High-Root Topological Insulators (HRTIs)

• High-root topological insulators are lattice models whose Hamiltonians are derived via root operations, imparting inherited finite-energy boundary modes from a well-characterized parent system.
• They use construction methods such as hopping renormalization and additional site incorporation to produce multiple spectral gaps while preserving locality and Hermiticity.
• Experimental implementations in photonic lattices and LC circuits confirm the models’ practical feasibility and demonstrate robust, tunable edge and corner states.

High-root topological insulators (HRTIs) are lattice models whose Hamiltonians are algebraically related to a simpler parent topological Hamiltonian by a root operation. In the most common case, a square-root model is engineered so that its square decomposes into one or more parent blocks with established topological character; more generally, a $2^n$-root model is connected to a source topological insulator or superconductor through repeated squaring intercalated with constant energy downshifts. The characteristic consequence is that boundary states of the parent are inherited by the root model at finite energies, often together with a multiplication of spectral gaps and a proliferation of boundary-state families (Mizoguchi et al., 2020, Marques et al., 2021, Marques et al., 2021). This notion must be distinguished from higher-order topological insulators (HOTIs), where the defining feature is codimension-two or codimension-three boundary localization—such as 2D corner states or 3D hinge states—without any root construction of the Hamiltonian (Schindler et al., 2017, Aggarwal et al., 2021).

1. Definition and nomenclature

In the HRTI literature, the basic object is a child Hamiltonian $H$ whose topology is not diagnosed directly in $H$ itself, but in a higher power $H^r$. For square-root topological insulators, $r=2$; for high-root constructions, $r$ may be $3$, $4$, or, in the systematic families developed so far, $2^n$. The defining mechanism is spectral inheritance: if a parent block in $H^r$ has topological boundary states, then the child $H$0 inherits corresponding finite-energy boundary modes whose energies follow the relevant root relation (Dias et al., 16 Aug 2025, Marques et al., 2021).

The terminology is easily confused with that of higher-order topology. HOTIs are classified by boundary codimension: a 3D second-order HOTI has gapped 2D surfaces but 1D hinge modes, while a 2D second-order HOTI has gapped edges but 0D corner modes. Schindler et al. formulated explicit chiral and helical 3D HOTI classes protected by spatio-temporal symmetries, mirror symmetries, or $H$1 (Schindler et al., 2017). The experimental work on Bi and $H$2 likewise concerns HOTIs with hinge-localized modes on $H$3 islands and explicitly notes that these phenomena are not “higher-root” constructions of Hamiltonians (Aggarwal et al., 2021). A square-root HOTI therefore lies at the intersection of the two notions: the parent $H$4 is higher-order, while the child $H$5 is high-root.

2. Algebraic construction and root hierarchies

The most systematic formulation of high-root topology in one dimension introduces a sequence of Hermitian, local, translationally invariant tight-binding Hamiltonians $H$6, $H$7, connected by

$H$8

Here $H$9 is a known source topological insulator or superconductor with protected zero-energy edge modes under open boundary conditions, and $H$0 is the $H$1-root model. The construction is organized by an “arborescence,” an outwards directed rooted tree whose hypotenuse branch $H$2 tracks the parent topological block, while off-hypotenuse branches generate residual blocks that must themselves be squared and shifted until leaves are reached (Marques et al., 2021).

Two explicit construction strategies were given. In method (ii), extra sites and couplings are added so that squaring equalizes squared on-site potentials on a selected sublattice while keeping hopping magnitudes fixed at $H$3 or $H$4. In method (i), selected hoppings are renormalized so that the same equalization occurs without extra sites, at the price of an overall energy rescaling after squaring. Both procedures preserve locality and Hermiticity, and both rely on bipartite, block-antidiagonal root Hamiltonians so that squaring yields block-diagonal parent and residual sectors (Marques et al., 2021).

The boundary-state spectrum follows a nested-radical law. If $H$5 is a source edge-state energy, then

$H$6

For source zero modes, $H$7. In the explicit one-dimensional constructions, the number of root edge states is $H$8 for topological insulators and $H$9 for topological superconductors (Marques et al., 2021). The same work also quantified the dilution of the topological component: each $H^r$0-root edge state carries weight $H^r$1 on the source topological mode, with the remaining weight distributed over impurity-like residual states.

Representative examples include the diamond-chain $H^r$2, whose square yields a Creutz-ladder parent block, and its quartic-root descendant; the Kitaev-chain-derived $H^r$3 and $H^r$4; and the corresponding $H^r$5-root SSH-like families (Marques et al., 2021). In all of them, topology is inherited rather than native to the child Hamiltonian.

3. Spectral structure and topological diagnostics

A central algebraic pattern in HRTIs is the chiral, block-off-diagonal form

$H^r$6

which implies spectral symmetry $H^r$7. Squaring gives

$H^r$8

thereby separating parent and residual sectors on different sublattices. This structure underlies the decorated-honeycomb square-root HOTI, the hybrid honeycomb–kagome circuit model, and related bipartite descendants (Mizoguchi et al., 2020, Song et al., 2020).

The topological diagnosis depends on the symmetry class of the parent block. In one dimension, inherited SSH-like parent blocks admit the usual chiral winding number

$H^r$9

or equivalently a Zak phase. In inversion-symmetric root chains, however, the literature notes that conventional winding or Zak-phase diagnostics may become ambiguous because unit-cell choices in the infinite limit are not physically fixed. For that reason, a later analysis proposed a pragmatic characterization based on the existence of edge-state bands in the infinite HRTI and on the restrictions imposed by generalized boundary conditions (Dias et al., 16 Aug 2025).

That same complex-band analysis reinterpreted one-dimensional HRTI edge states as sliced sections of impurity bands of a uniform tight-binding chain. For the parent uniform chain,

$r=2$0

with $r=2$1 giving Bloch states and $r=2$2 giving evanescent states. Edge states in finite or semi-infinite HRTIs were shown to be discrete subsets of these evanescent solutions, selected by an effective energy-dependent edge potential $r=2$3. The corresponding condition can be written as

$r=2$4

where $r=2$5 is the surface Green’s function of the semi-infinite uniform chain (Dias et al., 16 Aug 2025). This reformulation avoids real-space diagonalization and makes explicit why repeated root operations generate multiple finite-energy edge-state bands.

In two-dimensional square-root HOTIs, crystalline symmetry rather than winding number becomes the decisive invariant. For the decorated honeycomb model, $r=2$6 symmetry quantizes the band polarization through

$r=2$7

The nontrivial phase is $r=2$8 for $r=2$9, with a transition at $r$0; the child polarization is inherited entirely from the breathing-kagome block in $r$1 (Mizoguchi et al., 2020). In the martini-lattice square-root descendant, the corresponding bulk invariant is a non-trivial $r$2 topological invariant protected by $r$3 (2207.14540).

4. Higher-order descendants and representative lattice models

The explicit square-root HOTI proposed on a decorated honeycomb lattice remains the canonical two-dimensional example. Its five-site unit cell consists of two honeycomb backbone sites and three attached sites, with nearest-neighbor hoppings $r$4 and $r$5. Squaring produces a trivial honeycomb block and a breathing kagome block, and the higher-order topology of the child is inherited entirely from the latter. On triangular finite samples, the child exhibits threefold-degenerate in-gap corner states localized at the corners for $r$6, none for $r$7, and a bulk gap closing at $r$8 (Mizoguchi et al., 2020). Because of chiral symmetry, the corner modes appear at finite energies $r$9, rather than at zero energy as in many conventional HOTIs.

A related construction starts from the martini lattice, proposed as a HOTI in its own right, and then builds a square-root descendant from a honeycomb lattice with two-site decoration. In this case the squared Hamiltonian consists of two martini-lattice blocks, $3$0 and $3$1, and the corner states of the child again occur at finite energies $3$2. Both parent and child are diagnosed by a $3$3-protected bulk $3$4 invariant, and both support threefold-degenerate corner states on suitable triangular terminations (2207.14540).

The 2021 two-dimensional generalization of $3$5-root topology extended the root construction to weak, Chern, and higher-order topological insulators, and to topological semimetals. The paper explicitly constructed quartic-root versions of the asymmetric 2D SSH HOTI, the Haldane Chern insulator, the breathing-kagome HOTI, and a graphene-derived semimetal. In each case, squaring plus constant downshifts recovers a decoupled parent block carrying the relevant invariant, while residual blocks contribute impurity-like boundary states and degenerate spectra (Marques et al., 2021). A characteristic feature of these quartic-root descendants is dilution: the topological component of an edge or corner mode is one quarter, while the remaining three quarters reside in residual sectors.

This two-dimensional root program also sharpened a general distinction. The parent asymmetric 2D SSH HOTI has reflection symmetries, quantized weak polarizations, and a quadrupole moment $3$6, whereas its square-root and quartic-root descendants generally have non-quantized polarization and quadrupole moment at the root level because the relevant symmetries become $3$7-dependent in the enlarged unit cell. The invariant is therefore recovered only after squaring and downshifting to the parent block (Marques et al., 2021).

5. Experimental realizations

The first photonic realization of a second-order square-root HOTI was implemented in a laser-written decorated honeycomb lattice. The platform was a finite 2D array of single-mode optical waveguides written in an SBN:61 crystal, with 81 sites, open boundaries, and dimerization controlled by two nearest-neighbor distances $3$8 and $3$9. The nontrivial realization used $4$0, $4$1, with an estimated $4$2; the trivial realization reversed these distances. The experiment observed two sets of threefold-degenerate corner states in distinct bandgaps: top-gap modes with an out-of-phase structure and bottom-gap modes with an in-phase structure. Interferometry with an inclined quasi-plane wave directly resolved the phase structure of the localized corner states (Yan et al., 2021).

An independent experimental realization was achieved in topological LC circuits based on a hybrid honeycomb–kagome lattice with five nodes per unit cell. At the resonant design point $4$3, the circuit Laplacian realizes the same chiral block structure that leads, after squaring, to a honeycomb parent and a breathing-kagome parent. The higher-order topology is fully characterized by bulk polarization: the fifth band has $4$4 for $4$5 and $4$6 for $4$7, while the first-order Chern number vanishes for all capacitance ratios (Song et al., 2020). Because the corner modes lie at finite admittances rather than at zero, the experiment introduced identical grounded inductors $4$8 to all nodes, shifting the entire spectrum without changing eigenvectors and thereby enabling direct impedance detection of corner-localized modes.

The circuit experiment used $4$9, $2^n$0, $2^n$1, $2^n$2, and $2^n$3, with a designed corner-mode resonance at

$2^n$4

Corner, bulk, and edge nodes were probed by impedance spectroscopy, and the measured impedance map peaked at the corners in the topological regime. Adding next-nearest-neighbor capacitors near a corner broke chiral symmetry and blueshifted the corner resonance by approximately $2^n$5, whereas an analogous perturbation in the deep bulk preserved the corner frequency (Song et al., 2020).

Beyond these two demonstrations, the HRTI literature recurrently identifies photonic lattices, acoustic lattices, topolectrical circuits, cold-atom optical lattices, superconducting resonator chains, and Floquet systems as suitable platforms because they permit direct control of hopping magnitudes, phases, boundary terminations, and generalized boundary clusters (Marques et al., 2021, Dias et al., 16 Aug 2025).

6. Dynamics, transfer, robustness, and open directions

The finite-energy nature of root boundary modes has direct dynamical consequences. In the decorated-honeycomb square-root HOTI, preparing an initial single particle at a corner yields long-time corner retention in the nontrivial phase $2^n$6, whereas the trivial phase $2^n$7 shows spreading through the bulk and boundary. The site-resolved density

$2^n$8

remains strongly localized because the in-gap finite-energy corner states contribute non-decaying amplitudes at the corner (Mizoguchi et al., 2020).

A later one-dimensional development recast this dynamical viewpoint as a transport problem in sine–cosine high-root chains $2^n$9. These models possess multiple spectral gaps, multiple folding energies, and edge or domain-wall states in distinct gaps, enabling several transfer processes in the same device. For $H^r$0, the edge states lie at $H^r$1; fragmenting the chain into multiple domains creates additional domain-wall states and dramatically accelerates transfer. One-domain transfer times scale exponentially with length, two-domain fragmentation halves the exponent, and for many domains with fixed inner domain length the transfer time becomes linear in $H^r$2, with the explicit fit $H^r$3 reported for one parameter set. The same work derived exact relations between transfer times in different root models and in different gaps of the same model, and showed that increasing the number of domain-wall states improves transfer fidelity under a general disorder regime because protection is inherited from the parent chiral chain (Moreira et al., 19 Mar 2026).

Robustness remains symmetry-specific. In the decorated-honeycomb and martini-based square-root HOTIs, $H^r$4 symmetry quantizes the relevant polarization or $H^r$5 invariant; breaking $H^r$6 generally de-quantizes the invariant and can destabilize corner modes (Mizoguchi et al., 2020, 2207.14540). In one-dimensional HRTIs, angle disorder that preserves chiral symmetry can retain finite-energy edge states up to gap closing, while diagonal disorder or general disorder breaks the parent chiral structure after squaring and degrades protection (Moreira et al., 19 Mar 2026). The complex-band analysis likewise emphasized that boundary conditions are not a secondary detail: generalized boundary clusters act through an energy-dependent $H^r$7 and can create or destroy edge-state levels in individual gaps (Dias et al., 16 Aug 2025).

Several directions remain explicitly open. The square-root HOTI on the decorated honeycomb lattice established the $H^r$8 case and pointed to analogous square-root routes in three dimensions, such as decorated diamond lattices whose square includes a pyrochlore HOTI block, but it did not construct $H^r$9 descendants in 2D (Mizoguchi et al., 2020). The $H$00-root program anticipated extensions to higher-dimensional Chern, weak, and higher-order systems, and the complex-band framework stated that higher-dimensional generalizations are forthcoming (Marques et al., 2021, Dias et al., 16 Aug 2025). At the same time, the literature repeatedly underscores a conceptual boundary: HRTIs concern roots of Hamiltonians, whereas material systems such as Bi, $H$01, and SnTe are discussed within higher-order topology rather than high-root topology (Aggarwal et al., 2021, Schindler et al., 2017).

In this sense, HRTIs are best understood not as a separate replacement for existing topological classifications, but as an algebraic extension of them. Their defining operation is the passage from a parent topological block to a child Hamiltonian whose topology is spectrally inherited, symmetry-conditioned, and typically manifested through finite-energy boundary modes distributed across several gaps.