Topological SSH Arrays and Dimerized Couplings

• Topological SSH arrays are dimerized lattice systems where alternating couplings generate quantized band topology, symmetry-protected edge states, and bulk–boundary correspondence.
• These arrays are implemented in diverse platforms such as photonic waveguide lattices, acoustic systems, and elastic metamaterials, highlighting their versatility and practical control schemes.
• Recent studies extend SSH physics to nonlinear, non-Hermitian, and dynamically tuned regimes, opening new avenues in topological photonics, metamaterials, and quantum systems.

Topological Su–Schrieffer–Heeger arrays are dimerized lattice systems whose alternating couplings generate symmetry-protected band topology, quantized geometric phases, and boundary-localized states. In their canonical form they are one-dimensional bipartite chains with two sites per unit cell, but the same organizing principle now extends across photonic waveguide lattices, acoustic waveguides, elastic metamaterials, optomechanical arrays, Josephson metamaterials, circuit networks, synthetic-dimension pumps, and higher-order two-dimensional lattices (Ivanov et al., 2023, Chaplain et al., 2024, Coutant et al., 2021). Across these realizations, the defining feature is not the material platform but the controlled alternation of couplings and the resulting bulk–boundary correspondence.

1. Canonical structure and bulk invariants

The standard SSH array is a one-dimensional lattice with sublattices $a$ and $b$, intracell coupling $t_1$, and intercell coupling $t_2$. Its tight-binding Hamiltonian is

$$H = \sum_n \left(t_1 a_n^\dagger b_n + t_2 a_{n+1}^\dagger b_n + \mathrm{h.c.}\right),$$

and its Bloch form can be written as

$$H(k)=d_x(k)\sigma_x+d_y(k)\sigma_y, \quad d_x=t_1+t_2\cos k,\quad d_y=t_2\sin k,$$

with $d_z=0$ in the chiral-symmetric limit (Ivanov et al., 2023). The two bands are separated by a gap except at the transition $|t_1|=|t_2|$, where the gap closes at $k=\pi$ in the conventional gauge (Chaplain et al., 2024).

The bulk topology is encoded either by the winding of $d_x+i d_y$ around the origin or by the Zak phase of the occupied band. A standard winding-number expression is

$b$0

with $b$1 in the nontrivial phase for $b$2 and $b$3 otherwise (Ivanov et al., 2023). For inversion-symmetric SSH chains, the Zak phase is quantized to $b$4 or $b$5 and distinguishes the two dimerizations (Chaplain et al., 2024). Under open boundary conditions, bulk–boundary correspondence yields exponentially localized edge or interface modes in the nontrivial phase. Their localization length scales as

$b$6

in units of unit cells in the standard tight-binding description (Chaplain et al., 2024).

This canonical formulation remains the reference point even when the unit cell becomes more complex. Hidden-symmetry, decimated, spinful, or longer-range variants typically reduce back to an effective off-diagonal two-band form only under specific correlations among couplings or after projection onto a reduced subspace. In that sense, the SSH chain functions both as a microscopic model and as a normal form for broader classes of one-dimensional topological arrays (Röntgen et al., 2023, Bid et al., 2021).

2. Physical implementations and parameter mappings

In photonics, SSH arrays are commonly realized as femtosecond-laser-written waveguide lattices, where evanescent coupling depends exponentially on center-to-center spacing. Alternating intra-cell and inter-cell spacings implement $b$7 and $b$8, so a topological array corresponds to the geometry with stronger intercell than intracell coupling. In one reported fused-silica platform, representative topological parameters were $b$9 and $t_1$0, with refractive-index contrast $t_1$1 and mode-field diameter $t_1$2 at $t_1$3 (Ivanov et al., 2023).

An elastic-wave realization maps the SSH structure onto a 3D-printed photo-responsive polymer beam with periodically attached rods. There the rods act as sites and the beam-mediated interactions define the effective couplings, controlled geometrically by rod separations $t_1$4 and $t_1$5. The platform uses a beam thickness $t_1$6, lattice parameter $t_1$7, rod height $t_1$8, and radius $t_1$9, and supports a localized interface mode near $t_2$0–$t_2$1 inside the common bandgap of the two dimerizations (Chaplain et al., 2024).

Acoustic implementations admit an exact mapping to SSH eigenmodes. For an alternating-cross-section waveguide, the pressure amplitudes at discontinuities satisfy

$t_2$2

with

$t_2$3

so the effective SSH “energy” is $t_2$4 (Coutant et al., 2021). By inserting identical Helmholtz resonators in each segment, the same construction can place topological edge modes at prescribed frequencies, including in the subwavelength regime.

Josephson-junction metamaterials realize a bosonic SSH chain through electrostatically controlled Cooper-pair hopping. In the effective bosonic model, the hopping amplitudes and onsite potentials are

$t_2$5

providing a directly gate-tunable route to topological mid-gap “soliton” modes (Kuzmanovski et al., 2023).

What unifies these implementations is the same topological control logic: geometry, electrostatics, or resonant mediation sets an alternating coupling pattern; once the ratio $t_2$6 is driven across unity, the platform enters the SSH topological regime.

3. Boundary, interface, coupled-array, and higher-dimensional states

The simplest observable consequence of SSH topology is the appearance of edge-localized modes at a termination that begins with the weak bond. More generally, interfaces between domains with distinct winding numbers support exponentially localized mid-gap states. In photonic experiments based on two closely positioned SSH arrays, two interface configurations were distinguished: TT, where both arrays are topological and support a coupled interface-mode doublet, and TN, where only one array is topological and only a single interface mode persists on the topological side (Ivanov et al., 2023).

Composite SSH arrays sharpen this interface logic. A central defect waveguide coupled symmetrically to two SSH segments produces a single antisymmetric topological zero mode localized at the interface, with vanishing amplitude on the defect and opposite signs on the neighboring boundary sites (Tang et al., 2024). This construction suppresses trivial Tamm states and isolates a single controllable topological mode.

Coupling several SSH chains in parallel enriches the boundary spectrum. In two and three coupled photonic SSH arrays, the number of linear edge states at a given boundary equals the number of arrays that are in the topologically nontrivial phase, and the resulting edge modes organize into in-phase, out-of-phase, or symmetry-protected null-in-the-middle combinations depending on inter-array coupling and which constituent arrays are topological (Sabour et al., 2024). This is a direct extension of bulk–boundary correspondence from a single chain to a quasi-one-dimensional manifold of coupled SSH subsystems.

Two-dimensional SSH arrays further generalize the boundary hierarchy. A square 2D SSH photonic lattice with four waveguides per unit cell has quantized bulk polarizations $t_2$7 in the topological phase and supports corner-localized states as a higher-order topological insulator (Ivanov et al., 2023). Separate 2D SSH models on square lattices exhibit a different extension of the paradigm: tunable Dirac semimetals, Dirac-point merging transitions, weak topological insulating phases, and nodal-line metallic phases, with Dirac points not pinned to high-symmetry points in one model and constrained by glide-mirror symmetry in another (Li, 2022). In ribbon geometries, the topological transition point shifts with width and approaches the bulk 2D critical value as the ribbon becomes wider, establishing a dimensional crossover between 1D and 2D SSH physics (Obana et al., 2019).

A common misconception is that SSH arrays are intrinsically one-dimensional and limited to ordinary edge states. The reported literature instead shows three distinct extensions: interface doublets in composite 1D structures, multiplicity-enhanced edge spectra in coupled arrays, and higher-order corner states or weak-topology edge bands in 2D lattices (Sabour et al., 2024, Ivanov et al., 2023, Li, 2022).

4. Nonlinear and bosonic constructions

Nonlinearity does not merely perturb SSH boundary modes; in several platforms it generates entirely new topological nonlinear states. In a 2D SSH photonic topological insulator with focusing Kerr nonlinearity and anomalous group-velocity dispersion, stable three-dimensional light bullets bifurcate thresholdlessly from linear corner states. These inherit the topological protection of the parent corner modes, remain stable even when the branch penetrates the edge-state band, and survive under diagonal and off-diagonal disorder realizations tested in the study (Ivanov et al., 2023).

In coupled SSH waveguide arrays, the linear edge-state multiplicity described above continues into the nonlinear regime as families of multipole topological edge solitons. Two coupled arrays support dipole edge solitons, including robust out-of-phase branches and symmetry-breaking bifurcations to strongly asymmetric stable states. Three coupled arrays support tripole and dipole families with markedly different stability windows, including stable out-of-phase tripoles throughout the gap and stable in-phase three-peak states that increasingly concentrate power in the central array (Sabour et al., 2024).

A distinct route uses intrinsic anharmonicity to construct SSH physics dynamically. In a driven bosonic chain with Kerr nonlinear modes placed at every even site, linearization about the pumped steady state and Schrieffer–Wolff elimination of far-detuned nonlinear modes produce an effective SSH Hamiltonian for the remaining bosonic modes,

$t_2$8

with

$t_2$9

The squeezing parameter $$H = \sum_n \left(t_1 a_n^\dagger b_n + t_2 a_{n+1}^\dagger b_n + \mathrm{h.c.}\right),$$0, and hence the ratio $$H = \sum_n \left(t_1 a_n^\dagger b_n + t_2 a_{n+1}^\dagger b_n + \mathrm{h.c.}\right),$$1, is pump-controlled, so topology is generated optically rather than lithographically (Nair, 2022).

These nonlinear and bosonic constructions show that SSH topology is compatible with thresholdless spatiotemporal self-localization, symmetry-breaking nonlinear branches, and pump-induced effective dimerization. A plausible implication is that the SSH framework now functions as a design template for nonlinear spectral engineering rather than only as a linear band-topology model.

5. Non-Hermitian arrays and dissipation-assisted topology

Non-Hermiticity in SSH arrays appears in several distinct forms: controlled loss, asymmetric hopping, and periodically driven non-Hermitian Floquet dynamics. In a photonic topological insulator built from two coupled SSH arrays, controlled absorption was added only to the right array. In the TT configuration, increasing loss first decreases and then increases the normalized output power, producing loss-induced transparency identified as a macroscopic Zeno effect (Ivanov et al., 2023). The mechanism is subspace-selective: dissipation suppresses transfer into lossy states while retaining energy in the protected interface-mode subspace. The reported two-mode model yields decrements

$$H = \sum_n \left(t_1 a_n^\dagger b_n + t_2 a_{n+1}^\dagger b_n + \mathrm{h.c.}\right),$$2

and one branch acquires a reduced decay rate as $$H = \sum_n \left(t_1 a_n^\dagger b_n + t_2 a_{n+1}^\dagger b_n + \mathrm{h.c.}\right),$$3 increases. Crucially, this behavior does not require parity-time symmetry or exceptional points; the analysis explicitly notes that eigenvalues and eigenvectors do not coalesce for $$H = \sum_n \left(t_1 a_n^\dagger b_n + t_2 a_{n+1}^\dagger b_n + \mathrm{h.c.}\right),$$4 (Ivanov et al., 2023).

The contrast case is topologically informative. When a topological array is coupled to a trivial one, only a single interface edge mode survives, there is no doublet and no switching, and the output power decreases monotonically with increasing loss. Likewise, when both arrays are trivial, only weak and much smaller power variations occur. The reported macroscopic Zeno transparency is therefore tied specifically to the protected two-mode interface subspace (Ivanov et al., 2023).

A different non-Hermitian SSH generalization uses asymmetric intercell hoppings and periodic driving. Its Bloch Hamiltonian has off-diagonal elements

$$H = \sum_n \left(t_1 a_n^\dagger b_n + t_2 a_{n+1}^\dagger b_n + \mathrm{h.c.}\right),$$5

and its topology is characterized by a bi-orthonormal geometric phase rather than an ordinary Hermitian Zak phase (Vyas et al., 2020). In the undriven case the phase diagram contains a trivial insulator, a non-trivial insulator, and a topologically non-trivial Möbius metallic phase. Under high-frequency periodic driving, the intercell coupling is renormalized to $$H = \sum_n \left(t_1 a_n^\dagger b_n + t_2 a_{n+1}^\dagger b_n + \mathrm{h.c.}\right),$$6, so the drive amplitude acts as a direct topological control parameter (Vyas et al., 2020).

The resulting non-Hermitian SSH literature makes two points clear. First, dissipation can stabilize or reveal protected transport rather than merely destroy it. Second, the presence of non-Hermiticity does not by itself identify the relevant mechanism: macroscopic Zeno transparency, asymmetric-hopping topology, and Floquet bi-orthonormal phases are structurally different phenomena (Ivanov et al., 2023, Vyas et al., 2020).

6. Active tuning, pumping, and mode steering

A central historical limitation of topological waveguides is their narrow and fabrication-fixed operating window. Several SSH-array platforms now address this directly. In an elastic SSH beam fabricated from a photo-responsive polymer doped with azobenzene, local $$H = \sum_n \left(t_1 a_n^\dagger b_n + t_2 a_{n+1}^\dagger b_n + \mathrm{h.c.}\right),$$7 illumination reduces the Young’s modulus at the interface and shifts the edge-mode frequency continuously. Simulations showed a downshift of approximately $$H = \sum_n \left(t_1 a_n^\dagger b_n + t_2 a_{n+1}^\dagger b_n + \mathrm{h.c.}\right),$$8 for a $$H = \sum_n \left(t_1 a_n^\dagger b_n + t_2 a_{n+1}^\dagger b_n + \mathrm{h.c.}\right),$$9 modulus reduction, corresponding to about $$H(k)=d_x(k)\sigma_x+d_y(k)\sigma_y, \quad d_x=t_1+t_2\cos k,\quad d_y=t_2\sin k,$$0 of the bandgap width, while experiments showed a reversible shift of about $$H(k)=d_x(k)\sigma_x+d_y(k)\sigma_y, \quad d_x=t_1+t_2\cos k,\quad d_y=t_2\sin k,$$1 at $$H(k)=d_x(k)\sigma_x+d_y(k)\sigma_y, \quad d_x=t_1+t_2\cos k,\quad d_y=t_2\sin k,$$2 over an approximately $$H(k)=d_x(k)\sigma_x+d_y(k)\sigma_y, \quad d_x=t_1+t_2\cos k,\quad d_y=t_2\sin k,$$3 spot (Chaplain et al., 2024).

In composite photonic SSH arrays, coherent control can be achieved by interference rather than by structural reconfiguration. Two access arms feeding the interface sites accumulate a relative dynamic phase $$H(k)=d_x(k)\sigma_x+d_y(k)\sigma_y, \quad d_x=t_1+t_2\cos k,\quad d_y=t_2\sin k,$$4, and because the topological zero mode is antisymmetric across the two interface sites while nearby bulk modes are locally symmetric, the excitation weights follow

$$H(k)=d_x(k)\sigma_x+d_y(k)\sigma_y, \quad d_x=t_1+t_2\cos k,\quad d_y=t_2\sin k,$$5

For the simulated normalized interface amplitudes of the zero mode, the excitation efficiency was fitted as

$$H(k)=d_x(k)\sigma_x+d_y(k)\sigma_y, \quad d_x=t_1+t_2\cos k,\quad d_y=t_2\sin k,$$6

with a maximum $$H(k)=d_x(k)\sigma_x+d_y(k)\sigma_y, \quad d_x=t_1+t_2\cos k,\quad d_y=t_2\sin k,$$7 at $$H(k)=d_x(k)\sigma_x+d_y(k)\sigma_y, \quad d_x=t_1+t_2\cos k,\quad d_y=t_2\sin k,$$8 (Tang et al., 2024). Experimentally, changing the arm-length imbalance $$H(k)=d_x(k)\sigma_x+d_y(k)\sigma_y, \quad d_x=t_1+t_2\cos k,\quad d_y=t_2\sin k,$$9 between $d_z=0$0, $d_z=0$1, and $d_z=0$2 created wavelength-selective switching windows between bulk and topological-mode excitation in the $d_z=0$3–$d_z=0$4 range (Tang et al., 2024).

Adiabatic pumping provides a more global form of control. Cyclically modulated generalized SSH chains define a synthetic two-dimensional parameter space $d_z=0$5 with a Chern number

$d_z=0$6

and display Haldane-like phase diagrams when next-nearest-neighbor hoppings or on-site terms are modulated with relative phase $d_z=0$7 (Li et al., 2013). In optomechanical arrays, strong driving generates an effective generalized SSH Hamiltonian

$d_z=0$8

with three phases $d_z=0$9, four edge-state types, and explicit adiabatic pumping protocols that transfer edge excitations across the array through periodic modulation of $|t_1|=|t_2|$0, $|t_1|=|t_2|$1, $|t_1|=|t_2|$2, and $|t_1|=|t_2|$3 (Xu et al., 2018).

In Josephson metamaterials, topological control becomes fully electrostatic. Local gate detuning of a dimer creates mid-gap soliton states, and a time-dependent ramp transferring the detuning to a neighboring dimer shuttles the soliton by one lattice period with nearly perfect fidelity on timescales identified as experimentally reachable (Kuzmanovski et al., 2023). This suggests that SSH boundary modes can serve not only as static waveguiding states but also as mobile topological information carriers.

7. Generalized families, hidden SSH structures, and current directions

Much of the recent SSH literature concerns systems that are not manifestly canonical SSH chains but reduce to one after projection, symmetry reduction, or synthetic extension. Latent SSH models are a clear example: unit cells without explicit mirror or chiral symmetry can nevertheless possess a hidden reflection symmetry between two boundary sites. Isospectral reduction then yields an energy-dependent effective Hamiltonian

$|t_1|=|t_2|$4

whose chiralized form is algebraically SSH-like (Röntgen et al., 2023). Because $|t_1|=|t_2|$5 can have multiple roots $|t_1|=|t_2|$6, these systems may host multiple independent topological transitions and multiple edge-state pairs, even though the unreduced unit cell is highly asymmetric (Röntgen et al., 2023).

Quasiperiodic and re-entrant SSH arrays extend topology beyond periodic dimerization. In a generalized quasiperiodic model with

$|t_1|=|t_2|$7

the reported phase diagrams show topological re-entrant transitions. For bounded modulation ($|t_1|=|t_2|$8), the sequence can be TI $|t_1|=|t_2|$9 trivial $k=\pi$0 TAI $k=\pi$1 trivial, while for unbounded modulation ($k=\pi$2) the sequence can be trivial $k=\pi$3 TAI $k=\pi$4 trivial $k=\pi$5 TAI $k=\pi$6 trivial (Wang et al., 2024). The analysis uses a real-space winding number, the zero-energy Lyapunov exponent, and bulk-gap diagnostics, indicating that deterministic quasiperiodicity can induce topology rather than only disrupt it.

Higher winding numbers and richer degeneracies also arise in multilayer extensions. Rhombohedral-stacked $k=\pi$7-layer SSH networks have a lower-bidiagonal off-diagonal block $k=\pi$8 with $k=\pi$9, giving a winding number $d_x+i d_y$0 and $d_x+i d_y$1-fold degenerate zero-energy edge states under open boundaries (Lu et al., 11 Nov 2025). The same work proposes Wigner entropy as a phase-space diagnostic for boundary states, with topological boundary modes exhibiting enhanced Wigner entropy relative to bulk states (Lu et al., 11 Nov 2025).

Spin–orbit coupling, next-nearest-neighbor hopping, and symmetry reduction substantially enrich the classification. In multipartite spinful SSH lattices with an even number of sublattices per unit cell and conserved $d_x+i d_y$2 spin rotation, the winding number can take values $d_x+i d_y$3, corresponding to trivial, twofold-degenerate, and fourfold-degenerate zero-mode boundary states (Bahari et al., 2020). In generalized SSH models with same-sublattice next-nearest-neighbor hoppings, chiral symmetry is broken and the class shifts from BDI to AI, yet parity or $d_x+i d_y$4 symmetry can preserve quantized or diagnostically useful Zak phases in gapped regions (Ahmadi et al., 2020). In 2D SSH models with all second-nearest-neighbor interactions included, inversion-based polarization invariants continue to classify trivial, partially trivial, and fully topological phases even though chiral symmetry is broken (Niekerk et al., 2023).

A recurring misconception is that SSH topology requires an explicit two-site unit cell with exact chiral symmetry written in obvious form. The accumulated results show a broader statement: explicit dimerization is sufficient but not necessary. Latent symmetry, decimation, synthetic dimensions, quasiperiodic modulation, multilayer stacking, and pump-induced effective couplings can all realize SSH-type bulk–boundary correspondence when the reduced or effective problem retains the relevant topological structure (Röntgen et al., 2023, Bid et al., 2021, Li et al., 2013).

From this broader perspective, “Topological Su–Schrieffer–Heeger arrays” denotes not a single lattice geometry but a research program centered on dimerization, symmetry, and controllable boundary localization. Current directions identified in the cited literature include scaling to larger and higher-dimensional arrays, quantitative disorder thresholds, non-Hermitian skin effects in more asymmetric settings, bandwidth management of mid-gap states, interaction and nonlinearity effects beyond effective single-particle descriptions, and dynamically reconfigurable on-chip devices where geometric phase, dynamic phase, dissipation, and synthetic modulation are all used as independent control parameters (Ivanov et al., 2023, Chaplain et al., 2024, Tang et al., 2024).