Spectral Localizer Invariant

• The spectral localizer invariant is a real-space topological index built from a Dirac-type matrix and position operators, allowing local detection of topological phases in finite systems.
• It leverages index theory and finite-volume half-signature computations to robustly characterize even and odd index pairings under various symmetry classes.
• Its formulation enables analysis in disordered, non-Hermitian, and Floquet systems where traditional momentum-space topological measures falter.

The spectral localizer invariant is a real-space topological invariant extracted from a Dirac-type matrix built from a Hamiltonian or other local generator together with the position operators and a Clifford representation. In its standard Hermitian form, it is defined by the half-signature of a finite-volume spectral localizer; in symmetry-restricted settings it is given by the sign of a Pfaffian or determinant; and in K-theoretic formulations it computes even and odd index pairings. Because it is formulated directly on a Hilbert space with open boundaries and local operators, it is applicable to finite, disordered, radiative, nonlinear, Floquet, and non-Hermitian systems where momentum-space vector-bundle constructions are unavailable or insufficient (Cerjan et al., 2024, Dungen, 24 Feb 2026).

1. Construction and basic definition

The spectral localizer in $d$ spatial dimensions is a Hermitian matrix of the form

$$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$

where $H$ is a local Hamiltonian or generator, $X_1,\dots,X_d$ are the position operators, $\Gamma_j$ form an irreducible Clifford representation, $r_0$ is the probe point, $E$ is the target energy, and $\kappa>0$ balances position and energy scales (Cerjan et al., 2024). In two dimensions, with $\Gamma_1=\sigma_x$, $\Gamma_2=\sigma_y$, and $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$0, this becomes

$$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$1

which is the form used both in finite Chern-insulator studies and in tutorial treatments of local topological classification (Michala et al., 2020).

For a finite two-dimensional insulator, the spectral localizer index is

$$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$2

where $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$3 is the number of positive eigenvalues minus the number of negative eigenvalues (Michala et al., 2020). In class A, this is the local Chern invariant, and in crystalline systems with a bulk gap at $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$4 and $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$5 in the bulk, it agrees with the global Chern number up to a sign convention depending on the Clifford choice (Cerjan et al., 2024).

The construction was developed in the setting of the spectral localiser due to Loring and Schulz-Baldes, with origins of the Clifford-spectrum viewpoint traced back to Kisil, and it was designed precisely for finite and disordered systems where global Chern numbers are not directly available (Michala et al., 2020, Dungen, 24 Feb 2026). Its defining feature is locality in both space and energy: the invariant depends on the chosen center $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$6 and target energy $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$7, and therefore partitions heterogeneous samples into locally topological regions rather than assigning a single bulk number to the entire specimen (Cerjan et al., 2024).

2. Index-theoretic formulation and finite-volume half-signature

The spectral localizer admits a precise K-theoretic interpretation as an index-pairing formula. In the even case, for an even spectral triple $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$8 and an invertible self-adjoint $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$9 with $H$0, the infinite-volume spectral localiser is

$H$1

while in the odd case, for an invertible $H$2 with phase $H$3,

$H$4

A 2026 K-theoretic treatment shows that these localisers arise directly from the Kasparov product and that the corresponding index pairings can be written as spectral flow in infinite volume and as signatures after spectral truncation (Dungen, 24 Feb 2026).

For finite-volume truncation, one compresses to $H$5, obtaining

$H$6

Under the admissibility conditions

$H$7

the truncated localisers are invertible with spectral gap $H$8, and their signatures are independent of $H$9 and $X_1,\dots,X_d$0 (Dungen, 24 Feb 2026).

The finite-volume signature formulas are

$X_1,\dots,X_d$1

and

$X_1,\dots,X_d$2

The additional $X_1,\dots,X_d$3 term appears in the even case when $X_1,\dots,X_d$4 is not invertible; if $X_1,\dots,X_d$5 is invertible, the formula reduces to the earlier half-signature result (Dungen, 24 Feb 2026). In a complementary spectral-theoretic treatment of $X_1,\dots,X_d$6 and $X_1,\dots,X_d$7, the finite localizer

$X_1,\dots,X_d$8

was shown to satisfy

$X_1,\dots,X_d$9

with explicit bounds on $\Gamma_j$0 and $\Gamma_j$1 in terms of $\Gamma_j$2, locality constants, and $\Gamma_j$3 (Berkolaiko et al., 26 Dec 2025).

These formulations make precise that the spectral localizer invariant is not merely a heuristic local marker. It is an index pairing in K-theory and KK-theory, computed through a finite matrix whose signature is stable under the admissible parameter range (Dungen, 24 Feb 2026, Loring et al., 2018).

3. Local topology, localizer states, and the localizer gap

The localizer resolves topology because the triple $\Gamma_j$4 is only approximately commuting. In the basic two-dimensional setting of a finite Chern insulator, $\Gamma_j$5 and

$\Gamma_j$6

is small because $\Gamma_j$7 is finite range (Michala et al., 2020). If the localizer has an eigenvalue $\Gamma_j$8, then the associated vector produces an approximate simultaneous eigenvector for position and energy. For the untuned localizer, the bound

$\Gamma_j$9

holds, and with a $r_0$0-pseudospectral tolerance one obtains

$r_0$1

as an approximate-eigenvector error (Michala et al., 2020).

A central quantitative quantity is the localizer gap. For Hermitian localizers it is

$r_0$2

or, in the notation of finite Chern-insulator simulations,

$r_0$3

This gap is a local robustness margin: the invariant cannot change unless the localizer gap closes at the relevant $r_0$4 (Cerjan et al., 2024, Michala et al., 2020). Weyl’s inequality implies

$r_0$5

so if $r_0$6, the invariant is unchanged (Cerjan et al., 2024).

Recent work sharpened the validity criterion by replacing a global gap assumption with a properly defined local spectral gap of the Hamiltonian. In that formulation, the Hamiltonian has a $r_0$7-local gap at $r_0$8 if

$r_0$9

and the $E$0-local gap is

$E$1

The corresponding even localizer is

$E$2

with localizer gap

$E$3

and local index

$E$4

The improved criterion states that merely a properly defined local spectral gap of the Hamiltonian is required, only relative bounds on the Hamiltonian and its noncommutative derivative are relevant, and the tapering constant can be improved to $E$5 (Cerjan et al., 17 Jun 2025).

In two dimensions, a convenient explicit criterion is

$E$6

with $E$7 (Cerjan et al., 17 Jun 2025). Contrary to a common simplification, the current criterion is therefore local both in the spectral hypothesis and in the norm estimates.

4. Symmetry classes and generalized forms

The spectral localizer invariant extends well beyond the complex class-A half-signature. In one-dimensional chiral systems, the reduced chiral localizer

$E$8

is Hermitian at $E$9, and the local winding is

$\kappa>0$0

(Cerjan et al., 2024). In two-dimensional class AII, after a basis change rendering the localizer real skew-symmetric, the $\kappa>0$1 invariant is

$\kappa>0$2

while in class DIII one replaces the Pfaffian by a determinant on chiral blocks (Cerjan et al., 2024).

A systematic real-symmetry theory shows that for the $\kappa>0$3 $\kappa>0$4-valued real pairings, the invariant is computed from the skew localizer by

$\kappa>0$5

and in $\kappa>0$6 of those cases it reduces to determinant signs of an off-diagonal block,

$\kappa>0$7

This places the Pfaffian and determinant formulas on the same footing as the half-signature formulas for complex classes (Doll et al., 2021).

For short-ranged, line-gapped non-Hermitian Hamiltonians, the even non-Hermitian spectral localizer is

$\kappa>0$8

and the strong invariant is

$\kappa>0$9

where the signature counts eigenvalues with positive and negative real parts (Cerjan et al., 2023). For semimetals, a different localizer,

$\Gamma_1=\sigma_x$0

has a near-zero cluster whose multiplicity gives the total number of Dirac or Weyl points: $\Gamma_1=\sigma_x$1 (Schulz-Baldes et al., 2021). For time-quasiperiodic superconductors in class D, the one-dimensional non-Hermitian block

$\Gamma_1=\sigma_x$2

yields a $\Gamma_1=\sigma_x$3 invariant

$\Gamma_1=\sigma_x$4

with robustness controlled by the smallest singular value of $\Gamma_1=\sigma_x$5 (Qi et al., 2024).

These variants show that “spectral localizer invariant” is a family of symmetry-adapted real-space invariants rather than a single formula. The half-signature remains central, but Pfaffian signs, determinant signs, and near-zero multiplicities arise when the symmetry class or spectral regime changes.

5. Physical interpretation, transport, and applications

In finite Chern insulators, mapping $\Gamma_1=\sigma_x$6 at fixed $\Gamma_1=\sigma_x$7 partitions the sample into regions of constant localizer index. Regions of differing localizer index are separated by curves where the index jumps, and along any path connecting two such regions, the localizer must have an eigenvalue crossing zero (Michala et al., 2020). This is a local, finite-system form of bulk-boundary correspondence.

Numerical wave-packet studies in a finite $\Gamma_1=\sigma_x$8 Chern insulator show that wave-packets initialized on the boundary between regions of differing localizer index propagate along that boundary with minimal loss and essentially no backscattering in the clean case, even with strong defects such as missing sites and domain walls. With disorder, wave-packets still follow the boundary between regions of differing localizer index but lose significant mass into the bulk over time; the loss correlates with the appearance of small $\Gamma_1=\sigma_x$9 in the interior and along portions of the edge (Michala et al., 2020). The conjecture advanced there is that wave-packets propagating along boundaries between regions of differing spectral localizer index do not lose significant mass whenever the localizer gap is sufficiently large on both sides of the boundary (Michala et al., 2020).

In photonics, the framework has been extended to local nonlinearities, radiative environments, crystalline and higher-order topology, and Maxwell operators. For radiative/open systems with line-gapped non-Hermitian Chern phases, the tutorial formulation uses

$\Gamma_2=\sigma_y$0

with

$\Gamma_2=\sigma_y$1

(Cerjan et al., 2024). The same framework reformulates Maxwell’s equations either through

$\Gamma_2=\sigma_y$2

or through the effective Hamiltonian

$\Gamma_2=\sigma_y$3

(Cerjan et al., 2024).

In Floquet systems, one builds the standard localizer from the Floquet Hamiltonian

$\Gamma_2=\sigma_y$4

and computes

$\Gamma_2=\sigma_y$5

The localizer gap

$\Gamma_2=\sigma_y$6

then gives a quantitative robustness condition: if

$\Gamma_2=\sigma_y$7

the localizer invariant is unchanged (Wong et al., 2024). This connects experimentally accessible disorder in the instantaneous Hamiltonians to topological protection of the Floquet phase.

In the mobility-gap regime, spectral and skew localizers have been used to prove continuity of the probability distribution of strong invariants under homotopies preserving a mobility gap, and to show that interfaces between mobility-gapped systems with differing strong invariants must fail the fractional moments bound near the Fermi energy (Stoiber, 2024). The localizer therefore functions both as a finite-volume computational tool and as a framework for delocalization statements in disordered topology.

6. Relation to other markers, misconceptions, and open directions

The spectral localizer has often been discussed alongside other real-space markers such as Kitaev’s real-space formula and the Bianco–Resta marker, but an explicit equivalence to local Chern and winding markers was only made systematic recently. A 2025 derivation shows that, in the small-$\Gamma_2=\sigma_y$8 regime, the spectral localizer invariant

$\Gamma_2=\sigma_y$9

reduces exactly to the spatially averaged local Chern marker in even dimensions and to the winding marker in odd chiral dimensions: $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$00 with a $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$01-dependent bulk weight

$$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$02

(Jezequel et al., 31 Jul 2025). This removes the earlier situation in which equivalence was implicit rather than explicit.

A recurring misconception is that the spectral localizer requires a global bulk spectral gap. More recent results state instead that merely a properly defined local spectral gap of the Hamiltonian is required (Cerjan et al., 17 Jun 2025). Another common simplification is that the framework is restricted to Hermitian periodic band theory; the literature now includes non-Hermitian line-gapped systems, point-gapped defect problems, semimetals, Floquet systems, time-quasiperiodic Majoranas, and photonic radiative environments (Cerjan et al., 2023, Schulz-Baldes et al., 2021, Qi et al., 2024, Cerjan et al., 2024).

Several limitations remain explicit in the literature. No quantitative $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$03 threshold ensuring negligible loss of wave-packet mass is known; the predictive power of index and gap maps depends on $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$04; in strong disorder the localizer index pattern can become fragmented or trivial; and the interpretation of maps is more delicate when $$L(E, r_0, \kappa) = (H - E) \otimes \Gamma_{d+1} + \kappa \sum_{j=1}^d (X_j - r_{0j}) \otimes \Gamma_j,$$05 is small across large regions (Michala et al., 2020). In non-Hermitian classes beyond the line-gapped Chern setting, generalizations remain open (Cerjan et al., 2024). A plausible implication is that the spectral localizer will continue to develop along two parallel lines: sharper local validity criteria, and broader symmetry-adapted variants that preserve the finite-volume computability of the invariant.