The effect of staggered nonlinearity on the Su-Schrieffer-Heeger model

This paper investigates the Su-Schrieffer-Heeger model with staggered nonlinearity, revealing through both semi-analytical and numerical methods that strong nonlinearity induces topological phase transitions and unique spectral features, such as nonlinear Zak phase discontinuities and robust edge states, which highlight the complex interplay between topology and nonlinearity in lattice systems.

The Gist

Imagine a long, winding train track made of two parallel rails, labeled Rail A and Rail B. This is the Su-Schrieffer-Heeger (SSH) model, a famous toy model physicists use to understand "topology"—a branch of math that describes how things are connected.

In this model, the "trains" (which represent particles or light waves) hop between the rails. Sometimes the tracks are tight (easy to hop), and sometimes they are loose (hard to hop).

• The Trivial Phase: If the tracks are arranged one way, the train can't get stuck at the ends. It's just a normal, boring track.
• The Topological Phase: If you swap the tight and loose spots, something magical happens: the train gets "stuck" at the very ends of the track. These are called edge states. They are robust; you can shake the track, and the train stays at the end.

The Twist: Adding "Staggered Nonlinearity"

In the real world, things aren't perfectly linear. If you push a swing harder, it doesn't just go a little higher; it might start behaving wildly. In physics, this is called nonlinearity.

Previous studies looked at what happens if you add the same amount of "wildness" (nonlinearity) to both Rail A and Rail B.

This paper asks a new question: What if we add different amounts of wildness to each rail?

• Maybe Rail A gets a "super-charged" nonlinear boost.
• Maybe Rail B gets a "dampened" boost.
• Maybe one rail is pushed forward while the other is pulled back (positive vs. negative nonlinearity).

The authors call this staggered nonlinearity. It's like giving the left foot a heavy boot and the right foot a feather, then asking, "How does the person walk?"

The Two Ways They Studied It

The researchers used two different "lenses" to look at this problem:

1. The "Infinite Loop" View (Periodic Boundary Conditions)

Imagine the train track is a giant, endless loop with no start or end.

• The Finding: As they cranked up the nonlinearity, the energy levels of the train (the "bands") started to twist and turn.
• The "Loop" Structure: At low nonlinearity, the energy tracks are straight lines. But as nonlinearity increases, they start forming loops, like a rollercoaster doing a loop-de-loop.
• The Phase Transition: At a specific "tipping point" of nonlinearity, the tracks suddenly snap. The "gap" between energy levels closes, and the system undergoes a Topological Phase Transition.
• The "Zak Phase" (The Compass): To prove this transition is real, they used a mathematical compass called the Zak Phase. In the linear world, this compass points either North (0) or South (π). In their nonlinear world, they invented a "Nonlinear Zak Phase." They found that at the tipping point, this compass suddenly spins 180 degrees. This spin confirms that the system has fundamentally changed its nature, not just its speed.

2. The "Real Track" View (Open Boundary Conditions)

Now, imagine the track has actual ends (a start and a finish). This is where the real magic happens.

• The "Edge State" Independence: They found that the train stuck at the left end only cares about how "wild" the left rail is. It doesn't care what's happening on the right rail. It's like a person standing on a cliff; the wind on the other side of the world doesn't move them.
• The "Weyl Point" (The Magic Touch): In some cases, two energy tracks touch each other at a single point, like two roads merging. Usually, if you nudge the system, these roads would separate. But here, the roads just slide along each other without breaking apart. This is similar to "Weyl points" found in exotic materials called Weyl semimetals. It suggests a very robust, topological protection.
• The "Wave-Packet" (The Self-Made Cliff): This is the coolest discovery. Usually, if you make a system too "wild" (high nonlinearity), everything gets messy and localized (stuck in one spot). But here, they found a special state called a Wave-Packet.
  • Analogy: Imagine a surfer riding a wave. The surfer creates their own wave by moving, and then rides that wave. In this model, the particle creates its own "potential well" (a little valley) due to the nonlinearity, and then gets trapped in it. It looks like an edge state, but it's actually sitting right in the middle of the track!
• The "Balancing Act": When the nonlinearity on Rail A is positive (pushing) and Rail B is negative (pulling), the system finds a balance. Even at extreme levels of "wildness," the train doesn't get stuck; it stays delocalized (spread out). It's like a tightrope walker who, instead of falling, finds a way to balance perfectly because the forces on the left and right cancel each other out.

Why Does This Matter?

1. New Materials: This helps us design better materials for optical waveguides (fiber optics) or acoustic systems (sound waves). We can control how light or sound moves by tweaking these nonlinearities.
2. Quantum Computing: The ability to have an edge state that depends only on one side of the system could be used to store quantum information safely. You could move information from one end of a chip to the other without it getting lost.
3. Understanding Complexity: It shows that even in complex, interacting systems, the rules of topology (the "shape" of the physics) still hold strong, but they get dressed up in new, wilder clothes.

The Bottom Line

The authors took a simple, well-known model of a train track and added a "staggered" twist of chaos. They discovered that this chaos doesn't destroy the magic of topology; instead, it creates new, robust states, strange "self-made" traps for particles, and a new kind of compass to measure the system's shape. It's a reminder that even when things get messy, nature often finds a way to keep its balance.

Technical Summary

1. Problem Statement

The Su-Schrieffer-Heeger (SSH) model is a canonical 1D topological model describing particles on a lattice with alternating hopping amplitudes. While the linear SSH model is well-understood, real-world quantum systems often involve interactions. Incorporating full many-body interactions makes the SSH model analytically intractable due to exponential Hilbert space scaling.

• The Gap: Previous studies have explored uniform onsite nonlinearity (where the nonlinearity strength is the same on all lattice sites) as a mean-field approximation of interactions. However, the effect of staggered onsite nonlinearity—where the nonlinearity strength differs between the two sublattices ($A$ and $B$)—remains largely unexplored.
• Objective: This paper investigates how sublattice-dependent nonlinearity ($g_A \neq g_B$) alters the spectral properties, topological invariants (Zak phase), and edge state dynamics of the SSH model.

2. Methodology

The authors employ two complementary approaches to solve the state-dependent nonlinear Schrödinger equation (NLSE) derived from an effective nonlinear Hamiltonian:

A. Momentum Space Analysis (Periodic Boundary Conditions - PBC)

• Approach: Assumes Bloch state solutions ($\psi \sim e^{ikj}$) to reduce the system to a $2 \times 2$ effective Hamiltonian.
• Technique: Uses a self-consistent field (SCF) approach to solve the resulting nonlinear eigenvalue problem. By parametrizing the pseudo-spinor on the Bloch sphere, the authors derive a quartic polynomial equation for the energy eigenvalues.
• Topological Analysis: Derives a general expression for the "Nonlinear Zak Phase" that accounts for the geometric phase contributions from the state-dependent Hamiltonian. This extends previous work limited to uniform nonlinearity.
• Stability: Performs a dynamical stability analysis by constructing the $L$ matrix (linearized perturbation matrix) and examining its eigenvalues. Complex eigenvalues indicate dynamical instability.

B. Real Space Analysis (Open Boundary Conditions - OBC)

• Approach: Solves the full lattice model numerically without assuming Bloch periodicity, allowing for the study of edge states and solitons.
• Technique: Uses the Self-Consistent Field (SCF) Iterative Method. Initial trial states are taken from the corresponding linear model ($g_A=g_B=0$), and the system is iterated until convergence.
• Classification: Solutions are classified based on their spatial localization using the Inverse Participation Ratio (IPR) and specific amplitude thresholds at the lattice edges to distinguish between:
  • Left/Right Edge States (Solitons).
  • Bulk Localized States.
  • Delocalized Bulk States.

3. Key Contributions and Results

A. Momentum Space Findings

1. Nonlinear Zak Phase: The authors derived a general analytical formula for the nonlinear Zak phase applicable to any real-valued, well-behaved onsite nonlinearity.
2. Topological Phase Transition: At sufficiently high nonlinearity strengths, the system undergoes a nonlinearity-induced topological phase transition. This is marked by:
   • A closing of the energy gap.
   • A discontinuity (jump) in the nonlinear Zak phase.
   • The emergence of "incomplete energy bands" (loop structures) that exist only in specific regions of the Brillouin zone.
3. Dynamical Stability: The outer energy bands are generally dynamically stable. However, the additional bands forming loop structures are dynamically unstable. Near the phase transition point, the outer bands also exhibit instability near $k=0$ and $k=2\pi$.

B. Real Space Findings

1. Sublattice-Dependent Edge States: In the topologically nontrivial regime ($J_1 < J_2$) with high nonlinearity:
   • The energy of the left edge state depends only on $g_A$ and is independent of $g_B$.
   • Conversely, the right edge state depends only on $g_B$.
   • This decoupling allows for the potential control of edge states by tuning specific sublattice parameters.
2. Single Edge State Existence: The authors identified parameter regimes where only one edge state exists (e.g., the left state disappears while the right persists), suggesting potential applications in quantum state transfer.
3. Weyl-like Touching Points: At very large nonlinearities, two energy bands develop a touching point (gap closing). Unlike standard band gaps that open/close, this point shifts under perturbations (changes in onsite potential $v$) but does not disappear, mimicking the behavior of Weyl points in Weyl semimetals.
4. Persistence of Delocalized States:
   • In uniform nonlinearity ($g_A = g_B$), high nonlinearity typically leads to complete localization (solitons).
   • In the staggered case with opposite signs ($g_A$ and $g_B$ having different signs), delocalized bulk solutions persist even at extreme nonlinearity strengths. This is attributed to the interplay between attractive and repulsive nonlinearities.
5. Wave-Packet (WP) States: A specific localized, highly oscillatory state (termed a "Wave-Packet" state) is identified. It originates from the $k=0$ linear bulk state and becomes localized due to the self-induced potential peak created by the nonlinearity, acting as an effective edge.

4. Significance and Implications

• Theoretical Advancement: The work generalizes the understanding of topological phases in nonlinear systems, moving beyond uniform interactions to sublattice-dependent scenarios. The derivation of the general nonlinear Zak phase provides a robust tool for future topological studies in nonlinear media.
• Experimental Relevance: The findings are directly applicable to experimental platforms where nonlinearity can be engineered, such as photonic waveguides, acoustic systems, and mechanical metamaterials. The ability to control edge states via sublattice-specific nonlinearity offers new design principles for robust waveguiding.
• New Physics: The discovery of Weyl-like touching points in a 1D nonlinear lattice and the persistence of delocalized states under strong nonlinearity (when signs alternate) reveals rich interplay between topology and nonlinearity that is absent in linear or uniform nonlinear models.
• Future Directions: The paper suggests extending these studies to higher-order topological insulators, non-Hermitian systems, and time-periodic (Floquet) driven systems to further explore the physics of nonlinear topological matter.

In summary, this paper demonstrates that staggered nonlinearity acts as a powerful control knob for tuning topological properties, inducing phase transitions, and creating unique states (like Weyl-like points and persistent delocalized states) that are not possible in linear or uniformly nonlinear SSH models.