2D SSH Array: Topological Phases

• 2D Su-Schrieffer-Heeger (SSH) Array is a lattice model with alternating hopping terms that generalizes the 1D SSH chain to exhibit diverse topological phases.
• It features vectored Zak phases that classify bulk polarization, yielding robust edge, corner, and higher-order boundary states across various experimental platforms.
• Extensions incorporating additional hoppings, interactions, and disorder reveal rich phase transitions and potential applications in quantum information and topological devices.

A two-dimensional (2D) Su-Schrieffer-Heeger (SSH) array generalizes the well-known 1D dimerized chain by constructing a square- or rectangular-lattice model in which alternating (dimerized) hoppings extend along both spatial directions. The 2D SSH model provides a fertile platform for realizing a variety of topological phases, including edge, corner, and many-body bound states, and its rich physics underpins proposals and experiments in electronic, photonic, acoustic, and circuit systems. Below, key theoretical and physical properties of 2D SSH arrays are presented, emphasizing model construction, classification of topological invariants, bulk–boundary and higher-order correspondence, phase transitions, interactions, disorder, and representative experimental realizations.

1. Model Constructions and Extensions

The canonical 2D SSH array is defined on a square (or rectangular) lattice with a multipartite unit cell—typically with four sublattice sites per cell—enabling alternating strong/weak hopping amplitudes along $x$ and $y$. The standard Hamiltonian may be written as a sum of alternating intracell ($\gamma$) and intercell ($\gamma'$) hopping terms in both directions, generalizing the SSH dimerization:

$$\begin{align*} H_{2D} = & \sum_{i,j} \big[ \gamma\, a_{i,j}^\dagger b_{i,j} + \gamma'\, b_{i,j}^\dagger a_{i+1,j} + \gamma\, c_{i,j}^\dagger d_{i,j} + \gamma'\, d_{i,j}^\dagger c_{i,j+1} + \text{h.c.} \big]\,, \end{align*}$$

where $a,b,c,d$ denote sublattice sites in unit cell $(i,j)$. The model can be recast in a basis that allows coupling between SSH chains along orthogonal axes, and, in higher-dimensions, this construction is naturally extended:

$$H_{nD} = \sum_{l=1}^{n} I_2^{\otimes (l-1)} \otimes H_{1D} \otimes I_2^{\otimes (n-l)}\,,$$

with $H_{1D}$ being the SSH Hamiltonian.

Variants arise by inclusion of:

• Intracell and intercell next-nearest-neighbor (NNN/SNN) hoppings (Xu et al., 2020, Niekerk et al., 2023)
• Longer-range (third neighbor) hoppings that preserve or break chiral symmetry (Du et al., 2021)
• On-site staggered potentials (Geng et al., 2022)
• Periodically modulated hoppings with arbitrary periodicity (Kar, 2023)
• Imaginary on-site potentials to realize non-Hermitian extensions (Tyagi et al., 20 Jun 2025)
• Interactions: nearest-neighbor (NN) and/or Hubbard-type onsite terms (1710.09148, Feng et al., 2021, Xing et al., 2020)
• Experimental realizations via mapping to photonic, acoustic, circuit, or material platforms (Zheng et al., 2019, Geng et al., 2022, Buendía et al., 9 Oct 2025)

Such generalizations allow the exploration of a broad range of topological and non-topological phenomena.

2. Topological Invariants and Bulk–Boundary Correspondence

The topological classification of 2D SSH arrays is characterized not by a Chern number (which vanishes due to time-reversal symmetry), but by vectored Zak phases—one for each spatial direction:

$$\boldsymbol{\mathcal{Z}} = (\mathcal{Z}_x, \mathcal{Z}_y)\,, \quad \mathcal{Z}_l = \frac{1}{2} \Delta\phi_l(k_l)\,,$$

with $\phi_l(k_l)$ being the phase of the off-diagonal structure in the Bloch Hamiltonian along direction $l=x,y$. When inversion symmetry is present, these phases are quantized to $0$ or $\pi$ (modulo $2\pi$) (Obana et al., 2019, Liu, 2023). A nontrivial Zak phase implies quantized bulk charge polarization ($P_l = \mathcal{Z}_l/2\pi$) and, via bulk–boundary correspondence, the presence of robust edge (or higher-order) states localized at boundaries normal to direction $l$.

While the Chern number remains trivial, the underlying Berry curvature can be nonzero in systems with broken inversion symmetry or under non-Hermitian conditions, resulting in anomalous transverse responses such as a finite anomalous Nernst conductivity (Tyagi et al., 20 Jun 2025).

Topological invariants may also be evaluated using parity eigenvalues at high symmetry (TRIM) points:

$$(-1)^{\nu_x} = \prod_{j\,\text{occ.}} \frac{\xi_j(X)}{\xi_j(\Gamma)}, \quad (-1)^{\nu_y} = \prod_{j\,\text{occ.}} \frac{\xi_j(Y)}{\xi_j(\Gamma)},$$

with $(\nu_x, \nu_y)$ forming a vector invariant (Niekerk et al., 2023).

3. Bulk Bands, Edge, Corner, and Higher-Order Boundary States

The bulk spectrum of the 2D SSH model exhibits four bands (for a four-site unit cell) with explicit analytic expression:

$$\varepsilon_j(\mathbf{k}) = s_1 |\rho_x(k_x)| + s_2 |\rho_y(k_y)|,\qquad s_1,s_2 = \pm 1, \quad \rho_l(k_l) = \gamma + \gamma' e^{ik_l}$$

(Obana et al., 2019). The band structure undergoes inversion at $X$ and $Y$ high symmetry points when $\gamma'/\gamma>1$ (in the standard square geometry), driving a topological phase transition and yielding in-gap edge states.

In finite systems, the interplay of open boundaries along one or more directions leads to dimensional reduction; with $(\pi,\pi)$ Zak phase, the 2D array supports 0D states (corners), while e.g.\ $(\pi,0)$ or $(0,\pi)$ yields 1D edge states. This generalizes to the "hierarchical bulk–edge correspondence" or $n$–$(n-l)$ correspondence: the number of nontrivial Zak components dictates the codimension of boundary modes (Liu, 2023).

In models with NNN or SNN hoppings, energy bands become more complex, and additional boundary/corner states (e.g., "general corner states") may split from edge bands; these can be robust against disorder if isolated by a finite gap (Xu et al., 2020, Niekerk et al., 2023).

For periodic modulations beyond two-site periodicity, additional in-gap and zero-energy modes arise, exhibiting non-trivial localization at edges, corners, or even at intersection points of domain walls (Kar, 2023, Mandal et al., 21 Jun 2025).

4. Interactions, Many-Body Physics, and Disorder

Interactions introduce new physical phenomena and alter the nature of boundary states:

• Two-body SSH system: Mapping two-particle states in a 1D SSH chain with NN interactions onto a 2D SSH lattice produces intrinsically many-body topological bound states localized in the internal (relative) coordinate—these have no counterpart in single-particle models. Sharp potential walls (NN interaction) in the relative coordinate are essential for isolating the topological in-gap bands; Hubbard-type interactions do not support such bound states (1710.09148).
• Electron–phonon and electron–electron interactions: In the 2D SSH-Holstein model, competition between bond-dependent SSH and site-dependent Holstein couplings gives rise to antiferromagnetic (AF), bond-ordered wave (BOW), and charge-density wave (CDW) phases. The SSH model by itself can support BOW order with a critical electron–phonon coupling $g_c$, and the addition of the Holstein term can drive a first-order BOW–CDW transition (Xing et al., 2020, Casebolt et al., 22 Mar 2024). In the SSHH model (SSH plus Hubbard U), a first-order transition occurs from AF to BOW as the SSH coupling is increased (Feng et al., 2021).
• Correlations and longer-range hopping: Longer-range hopping (e.g., third neighbor) can enlarge the topological invariant (winding number/Berry phase), producing new phase boundaries. On-site electron–electron correlations (Hubbard U) generically renormalize effective hopping parameters and can drive correlation-induced topological transitions even from trivial to non-trivial phases when analyzed in a slave-rotor mean-field framework (Du et al., 2021).
• Quasiperiodic disorder: Diagonal quasiperiodic disorder localizes states and shifts low-energy spectral weight away from zero energy. Detailed participation ratio studies reveal transitions from extended to partially extended/localized regimes, contingent on disorder strength (Mandal et al., 21 Jun 2025).

5. Phase Transitions, Dirac Points, and Phase Diagrams

2D SSH arrays exhibit a variety of topological phase transitions:

• Dimerization-driven transitions between trivial and topological phases ($\gamma'/\gamma=1$) are marked by closure of bandgaps at high-symmetry points and parity inversion (Obana et al., 2019, Geng et al., 2022).
• The presence of Dirac points (robust in the presence of chiral symmetry) gives rise to semimetallic phases; their position and merger in the Brillouin zone are tunable by model parameters. Merging of Dirac points (of opposite topological charge) leads to transitions into weak topological insulators or to nodal-line metallic phases, with the winding numbers distinguishing the resulting edge-mode structure. For type-II SSH models with glide symmetry, Dirac points can be strictly pinned (Li, 2022).
• Addition of SNN or NNN hoppings, broken symmetries (chiral, TR, PH), or interactions can alter the location and nature of phase boundaries. Quantized or fractional Zak phases, Chern numbers (in Chern insulator variants), or parity-based invariants are used to classify and track such transitions (Xu et al., 2020, Agrawal et al., 2022, Du et al., 2021, Tyagi et al., 20 Jun 2025).
• The 2D SSH array in a perpendicular magnetic field displays transitions from a gapped phase to a flat-band regime with emergence of edge-localized and Landau level states, realizing unconventional quantum Hall phases, including odd-integer quantized transmission due to competition among intra-, inter-cell, and diagonal hopping (Gupta et al., 25 Sep 2024).

6. Physical Realizations and Applications

Physical realization of 2D SSH arrays spans multiple experimental platforms:

• Acoustic and photonic crystals: Arrays of air channels, dielectric pillars, or plasmonic nanoparticles can be configured to map precisely onto SSH Hamiltonians, with coupling strengths tunable via geometry. Topological edge, corner, and higher-order modes can be directly excited and measured via pressure, electromagnetic field, or LDOS measurements. The observed edge and corner states are robust even with open boundaries and weak disorder, and higher-order phases—essential for topological photonic device design—are accessible (Zheng et al., 2019, Buendía et al., 9 Oct 2025, Yu et al., 2022).
• Electronic systems: A rectangular Si lattice grown on Ag(001) manifests gapped Dirac cones and edge modes in direct ARPES and DFT measurements, evidencing SSH-type anisotropic polarization and edge state protection not present in hexagonal lattices (Geng et al., 2022).
• Circuit QED/topolectric circuits: RLC networks serve as direct analogs of SSH arrays, with resonance conditions mapping onto topological boundary conditions. Edge resonance modes and anomalous Hall-type response can be probed by tuning the loss and impedance across the lattice (Tyagi et al., 20 Jun 2025).
• Cold atoms and nanomechanics: Synthetic SSH arrays can be constructed in cold-atom optical lattices and nanomechanical metamaterials, by stacking 1D elements and modulating hopping or applying artificial gauge fields (Agrawal et al., 2022).
• Quantum computation and information: Robust Majorana zero modes and corner states, accessible via modulation or engineered domain walls, are proposed for topological quantum information processing and topological shielding in decoherence-prone environments (Kar, 2023, Mandal et al., 21 Jun 2025).

7. Outlook: Higher-Order Topology, Disorder, and Further Generalizations

The hierarchical (or recursive) construction of 2D SSH arrays enables analytic insight into the full $n$-dimensional model class, linking topology, symmetry, and higher-order phenomena such as hinge, corner, and "fractional" interface states protected by partial subsymmetries even when the full chiral symmetry is broken (Liu, 2023). Ongoing research explores:

• The role of bond and site disorder, domain wall engineering, and quasiperiodic potentials in localizing and protecting states;
• Non-Hermitian extensions (gain/loss) with exotic singularities (exceptional points) and generalized bulk-boundary correspondence (Tyagi et al., 20 Jun 2025);
• Tunable protocols for transitions between topological, Chern, and weak/strong insulator phases via asymmetric hoppings, phases, and interaction strengths (Agrawal et al., 2022, Du et al., 2021).
• Experimental device design in photonic/plasmonic platforms aiming for on-chip robust signal routing, sensors, and compact topological lasers (Yu et al., 2022, Buendía et al., 9 Oct 2025).

The 2D SSH array thus stands as a central example in the paper of higher-dimensional topology and boundary phenomena, with direct relevance and adaptability in both fundamental and applied quantum, photonic, and electronic systems.