Weakly interacting one-dimensional topological insulators: a bosonization approach

This paper employs bosonization to investigate weakly interacting one-dimensional topological insulators, demonstrating that chiral symmetry protects edge state degeneracy, that topological indices are determined by inter-chain coupling types, and that general topological phases with index $\nu$ are equivalent at low energies to theories of at least $\nu$ Su-Schrieffer-Heeger chains.

The Gist

Imagine you are walking through a long, narrow hallway made of a special kind of floor tiles. This hallway represents a one-dimensional topological insulator.

In the world of physics, "topological" means the material has a hidden, global property that makes it very robust. Think of it like a knot in a rope: you can wiggle the rope, stretch it, or twist it, but the knot stays a knot unless you cut the rope. In these materials, this "knot" protects certain special states that live only at the very ends (the edges) of the hallway.

This paper is about what happens when you add interactions to this hallway. In the real world, electrons (the particles moving through the hallway) don't just walk alone; they bump into each other, push, and pull. The authors wanted to understand: If the electrons start arguing with each other, does the special "knot" protection at the edges still hold? And how does the hallway change?

To answer this, they used a mathematical tool called Bosonization.

The Magic Trick: Turning Particles into Waves

Imagine you are trying to describe a chaotic crowd of people running down a hallway. It's hard to track every single person (electron). Instead, imagine you stop looking at individuals and start looking at the waves they create as they run.

"Bosonization" is the mathematical magic trick that turns the messy problem of interacting particles into a much cleaner problem of interacting waves. The authors used this trick to turn the complex electron hallway into a simpler "wave hallway" to see what happens at the edges.

The Key Findings (Explained Simply)

1. The Edge is a "Kink" in the Wave

In the non-interacting world (where electrons ignore each other), the special edge state looks like a smooth transition or a "kink" in the wave at the end of the hallway. It's like a step in a staircase that only exists at the very end.

• The Discovery: Even when the electrons start interacting (pushing and pulling), this "kink" doesn't disappear. It just gets a little wider or narrower depending on how strong the push is. The authors calculated exactly how wide this kink gets.

2. The "Double Hallway" Experiment

The authors then imagined two hallways running side-by-side, connected by a weak bridge (interaction).

• Scenario A (Identical Hallways): If both hallways are in the same "topological state," they have a lot of special edge states. When the electrons start interacting, they act like a crowd trying to sit in a small room. They can't all fit comfortably in the same way. The interaction forces some of these edge states to merge or disappear, reducing the number of unique "seats" available at the edge.
• Scenario B (Different Hallways): If one hallway is "topological" (has the knot) and the other is "trivial" (no knot), the interaction doesn't destroy the edge states as easily. The "knot" is still protected.

3. The Bodyguard: Chiral Symmetry

Why don't the interactions destroy the edge states completely? The authors found a "bodyguard" protecting them.

• The Analogy: Imagine the edge state is a VIP guest. The interactions are a rowdy crowd trying to push the VIP out. But there is a strict bouncer named Chiral Symmetry. As long as the bouncer is on duty, the rowdy crowd cannot kick the VIP out, even if they push hard. The paper proves that this "bouncer" (chiral symmetry) is what keeps the edge states safe, even when the electrons are interacting.

4. The "Super-Hallway" (Longer Range Hopping)

Finally, they looked at a single hallway that has a more complex structure, allowing electrons to jump further down the line (not just to the next tile, but to the one after that).

• The Surprise: Even though this is just one hallway, it behaves mathematically like two hallways stuck together. The authors showed that if a hallway has a "winding number" of 2 (a double knot), it is effectively equivalent to having two separate chains of electrons. This means the rules for two chains apply to this single, complex chain.

The Big Picture

This paper is like a manual for engineers building quantum computers.

• The Problem: Quantum computers are fragile. If particles interact, the delicate information stored at the edges (the qubits) might get scrambled.
• The Solution: This paper tells us that as long as we keep the "bouncer" (chiral symmetry) on duty, the information at the edges remains safe, even if the particles are interacting.
• The Method: By using the "wave" analogy (Bosonization), they showed us exactly how to predict how these edge states will behave, making it easier to design future materials that are robust against the chaos of the real world.

In short: Interactions make the edge states wiggle and change size, but they don't break the knot. As long as the symmetry rules are followed, the special edge states survive.

Technical Summary

1. Problem Statement

The paper addresses the challenge of understanding how weak electron-electron interactions affect the topological properties of one-dimensional (1D) topological insulators, specifically those based on the Su-Schrieffer-Heeger (SSH) model.

• Context: While non-interacting topological phases are well-classified (e.g., by winding numbers $\nu$), the introduction of interactions can alter these classifications. In 1D, weak interactions are known to reduce the topological classification from $\mathbb{Z}$ to $\mathbb{Z}_n$ (e.g., $\mathbb{Z}_8$ for Majorana chains).
• Gap: Existing methods often rely on numerical techniques or adiabatic continuity arguments. There is a need for an analytical framework that can quantitatively derive properties like edge state degeneracy, localization length, and symmetry protection in interacting systems.
• Specific Challenge: Describing topological edge states in bosonization is non-trivial, particularly when dealing with open boundary conditions or strong impurities, and extending these descriptions to coupled chains or higher winding numbers.

2. Methodology

The authors employ bosonization, a powerful technique for analyzing 1D interacting fermion systems, to map fermionic models onto effective bosonic field theories.

• Core Approach: They treat the physical edge of a topological insulator not as a hard open boundary, but as a strong impurity (or a domain wall between topological and trivial phases).
• Key Insight: In the bosonic language, topological edge states manifest as degenerate kinks (solitons) in the bosonic fields at the boundary.
• Models Studied:
  1. Single Interacting SSH Chain: Including nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions.
  2. Capacitively Coupled SSH Chains: Two chains coupled via Hubbard interaction (on-site repulsion/attraction), both in identical and distinct topological phases.
  3. Coupled Chain Models with Hopping: Models with inter-chain hopping that realize different chiral symmetry classes (BDI, CII, AIII, CI, DIII).
  4. Extended SSH Model: A single chain with longer-range hopping capable of supporting higher winding numbers ($\nu=2$).
• Symmetry Analysis: The authors explicitly derive the transformation rules of bosonic fields ($\phi, \theta$) under time-reversal ($T$), particle-hole ($P$), and chiral ($C$) symmetries to determine which symmetries protect edge state degeneracy in the presence of interactions.

3. Key Contributions and Results

A. Single SSH Chain and Edge States

• Boundary Conditions: The authors demonstrate that modeling a physical edge as a strong impurity with an added scattering phase shift ($\delta = \pi/2$) reproduces the known results of open boundary bosonization but with simpler mathematical treatment.
• Localization Length: They compute the localization length ($l_{loc}$) of edge states as the width of a half-soliton in the sine-Gordon model.
  • Result: $l_{loc}$ is non-monotonic with respect to interaction strength.
  • Umklapp Effects: Including umklapp scattering (relevant for strong repulsion) increases the localization length by reducing the energy gap, consistent with the fact that repulsion tends to close the gap.
• Symmetry Protection: They prove that the degeneracy of edge states in the interacting SSH model is protected by chiral symmetry (or particle-hole symmetry), similar to the non-interacting case. Perturbations breaking these symmetries lift the degeneracy.

B. Capacitively Coupled SSH Chains (Hubbard Interaction)

• Identical Chains ($\nu=1$ each, total $\nu=2$):
  • Non-interacting: The ground state has a 4-fold degeneracy per edge (due to two independent edge modes).
  • Interacting: Weak Hubbard interactions reduce this degeneracy to 2-fold.
  • Mechanism: The interaction creates a gap in the spin sector (for repulsive $U>0$) or charge sector (for attractive $U<0$). The energy cost to create kinks differs between sectors, splitting the 4-fold degeneracy.
  • Protection: The remaining 2-fold degeneracy is protected by chiral symmetry. Interestingly, depending on the sign of the interaction, the remaining degeneracy may be protected by a different set of symmetries (e.g., particle-hole symmetry) rather than just chiral symmetry.
• Distinct Phases (One trivial, one topological):
  • The ground state remains 2-fold degenerate. Weak interactions do not break this degeneracy because the kink magnitudes in the relevant sectors remain equal.

C. Coupled Chains with Inter-Chain Hopping

• Symmetry Classes: The authors show that adding inter-chain hopping breaks additional unitary symmetries, placing the system into specific topological classes (e.g., BDI, CII, AIII, CI, DIII).
• Winding Number Determination: In the bosonic language, the winding number of a weakly coupled system is determined solely by the type of chiral symmetry preserved by the coupling terms.
  • Chiral Symmetry $C_1$: Preserves the sum of winding numbers (e.g., $\nu=2$ for two $\nu=1$ chains).
  • Chiral Symmetry $C_2$: Leads to a difference of winding numbers (e.g., $\nu=0$), effectively breaking the edge state degeneracy unless protected by time-reversal symmetry (as in class DIII).

D. Extended SSH Model and Higher Winding Numbers

• Mapping to Coupled Chains: The authors prove a general principle: A 1D model with a topological index $\nu$ is, at low energies near a phase transition, equivalent to a theory of at least $\nu$ coupled SSH chains.
• Example ($\nu=2$): An extended SSH model with next-nearest-neighbor hopping (single band, $\nu=2$) has two Fermi points at the critical line. The authors show this maps to the bosonic theory of two coupled chains.
• Implication: This explains why interactions in a single chain with $\nu=2$ reduce the degeneracy in the same way as in two coupled chains (reducing $2^\nu$ to $2^{\nu-1}$ or similar, depending on symmetry).

4. Significance

• Analytical Framework: The paper establishes a robust bosonization framework for studying interacting topological phases. It simplifies the treatment of boundaries by using the "strong impurity" analogy, making it easier to extend to complex coupled systems.
• Quantitative Predictions: It provides quantitative predictions for the localization length of edge states and the specific reduction of ground state degeneracy due to interactions, confirming and extending previous numerical and perturbative results.
• Symmetry Insights: The work clarifies the role of emergent symmetries in protecting topological phases. It highlights that the set of protective symmetries in interacting systems can differ from the non-interacting case and depends on the sign of the interaction.
• Universality: The finding that a model with winding number $\nu$ maps to $\nu$ coupled chains provides a unifying perspective for understanding topological insulators with higher indices, bridging the gap between single-chain and multi-chain models.

In summary, this paper successfully applies bosonization to demonstrate how weak interactions modify the topological landscape of 1D systems, specifically quantifying the reduction of edge state degeneracy and identifying the precise symmetry mechanisms that protect the remaining topological features.