Topological transitions controlled by the interaction range

This paper demonstrates that in a one-dimensional Su-Schrieffer-Heeger model, even relatively weak long-range interactions with a sufficiently large decay range can trigger topological phase transitions, offering a new mechanism for controlling topological phases.

The Gist

Imagine you are walking down a long, narrow hallway lined with doors. This hallway represents a topological material—a special kind of substance where energy (like light or electrons) can travel along the edges without getting stuck or bouncing back, much like a ghost that can't be stopped by walls.

For decades, scientists studied these hallways assuming that people could only open the door right next to them. This is the "nearest-neighbor" rule. If you are in room 1, you can only talk to room 2. This simple rule creates a predictable pattern: sometimes the hallway is "open" (topological), and sometimes it's "closed" (trivial).

The New Discovery: The "Long-Reach" Whisper

This paper introduces a twist to that story. The researchers, Vlad Simonyan and Maxim Gorlach, asked: What if people could whisper to rooms far down the hall, not just the one next door?

They imagined a hallway where, in addition to talking to your neighbor, you can also whisper to someone 10, 20, or even 100 doors away. However, the further away the person is, the quieter your whisper becomes. This is called an exponentially decaying interaction.

The Big Surprise: It's Not About Loudness, It's About Reach

Here is the counter-intuitive magic of their discovery:

Old Thinking: To change the hallway from "closed" to "open" (triggering a topological transition), you needed to shout very loudly to the distant rooms. The whisper had to be almost as loud as the conversation with your immediate neighbor.

New Discovery: You don't need to shout at all! Even if your whispers to distant rooms are extremely quiet (very weak), you can still flip the switch and change the nature of the hallway—as long as you can reach far enough.

The Analogy: The "Crowd Effect"

Think of it like this:
Imagine you are trying to tip over a heavy table.

• The Old Way: You need one giant person (a strong interaction) to push the table hard enough to move it.
• The New Way: You don't need a giant. You just need a lot of tiny ants. If each ant pushes with a tiny, almost invisible force, but there are thousands of them reaching out from far away, their combined effort can still tip the table over.

In the paper's model, even though each "long-range whisper" is weak, the sheer number of them (because the reach is so long) adds up to a massive, collective force. This collective force is enough to close the "gap" in the energy levels and flip the system into a new topological state.

The "Range" Knob

The researchers found that you can control this system with a special "Range Knob" (called $\lambda$ in the paper).

• If you turn the knob to make the range short, you need a very strong whisper to change anything.
• If you turn the knob to make the range long, even a tiny, barely audible whisper is enough to trigger a massive change in the system.

Why Does This Matter?

This is like discovering a new way to control traffic in a city. Previously, to change traffic flow, you had to build massive, expensive highways (strong interactions). Now, scientists realize that a vast network of tiny, quiet side streets (weak but long-range interactions) can actually redirect the entire flow of traffic just as effectively.

In the real world, this applies to:

• Trapped Ions: Tiny atoms held in place by lasers that can "feel" each other over long distances.
• Photonic Circuits: Light traveling through glass fibers that can interact with fibers far away.
• New Materials: Designing materials that are robust against defects, where the "protection" comes not just from local connections, but from a global web of weak connections.

The Bottom Line

The paper teaches us that in the quantum world, distance matters more than strength. A weak connection that stretches far can be more powerful than a strong connection that only reaches a short distance. By tuning how far these interactions reach, we can unlock new, protected states of matter that were previously hidden from us.

Technical Summary

1. Problem Statement

Topological systems, such as the Su-Schrieffer-Heeger (SSH) model, are typically analyzed using tight-binding Hamiltonians that consider only nearest-neighbor interactions. While it is known that strong long-range interactions can alter topological phase diagrams, the specific role of the interaction range (decay length) when the interaction strength is weak remains underexplored. Most extended models either truncate interactions to a finite set of neighbors or focus on strong coupling regimes. The authors aim to fill this gap by investigating whether weak but long-range couplings can induce topological phase transitions, treating the interaction range as a primary control parameter alongside coupling strength.

2. Methodology

The authors propose and analyze an extended one-dimensional (1D) topological model with the following characteristics:

• Model Structure: An extended SSH model featuring alternating nearest-neighbor couplings ($J_1$ and $J_2$) within and between unit cells.
• Long-Range Couplings: The model includes additional couplings connecting sites on different sublattices ($A_n$ to $B_m$ where $m \neq n$). These couplings decay exponentially with distance:
  $$J(s) = J e^{-\lambda s}$$
  where $J$ is the overall strength, $s = |n-m|$ is the distance, and $\lambda$ is the inverse characteristic interaction length (controlling the range).
• Symmetry Preservation: The specific choice of connecting different sublattices preserves chiral symmetry ($\{ \Gamma, \hat{H}(k) \} = 0$), ensuring a symmetric energy spectrum and the existence of zero-energy edge modes.
• Analytical Approach:
  • The Hamiltonian is transformed into Bloch representation $H(k)$.
  • Due to the exponential decay, the infinite sums of couplings are evaluated analytically to derive a closed-form expression for the off-diagonal element $h(k)$.
  • The winding number ($w$) is calculated as the topological invariant:
    $$w = \frac{1}{2\pi i} \int_{-\pi}^{\pi} dk \frac{d}{dk} \ln h(k)$$
  • Phase boundaries are determined analytically by finding conditions where $h(k)$ touches the origin in the complex plane (band gap closing).
• Numerical Verification: The authors perform numerical diagonalization of the Hamiltonian for finite systems with open boundary conditions. They calculate the Inverse Participation Ratio (IPR) to distinguish between localized edge states (topological phase) and extended bulk states (trivial phase).

3. Key Contributions

• Interaction Range as a Control Parameter: The paper establishes that the interaction range ($\lambda^{-1}$) is a critical tuning parameter. Even if the long-range coupling strength $J$ is very weak (or vanishingly small), a sufficiently large interaction range can trigger a topological phase transition.
• Analytical Phase Boundaries: The authors derive exact analytical expressions for the phase transition curves in the $(J, \lambda)$ parameter space, identifying three distinct boundary conditions:
  1. $J = (J_1 - J_2)e^\lambda / \tanh(\lambda/2)$
  2. $J = -(J_1 + J_2)e^\lambda \tanh(\lambda/2)$
  3. $J = J_2 e^\lambda$
• Discovery of a New Topological Sector: The model reveals a phase with winding number $w = -1$. While $w=1$ and $w=-1$ are topologically equivalent regarding the Zak phase ($\pi$) and edge state existence, the transition between them involves a gap closing at intermediate momentum, a feature absent in the canonical SSH model.
• Mechanism of Transition: The authors explain that many weak long-range links contribute coherently to the $k=0$ Bloch state. Their cumulative effect creates an effective coupling of order $2J/\lambda$. When this compensates the nearest-neighbor sum ($J_1 + J_2$), the band gap closes, inducing the transition.

4. Key Results

• Phase Diagrams: The phase diagrams in the $(J, \lambda)$ space show regions with winding numbers $w \in \{0, 1, -1\}$.
  • Canonical Limit: As $\lambda \to \infty$ (short-range limit), the model recovers the standard SSH behavior.
  • Long-Range Regime: Decreasing $\lambda$ (increasing range) deforms the winding curve $h(k)$ in the complex plane. If the curve crosses the origin, a topological transition occurs.
• Gap Closing Locations: Unlike the canonical SSH model where gaps close only at the center ($k=0$) or edges ($k=\pi$) of the Brillouin zone, the extended model allows gap closing at intermediate momenta (e.g., $k \approx \pi/2$). This leads to transitions between $w=1$ and $w=-1$ or between $w=0$ and $w=\pm 1$.
• Edge State Localization:
  • Numerical simulations confirm that in the topological phases ($w \neq 0$), zero-energy edge states are exponentially localized at the boundaries.
  • In the trivial phase, states that appear localized for finite systems are shown to be extended (IPR scales as $1/N$) as system size increases.
  • The spatial structure of the edge states remains exponentially localized, with a localization length determined by the renormalized bulk gap.

5. Significance and Implications

• New Control Mechanism: The work demonstrates that topological phases can be controlled not just by tuning coupling strengths, but by engineering the spatial decay length of interactions. This offers a new degree of freedom for designing topological materials.
• Experimental Relevance: The findings are directly applicable to various physical platforms where long-range interactions are inherent or tunable, including:
  • Arrays of trapped ions.
  • Dipolar scatterers and meta-atoms.
  • Optical waveguide lattices coupled via evanescent fields.
  • Photonic molecules.
• Theoretical Insight: It challenges the conventional truncation of hopping terms in topological models, showing that the "tail" of long-range interactions, even if weak, plays a decisive role in the global topological properties of the system.

In conclusion, Simonyan and Gorlach provide a rigorous theoretical framework showing that weak, long-range interactions can drive topological phase transitions, fundamentally expanding the understanding of how interaction range influences topological order in 1D systems.