2023, Chadha et al., 2023, Wong et al., 2024).

For Floquet systems, quantitative bounds link the protection of the local Chern marker (from the localizer) to time-integrated disorder in the driven Hamiltonian, giving a practical measure of topological robustness (Wong et al., 2024).

5. Applications: Numerical Implementation, Wave-Function Dynamics, and Physical Systems

Numerical Procedure

The practical algorithm involves the following steps (Berkolaiko et al., 26 Dec 2025, Viesca et al., 2019):

• Form the finite-dimensional Hamiltonian $H$, Dirac operator $D$, and project to a finite window via an appropriate cutoff projector.
• Construct the localizer matrix $L_{\kappa,\rho}$ for chosen $(\kappa, \rho)$.
• Diagonalize $L_{\kappa,\rho}$ and compute the signature.
• For real symmetry classes, extract the Pfaffian sign as a $\mathbb Z_2$ invariant.

This approach converges exponentially fast in box size for gapped phases and is robust to moderate disorder. It is numerically efficient, requiring only local data, and is independent of boundary conditions (Berkolaiko et al., 26 Dec 2025, Schulz-Baldes et al., 2021, Cerjan et al., 17 Jun 2025).

Wave-Packet Propagation

The spatial map of the spectral localizer index predicts the localization and robustness of boundary or defect modes; wave packets initiated near boundaries between regions of differing index propagate with minimal loss as long as the localizer gap remains open, even in the presence of disorder or defects (Michala et al., 2020).

Experimental and Theoretical Utility

The method is widely applicable in the study of topological insulators, semimetals (where the number of zero-modes of the localizer yields the Dirac/Weyl point count), higher-order topological states, non-Hermitian systems with skin effects, Floquet topological phases, and photonic/metamaterial platforms (Franca et al., 2023, Cerjan et al., 2024, Chadha et al., 2023, Wong et al., 2024).

6. Extensions and Open Directions

The spectral localizer framework is continuously evolving. Recent developments include:

• Improved locality criteria and refined bounds on the admissible $(\kappa, \rho)$, with demonstrated utility for heterostructures, quasicrystals, amorphous media, and mobility-gapped systems (Cerjan et al., 17 Jun 2025).
• Generalization to semifinite von Neumann algebras for weak invariants and large systems (Schulz-Baldes et al., 2020).
• Systematic equivalence proofs connecting the localizer signature to standard topological markers at leading order (Jezequel et al., 31 Jul 2025).
• Analytical understanding of topological zero-modes in metals versus topological semimetals (Franca et al., 2023).
• Applications to non-Hermitian line- and point-gapped systems, including topological lasers and radiative photonic platforms (Cerjan et al., 2023, Chadha et al., 2023).

Future directions encompass interaction effects, higher dimensions, and advanced mathematical questions in non-commutative geometry.

7. Summary Table: Key Localizer Constructions

Setting	Localizer block matrix form	Invariant computed
Odd index pairing	$$\begin{pmatrix} K D & A^* \ A & -K D \end{pmatrix}$$	Winding number, $\mathbb Z$
Even index pairing	$$\begin{pmatrix} -H & K D_0^* \ K D_0 & H \end{pmatrix}$$	Chern number, $\mathbb Z$
Non-Hermitian line-gap	$$\begin{pmatrix} -H & \kappa D_0^* \ \kappa D_0 & H^* \end{pmatrix}$$	Strong index
Floquet system	$$\begin{pmatrix} H_F - E & \kappa(Q - q) - i\kappa(P - p) \ \kappa(Q - q) + i\kappa(P - p) & -(H_F - E) \end{pmatrix}$$	Floquet Chern marker

The spectral localizer thus bridges operator-algebraic topology, numerical practicality, and real-space analytic control, serving as a central tool in modern topological condensed matter physics and beyond (Berkolaiko et al., 26 Dec 2025, Loring et al., 2018, Loring et al., 2018, Cerjan et al., 17 Jun 2025, Viesca et al., 2019, Jezequel et al., 31 Jul 2025).