🔬 condensed matter

Exact analytical edge states in the extended Su-Schrieffer-Heeger model

This paper investigates the topological phases of the extended Su-Schrieffer-Heeger model by deriving exact analytical expressions for semi-infinite edge states and establishing their correspondence with bulk winding numbers, while also providing highly accurate approximate solutions for finite chains.

Original authors: P. A. Grizzi, A. A. Aligia, P. Roura-Bas

Published 2026-04-23

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CC BY 4.0

Original authors: P. A. Grizzi, A. A. Aligia, P. Roura-Bas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a long, endless train track made of alternating types of train cars: Type A and Type B.

In the classic version of this story (the original SSH model), the cars are linked by springs. Some springs are tight (strong connection), and some are loose (weak connection). If the loose springs are on the outside of the train cars and the tight ones are on the inside, the ends of the train become "special." They hold onto a little bit of energy that doesn't want to go anywhere. These are called edge states.

Now, imagine we upgrade this train. We don't just connect a car to its immediate neighbor; we add long-range springs that jump over a car to connect to the one two or three spots away. This is the Extended Su-Schrieffer-Heeger (eSSH) model.

This paper is like a master engineer's manual that explains exactly how these "special ends" behave when you add those long-range springs. Here is the breakdown in simple terms:

1. The "Winding Number" (The Secret Code)

In the world of physics, materials are often classified by a secret code called a topological invariant. Think of this as a "twist count."

If you draw a map of how the train cars are connected as you go down the line, does the path loop around a central point once? Twice? Or not at all?
This count is called the Winding Number ( $\delta$ $δ$ ).
- $\delta = 0$ : The path doesn't loop. The train is "boring" (trivial). No special energy sits at the ends.
- $\delta = 1$ : The path loops once. One special energy state appears at each end.
- $\delta = 2$ : The path loops twice. Two special energy states appear at each end.

The authors calculated exactly how changing the strength of the springs (the "hopping" parameters) changes this winding number. They drew a map (a phase diagram) showing exactly where the train switches from "boring" to "special."

2. The Magic of the "Edge"

The most exciting part of the paper is finding the exact mathematical formula for these special states at the ends of the train.

Usually, when you have a long chain, the "special energy" at the end fades away as you move toward the middle, like a sound getting quieter. The authors found that this fading happens in a very specific, predictable pattern. They call this pattern $z$ .

If $|z| < 1$ , the energy dies out quickly as you move into the chain. This means the state is trapped at the edge.
If $|z| = 1$ , the energy doesn't die out; it spreads everywhere. This is the moment the "specialness" disappears, and the material becomes "boring" again.

The paper proves a beautiful rule: The moment the "twist count" (winding number) changes, the "fading factor" ( $z$ ) hits exactly 1. It's like a light switch: when the bulk of the material changes its nature, the edge states either appear or vanish instantly.

3. The "Ghost" States

In the extended model (with long-range springs), things get weird.

In the old model, you usually got one special state at the end.
In this new model, you can get two special states at the end.
Even cooler: These two states can be "ghosts" of each other. They might look like they are sitting on the very first car, or they might sit on the third car, depending on how you tune the springs.

The authors derived formulas that tell you exactly where these ghosts sit and how strong they are. They showed that even if the train is finite (not infinite), these formulas work almost perfectly. It's like having a blueprint that predicts exactly where the "magic" will happen in a real-world experiment, without needing to run a supercomputer simulation.

4. Why This Matters (The Real World)

You might ask, "Who cares about imaginary train tracks?"
Well, scientists have actually built these tracks using light (photonic lattices) and atoms (cold atoms).

Experiment 1: Researchers used lasers to write these tracks in glass. They saw the "ghost states" exactly where the paper predicted.
Experiment 2: Other researchers used atoms to create a "superradiance lattice" and watched the system switch from having 0 special states to 2 special states.

This paper connects the math to the reality. It explains why the experiments worked and gives a precise recipe for how to tune the system to get the number of special states you want.

The Big Takeaway

Think of the material as a knot.

If the knot is loose (trivial), the ends are just normal rope.
If the knot is tight and twisted (topological), the ends become magic handles that hold energy.
This paper figured out the exact geometry of the knot when you add extra twists (long-range connections). It tells us that if you pull the string just right, you can make two magic handles appear at the end instead of one, and it gives you the exact formula to predict where they will sit.

This is a huge step forward because it moves us from "guessing" what happens in these complex materials to knowing exactly what will happen, allowing us to design better quantum computers and sensors in the future.

1. Problem Statement

The paper addresses the need for exact analytical descriptions of topological edge states in the Extended Su–Schrieffer–Heeger (eSSH) model. While the standard SSH model (nearest-neighbor hopping) is well-understood, recent experimental realizations (e.g., in photonic lattices and superradiance lattices) involve long-range hopping processes (beyond nearest neighbors) between translationally inequivalent atoms.

Existing studies on these extended models often rely on numerical diagonalization of finite chains. The authors aim to:

Derive exact analytical expressions for the zero-energy edge states of a semi-infinite eSSH chain.
Establish a rigorous bulk-boundary correspondence (BBC) linking the bulk winding number to the existence and number of edge states.
Provide highly accurate approximate analytical expressions for low-energy edge states in finite chains, explaining experimental observations that were previously only understood numerically.

2. Methodology

The authors employ a specific analytical technique introduced by Alase et al., adapted for the eSSH model with hopping up to the second nearest neighbor ( $M=2$ ).

Model Definition: The system is a 1D dimerized lattice with intra-cell coupling $u_0$ and inter-cell couplings $u_1$ (nearest) and $u_2$ (next-nearest). The Hamiltonian preserves chiral, time-reversal, and particle-hole symmetries, placing it in the BDI topological class.
Bulk Analysis (Periodic Boundary Conditions):
- The Bloch Hamiltonian $H(k)$ is constructed with off-diagonal elements $Q(k) = d_1(k) - i d_2(k)$ .
- The winding number $\delta$ is calculated by integrating the phase change of $Q(k)$ over the Brillouin zone. This integer invariant ( $\delta \in \{0, 1, 2\}$ ) determines the topological phase.
Edge State Derivation (Semi-Infinite Chain):
- The authors split the Schrödinger equation into bulk and edge components using projection operators ( $P_b$ and $P_e$ ).
- They introduce generalized Bloch states with a complex wave vector $z$ (where $z = e^{-ik}$ in the bulk, but $|z| \neq 1$ for edge states):
  $|\psi_A(z)\rangle = \sum_{j=1}^{\infty} z^{j-1} |A_j\rangle, \quad |\psi_B(z)\rangle = \sum_{j=1}^{\infty} z^{j-1} |B_j\rangle$
- For zero-energy states ( $E=0$ ), the condition reduces to finding roots of the polynomial $Q_1(z) = u_0 + u_1 z + u_2 z^2 = 0$ .
- Localization Condition: A state is localized at the left edge if the roots $z$ satisfy $|z| < 1$ . The number of such roots corresponds to the number of edge states.
Finite Chain Approximation:
- For finite chains of length $N$ , the authors construct a low-energy subspace spanned by edge states decaying from both ends.
- They solve a generalized eigenvalue problem within this subspace to obtain analytical expressions for the energy splitting and wavefunction mixing, with errors scaling as $|z|^{2N}/\Delta_g$ .

3. Key Contributions

A. Exact Analytical Solutions for Semi-Infinite Chains

The paper derives closed-form expressions for the decay factor $z$ and the wavefunctions.

Root Analysis: The number of edge states is determined by the number of roots of $Q_1(z)$ inside the unit circle ( $|z|<1$ ).
Phase Transitions: The authors show that a change in the winding number $\delta$ (bulk gap closing) coincides exactly with the condition $|z|=1$ for the edge state solutions.
Complex Roots: In the $\delta=2$ phase, the roots $z_\pm$ are complex conjugates. The authors demonstrate how to construct two orthonormal real edge states from these complex roots, even when the roots approach the real axis (a critical point often missed in simpler analyses).

B. Bulk-Boundary Correspondence (BBC)

The study rigorously confirms the BBC for the eSSH model:

$\delta = 0$ : No roots with $|z|<1$ ; no edge states.
$\delta = 1$ : One root with $|z|<1$ ; one edge state (analogous to standard SSH).
$\delta = 2$ : Two roots with $|z|<1$ ; two edge states per boundary.
The transition between phases occurs when the bulk gap closes ( $Q(k)=0$ ), which implies $|z|=1$ for the semi-infinite solution.

C. Application to Finite Chains and Experiments

The authors apply their theory to two specific experimental setups:

Photonic Lattices (Ref [8]): They analyze a 32-site chain with parameters $(u_0, u_1, u_2) = (1, 1.5, 4.8)$ $(u_{0}, u_{1}, u_{2}) = (1, 1.5, 4.8)$ and $(1, 0.6, 4.8)$ $(1, 0.6, 4.8)$ .
- Result: Their analytical formulas predict two pairs of low-energy states. They successfully explain the spatial distribution of these states: one state peaks at the first site ( $A_1$ ) and the other at the third site ( $A_2$ ), matching numerical results with high precision.
Superradiance Lattices (Ref [9]): They analyze a tunable system where the winding number transitions from $\delta=0$ $δ = 0$ to $\delta=2$ $δ = 2$ via a parameter $\eta$ $η$ .
- Result: They derive the exact form of the edge states along the transition path, showing how the weight distribution evolves as the system moves from trivial to non-trivial topology.

4. Key Results

Phase Diagram: A complete phase diagram is mapped in the $(u_0, u_2)$ plane (with $u_1=1$ ), identifying regions for $\delta=0, 1, 2$ .
Wavefunction Structure:
- For $\delta=2$ , the edge states are not simple exponential decays but involve oscillatory components due to complex $z$ .
- The authors provide explicit formulas for the orthonormalized real wavefunctions $|\tilde{\psi}_+\rangle$ and $|\tilde{\psi}_-\rangle$ , which are linear combinations of the complex exponential solutions.
Energy Splitting: For finite chains, the energy splitting between the two edge states at each end is derived analytically. The splitting scales as $|z|^{N-1}$ , confirming that the states are exponentially localized and decoupled for large $N$ .
Accuracy: The analytical approximations for finite chains show "excellent agreement" with numerical diagonalization, with relative errors on the order of $10^{-5}$ to $10^{-6}$ for the parameters tested.

5. Significance

Theoretical Rigor: This work moves beyond numerical observation to provide a first-principles analytical framework for higher-winding-number topological phases. It clarifies the mathematical structure of edge states when multiple roots exist.
Experimental Validation: The paper bridges the gap between abstract topological invariants and experimental observables. By reproducing the specific intensity profiles and energy splittings observed in recent photonic and atomic experiments, it validates the eSSH model as the correct description for these systems.
Computational Efficiency: The derived analytical expressions offer a computationally cheap alternative to numerical diagonalization for long chains, which is crucial for studying large-scale topological systems or performing parameter sweeps.
Generalizability: While focused on $M=2$ , the authors note that the method extends to $M$ -th nearest neighbors, allowing for the prediction of up to $M$ edge states, providing a roadmap for analyzing even more complex topological lattices.

In summary, the paper successfully establishes a rigorous analytical link between the bulk winding number and the specific structure of edge states in extended topological models, providing a powerful tool for interpreting and designing future topological experiments.