Extended SSH Model

• Extended SSH model is a generalized framework that incorporates long-range hoppings, enlarged unit cells, and electron-electron interactions to reveal novel topological phases.
• It employs multi-site tunneling and additional interaction terms to generate composite excitations, multi-band invariants, and robust edge states observable in various experiments.
• Incorporating both on-site and nearest-neighbor interactions, the model supports emergent dimer states and topological-CDW transitions verified in ultracold atomic, photonic, and ferromagnetic systems.

The extended SSH model refers to a broad class of generalizations of the Su–Schrieffer–Heeger (SSH) model that incorporate additional internal structure, longer-range hoppings, and electron-electron interactions beyond the minimal two-site unit cell and nearest-neighbor hopping of the original SSH chain. These extensions enable the realization and characterization of novel topological phases, emergent composite excitations, multi-band bulk invariants, and robust edge states—not only in idealized models but also in experimental platforms such as ultracold atoms, optical superlattices, engineered photonic systems, ferromagnetic materials, and designed quantum simulators.

1. Model Structure and Extended Hamiltonians

The archetypal SSH model describes a one-dimensional dimer chain with alternating hopping amplitudes between two sublattices (A/B), supporting two distinct topological phases distinguished by a bulk winding number (or equivalently, the Zak phase) and the presence of midgap edge states (Batra et al., 2019). The extended SSH model generalizes this framework in several directions:

• SSH4 and SSH-n models: The unit cell is enlarged to include four or more sublattices (e.g., A₁, B₁, A₂, B₂), with multiple distinct tunneling amplitudes (e.g., tₐ, t_b, t_c, t_d in SSH4), and hopping periodicity >2 sites (Xie et al., 2019, Zhou et al., 2022, Joshi et al., 23 Mar 2025).
• Long-range (LR) hoppings: Inclusion of next-nearest and further neighbor terms (with coefficients Jₙⱼ, up to some range N) yields a tight-binding Hamiltonian of the form

$$H_N = \sum_{|i-j| \leq N} J_{ij} c_i^\dagger c_j + \text{h.c.},$$

with odd (inter-sublattice) and even (intra-sublattice) hopping terms (Pérez-González et al., 2018, Joshi et al., 23 Mar 2025).

• Interaction terms: Extended SSH models may include on-site ($U$) and nearest-neighbor ($V$) density-density interactions (Hubbard- or Bose-Hubbard-type), supporting nontrivial two-body (dimer) physics, charge-density wave (CDW) order, and interaction-induced edge states (Liberto et al., 2016, Zhou et al., 2022).
• Non-Hermitian generalizations and driven systems: By engineering gain/loss, periodic driving (Floquet engineering), or non-Hermitian hopping, extended SSH models realize PT-symmetric, quasi-Hermitian, or Floquet topological phases (Turker et al., 2018, Io et al., 2023, Rottoli et al., 7 Feb 2024).

2. Topological Invariants and Bulk–Edge Correspondence

In extended SSH models, topological invariants generalize naturally to accommodate higher-dimensional unit cells and longer-range hopping:

• Momentum-space winding numbers: For a generic chiral-symmetric, block-off-diagonal Hamiltonian, the invariant

$$W = \frac{1}{2\pi i} \int_0^{2\pi} dk\, \partial_k \log \det[h(k)]$$

counts the winding of the off-diagonal block $h(k)$ (Joshi et al., 23 Mar 2025, Xie et al., 2019). In SSH4 and SSHLR models, winding numbers of |W| = 0, 1, or 2 are possible, with W = 2 corresponding to four zero-energy edge states.

• Zak phase and mean chiral displacement: The quantized Berry (Zak) phase and the time-averaged mean chiral displacement provide robust measures of topological phase, directly measurable in cold atom and photonic experiments (Xie et al., 2019).
• Chern number and Bott index: For periodically driven models (with time as an extra coordinate), adiabatic charge pumping enables the use of a 2D Chern number and real-space Bott index, capturing dynamical topology in (k, t) space (Joshi et al., 23 Mar 2025).
• Multi-band generalizations: For n-band models, the proper invariant is the sum of Berry phases (or winding numbers) for the occupied bands, requiring careful gauge-fixing to ensure correspondence with edge state counting in both symmetric and inversion-symmetry-broken cases (Lee et al., 2022).
• Bulk–edge correspondence: The number of robust midgap (zero-energy) edge states matches the winding invariant, and this relation persists in multi-band and interacting cases, with the possible exception that left/right boundaries yield different edge counts if inversion symmetry is broken (Lee et al., 2022, Xie et al., 2019).

3. Role of Interactions: Two-body Bound States, Phase Transitions, Edge Phenomena

Interactions dramatically enrich the physics of extended SSH models, enabling emergent composite objects and novel correlated phases:

• Dimer (bound) states: With both on-site ($U$) and nearest-neighbor ($V$) interactions, two classes of bound states exist—(i) in-cell dimers, akin to traditional doublons, and (ii) out-of-cell (intercell) dimers, stabilized only when $V \neq 0$ (Liberto et al., 2016).
• Resonant hybridization and doublon mobility: When the energies of in-cell and out-of-cell dimers cross (as functions of $U$ and $V$), the effective doublon hopping is dramatically enhanced at resonance, exhibiting first-order $J_2$-dependence rather than the usual $J_2^2$ scaling.
• Topological–CDW transitions: Increasing $V$ in SSH4-type models induces continuous (third-order) transitions from topological insulating phases (with protected edge states and twofold degenerate entanglement spectra) to CDW phases characterized by local density modulations, scaling of the entanglement entropy, and critical exponents matching the Luttinger liquid universality class (Zhou et al., 2022).
• Edge bound states and strong boundary localization: Nearest-neighbor interactions ($V$) generically stabilize strongly localized edge states, which persist even in the strong coupling limit ($U, V \gg J$), in contrast with systems having only on-site interactions, where large $U$ delocalizes edge states (Liberto et al., 2016).

4. Long-Range Hopping, Disorder, and Floquet Engineering

• Odd/even hopping dichotomy: Odd-length hopping (n odd, inter-sublattice) preserves chiral and particle–hole symmetry, enabling topological phases with large bulk winding numbers and multiple edge-state pairs, while even-length (intra-sublattice) hopping breaks these symmetries and decouples edge states from the winding number, though inversion symmetry can preserve a quantized Zak phase (Pérez-González et al., 2018).
• Disorder effects: Diagonal (on-site) disorder breaks sublattice symmetry and erases protected zero modes. Off-diagonal (hopping) disorder can leave topological edge states robust, though in models supporting multiple edge pairs, disorder can distinguish between different classes of edge states (Pérez-González et al., 2018).
• Periodic (Floquet) driving: Time-periodic fields can be used to renormalize selective hopping amplitudes, either restoring chiral symmetry (by suppressing certain hoppings via Bessel-zero engineering) or inducing new Floquet topological phases. In PT-symmetric Floquet SSH models, a stable topological phase is realized provided the driving frequency is high and the gain/loss parameter is below a critical threshold (Turker et al., 2018).

5. Physical Implementations and Measurement Protocols

• Ultracold atoms and optical superlattices: The SSH4 model has been realized with ultracold $^{87}$Rb in a momentum lattice, with independent control over four-site tunneling amplitudes via tailored Raman beam patterns. Observables such as the mean chiral displacement and quench dynamics from edge and bulk initial states provide direct signatures of topological invariants and edge state localization (Xie et al., 2019, Zhou et al., 2022).
• Photonic and optoelectronic lattices: Extended SSH and Floquet SSH models are implemented in arrays of optical waveguides, where periodic spatial modulation encodes time-periodic (driven) Hamiltonians. PT symmetry and gain/loss are realized via controlled optical pumping or absorption (Turker et al., 2018).
• Ferromagnetic systems: The “double independent SSH” (DISSH) model leverages magnetic point group symmetries (combined unitary and anti-unitary operators) to achieve SSH-like topological phases even when chiral symmetry is absent, enabling tunable topological transitions in 2D ferromagnetic tight-binding bands by controlling the Zeeman term (Cheung et al., 2020).
• Defect engineering and quantum information: By introducing controlled defects (site-specific hopping modulation), chains can be segmented into smaller topological blocks, each hosting localized spin centers (qubits). The interplay of defect-induced edge states and on-site Hubbard interaction provides a means to engineer singlet/triplet pairings with applications in spin-based quantum technologies (Wang et al., 24 Jul 2025).

6. Non-Hermitian, Continuum, and Generalized Models

• Non-Hermitian and quasi-Hermitian topological phases: In models with non-Hermitian hopping/amplitude, the concept of a modified winding number (or spectral winding number in the presence of the skin effect) classifies the existence and spatial distribution of edge states. The quasi-Hermitian limit enables the mapping to a Hermitian model with a real spectrum, preserving bulk–boundary correspondence (Io et al., 2023).
• Entanglement Hamiltonian and criticality: In non-Hermitian SSH chains, the entanglement Hamiltonian at criticality features a novel imaginary chemical potential (absent in unitary systems), directly causing negative entanglement entropy—an effect rooted in the associated $c=-2$ ghost CFT structure (Rottoli et al., 7 Feb 2024).
• Continuous non-local models: The recently developed continuous non-local Dirac-type SSH model with translation operators reproduces all topological features of the lattice SSH model (energy bands, Zak phase), preserves exact chiral symmetry without external potentials, and exhibits an infinite flat band of exponentially localized zero modes when defined on a finite domain. A tunable length scale $a$ enables interpolation between local and non-local regimes and facilitates the construction of continuous analogues for a broad class of bipartite/multipartite lattices (Diakonos et al., 15 Sep 2025).

7. Dynamical and Quantum Geometric Diagnostics

• Adiabatic charge pumping and higher invariants: By cyclically modulating parameters over time, the driven extended SSH (e.g., SSH4, SSHLR) models map to 2D (k, t)-spaces with topological quantization encoded in the Chern number (evaluated as an integral of the Berry curvature) and the Bott index in real space. The net pumped charge in a cycle corresponds to these invariants and can exceed the static winding number, reflecting the presence of dynamically created zero modes (Joshi et al., 23 Mar 2025).
• Quantum metric and phase boundary detection: The symmetric (quantum metric) part of the quantum geometric tensor is highly sensitive to phase transitions not necessarily captured by the quantized Berry curvature; fluctuations in the metric provide a practical tool to identify topological phase boundaries and criticality, complementing invariants derived from the antisymmetric part (Cheng et al., 2023, Joshi et al., 23 Mar 2025).

---

The extended SSH model framework encompasses a diversity of physical realizations, topological phenomena, and experimental protocols. Its importance is underscored by the direct correspondence between the analytical structure of multi-band and long-range Hamiltonians, the existence and robustness of edge and defect-induced states, the tunability via interactions, disorder, and driving, and the realization of bulk–boundary correspondence through a hierarchy of topological invariants and dynamical observables.