Superconducting Su-Schrieffer-Heeger Model

• The superconducting SSH model is defined by extending bond-modulated electron hopping to include either explicit BdG pairing or emergent SC-like behavior from electron–phonon coupling.
• It encompasses diverse regimes including topological superconductivity with Majorana and odd-frequency pairing, Luther–Emery liquids, and d-wave/s-wave instabilities.
• Research shows that lattice geometry, interaction strength, and bond order critically influence pairing symmetry and superconducting scales in these SSH-inspired systems.

The superconducting Su–Schrieffer–Heeger model denotes a family of theories built around the SSH motif of bond-modulated hopping, but extended in two distinct directions. In one direction, the SSH chain is promoted to a Bogoliubov–de Gennes system by adding explicit superconducting pairing, yielding dimerized topological superconductors with Majorana, soliton, and odd-frequency phenomena. In the other, SSH electron–phonon coupling is retained as the microscopic interaction, and superconductivity or superconducting-like behavior emerges from the competition between bond phonons, electron correlations, and lattice geometry in one and two dimensions (Tamura et al., 2020, Piccioni et al., 2024, Xing et al., 2023).

1. Model classes and canonical Hamiltonians

A central SSH construction couples fermions to bond distortions rather than to onsite density. In the one-dimensional Hubbard–SSH model with optical phonons, the hopping on bond $i,i+1$ is modulated by the displacement difference,

$$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$

with dimensionless coupling

$$\lambda = \frac{t\alpha^2}{m\omega^2} = \frac{t\tilde{\alpha}^2}{\hbar \omega}.$$

This bond-centered structure is the defining distinction from Holstein-type density coupling and underlies the preference for Peierls or bond-order instabilities, as well as several superconducting regimes after doping (Piccioni et al., 2024).

On the square lattice, the attractive SSHH model uses bond phonons $\hat X_{ij}$ that modulate nearest-neighbor hopping as $t(1-\lambda \hat X_{ij})$, together with an onsite Hubbard term $U<0$. The same bond-phonon logic appears in the optical SSH model on square and triangular lattices, where phonons live on sites or bonds but always enter through differences of neighboring displacements in the kinetic term (Xing et al., 2023, Cai et al., 2023, Casebolt et al., 5 Apr 2026).

A second major class is the explicitly superconducting SSH chain. In that setting, the normal SSH Bloch Hamiltonian $H_0(k)$ is embedded into a BdG matrix,

$$\mathcal{H}(k)= \begin{pmatrix} H_0(k) & \Delta(k)\ \Delta^\dagger(k) & -H_0^*(-k) \end{pmatrix},$$

with staggered hopping $t_1,t_2$, possibly sublattice-dependent chemical potentials $\mu_A,\mu_B$, and inter-sublattice pairing amplitudes that inherit the dimerized structure (Tamura et al., 2020). Closely related dimerized p-wave chains also arise as the $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$0 commensurate limit of a generalized Aubry–André–Harper model with p-wave pairing, where the superconducting SSH limit has dimerized hoppings $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$1 and equal bond pairing on intracell and intercell links (Zeng et al., 2016).

Model class	Defining ingredient	Superconducting manifestation
Electron–phonon SSH	Bond-modulated hopping from dynamical phonons	Emergent s-wave, d-wave, or Luther–Emery behavior
BdG SSH chain	Explicit pairing added to dimerized SSH band structure	Topological superconductivity, Majorana and odd-frequency pairing
Interacting extended SSH	Density interactions and longer-range hoppings in number-conserving SSH variants	“SC-like” algebraic pair correlations without anomalous averages

The phrase “superconducting SSH model” therefore does not identify a single Hamiltonian. It identifies a structural principle—SSH bond modulation—realized either as a microscopic pairing mechanism or as part of an already superconducting topological band theory.

2. One-dimensional superconducting and superconducting-like regimes

In the one-dimensional repulsive Hubbard–SSH model, half-filling supports either a translationally invariant Mott insulator, with charge gap and gapless spin sector, or a spontaneously dimerized Peierls insulator, with fully gapped charge and spin excitations. The Peierls state is diagnosed by a nonzero dimerization parameter $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$2 and by the bond-order parameter

$$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$3

At $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$4, the Peierls phase exhibits a large $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$5 peak, with $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$6 for $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$7, whereas the Mott phase is characterized by $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$8 and $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$9 at small $$\lambda = \frac{t\alpha^2}{m\omega^2} = \frac{t\tilde{\alpha}^2}{\hbar \omega}.$$0 (Piccioni et al., 2024).

Doping qualitatively separates the two parent insulators. Doping the Mott phase immediately yields a conventional Luttinger liquid with gapless charge and spin sectors. Doping the Peierls phase instead produces a metallic state with gapless charge but a finite spin gap at small $$\lambda = \frac{t\alpha^2}{m\omega^2} = \frac{t\tilde{\alpha}^2}{\hbar \omega}.$$1, i.e. a Luther–Emery liquid. In the notation of that study, the natural bond-centered singlet pair field is

$$\lambda = \frac{t\alpha^2}{m\omega^2} = \frac{t\tilde{\alpha}^2}{\hbar \omega}.$$2

and the long-distance asymptotics obey

$$\lambda = \frac{t\alpha^2}{m\omega^2} = \frac{t\tilde{\alpha}^2}{\hbar \omega}.$$3

The spin-gapped metal is explicitly identified as the one-dimensional counterpart of a superconductor, but the same work also reports that bond–bond correlations are the most enhanced in the Luther–Emery regime, implying that bond-order or CDW tendencies remain dominant in the parameter sets studied, consistent with $$\lambda = \frac{t\alpha^2}{m\omega^2} = \frac{t\tilde{\alpha}^2}{\hbar \omega}.$$4 (Piccioni et al., 2024).

A distinct one-dimensional route appears in the interacting extended SSH model with spinless fermions, extended-range hoppings $$\lambda = \frac{t\alpha^2}{m\omega^2} = \frac{t\tilde{\alpha}^2}{\hbar \omega}.$$5, and density interactions $$\lambda = \frac{t\alpha^2}{m\omega^2} = \frac{t\tilde{\alpha}^2}{\hbar \omega}.$$6. That model preserves global $$\lambda = \frac{t\alpha^2}{m\omega^2} = \frac{t\tilde{\alpha}^2}{\hbar \omega}.$$7 particle number and contains no explicit pairing term, yet it supports two superconducting-like phases. The corresponding number-conserving bond operators are

$$\lambda = \frac{t\alpha^2}{m\omega^2} = \frac{t\tilde{\alpha}^2}{\hbar \omega}.$$8

with algebraically decaying correlators $$\lambda = \frac{t\alpha^2}{m\omega^2} = \frac{t\tilde{\alpha}^2}{\hbar \omega}.$$9 and $\hat X_{ij}$0. These phases arise from the interplay of extended hoppings and imbalanced attractive interactions $\hat X_{ij}$1, and the paper emphasizes that they are “SC-like” precisely because $\hat X_{ij}$2 remains intact and no anomalous expectation value is introduced (Hui et al., 8 Jun 2026).

A third one-dimensional construction is the interacting staggered SSH ladder. In the noninteracting limit it is exactly equivalent to two decoupled Kitaev–Majorana chains, or equivalently two one-dimensional p-wave superconductors. Near the Gaussian critical point, interactions bosonize to a double-frequency sine-Gordon model, and the phase diagram includes a Tomonaga–Luttinger liquid, spontaneous dimerization, CDW order, and a mixed phase. Topological distinction between thermodynamically equivalent phases requires nonlocal parity and string order parameters, and the elementary bulk solitons can carry $\hat X_{ij}$3 or continuously varying charges in the mixed phase (Nersesyan, 2020).

3. BdG SSH chains, topological phases, and odd-frequency pairing

In explicitly superconducting SSH chains, dimerization and pairing coexist at the single-particle Hamiltonian level. One representative realization uses staggered nearest-neighbor hoppings $\hat X_{ij}$4, sublattice-dependent chemical potentials $\hat X_{ij}$5, and inter-sublattice pairing amplitudes whose momentum dependence contains both even- and odd-parity components. In that framework, the sublattice index acts as an additional Cooper-pair quantum number, and the anomalous Green’s function acquires intra-sublattice and inter-sublattice odd-frequency components in the bulk. Intra-sublattice odd-frequency pairing requires staggering and $\hat X_{ij}$6, whereas inter-sublattice odd-frequency pairing requires $\hat X_{ij}$7 and $\hat X_{ij}$8 (Tamura et al., 2020).

That same model behaves as a two-band superconductor, and the odd-frequency amplitudes are strongly enhanced at topological phase transitions where the bulk gap closes. The analysis further correlates bulk odd-frequency pairing with higher-energy pseudogaps in the density of states and with a CDW generated by chemical-potential imbalance between sublattices. This establishes the superconducting SSH chain as a bulk odd-frequency system, rather than one in which odd-frequency amplitudes occur only at interfaces (Tamura et al., 2020).

The generalized Aubry–André–Harper realization provides a complementary classification. For the commensurate case $\hat X_{ij}$9, the model becomes a superconducting dimerized chain with intracell and intercell hoppings

$t(1-\lambda \hat X_{ij})$0

together with p-wave pairing $t(1-\lambda \hat X_{ij})$1. With $t(1-\lambda \hat X_{ij})$2, the topological transition between SSH-like and Kitaev-like regimes occurs at

$t(1-\lambda \hat X_{ij})$3

The resulting phases are SSH-like trivial, SSH-like topological, and Kitaev-like topological; in the SSH-like topological phase there are two Majorana zero modes per edge that pair into a Dirac end state, while in the Kitaev-like phase there is one Majorana zero mode per edge (Zeng et al., 2016).

An extended spinless superconducting SSH model with staggered hopping, p-wave pairing, and uniform potential $t(1-\lambda \hat X_{ij})$4 sharpens the operational role of these topological defects. Its phase boundaries are given by $t(1-\lambda \hat X_{ij})$5 and $t(1-\lambda \hat X_{ij})$6, separating an SSH-like topological phase from a Kitaev-like topological phase and a trivial phase. In a heterostructure containing both regions, a mobilizable SSH soliton can be adiabatically moved through Majorana modes to implement a NOT operation on a Majorana qubit. The same study shows that superconducting proximity breaks the strict $t(1-\lambda \hat X_{ij})$7 quantization of the soliton’s electric charge, allowing “decimal” values such as $t(1-\lambda \hat X_{ij})$8, even though the BdG counting charge remains $t(1-\lambda \hat X_{ij})$9 (Xiong et al., 2014).

4. Square-lattice SSH electron–phonon models and pairing competition

On the square lattice with attractive $U<0$0, the SSHH model exhibits a competition among charge order, bond order, and onsite s-wave pairing. At half-filling and small SSH coupling $U<0$1, the ground state reduces to attractive-Hubbard physics with degenerate CDW and s-wave pairing correlations, whereas large $U<0$2 yields a gapped $U<0$3 bond-order wave. Upon doping, the strong-coupling BOW undergoes a first-order transition to an s-wave paired state at a finite $U<0$4; larger $U<0$5 enhances the pairing correlations in the doped phase, but increasing SSH coupling suppresses both the half-filled CDW and the s-wave channel because bond-modulating electron–phonon coupling competes with onsite pairing symmetry (Xing et al., 2023).

Within singular-mode functional renormalization group, the square-lattice SSH–Hubbard model yields a different but compatible hierarchy of instabilities. At half-filling and $U<0$6, smaller $U<0$7 and larger $U<0$8 produce a degenerate SDW/CDW/sSC state, whereas larger $U<0$9 and smaller $H_0(k)$0 favor a valence-bond solid. At finite doping $H_0(k)$1, SSH phonons favor s-wave superconductivity when $H_0(k)$2; increasing positive $H_0(k)$3 generates d-wave superconductivity and then incommensurate SDW order, with a narrow incommensurate VBS window at moderate $H_0(k)$4 and $H_0(k)$5. In this formulation, SSH phonons enhance both charge and spin fluctuations because the bond vertex carries nontrivial form factors, and the sSC and dSC instabilities are interpreted as being driven by CDW and SDW fluctuations, respectively (Yang et al., 2022).

Numerically exact DQMC comparisons between Holstein and SSH models further clarify the role of bond coupling. In the two-dimensional optical-SSH model, s-wave pairing correlations remain robust to relatively large $H_0(k)$6 and to densities near half-filling because SSH coupling supports light bipolarons and avoids the heavy local self-trapping characteristic of Holstein coupling. At $H_0(k)$7 and $H_0(k)$8, Holstein pairing grows up to $H_0(k)$9 and then collapses as the effective mass rises sharply, while optical-SSH pairing saturates and the mass enhancement remains mild. The same study also finds that a weak onsite Hubbard repulsion $$\mathcal{H}(k)= \begin{pmatrix} H_0(k) & \Delta(k)\ \Delta^\dagger(k) & -H_0^*(-k) \end{pmatrix},$$0 suppresses both uniform and extended s-wave susceptibilities in the SSH models (Ly et al., 2023).

A separate DQMC study on the square-lattice SSH model reports markedly larger superconducting scales in a different regime. At hole doping $$\mathcal{H}(k)= \begin{pmatrix} H_0(k) & \Delta(k)\ \Delta^\dagger(k) & -H_0^*(-k) \end{pmatrix},$$1, the anti-adiabatic limit generates an effective pair-hopping amplitude $$\mathcal{H}(k)= \begin{pmatrix} H_0(k) & \Delta(k)\ \Delta^\dagger(k) & -H_0^*(-k) \end{pmatrix},$$2, and the superconducting $$\mathcal{H}(k)= \begin{pmatrix} H_0(k) & \Delta(k)\ \Delta^\dagger(k) & -H_0^*(-k) \end{pmatrix},$$3 grows roughly linearly with $$\mathcal{H}(k)= \begin{pmatrix} H_0(k) & \Delta(k)\ \Delta^\dagger(k) & -H_0^*(-k) \end{pmatrix},$$4. At finite $$\mathcal{H}(k)= \begin{pmatrix} H_0(k) & \Delta(k)\ \Delta^\dagger(k) & -H_0^*(-k) \end{pmatrix},$$5, the SSH model displays a superconducting dome with $$\mathcal{H}(k)= \begin{pmatrix} H_0(k) & \Delta(k)\ \Delta^\dagger(k) & -H_0^*(-k) \end{pmatrix},$$6, peaking near the AFM–VBS quantum critical point at $$\mathcal{H}(k)= \begin{pmatrix} H_0(k) & \Delta(k)\ \Delta^\dagger(k) & -H_0^*(-k) \end{pmatrix},$$7, whereas the corresponding Holstein model has $$\mathcal{H}(k)= \begin{pmatrix} H_0(k) & \Delta(k)\ \Delta^\dagger(k) & -H_0^*(-k) \end{pmatrix},$$8 under the same conditions (Cai et al., 2023).

Taken together, these square-lattice results show that the superconducting SSH model is strongly regime-dependent. In weak-coupling dilute settings, the gains relative to Holstein can be small; near half-filling and in stronger-coupling or anti-adiabatic regimes, bond-modulated hopping can sustain large pairing correlations and, in some formulations, substantially larger $$\mathcal{H}(k)= \begin{pmatrix} H_0(k) & \Delta(k)\ \Delta^\dagger(k) & -H_0^*(-k) \end{pmatrix},$$9 than site-coupled phonons (Ly et al., 2023, Cai et al., 2023).

5. Repulsive SSHH models, d-wave superconductivity, and lattice-geometry dependence

With repulsive $t_1,t_2$0, the square-lattice SSHH model can favor d-wave rather than s-wave pairing. DMRG on four-leg cylinders at doping $t_1,t_2$1 and phonon frequency $t_1,t_2$2 finds that the anti-adiabatic limit generates an effective interaction

$t_1,t_2$3

with $t_1,t_2$4. Large onsite $t_1,t_2$5 suppresses onsite s-wave pairing but leaves the SSH-induced nearest-neighbor antiferromagnetic exchange effective, and this promotes bond-centered d-wave pairing (Wang et al., 2022).

Numerically, for $t_1,t_2$6, d-wave superconductivity appears when $t_1,t_2$7 with $t_1,t_2$8. In the d-wave phase at $t_1,t_2$9, bond pair correlations decay as power laws with $\mu_A,\mu_B$0 on cylinders and $\mu_A,\mu_B$1 on strips; the charge sector is gapless with $\mu_A,\mu_B$2, while the spin sector is gapped, with exponential decay length $\mu_A,\mu_B$3. For smaller $\mu_A,\mu_B$4, the same geometry supports a filled-stripe CDW with short-range superconducting correlations (Wang et al., 2022).

This mechanism differs sharply from the attractive-SSH case. There, onsite s-wave pairing competes directly with bond order; here, strong $\mu_A,\mu_B$5 filters out the onsite channel and converts the SSH-induced exchange into a route toward d-wave order. A plausible implication is that the leading superconducting symmetry in SSH systems is not fixed by the bond phonon alone; it is selected by the combination of correlation strength, doping, and Fermi-surface geometry.

Geometry can redirect the instability even more strongly. On the triangular lattice, DQMC for the optical SSH model identifies two distinct doping regimes. At one-quarter filling $\mu_A,\mu_B$6, where the noninteracting Fermi surface is circular, the system undergoes a metal-to-insulating BOW transition that breaks local $\mu_A,\mu_B$7 symmetry, with $\mu_A,\mu_B$8 at $\mu_A,\mu_B$9. At three-quarters filling $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$00, where the Fermi surface is hexagonal, the model exhibits a BOW phase for $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$01 and an s-wave superconducting phase for sufficiently large $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$02, robustly for $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$03, with $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$04 at $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$05 (Casebolt et al., 5 Apr 2026).

The triangular-lattice study is also notable for what it does not find: enhanced magnetic correlations are absent, contrary to what had been reported for square-lattice SSH models. In that setting, the tendency toward pairing is instead associated with the possibility of a sign change in the effective intersite hopping induced by large bond displacements (Casebolt et al., 5 Apr 2026). This emphasizes that the superconducting SSH problem is not universal across lattices; frustration, nesting, and phonon kinematics alter the dominant instability in essential ways.

6. Higher-dimensional topological extensions and conceptual distinctions

The SSH motif also generates topological superconductivity beyond one-dimensional chains. A three-dimensional model of coupled SSH chains reproducing crystalline polyacetylene develops a Weyl nodal ring when the interchain coupling $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$06 exceeds the threshold

$$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$07

with the nodal ring lying at $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$08 and satisfying

$$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$09

Adding an intra-orbital s-wave pairing term yields a BdG superconductor with drumhead-derived annular Majorana surface states and split Bogoliubov nodal rings. In that formulation, the superconducting phase belongs to class DIII when time-reversal symmetry is retained (Rosenberg et al., 2021).

Across this literature, three meanings of “superconducting” must be kept distinct. First, a Luther–Emery liquid in one dimension is the counterpart of a superconductor because it has a spin gap and gapless charge, but it still lacks true off-diagonal long-range order; its pair correlations are algebraic, and in the doped Peierls regime of the Hubbard–SSH chain bond-order correlations remain dominant in the parameter space studied (Piccioni et al., 2024). Second, the “SC-like” phases of the interacting extended SSH model are strictly number-conserving and do not imply anomalous expectation values, BdG quasiparticles, or Majorana zero modes (Hui et al., 8 Jun 2026). Third, explicit BdG SSH models genuinely break $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$10 at the mean-field level and can support Majorana edge or surface excitations, odd-frequency bulk pairing, and topological winding-number classifications (Tamura et al., 2020, Rosenberg et al., 2021).

These distinctions also frame the main controversy in the field. Some SSH studies emphasize bond order as the dominant one-dimensional instability and view superconductivity primarily as a higher-dimensional implication; others report large two-dimensional superconducting scales, including robust d-wave order with repulsive $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$11 or large s-wave $$\hat{\cal H} = -t\sum_{i,\sigma} \left[1-\tilde{\alpha}\left(\hat{X}_{i+1}-\hat{X}_{i}\right)\right] \left(\hat{c}^\dagger_{i,\sigma}\hat{c}_{i+1,\sigma} + \text{h.c.}\right) + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} + \frac{\hbar \omega}{2}\sum_i\left(\hat{P}_i^2 + \hat{X}_i^2\right),$$12 in anti-adiabatic bond-phonon regimes (Piccioni et al., 2024, Wang et al., 2022, Cai et al., 2023). The consistent element is not a universal pairing symmetry or critical scale, but a common mechanism: SSH coupling reshapes the kinetic term itself. That bond-centered structure can favor Peierls order, light bipolarons, pair hopping, antiferromagnetic exchange, topological superconductivity, or number-conserving superconducting-like phases, depending on dimensionality, filling, interaction sign, and lattice geometry.