Non-Bloch Higher-Order Construction

• Non-Bloch higher-order construction is a framework that replaces conventional Bloch bands with complex momenta and generalized Brillouin zones to describe codimension >1 boundary modes.
• It combines Hermitian higher-order topology principles with layered non-Bloch spectral techniques to address geometry-dependent skin effects and directional localization.
• Recent advances introduce biorthogonal eigenstates and non-Bloch Wannier functions, establishing groundwork for multidirectional bulk-boundary correspondence in non-Hermitian systems.

Searching arXiv for recent and foundational papers on non-Bloch higher-order constructions, higher-dimensional non-Bloch band theory, and related Hermitian HOTI baselines. Search query: "non-Bloch higher-order topology generalized Brillouin zone higher dimensions non-Hermitian skin effect" Non-Bloch higher-order construction denotes the effort to formulate higher-order boundary phenomena—especially corner-, hinge-, and other codimension-$>1$ effects—in the non-Hermitian regime where ordinary Bloch band theory fails under open boundary conditions. In this setting, the relevant objects are not ordinary real-momentum Bloch bands but complexified momenta, generalized Brillouin zones (GBZs), biorthogonal eigenstates, open-boundary projectors, and geometry-dependent spectral data. Across the present literature, the subject is best understood as a layered program rather than a single closed formalism: Hermitian higher-order topology provides the construction logic; higher-dimensional non-Bloch band theory provides the open-boundary spectral framework; and recent work on geometry dependence, strip GBZs, and non-Bloch quantum geometry supplies the technical ingredients needed for genuinely higher-order non-Hermitian formulations (Calugaru et al., 2018, Yokomizo et al., 2022, Wang et al., 2022, Xiong et al., 2024, Wang et al., 28 Jun 2025, Sun et al., 19 May 2026).

Paper	Direct contribution	Relation to non-Bloch higher-order construction
(Calugaru et al., 2018)	Anti-commuting Wilsonian mass hierarchy	Hermitian HOT baseline
(Wang et al., 2020)	Separable product construction from 1D edge states	Bottom-up HOT baseline
(Yokomizo et al., 2022)	2D non-Bloch band theory in two restricted classes	First 2D GBZ reductions
(Wang et al., 2022)	Amoeba/Ronkin formulation in arbitrary dimensions	Higher-dimensional GBZ backbone
(Kawabata et al., 2020)	Higher-order NHSE and modified higher-dimensional non-Bloch logic	Codimension-$>1$ skin physics
(Xiong et al., 2024)	Geometry-adaptive spectral-potential theory in arbitrary dimensions	Geometry-sensitive NHSE classification
(Wang et al., 28 Jun 2025)	Strip generalized Brillouin zone (SGBZ) for 2D GDSE	Sequential non-Bloch construction
(Sun et al., 19 May 2026)	Open-boundary non-Bloch quantum geometry and Wannier functions	Wannier-sector ingredients
(Liberto et al., 2020)	Non-Abelian Bloch oscillations in Hermitian BBH model	Important contrast, not true non-Bloch

1. Conceptual domain and terminology

The phrase combines two ideas that must be sharply separated. “Higher-order” refers to codimension-$n$ boundary structure: in a $D$-dimensional $n$th-order topological phase, zero modes occur on a boundary of codimension $d_c=n$, equivalently on a manifold of dimension $D-n$ (Calugaru et al., 2018). “Non-Bloch” refers to the non-Hermitian regime in which open-boundary spectra are not described by ordinary Bloch momenta, so one replaces $e^{ik}$ by complex variables such as $\beta$, works on a generalized Brillouin zone, and uses left-right biorthogonal states (Yokomizo et al., 2022, Wang et al., 2022).

This immediately excludes a common conflation. Non-Abelian multi-band Bloch dynamics is not, by itself, non-Bloch band theory. The BBH-based study of non-Abelian Bloch oscillations is entirely Hermitian, concerns a degenerate occupied-band $SU(2)$ Berry structure, and explicitly does not formulate a non-Bloch generalized Brillouin zone, complex momentum, skin effect, or non-Hermitian bulk-boundary correspondence (Liberto et al., 2020). Its relevance is only that it goes beyond the simplest Abelian single-band Bloch picture.

A second conceptual distinction concerns scope. Some papers study higher-order skin accumulation, i.e. macroscopic codimension-$>1$0 NHSE, whereas others provide the open-boundary geometry and Wannier machinery from which a non-Bloch higher-order topological theory could later be built. The existing literature therefore contains both direct higher-order non-Hermitian boundary phenomena and foundational constructions that stop short of full higher-order topological invariants (Kawabata et al., 2020, Sun et al., 19 May 2026).

2. Hermitian higher-order baselines

The most explicit Hermitian baseline is the general construction principle based on anti-commuting Wilsonian masses. If a $>1$1-dimensional first-order or regular topological phase involves $>1$2 Hermitian matrices and there exist $>1$3 additional mutually anti-commuting Hermitian matrices that anti-commute with the parent set, then one may realize $>1$4 order topology with suitable symmetry-breaking Wilsonian masses (Calugaru et al., 2018). The organizing relation is the codimension rule $>1$5. This produces the familiar hierarchy surface $>1$6 hinge $>1$7 corner in $>1$8, and in the nodal-loop case even a “fourth-order” phase without localized boundary zero modes because $>1$9 (Calugaru et al., 2018).

The significance of this Hermitian construction is that it is algebraic rather than specifically Bloch-theoretic. The central mechanism is a nested domain-wall reduction generated by mutually anti-commuting masses. This suggests that, in a future non-Bloch setting, one should preserve the mass-hierarchy and boundary-reduction logic even when ordinary Bloch momentum is replaced by complexified open-boundary data. That implication is structural rather than explicit in the Hermitian paper.

A second Hermitian baseline is the separable Kronecker-sum construction. In the off-diagonal Aubry–André–Harper setting, the 2D Hamiltonian takes the form

$n$0

so every 2D eigenstate factorizes into a product of 1D eigenstates, and a corner state is exactly

$n$1

with energy $n$2 (Wang et al., 2020). This is not non-Bloch theory, but it provides a clean directional blueprint: higher-order localization can be assembled from lower-dimensional boundary-localized building blocks. A plausible implication is that a non-Bloch higher-order construction will often need an analogous direction-by-direction assembly, with generalized non-Bloch edge solutions replacing Hermitian edge states.

The Hermitian BBH quadrupole model enters the subject in a different way. In the non-Abelian Bloch-oscillation analysis, the model is a 2D Hermitian higher-order topological insulator with non-commuting mirrors, quantized quadrupole moment, vanishing bulk polarization, and corner charges, but the dynamical focus is on degenerate-band Wilson loops, non-Abelian Berry curvature, and Wannier-center dynamics rather than on non-Hermitian non-Bloch physics (Liberto et al., 2020). This paper therefore serves as a cautionary reference: “beyond simple Bloch” does not automatically mean “non-Bloch.”

3. Emergence of higher-dimensional non-Bloch band theory

The first substantial extension of non-Bloch band theory beyond one dimension was achieved only in restricted settings. In two classes of 2D non-Hermitian systems, the problem can be reduced to effective 1D non-Hermitian problems. In one class, a symmetry suppresses the skin effect in one direction, so one direction remains real-momentum and the other is complexified fiberwise. In the other, the characteristic equation separates into $n$3- and $n$4-dependent parts, so one builds the 2D GBZ direction by direction from two 1D equal-modulus conditions (Yokomizo et al., 2022). The corresponding non-Bloch Chern number is defined on the generalized Brillouin zone $n$5 using biorthogonal left/right eigenvectors,

$n$6

and restores bulk-edge correspondence for the non-Hermitian Chern insulator on a rectangle (Yokomizo et al., 2022).

A more general arbitrary-dimensional framework was then built from the amoeba of the characteristic polynomial. For

$n$7

the amoeba is the image of the zero set under the logarithmic map $n$8, and the physically selected decay exponents are determined by the minimizer $n$9 of the Ronkin function (Wang et al., 2022). The higher-dimensional GBZ is then defined by

$D$0

In this formulation, the 1D equal-modulus rule is reinterpreted as central-hole closure of the amoeba, and the open-boundary bulk spectrum is confined to energies for which the amoeba has no central hole (Wang et al., 2022).

An alternative arbitrary-dimensional route uses spectral potential rather than direct root matching. In this framework the thermodynamic spectral potential is obtained recursively from lower-dimensional slices, and in $D$1 dimensions the result is

$D$2

This geometry-adaptive theory emphasizes that higher-dimensional non-Bloch spectra are geometry-dependent, that the relevant generalized Brillouin zone is determined by minimizing complex deformations, and that net winding numbers classify NHSE into critical and non-reciprocal types (Xiong et al., 2024). Its practical importance for higher-order work is that codimension-$D$3 localization is encoded by simultaneous nonzero inverse localization lengths $D$4 along multiple open directions.

These constructions jointly establish that higher-order non-Bloch theory cannot be a naive tensor product of 1D GBZ rules. In higher dimensions, geometry, boundary orientation, and the order in which open limits are taken become intrinsic parts of the spectral problem (Yokomizo et al., 2022, Wang et al., 2022, Xiong et al., 2024).

4. Sequential non-Bloch construction and the strip generalized Brillouin zone

The most explicit 2D geometry-sensitive construction is the strip generalized Brillouin zone (SGBZ) theory for geometry-dependent skin effect. Its central move is to recover a controlled 2D non-Bloch framework by taking two sequential 1D thermodynamic limits: first along a chosen major axis and then along a minor axis (Wang et al., 28 Jun 2025). Starting from a general non-Bloch Hamiltonian $D$5, one first forms a strip Hamiltonian $D$6, open in the minor direction and parametrized by the complex major-axis factor $D$7. For finite strip width, standard 1D non-Bloch theory yields a quasi-1D major-axis GBZ (QMGBZ). In the thermodynamic strip limit, this is replaced by a parametric minor-axis GBZ (PMGBZ) and a strip winding number $D$8, whose sign change selects the SGBZ (Wang et al., 28 Jun 2025).

The construction is explicitly directional. The paper proves that the SGBZ depends only on the major axis, not on the minor-axis choice. Different major axes can therefore generate inequivalent SGBZs, and the competition between incompatible SGBZs is identified as the origin of geometry-dependent skin effect in fully open 2D samples (Wang et al., 28 Jun 2025). This is the strongest existing formal statement that a higher-dimensional non-Bloch theory must be geometry-adapted.

The same paper provides a sufficient condition for geometry-dependent skin effect in terms of non-Bloch dynamical degeneracy splitting. Along an equal-frequency contour on the SGBZ, if $D$9, $n$0, or $n$1 is nonuniform, then the continuum degeneracy of non-Bloch states breaks down into a discrete set, and GDSE follows (Wang et al., 28 Jun 2025). This result is especially suggestive for higher-order constructions because corner or hinge matching depends not only on equal complex energy but also on compatible decay moduli in multiple directions.

The paper does not itself construct a corner SGBZ, a nested Wilson loop, or a corner invariant. Nevertheless, its sequential logic is already “nested” in an operational sense: open one direction, construct a strip non-Bloch object, then open the second direction and study compatibility. This suggests that a non-Bloch higher-order theory in 2D may have to be formulated as a matching problem between multiple edge-adapted SGBZs rather than from a single universal 2D momentum manifold. That is an inference from the formalism rather than a theorem stated in the paper.

5. Higher-order skin effects and the need to modify non-Bloch theory

Higher-order non-Hermitian skin effects show directly that codimension-$n$2 localization can be a macroscopic non-Bloch phenomenon. In two dimensions, the second-order skin effect yields $n$3 corner skin modes rather than the $n$4 corner zero modes of Hermitian second-order topological insulators; in three dimensions, the third-order skin effect yields $n$5 corner skin modes out of $n$6 total modes (Kawabata et al., 2020). This establishes that higher-order boundary accumulation in non-Hermitian systems is not merely a perturbation of Hermitian HOTI phenomenology.

The same work demonstrates why standard 1D non-Bloch logic fails in higher dimensions. In the 2D model, ordinary first-order skin effect is suppressed along $n$7 and $n$8 separately by transposition-associated mirror symmetries, so directional 1D winding numbers vanish, yet corner skin accumulation still occurs when both directions are open (Kawabata et al., 2020). To solve the open-boundary problem, the paper uses a genuinely two-dimensional ansatz with two coupled complex variables $n$9 and four reflected sectors. The resulting corner-skin branch is fixed not by independent 1D GBZ rules but by simultaneous conditions such as

$d_c=n$0

which produce simultaneous decay in both directions (Kawabata et al., 2020).

Topologically, the higher-order skin effect is tied to intrinsic non-Hermitian point-gap topology rather than to occupied-band line-gap invariants. In the 2D case, the relevant bulk invariant is a $d_c=n$1-quantized Wess-Zumino term,

$d_c=n$2

protected by a rotation-type symmetry, and $d_c=n$3 for $d_c=n$4 (Kawabata et al., 2020). The importance of this result is not that it already furnishes a universal higher-order non-Bloch band theory, but that it proves such a theory must incorporate multidirectional complex momenta and simultaneous boundary matching.

A later arbitrary-dimensional spectral-potential theory clarifies the status of these modes. It explicitly captures the dominant $d_c=n$5 skin sector and classifies NHSE into critical and non-reciprocal types using net winding numbers, but it also states that subleading modes—“hybrid skin-topological modes, topological boundary states, and higher-order skin modes, which are of order $d_c=n$6 ($d_c=n$7)”—are not explicitly included in the spectral potential (Xiong et al., 2024). This marks an important boundary of the present theory: higher-order non-Bloch skin physics is partly accessible, but a complete non-Bloch treatment of subleading higher-order topological modes remains open.

6. Non-Bloch quantum geometry, Wannier objects, and current limitations

A fully open-boundary, non-Bloch quantum-geometric framework has recently supplied a different set of ingredients. For a 1D non-Hermitian system under open boundary conditions, the real-space integrated quantum metric

$d_c=n$8

is shown to equal exactly the GBZ integral of the left-right non-Bloch quantum metric,

$d_c=n$9

(Sun et al., 19 May 2026). Here $D-n$0 is an open-boundary biorthogonal projector, either built directly from left/right OBC eigenstates or represented in the non-Bloch basis on the GBZ.

The same work defines non-Bloch Wannier functions adapted to the skin-effect regime,

$D-n$1

and proves that the non-Bloch integrated quantum metric is exactly the gauge-invariant part of their spread functional (Sun et al., 19 May 2026). It also introduces projected-position Wannier functions by diagonalizing

$D-n$2

which is the precise object ordinarily used to construct Wannier sectors.

This paper does not explicitly construct higher-order topology, corner states, multipole moments, or nested Wilson loops. Its value lies instead in providing the OBC-compatible projector, GBZ measure, non-Bloch Berry phase, and non-Bloch Wannier basis from which such a theory could plausibly be built (Sun et al., 19 May 2026). A plausible implication is that a non-Bloch higher-order theory should proceed by replacing ordinary Bloch projectors and Wannier-sector constructions with biorthogonal OBC/non-Bloch analogues, then iterating projected-position constructions in multiple directions.

The main limitations of the present field follow directly from the cited works. There is still no universally accepted non-Bloch nested Wilson-loop formalism for higher-order topology, no general corner or hinge invariant on a geometry-dependent higher-dimensional GBZ, and no complete treatment of the interplay among corner topology, skin accumulation, and scale-free modes (Yokomizo et al., 2022, Xiong et al., 2024, Wang et al., 28 Jun 2025). Just as importantly, several influential papers near this subject are not themselves non-Bloch higher-order constructions: the Hermitian mass-hierarchy and separable HOT baselines provide construction logic (Calugaru et al., 2018, Wang et al., 2020); the BBH non-Abelian Bloch-oscillation work provides a Hermitian multi-band dynamical diagnostic but explicitly not a non-Bloch generalized Brillouin-zone theory (Liberto et al., 2020).

Taken together, the literature supports a precise but limited synthesis. Non-Bloch higher-order construction presently consists of three compatible layers: a Hermitian higher-order codimension-reduction logic, a higher-dimensional non-Bloch open-boundary spectral formalism, and an emerging OBC-compatible quantum geometry/Wannier language. What remains incomplete is their full integration into a universal theory of non-Bloch higher-order topological invariants and codimension-$D-n$3 bulk-boundary correspondence (Calugaru et al., 2018, Wang et al., 2022, Xiong et al., 2024, Wang et al., 28 Jun 2025, Sun et al., 19 May 2026).