Exploring topological phases with extended Su-Schrieffer-Heeger models

This paper reviews various extensions of the Su-Schrieffer-Heeger (SSH) model—including increased dimensionality, enlarged unit cells, and added physical terms—by discussing specific case studies and comprehensively elaborating on their resulting topological properties.

The Gist

Imagine a long, narrow hallway made of tiles. This hallway represents a Su-Schrieffer-Heeger (SSH) model, which is the simplest possible "toy" physicists use to understand topological phases of matter.

In this hallway, the tiles are arranged in pairs. Sometimes the gap between the two tiles in a pair is small (strong connection), and the gap between the pairs is large (weak connection). Or, it could be the other way around.

The Magic of the "Edge"

Here is the cool part: If the gaps between the pairs are larger than the gaps inside the pairs, something magical happens at the very ends of the hallway. A tiny, invisible "ghost" particle gets stuck at the left end and another at the right end. These are called zero-energy edge modes.

If you swap the gaps (making the inside gaps larger), these ghosts disappear. The hallway is now "boring" (topologically trivial).

Why is this important? Because these edge ghosts are topologically protected. Imagine trying to push the ghost off the edge. You can shake the floor, paint the tiles, or throw a pebble at it, but as long as you don't completely break the pattern of the hallway, the ghost cannot be removed. It's like a knot in a rope; you can wiggle the rope all you want, but the knot stays until you untie it completely.

The Paper's Mission: Building Bigger Castles

This paper is a review article. It doesn't just look at this simple hallway; it asks: "What happens if we make this hallway more complex?" The author, Raditya Weda Bomantara, explores four main ways to upgrade this simple model into something much more sophisticated.

1. Going 3D: From Hallways to Skyscrapers

Imagine taking many of these 1D hallways and stacking them on top of each other to build a 2D floor or a 3D skyscraper.

• The Result: You get Topological Insulators and Weyl Semimetals.
• The Analogy: Think of a 3D block of cheese. Inside the cheese, electricity can't flow (it's an insulator). But if you slice the cheese, the surface becomes a superhighway where electricity flows without resistance. In the 3D version, these "superhighways" can even form weird shapes called Fermi arcs (like bridges connecting two islands of energy) that don't exist in the simple hallway.

2. Changing the Pattern: The "Corner" Ghosts

What if, instead of just pairs of tiles, we arrange them in groups of three (a trimer)?

• The Result: Higher-Order Topological Phases.
• The Analogy: In the simple hallway, the ghosts hide at the ends (1D boundaries). In these complex 3D stacks, the ghosts don't hide on the walls or the floor; they hide in the corners. Imagine a cube where the only place a ghost can hide is the very tip of the corner. This is a "second-order" topological phase. It's like finding a treasure not at the edge of the map, but at the very corner of the island.

3. Making the Tiles Bigger: The "Square Root" Trick

What if we keep the hallway 1D but make the repeating pattern bigger? Instead of a pattern of 2 tiles, we use a pattern of 4, 6, or even more.

• The Result: Extended SSH Models (like the SSH3 or Square-Root models).
• The Analogy: Imagine the hallway has a secret rhythm. In the simple version, the rhythm is "Left-Right, Left-Right." In the extended version, the rhythm is "Left-Right-Left-Right-Left-Right." This creates more "lanes" for the ghosts to hide in. Sometimes, the ghosts aren't at zero energy anymore; they hide at specific, non-zero energy levels, creating a richer, more colorful spectrum of possibilities.

4. Adding Chaos and Magic: Driving, Loss, and Interaction

Finally, the paper looks at what happens if we stop treating the hallway as a static, perfect thing and add real-world messiness.

• Periodic Driving (The Shaking Floor): Imagine shaking the hallway back and forth rhythmically. This creates Floquet phases. You can get "ghosts" that appear and disappear in sync with the shaking, including a weird type called $\pi$-modes that have no static equivalent. It's like a magic trick where the ghost only appears when you blink.
• Non-Hermiticity (The Leaky Floor): Imagine the hallway has a leak. Particles can enter or leave (gain and loss). In this case, the "bulk-boundary correspondence" breaks. Instead of ghosts hiding at the edges, all the particles in the hallway might suddenly rush to one end and pile up there. This is called the Non-Hermitian Skin Effect. It's like a crowd of people in a hallway suddenly all running to the exit because the floor is slippery on one side.
• Interactions (The Crowd): If the particles in the hallway can talk to each other (interact), the math gets incredibly hard. But it turns out, these interactions can create new types of ghosts that wouldn't exist if the particles were alone.

Why Should You Care?

You might think, "Who cares about a hallway with ghosts?"

But this simple model is the Lego brick of modern physics.

1. Quantum Computing: These protected edge states are perfect for storing quantum information. Because they are so hard to disturb, they could be the key to building stable quantum computers that don't crash from tiny errors.
2. New Materials: By understanding these patterns, scientists can design new materials (like acoustic waveguides or photonic crystals) that guide sound or light in impossible ways, creating perfect insulators or one-way traffic for waves.
3. Future Tech: The paper suggests that by mixing these extensions (stacking them, shaking them, and making them leaky), we can discover entirely new states of matter that we haven't even imagined yet.

The Takeaway

The SSH model is like a simple melody. This paper is a guide to how that melody can be rearranged, sped up, layered, and distorted to create a symphony of new physical phenomena. It shows that even the simplest starting point can lead to the most complex and exotic discoveries in the universe.

Technical Summary

1. Problem Statement

The Su-Schrieffer-Heeger (SSH) model is the prototypical one-dimensional (1D) topological system, characterized by alternating nearest-neighbor hopping amplitudes. While the standard SSH model successfully demonstrates the bulk-boundary correspondence and the existence of zero-energy edge modes (zero modes) in its topological phase, it is limited to a simple 1D lattice with a two-site unit cell.

The central problem addressed in this review is how to systematically generalize the SSH model to explore more complex and exotic topological phenomena. Specifically, the paper investigates how extending the SSH model through various methods—such as increasing dimensionality, enlarging the unit cell, or incorporating non-standard physical effects (driving, non-Hermiticity, interactions)—can generate higher-order topological phases, higher-dimensional topological insulators, Weyl semimetals, and novel non-equilibrium topological states.

2. Methodology

The paper employs a comprehensive review and analytical approach, categorizing extensions of the SSH model into three primary methodological frameworks:

• Dimensional Extension via Continuum Limit: The authors map the discrete SSH Hamiltonian to a modified 1D Dirac equation with a Wilson mass term. By generalizing this continuum Hamiltonian to 2D and 3D and then discretizing them back onto lattices, they derive higher-dimensional topological models (e.g., Qi-Wu-Zhang model, 3D Topological Insulators, Weyl Semimetals).
• Real-Space and Momentum-Space Structural Modifications:
  • Unit Cell Enlargement: The authors analyze models where the unit cell size is increased (e.g., trimer models) either by changing the periodicity of hopping amplitudes in real space or by replacing Pauli matrices with higher-dimensional matrices in momentum space.
  • Square-Root Construction: A mathematical procedure is reviewed where a new Hamiltonian is constructed such that its square yields the standard SSH Hamiltonian, leading to multi-band structures with unique symmetries.
• Incorporation of Physical Effects: The review analyzes the SSH model under non-equilibrium and non-conservative conditions:
  • Periodic Driving (Floquet Theory): Analyzing time-periodic hopping amplitudes to derive Floquet operators and quasienergy spectra.
  • Non-Hermiticity: Introducing non-reciprocal hopping terms to study the interplay between topology and gain/loss mechanisms.
  • Interactions and Nonlinearity: Briefly discussing extensions involving Hubbard interactions and state-dependent nonlinear terms.

3. Key Contributions and Results

A. Higher-Dimensional Topological Phases

• 2D Topological Insulators: By discretizing a 2D Dirac Hamiltonian, the authors derive the Qi-Wu-Zhang (QWZ) model. They demonstrate that this model supports chiral edge modes characterized by a non-zero Chern number ($C = \pm 1$) in the non-trivial regime.
• 3D Topological Insulators: Generalizing to 3D yields a model belonging to the DIII symmetry class (protected by time-reversal symmetry). The system is characterized by a $\mathbb{Z}_2$ invariant. The authors distinguish between Strong Topological Insulators (STI), which host robust helical surface modes insensitive to boundary details, and Weak Topological Insulators (WTI), which host boundary modes only on specific surfaces and are essentially stacked 2D layers.
• Weyl Semimetals: A variation of the 3D model breaks time-reversal or inversion symmetry, resulting in band touching points (Weyl points) with linear dispersion. These points possess a topological charge (chirality), and the system hosts Fermi arcs on the surface connecting the projections of Weyl points with opposite chirality.

B. Higher-Order Topological Phases

• By stacking multiple SSH chains and coupling them orthogonally, the authors construct a Second-Order Topological Insulator (SOTI).
• Unlike first-order phases (which have edge states), this 2D model hosts zero-energy corner modes (boundaries of boundaries).
• The topology is protected by a combination of chiral and spatial symmetries (inversion/mirror). The topological invariant is the product of the winding numbers of the constituent 1D chains ($W_{xy} = W_x \times W_y$).

C. Extended Unit Cell Models

• SSH3 Model (Trimer): Increasing the unit cell to three sites (A, B, C) creates a 3-band system. While it lacks standard chiral symmetry, it possesses point chiral symmetry. The topological invariant is a normalized sublattice Zak phase, which predicts the existence of non-zero energy edge modes (unlike the zero modes in standard SSH).
• Square-Root SSH Model: By defining a Hamiltonian $H_{sq}$ such that $H_{sq}^2 \approx H_{SSH}$, the authors generate a 4-band model. This model exhibits two distinct topological gaps (one at zero energy, one at finite energy) and supports edge modes in both gaps, characterized by sums of Zak phases.
• SSH3m Model: A momentum-space extension replacing Pauli matrices with $3 \times 3$ matrices. This model preserves chiral symmetry but features a pinned zero-energy bulk band. Consequently, it cannot support zero-energy edge modes but instead supports pairs of non-zero energy edge modes symmetric about $E=0$.

D. Non-Equilibrium and Non-Hermitian Extensions

• Periodically Driven (Floquet) SSH: Time-periodic hopping leads to a Floquet spectrum with quasienergies. The system supports two types of edge modes: zero modes and $\pi$ modes (pinned at half the driving frequency). Two winding numbers ($\nu_0, \nu_\pi$) are required to fully characterize the topology.
• Non-Hermitian SSH: Introducing non-reciprocal hopping ($\gamma$) leads to the Non-Hermitian Skin Effect (NHSE), where all bulk eigenstates localize at one boundary. The topology is characterized by an energy winding number in the complex plane, which differs fundamentally from the eigenstate-based invariants of Hermitian systems.

4. Significance

• Unifying Framework: The paper establishes the SSH model not just as a standalone system, but as a fundamental "building block" for a vast landscape of topological matter. It demonstrates how complex phenomena in 2D/3D, higher-order phases, and non-equilibrium systems can be traced back to the simple physics of the 1D SSH chain.
• Experimental Relevance: The review highlights that these extended models are not merely theoretical constructs but have been realized in diverse experimental platforms, including photonic waveguides, acoustic metamaterials, ultracold atoms, and superconducting circuits.
• Technological Potential: The ability to engineer specific edge modes (e.g., $\pi$ modes for qubit transfer, corner modes for robust storage) suggests that extended SSH models are promising candidates for topological quantum computing and robust quantum transport devices.
• Methodological Insight: By detailing the mathematical tools (winding numbers, Zak phases, Chern numbers, $\mathbb{Z}_2$ invariants) required for each extension, the paper serves as a technical guide for researchers aiming to design new topological materials or simulate them in synthetic quantum systems.

In conclusion, this review comprehensively maps the "topological zoo" accessible through the extension of the SSH model, bridging the gap between simple 1D lattice models and complex, high-dimensional, and non-equilibrium topological phases of matter.