Topological transitions controlled by the interaction range

TL;DR

The paper analyzes a one-dimensional SSH-like lattice with long-range, exponentially decaying couplings that preserve chiral symmetry, introducing a range parameter $\lambda$ in addition to the coupling strength $J$. By deriving a closed-form off-diagonal Bloch term $h(k)$, the authors map topology to the winding number $w$ and derive analytic phase boundaries in the $(J,\lambda)$ plane, revealing phases with $w\in\{-1,0,1\}$ and multiple gap-closing scenarios, including at intermediate momenta. A key finding is that increasing the interaction range (decreasing $\lambda$) can induce topological transitions even for arbitrarily small $J$, with a threshold $\lambda_{cr} \approx -\frac{2J}{J_1+J_2}$ arising from constructive contributions at $k=0$. Numerical edge-state analysis via open boundaries confirms zero-energy edge modes in the topological sectors, while IPR scaling shows true localization only in the topological phase. These results establish the interaction range as a bona fide control parameter for topological phases, with potential relevance to trapped ions, dipolar lattices, and extended photonic systems.

Abstract

We study a one-dimensional topological model featuring a Su-Schrieffer-Heeger type pattern of nearest-neighbor couplings in combination with the longer-range interactions exponentially decaying with the distance. We demonstrate that even relatively weak long-range couplings can trigger the topological transition if their range is large enough. This provides an additional facet in the control of topological phases.

Figures (4)

• Figure 1: Schematic illustration of the Su–Schrieffer–Heeger model (SSH) extended by the long-range couplings. The coupling amplitudes decay exponentially with the distance as $J_{(n)} = Je^{-n\lambda}$, where $\lambda$ controls the interaction range and $n$ is the difference between the unit cells coordinates. Red and blue sites depict the two sublattices.
• Figure 2: (a) Winding number phase diagrams in the $(J, \lambda)$ parameter space for the different dimerizations of the lattice: (left) $J_1=1$, $J_2=3$; (right) $J_1 = 3$, $J_2 = 1$. $J_1$ link is inside the unit cell. Distinct topological phases characterized by the different winding numbers $w \in \{ -1, 0, 1 \}$ are color-coded. The phase boundaries corresponding to band gap closing are marked by red, blue and black curves. (b) Band diagrams calculated for the parameters indicated by the points $\alpha,\beta,\gamma$ and $\delta$ in panel (a). Dashed lines showing the respective dispersion for the canonical SSH model are provided for reference.
• Figure 3: Evolution of the winding curve as a function of $\lambda$. The black dashed circle corresponds to the case without long-range couplings, which is the canonical SSH model. The colored curves show the trajectories of $h(k)$ for $J_1=1, J_2=3,J=-5$ and different values of $\lambda$ which are color-coded. The blue star marks the origin $h(k)=0$.
• Figure 4: (a) The map of inverse participation ratio (IPR) for the zero-energy mode in an extended SSH model with the nearest-neighbor couplings $J_1 = 1$, $J_2 = 3$ and system size $N = 100$. The red, blue and black curves indicate bandgap closing points, the silver line illustrates the phase transition driven by the interaction length. (b) and (c) are the spatial profile of the zero energy mode with energy $\varepsilon\approx0$ and IPR dependence on the array length $N$ for the parameters shown by the red star in panel a: $\lambda = 1.5, J = -5$. (d,e) present the same data for another set of parameters, gold star in panel a, $\lambda = 0.5, J = -5$, $\varepsilon \approx 1.23$ which is the closest to the zero energy .