Dark solitons in nonlinear Su-Schrieffer-Heeger lattices

This paper systematically investigates dark solitons in nonlinear Su-Schrieffer-Heeger lattices, revealing that these intensity dips on a nonzero background are robust against the linear band structure and can exhibit linear stability when intracell coupling significantly exceeds intercell coupling.

The Gist

Imagine a long, winding train track made of tiny, connected train stations. In the world of physics, this is called a lattice. Usually, these tracks are perfectly symmetrical, but in a special design called the SSH lattice (named after three scientists), the tracks are "staggered." Some connections between stations are tight and strong, while others are loose and weak. This specific pattern creates a hidden "topological" magic: it forces special waves to travel only along the very edges of the track, ignoring the middle.

Now, imagine you add a twist: the tracks themselves change shape depending on how heavy the train is. This is nonlinearity. In the past, scientists studied "bright" solitons in these systems—think of these as bright, glowing humps of energy that travel along the track without spreading out.

This paper is about discovering something different: Dark Solitons.

The Analogy: The "Silent Spot" on a Busy Highway

To understand a Dark Soliton, imagine a busy highway where cars are driving at a steady, constant speed (this is the "background").

• A Bright Soliton would be a massive, glowing truck driving down the road, adding extra traffic.
• A Dark Soliton is different. Imagine the highway is full of cars, but suddenly, there is a perfectly empty gap in the middle of the traffic. The cars on either side of the gap are still there, but the spot in the middle is "dark" or empty.

The researchers in this paper asked: Can we create and control these "empty gaps" (dark solitons) in our special, staggered train tracks, even when the tracks change shape based on the train's weight?

The Journey of Discovery

The team, led by Rujiang Li, explored two main scenarios:

1. The "Non-Trivial" Track (The Magic Edge)
In the original design, the track has a "topological" property. It's like a track that naturally wants to keep waves on the edges.

• What they found: They successfully created dark solitons in the middle of the track (bulk) and at the very ends (edges).
• The Catch: These "empty gaps" were very fragile. Like a soap bubble, they tended to pop and collapse quickly. They were unstable. No matter where you put the gap, it eventually broke apart.

2. The "Trivial" Track (The Normal Track)
They then flipped the design of the track. Now, the "magic" edge effect was gone; it was just a normal, boring track.

• What they found: Surprisingly, they found that in this "boring" track, the dark solitons could actually stay stable under certain conditions.
• The Secret Sauce: If the "tight" connections (intracell) were much stronger than the "loose" connections (intercell), the empty gaps became rock-solid. They could travel for a long time without breaking.

The Big Surprise: The Gap Doesn't Care About the Map

Here is the most fascinating part of their discovery.

Usually, in physics, if you change the shape of the track (the band structure), the waves change completely. If you move a bright wave into the wrong part of the track, it dissolves.

But these Dark Solitons were tough. The researchers found that the "empty gap" (the intensity dip) remained perfectly preserved, even if the wave was traveling through a part of the track where, in a normal world, it shouldn't exist.

• Analogy: Imagine a hole in a blanket. Usually, if you stretch the blanket, the hole stretches and distorts. But these holes were like magic cutouts; no matter how much you stretched or twisted the blanket, the hole stayed perfectly round and deep.

Why Does This Matter?

1. New Types of Data Storage: In fiber optics (the internet cables), we use light to send data. "Bright" pulses are like sending a "1". "Dark" solitons (the gaps) could be a new way to send "0s" or complex codes that are harder to disrupt.
2. Robustness: The fact that these gaps stay stable even when the system changes suggests we could build communication systems that are very resistant to noise or damage.
3. Experimental Possibility: The paper suggests we could build these systems using electric circuits (like a giant circuit board with resistors and capacitors) or lasers in glass waveguides.

The Bottom Line

The researchers found a way to create "holes" in a flow of energy that are surprisingly tough. While these holes were wobbly in the "magical" version of the track, they became super-stable in the "normal" version if the connections were tuned just right. This opens the door to new ways of manipulating light and electricity, proving that sometimes, the most interesting things in physics aren't the bright spots, but the dark ones.

Technical Summary

1. Problem Statement

While nonlinear topological lattices, specifically the Su-Schrieffer-Heeger (SSH) model, have been extensively studied for bright solitons (intensity humps on a zero background) and topological edge states, dark solitons (intensity dips on a non-zero continuous wave background) remain under-explored in one-dimensional nonlinear SSH systems.
Key challenges addressed in this work include:

• Delocalization: Unlike bright solitons which often delocalize and lose their structure when entering linear bulk bands, dark solitons are inherently delocalized. It is unclear if their characteristic intensity dips can be preserved across different spectral regions (gaps vs. bulk bands) and topological phases.
• Boundary Effects: Standard open boundary conditions in finite lattices perturb the constant background required for dark solitons, making their theoretical definition and numerical observation difficult.
• Stability: The stability of dark solitons in nonlinear topological systems is not well understood, particularly regarding how topological phases (trivial vs. non-trivial) influence their linear and dynamic stability.

2. Methodology

The authors employed a systematic theoretical and numerical approach:

• Model: They utilized a nonlinear SSH lattice model with Kerr-type onsite nonlinearity. The Hamiltonian includes staggered intracell ($J$) and intercell ($J'$) couplings.
• Anti-Continuum (AC) Limit Approach: To find exact solutions, they started with the AC limit ($J' = 0$ for non-trivial lattices, $J=0$ for trivial lattices), where unit cells are decoupled. In this limit, they identified symmetric and antisymmetric solutions for single unit cells.
• Continuation Method: Using Newton's method, they continued these AC limit solutions into the regime of finite coupling ($J' > 0$ or $J > 0$) to generate dark soliton solutions in the full lattice.
• Modified Boundary Conditions: To ensure the soliton background remained constant (a requirement for dark solitons) even at the lattice edges, the authors introduced specific modified boundary conditions:
  • Periodic-like coupling: Connecting the leftmost and rightmost sites with coupling $J'$ (or $-J'$) to maintain inversion symmetry or antisymmetry.
  • Unidirectional coupling: For edge solitons, coupling the rightmost site to a specific sublattice site in the middle unit cell to prevent boundary perturbations.
• Stability Analysis:
  • Linear Stability: They performed a linear stability analysis by introducing infinitesimal perturbations to the soliton solutions and calculating the eigenvalues ($\delta$). A positive imaginary part indicates instability.
  • Direct Numerical Simulation: They simulated the temporal evolution of the solitons using the Runge-Kutta algorithm to verify dynamic stability.

3. Key Contributions

• Systematic Classification: The paper provides the first systematic investigation of dark solitons in 1D nonlinear SSH lattices, categorizing them into dark bulk solitons (dips in the lattice interior) and dark edge solitons (dips at the boundaries).
• Spectral Versatility: The study demonstrates that dark solitons can exist in the semi-infinite gap, the middle finite gap, and even within the linear bulk bands. Crucially, unlike bright solitons, the intensity dips of dark solitons remain well-preserved regardless of whether the frequency lies in a gap or a band.
• Topological Phase Dependence: The authors mapped the existence and stability of these solitons across both topologically non-trivial ($J < J'$) and topologically trivial ($J > J'$) phases, revealing distinct behaviors for each.
• Stability Regimes: A significant finding is the identification of linearly stable dark soliton regimes. While most dark solitons are dynamically unstable, specific types become linearly stable when the intracell coupling is significantly larger than the intercell coupling (deep in the topologically trivial phase).

4. Key Results

A. Dark Bulk Solitons

• Non-Trivial Lattice ($J < J'$):
  • Solitons derived from symmetric AC solutions reside in the semi-infinite gap.
  • Solitons derived from antisymmetric solutions can reside in both the semi-infinite gap and the middle finite gap.
  • Stability: All identified dark bulk solitons in the non-trivial phase are linearly and dynamically unstable.
• Trivial Lattice ($J > J'$):
  • Solitons can exist in the semi-infinite gap, middle finite gap, and even extend into the linear bulk bands.
  • Stability: While generally unstable, specific branches of dark bulk solitons exhibit linear stability when the ratio $J/J'$ is sufficiently large (intracell coupling dominates). Numerical simulations confirm these stable solitons maintain their intensity dips for long durations ($t \ge 10^5$).

B. Dark Edge Solitons

• Non-Trivial Lattice:
  • Edge solitons exist only in the semi-infinite gap.
  • They are always dynamically unstable.
• Trivial Lattice:
  • Edge solitons can exist in both the semi-infinite gap and the middle finite gap.
  • Stability: Similar to bulk solitons, specific edge soliton branches become linearly stable in the regime where intracell coupling is much larger than intercell coupling.

C. Robustness of Intensity Dips

A central result is that the intensity dip (the core feature of the dark soliton) is robust. It does not delocalize or disappear when the soliton frequency enters the linear bulk band, a behavior that contrasts sharply with bright solitons which lose their hump structure in the bulk band.

5. Significance and Implications

• Theoretical Insight: The work bridges the gap between conventional dark soliton theory and topological physics, showing that topological band structures do not necessarily destroy the integrity of dark soliton profiles.
• Stability Control: The discovery of linearly stable dark solitons in the topologically trivial phase (specifically when $J \gg J'$) offers a new mechanism for controlling soliton stability in nonlinear lattices, potentially enabling long-lived nonlinear states.
• Experimental Realization: The authors suggest that these states can be realized in various physical platforms, including:
  • Photonic waveguide arrays (using evanescent coupling).
  • Polaritonic systems.
  • Bose-Einstein condensates.
  • Electrical circuit lattices, which are particularly suited for implementing the required long-range and negative/unidirectional couplings used in the modified boundary conditions.

In summary, this paper expands the landscape of nonlinear topological physics by proving that dark solitons are a viable and robust class of states in SSH lattices, with unique stability properties that depend on the underlying topological phase and coupling ratios.