Non-Hermitian Topological Square-Root Insulators

• The paper demonstrates that square-root constructions of non-Hermitian Hamiltonians map to parent topological models, producing finite-energy edge and corner states.
• Non-Hermitian square-root insulators are built via graph-theoretic spectral descent, enabling inherited topological invariants and accommodating both skin-effect-free and nonreciprocal regimes.
• The framework enforces exceptional deficiency through lattice sum rules, leading to system-wide amplification and a breakdown of conventional bulk-boundary correspondence.

to=shell 天天中彩票提款 code: {"command":["bash","-lc","python - <<'PY'\nimport urllib.request\nurl='http://export.arxiv.org/api/query?search_query=all:non-Hermitian+topological+square-root+insulators&start=0&max_results=3'\ntry:\n print(urllib.request.urlopen(url,timeout=20).read(300).decode('utf-8','ignore'))\nexcept Exception as e:\n print('ERR',e)\nPY"]} to=shell 天天中彩票不中返 code: {"command":["bash","-lc","python - <<'PY'\nimport urllib.request\nurl='https://export.arxiv.org/api/query?search_query=all:non-Hermitian+topological+square-root+insulators&start=0&max_results=3'\ntry:\n data=urllib.request.urlopen(url,timeout=20).read(500)\n print(data.decode('utf-8','ignore'))\nexcept Exception as e:\n print('ERR',repr(e))\nPY"]} to=shell 彩神争霸邀请码 code: {"command":["bash","-lc","echo hello"]} Non-Hermitian topological square-root insulators are rooted lattice systems in which a non-Hermitian Hamiltonian is constructed so that its square reproduces, up to block decomposition and possible energy shifts, a parent topological model. In this framework, topological boundary modes of the parent reappear in the rooted system at finite energy, while non-Hermiticity introduces complex spectra, exceptional-point physics, nonreciprocity, and modified bulk-boundary correspondence. The subject spans graph-theoretic square-root constructions, higher-order and Floquet generalizations, and, in a recent development, an exceptionally deficient regime in which every energy level of a finite open system is an exceptional point enforced by a lattice sum rule (Ezawa, 2020, Bid et al., 15 Aug 2025).

1. Square-root topology and its parent-child structure

The defining operation of a square-root topological insulator is spectral descent: a Hamiltonian $H_{\text{root}}$ is engineered so that

$$(H_{\text{root}})^2 = H_{\text{par}} \oplus H_{\text{res}},$$

where $H_{\text{par}}$ reproduces the parent topological Hamiltonian, up to a constant shift, and $H_{\text{res}}$ is a residual block. A systematic graph-theoretic construction realizes this by subdividing the graph of an original tight-binding model, inserting one vertex on each link, and defining a bipartite rooted Hamiltonian on the subdivided graph. In block form,

$$H_{\text{root}}= \begin{pmatrix} O_{N\times N} & H^{\text{left}}_{N\times M}\ H^{\text{right}}_{M\times N} & O_{M\times M} \end{pmatrix}, \qquad H_{\text{par}}=H^{\text{left}}H^{\text{right}}, \qquad H_{\text{res}}=H^{\text{right}}H^{\text{left}},$$

with $H_{\text{par}}=C+H_{\text{original}}$ for a constant shift $C$ (Ezawa, 2020).

This construction has several immediate consequences. Zero-energy topological edge states of the parent system become in-gap edge states at nonzero energies $E=\pm \sqrt{C}$ in the rooted system. Bulk eigenvalues map as $E=\pm \sqrt{\varepsilon+C}$, where $\varepsilon$ is an eigenvalue of the original model. Because the subdivided graph is bipartite, protected zero-energy flat bands also appear, with multiplicity $$(H_{\text{root}})^2 = H_{\text{par}} \oplus H_{\text{res}},$$0. The eigenvector correspondence implies that topological invariants of the parent model are inherited by the rooted model, even though the spectral organization is different (Ezawa, 2020).

The square-root idea is not limited to first-order boundary phenomena. A photonic realization of a second-order square-root topological insulator used a decorated honeycomb lattice whose squared Hamiltonian decomposes as a direct sum of a honeycomb sector and a breathing Kagome sector,

$$(H_{\text{root}})^2 = H_{\text{par}} \oplus H_{\text{res}},$$1

thereby embedding higher-order topology in the rooted lattice. In that experiment, the rooted system exhibited paired non-zero-energy corner states, including an out-of-phase corner state analogous to the breathing Kagome parent and an in-phase corner state unique to the square-root structure (Yan et al., 2021). This established that square-root topology can generate boundary phenomenology absent from the parent even when the parent remains the source of the underlying topological content.

2. Non-Hermitian extensions of rooted topology

The square-root scheme generalizes naturally to non-Hermitian systems. In the graph-theoretic formulation, the non-Hermitian extension was illustrated using a nonreciprocal SSH parent Hamiltonian,

$$(H_{\text{root}})^2 = H_{\text{par}} \oplus H_{\text{res}},$$2

with $$(H_{\text{root}})^2 = H_{\text{par}} \oplus H_{\text{res}},$$3 encoding nonreciprocity. The corresponding rooted model retains the block-square relation, but $$(H_{\text{root}})^2 = H_{\text{par}} \oplus H_{\text{res}},$$4, and the rooted spectrum is generally complex. The topological edge states persist as finite-energy in-gap states, now at

$$(H_{\text{root}})^2 = H_{\text{par}} \oplus H_{\text{res}},$$5

provided $$(H_{\text{root}})^2 = H_{\text{par}} \oplus H_{\text{res}},$$6 (Ezawa, 2020).

A distinct non-Hermitian route constructs rooted topological operators by adding anti-Hermitian terms that anticommute with the Dirac kinetic term and higher-order Wilson masses,

$$(H_{\text{root}})^2 = H_{\text{par}} \oplus H_{\text{res}},$$7

In this class, the systems are always devoid of non-Hermitian skin effects. For $$(H_{\text{root}})^2 = H_{\text{par}} \oplus H_{\text{res}},$$8, all eigenvalues remain real and zero-energy topological boundary modes persist; for $$(H_{\text{root}})^2 = H_{\text{par}} \oplus H_{\text{res}},$$9, generic points in the spectrum become complex, topological boundary states disappear, and the band topology is trivial. These models were explicitly developed for first-order, higher-order, and semimetallic phases in one, two, and three dimensions, and their squared spectrum was shown to connect directly to Hermitian parent models with shifted parameters (Salib et al., 2023).

Taken together, these constructions show that non-Hermitian square-root topology is not a single mechanism but a family of mechanisms. Some rooted models exploit nonreciprocity and complex eigenvalues; others preserve a real line-gapped spectrum over an extended parameter regime and suppress the skin effect by construction. This suggests that the rooted relation $H_{\text{par}}$0 is the unifying feature, whereas the spectral and transport phenomenology depends sensitively on how non-Hermiticity is introduced.

3. Exceptional deficiency from a lattice sum rule

A recent development introduces a particularly stringent non-Hermitian square-root regime: exceptional deficiency, defined as the situation in which every energy level is an exceptional point, so eigenvalues are degenerate and the corresponding eigenstates coalesce across the full spectrum (Bid et al., 15 Aug 2025). In this setting, exceptional-point behavior ceases to be a finely tuned spectral accident and becomes an enforced structural property.

The construction starts from a non-Hermitian, chiral-symmetric Hamiltonian

$H_{\text{par}}$1

with chiral symmetry

$H_{\text{par}}$2

Its square defines a parent problem on decoupled sublattices $H_{\text{par}}$3, and the crucial ingredient is the lattice sum rule

$H_{\text{par}}$4

This condition removes coupling from one subpart of the squared parent system to another and thereby enforces a specific self-orthogonality structure. Right eigenstates satisfy

$H_{\text{par}}$5

with an analogous equation for left eigenstates. The sum rule then guarantees

$H_{\text{par}}$6

for all $H_{\text{par}}$7, including the degenerate case $H_{\text{par}}$8. In the non-Hermitian setting, this self-orthogonality is the criterion for exceptional points. Because it holds throughout the spectrum, the full finite open system is exceptionally deficient (Bid et al., 15 Aug 2025).

The significance of this result lies in how it reinterprets rooted topology. The square-root architecture is not merely a way of translating a parent topology into finite-energy boundary modes; with the lattice sum rule, it becomes a mechanism for enforcing a spectrum made entirely of defective eigenvalues. In the example emphasized in the paper, a non-Hermitian quadrupole insulator realizes this condition, and all states, including corner states, are exceptional points (Bid et al., 15 Aug 2025).

4. Dynamical signatures: broadband and adiabatic amplification

The exceptional deficiency of these rooted systems has direct dynamical consequences. For a static Hamiltonian at an exceptional point, time evolution necessarily involves generalized eigenstates. If $H_{\text{par}}$9 is a generalized eigenvector satisfying $H_{\text{res}}$0, the state evolves as

$H_{\text{res}}$1

When the initial state lies outside the exceptional-point eigenspace, the total intensity obeys

$H_{\text{res}}$2

Because every energy level is an exceptional point, this amplification is broadband rather than frequency-selective: it is not confined to a narrow spectral neighborhood but occurs over a wide range of initial conditions (Bid et al., 15 Aug 2025).

The same work identifies a second dynamical signature under slow parameter driving. The adiabatic intensity change is governed by a geometric amplification factor

$H_{\text{res}}$3

where

$H_{\text{res}}$4

In Hermitian systems, a closed path yields $H_{\text{res}}$5. In the exceptionally deficient case, however, the non-Abelian Berry structure at the exceptional degeneracy gives robust nonzero amplification even for retraced paths. The reported adiabatic scaling law,

$H_{\text{res}}$6

parallels the static $H_{\text{res}}$7 law and identifies adiabatic state amplification as a geometric manifestation of the same defectiveness (Bid et al., 15 Aug 2025).

These results sharpen the distinction between ordinary non-Hermitian amplification and rooted exceptional amplification. The mechanism is not simply gain and loss, but the enforced presence of generalized eigenvectors throughout the spectrum. A plausible implication is that square-root topology furnishes an architectural route to system-wide polynomial amplification that would otherwise require repeated fine-tuning of isolated exceptional points.

5. Bulk-boundary dichotomy, skin effect, and related non-Hermitian topologies

A central subtlety is that exceptional deficiency is exact in finite open systems but not generically in the periodic bulk. In the exceptionally deficient quadrupole construction, open boundary conditions preserve the lattice sum rule exactly, and every state is an exceptional point. In the infinite periodic system, the same symmetry structure protects degeneracies, but not all of them remain exceptional: for $H_{\text{res}}$8, exceptional deficiency is partially lost, whereas extended Bloch states remain exceptional only along $H_{\text{res}}$9 (Bid et al., 15 Aug 2025).

The periodic Bloch Hamiltonian exhibited for the quadrupole example is

$$H_{\text{root}}= \begin{pmatrix} O_{N\times N} & H^{\text{left}}_{N\times M}\ H^{\text{right}}_{M\times N} & O_{M\times M} \end{pmatrix}, \qquad H_{\text{par}}=H^{\text{left}}H^{\text{right}}, \qquad H_{\text{res}}=H^{\text{right}}H^{\text{left}},$$0

with bands

$$H_{\text{root}}= \begin{pmatrix} O_{N\times N} & H^{\text{left}}_{N\times M}\ H^{\text{right}}_{M\times N} & O_{M\times M} \end{pmatrix}, \qquad H_{\text{par}}=H^{\text{left}}H^{\text{right}}, \qquad H_{\text{res}}=H^{\text{right}}H^{\text{left}},$$1

$$H_{\text{root}}= \begin{pmatrix} O_{N\times N} & H^{\text{left}}_{N\times M}\ H^{\text{right}}_{M\times N} & O_{M\times M} \end{pmatrix}, \qquad H_{\text{par}}=H^{\text{left}}H^{\text{right}}, \qquad H_{\text{res}}=H^{\text{right}}H^{\text{left}},$$2

The paper interprets this finite-open versus periodic mismatch as a manifestation of the breakdown of bulk-boundary correspondence typical of non-Hermitian systems and relates it to the non-Hermitian skin effect (Bid et al., 15 Aug 2025).

This point prevents a common conflation. Non-Hermitian square-root insulators are not synonymous with exceptional topological insulators. The latter are three-dimensional non-Hermitian phases characterized by a bulk point gap, require no symmetry for stability, and support anomalous surface states that either form a single sheet covering the point gap or host a single surface exceptional point. Their bulk invariant is a three-dimensional winding number defined with respect to the point gap, and the associated surface anomaly cannot exist in purely two-dimensional systems (Denner et al., 2020). By contrast, the exceptionally deficient square-root construction is based on chiral symmetry, a square-root parent-child relation, and a lattice sum rule, and its strongest statement applies to finite open systems rather than to a symmetry-free three-dimensional point-gap phase (Bid et al., 15 Aug 2025).

A second misconception is that non-Hermitian square-root topology must entail skin accumulation. The skin-effect-free constructions of non-Hermitian topological operators demonstrate the opposite: one can realize rooted topological phases with real spectra and conventional left/right boundary localization, preserving bulk-boundary correspondence in a Hermitian-like manner as long as the spectrum remains real (Salib et al., 2023). Non-Hermitian rooted topology therefore spans both skin-dominated and skin-free regimes.

6. Experimental platforms and extensions beyond the static square root

Several implementation routes have been proposed. For exceptionally deficient square-root insulators, the suggested platforms are topolectric circuits, active acoustic and mechanical metamaterials, and quantum-optical systems, including cold atom setups with engineered dissipation or synthetic dimensions (Bid et al., 15 Aug 2025). The appeal of these platforms is that they allow direct engineering of nonreciprocal couplings, gain and loss, and lattice connectivity, which are the ingredients entering the square-root construction and the lattice sum rule.

Photonic systems supply a broader experimental landscape for rooted non-Hermitian topology. General $$H_{\text{root}}= \begin{pmatrix} O_{N\times N} & H^{\text{left}}_{N\times M}\ H^{\text{right}}_{M\times N} & O_{M\times M} \end{pmatrix}, \qquad H_{\text{par}}=H^{\text{left}}H^{\text{right}}, \qquad H_{\text{res}}=H^{\text{right}}H^{\text{left}},$$3-root SSH models have been designed in non-Hermitian photonic ring systems using loops of unidirectional couplings as building blocks. In these models, the Hamiltonian satisfies a generalized chiral symmetry,

$$H_{\text{root}}= \begin{pmatrix} O_{N\times N} & H^{\text{left}}_{N\times M}\ H^{\text{right}}_{M\times N} & O_{M\times M} \end{pmatrix}, \qquad H_{\text{par}}=H^{\text{left}}H^{\text{right}}, \qquad H_{\text{res}}=H^{\text{right}}H^{\text{left}},$$4

and the complex spectrum develops a “ring gap,” described as irreducible to the usual point or line gaps. The resulting edge states appear on the $$H_{\text{root}}= \begin{pmatrix} O_{N\times N} & H^{\text{left}}_{N\times M}\ H^{\text{right}}_{M\times N} & O_{M\times M} \end{pmatrix}, \qquad H_{\text{par}}=H^{\text{left}}H^{\text{right}}, \qquad H_{\text{res}}=H^{\text{right}}H^{\text{left}},$$5 branches of the complex spectrum, and near-perfect unidirectional effective couplings can be generated by auxiliary rings with modulated gain and loss that realize high imaginary gauge fields (Viedma et al., 2023).

Floquet generalizations extend rooted topology to periodically driven open systems. A generic $$H_{\text{root}}= \begin{pmatrix} O_{N\times N} & H^{\text{left}}_{N\times M}\ H^{\text{right}}_{M\times N} & O_{M\times M} \end{pmatrix}, \qquad H_{\text{par}}=H^{\text{left}}H^{\text{right}}, \qquad H_{\text{res}}=H^{\text{right}}H^{\text{left}},$$6th-root procedure enlarges the Hilbert space so that the rooted Floquet operator $$H_{\text{root}}= \begin{pmatrix} O_{N\times N} & H^{\text{left}}_{N\times M}\ H^{\text{right}}_{M\times N} & O_{M\times M} \end{pmatrix}, \qquad H_{\text{par}}=H^{\text{left}}H^{\text{right}}, \qquad H_{\text{res}}=H^{\text{right}}H^{\text{left}},$$7 satisfies a cyclic block relation upon taking the $$H_{\text{root}}= \begin{pmatrix} O_{N\times N} & H^{\text{left}}_{N\times M}\ H^{\text{right}}_{M\times N} & O_{M\times M} \end{pmatrix}, \qquad H_{\text{par}}=H^{\text{left}}H^{\text{right}}, \qquad H_{\text{res}}=H^{\text{right}}H^{\text{left}},$$8th power. Applied to non-Hermitian Floquet topological insulators, this yields $$H_{\text{root}}= \begin{pmatrix} O_{N\times N} & H^{\text{left}}_{N\times M}\ H^{\text{right}}_{M\times N} & O_{M\times M} \end{pmatrix}, \qquad H_{\text{par}}=H^{\text{left}}H^{\text{right}}, \qquad H_{\text{res}}=H^{\text{right}}H^{\text{left}},$$9-th- and $H_{\text{par}}=C+H_{\text{original}}$0-th-root first- and second-order phases with edge or corner modes at fractional quasienergies $H_{\text{par}}=C+H_{\text{original}}$1, while allowing coexistence of non-Hermitian skin effect with fractional-quasienergy boundary states (Zhou et al., 2022).

These extensions indicate that non-Hermitian topological square-root insulators now form part of a larger rooted-topology program. Static square roots, higher roots, higher-order lattices, and Floquet descendants all preserve the central rooted relation to a parent system, but they diversify the admissible gap notions, boundary spectra, and dynamical responses. Within this broader program, the exceptionally deficient square-root insulator occupies a distinctive position: it uses the rooted architecture not only to inherit topology, but also to enforce defectiveness throughout the spectrum and thereby generate system-wide amplification phenomena (Bid et al., 15 Aug 2025).