Quantum Tunneling-induced Hybridization and Coherent Dynamics of Jackiw-Rebbi Zero Modes in a Modified Su-Schrieffer-Heeger Chain

Technical Summary: Quantum Tunneling-induced Hybridization and Coherent Dynamics of Jackiw-Rebbi Zero Modes in a Modified Su-Schrieffer-Heeger Chain

Problem Statement
The paper investigates the behavior of Jackiw-Rebbi (JR) zero modes in a modified Su-Schrieffer-Heeger (SSH) chain. While the conventional SSH model (with a two-sublattice basis) supports a single topological domain wall (DW) and a corresponding zero-energy mode at a mass kink, the modified model considered here features a four-sublattice unit cell. This modification introduces two distinct bulk gap-closing points: a quadratic-type point at $k=0$ and Dirac-type points at $k=\pm \pi/4a$. Previous numerical studies (e.g., Ref. [29]) observed that for specific parameters, zero-energy states (ZESs) did not appear at the domain wall but remained at the chain edges, yet the physical origin of this absence was unexplained. Furthermore, the interplay between quantum tunneling and the hybridization of JR modes in this specific modified regime, and the resulting coherent dynamics, had not been explored.

Methodology
The authors employ a dual approach combining analytical continuum theory and exact numerical diagonalization:

1. Analytical Continuum Theory: The Bloch Hamiltonian is expanded around the critical momentum points ($k=0$ and $k=\pm \pi/4a$) to derive effective low-energy Dirac Hamiltonians.
   • Around $k=0$, the effective mass is derived as $M(x) = -\Delta^2(x)$.
   • Around $k=\pm \pi/4a$, the effective mass is derived as $M(x) = 2t^2 - \Delta^2(x)$, where $\Delta(x)$ follows a kink profile.
   • The authors analyze the sign inversion of these mass terms to determine the existence and location of topological interfaces.
   • An effective two-state Hamiltonian is constructed by projecting the Dirac Hamiltonian onto the subspace of the two localized zero modes, treating the system as a quantum mechanical double-well.
2. Numerical Diagonalization: The real-space Hamiltonian for the modified SSH chain (with open boundary conditions) is diagonalized to obtain energy spectra and probability densities. This validates the analytical predictions regarding the existence of ZESs and their hybridization.
3. Real-Time Evolution: The coherent dynamics are studied by evolving an initial state localized at one interface and calculating the time-dependent probability amplitudes, expectation values of position, and sublattice polarization density.

Key Contributions and Results

• Origin of Zero Modes and Mass Inversion:

  • The study analytically demonstrates that expanding the Hamiltonian around the quadratic point ($k=0$) yields an effective mass that does not change sign under a kink profile. Consequently, this regime cannot support a JR zero mode, explaining why previous numerical studies found no ZESs at the domain wall for these parameters.
  • Conversely, expanding around the Dirac points ($k=\pm \pi/4a$) yields an effective mass $M(x) = 2t^2 - \Delta^2(x)$. When the domain wall amplitude $\Delta_0$ exceeds the critical value $\sqrt{2}t$, this mass profile undergoes two successive sign inversions near the chain center. This creates two spatially separated interfaces (a "double interface" or kink-antikink pair), each supporting a localized JR zero mode.

• Tunneling-Induced Hybridization:

  • When the two interfaces are infinitely separated, the two JR zero modes are degenerate at zero energy.
  • For finite separation, the exponentially decaying tails of the wavefunctions overlap, leading to quantum tunneling. This lifts the zero-energy degeneracy, splitting the modes into symmetric ($\Psi_+$) and antisymmetric ($\Psi_-$) hybridized states with energies $E = \pm |T|$, where $T$ is the tunneling matrix element.
  • The energy splitting $\Delta E$ scales exponentially with the interface separation $D_s$ and the domain wall width $\xi$, characteristic of quantum tunneling in double-well potentials.

• Coherent Dynamics and "Jackiw-Rebbi Molecule":

  • The system exhibits coherent oscillations of occupation probability between the left and right interfaces, analogous to Rabi oscillations in a two-level system.
  • The oscillation period is governed by the hybridization gap ($\Delta E$).
  • Crucially, the dynamics involve a periodic transfer of sublattice polarization. The left interface mode is polarized on the (A, C) sublattices, while the right interface mode is polarized on the (B, D) sublattices. As the particle tunnels, the sublattice character oscillates between these two sectors.
  • The authors propose viewing this pair of tunneling-coupled JR modes as a "Jackiw-Rebbi molecule," a topological analogue of the bonding and antibonding orbitals in the hydrogen molecular ion ($H_2^+$).

• Numerical Verification:

  • Numerical results confirm the analytical findings: for $\Delta_0 > \sqrt{2}t$, near-zero-energy states appear with finite splitting, and their probability densities are localized at the two mass-inversion interfaces.
  • The numerical plots explicitly show the sublattice polarization (odd sites for the left interface, even sites for the right), confirming the distinct sector support predicted by the continuum theory.

Significance and Claims
The paper claims to establish a unified framework connecting topological defect states, quantum tunneling, and coherent dynamics within modified SSH systems. Its primary significance lies in:

1. Theoretical Clarification: Providing a clear physical explanation for the absence of ZESs at the quadratic point and their emergence at the Dirac points in modified SSH chains, resolving ambiguities from prior numerical work.
2. New Paradigm: Demonstrating that a single domain wall in a modified lattice can generate two interfaces and a pair of coupled zero modes, effectively creating a tunable "Jackiw-Rebbi molecule."
3. Controllability: Showing that the hybridization gap and tunneling dynamics can be tuned via domain wall parameters (amplitude $\Delta_0$, separation $D_s$, and width $\xi$).
4. Experimental Relevance: The authors suggest that these phenomena are accessible in synthetic topological platforms such as photonic waveguide arrays, cold-atom lattices, and engineered Dirac systems, where spatially varying hopping amplitudes can be precisely controlled. The results offer a route for controllable topological quantum-state transfer and the study of coherent dynamics in engineered lattice structures.