Interaction-controlled localization in one-dimensional chain: From edges to domain walls

Using a Hartree-Fock mean-field approach, this study demonstrates that in a half-filled Su-Schrieffer-Heeger chain, the localization of bound states is governed by the ratio of extended to on-site Hubbard interactions ($2V/U$), which determines whether edge spin-density-wave modes or mid-chain charge-density-wave domain walls emerge, independent of the system's band topology.

The Gist

Imagine a long, narrow hallway lined with 100 pairs of lockers. In physics, this is a model called the SSH chain, a simplified way to study how electrons move through a one-dimensional material. Usually, scientists study this hallway when the electrons don't talk to each other. In that quiet scenario, if the hallway has a specific "twisted" pattern, two special electrons get stuck right at the very ends of the hall (the edges), like guests waiting at the front and back doors.

But in this paper, the authors ask: What happens when the electrons start arguing with each other?

They introduce two types of "arguments" (interactions) between the electrons:

1. The "Personal Space" Argument ($U$): Electrons hate sharing the same locker. If two electrons try to squeeze into one spot, they push each other away hard.
2. The "Neighborly Grudge" Argument ($V$): Electrons also dislike having a neighbor with too much stuff. If the locker next to you is full, you get annoyed.

The researchers used a computer simulation (a "mean-field" approach, which is like taking an average of everyone's mood) to see how these arguments change where the electrons hang out. They discovered a simple rule that dictates the behavior, regardless of whether the hallway was originally "twisted" (topological) or "straight" (trivial).

The Golden Rule: The Ratio of Arguments

The location of the stuck electrons depends entirely on the ratio of the two arguments: Is the "Personal Space" grudge ($U$) stronger than twice the "Neighborly Grudge" ($2V$)?

Scenario A: The "Personal Space" Wins ($U > 2V$)

Imagine the electrons are extremely introverted. They care more about not sharing their own locker than about their neighbor's locker.

• The Result: The electrons stay stuck at the ends of the hallway (the edges).
• The Analogy: Think of a shy person at a party who only feels comfortable standing by the exit doors. Because they are so focused on their own space, they develop a strong "spin" (a magnetic personality) right at the doors, while the middle of the room remains calm.
• The Finding: Even if the hallway wasn't originally "twisted" to have edge guests, the strong personal space argument creates these edge guests. They are essentially "Spin Density Waves" (SDW) stuck at the boundaries.

Scenario B: The "Neighborly Grudge" Wins ($U < 2V$)

Now imagine the electrons are very sensitive to their neighbors. They don't mind sharing a locker as much as they mind having a full locker next to them.

• The Result: The electrons stop hanging out at the doors. Instead, they get stuck right in the middle of the hallway.
• The Analogy: Imagine a long line of people trying to alternate between holding a red ball and a blue ball. If the line is even, everyone is happy. But if the line is odd (like 100 lockers with an odd number of pairs), someone has to break the pattern in the middle. This creates a "fault line" or a Domain Wall right in the center. The electrons get trapped at this fault line because it's the only place where the "neighborly grudge" can be satisfied.
• The Finding: These are "Charge Density Waves" (CDW). The electrons form a pattern of alternating full and empty lockers, and the "stuck" state is the glitch in the middle of that pattern.

The "Magic" Tipping Point

The paper found something fascinating at the exact moment the two arguments are balanced ($2V \approx U$).

• This is where the "edge localization" (staying at the doors) is at its absolute peak.
• It's like a seesaw. As long as one side is heavier, the system is stable. But right at the balance point, the system is most sensitive, and the electrons are most likely to be found at the edges before they suddenly jump to the middle if the "neighborly grudge" gets even slightly stronger.

Does the Hallway Shape Matter?

Usually, in physics, the shape of the hallway (topology) determines if you get guests at the door.

• The Big Surprise: The authors found that the shape doesn't matter.
• Whether the hallway is "twisted" (Topological), "straight" (Trivial), or "perfectly balanced" (Critical), the electrons follow the same rule:
  • Strong Personal Space ($U$) $\rightarrow$ Guests at the Edges.
  • Strong Neighborly Grudge ($V$) $\rightarrow$ Guests in the Middle.

Summary

The paper concludes that in a one-dimensional chain of electrons, interactions (arguments) are the real boss, not the underlying structure of the material.

• If electrons are selfish ($U$ dominates), they hide at the edges.
• If electrons are sensitive to neighbors ($V$ dominates), they hide in the middle of the chain.
• This behavior is driven purely by the electrons' relationship with each other, creating new types of "stuck" states that exist independently of the material's original design.

In short: It's not where you live (the topology); it's who you argue with (the interactions) that decides where you end up.

Technical Summary

Technical Summary: Interaction-Controlled Localization in One-Dimensional Chains

Problem Statement
The Su-Schrieffer-Heeger (SSH) model is a paradigmatic framework for studying topological phenomena in one-dimensional dimerized lattices, specifically hosting zero-energy edge modes in its topologically nontrivial phase. While the non-interacting limit of the SSH model is well-understood through band theory and symmetry, the role of electron-electron interactions in modifying these topological properties remains less explored. Specifically, there is a lack of systematic understanding regarding the combined effects of on-site ($U$) and nearest-neighbor ($V$) Hubbard interactions on the existence, evolution, and spatial localization of bound states. Previous studies have addressed these interactions individually or in specific contexts (e.g., entanglement spectra, bosonization), but a comprehensive analysis of how the ratio of these interactions dictates real-space localization—distinguishing between edge modes and mid-chain domain walls across different hopping regimes—is required.

Methodology
The authors investigate a half-filled extended Hubbard model on a dimerized SSH chain using a generalized Hartree-Fock mean-field approach. The system is described by a Hamiltonian containing alternating hopping amplitudes ($t_1$ and $t_2$), on-site repulsion ($U$), and nearest-neighbor density-density interaction ($V$).

The study employs a self-consistent mean-field procedure to decouple the interaction terms:

1. Decoupling: The interaction terms are approximated by separating them into spin-dependent effective site energies ($\epsilon_{i,\sigma}^{\text{eff}}$).
2. Diagonalization: The Hamiltonian is split into independent up- and down-spin sectors, which are diagonalized to obtain eigenvalues and eigenvectors.
3. Iteration: Starting with initial guesses for local spin and charge densities, the procedure iterates until convergence is achieved, yielding the final energy spectrum and eigenstate distributions.
4. Regimes: The analysis covers three distinct hopping regimes: the topological phase ($t_1 < t_2$), the critical point ($t_1 = t_2$), and the trivial phase ($t_1 > t_2$).

Key Results
The study reveals that the spatial character of localized states is governed primarily by the interaction ratio $2V/U$, rather than the underlying band topology of the SSH chain.

• Edge vs. Mid-Chain Localization:

  • $U > 2V$ (SDW Dominance): In this regime, the system favors spin-density-wave (SDW) ordering. Localized states appear at the edges of the chain. These edge states exhibit enhanced spin polarization (SDW order) and are associated with Friedel oscillations in the charge density. This behavior is observed across topological, critical, and trivial hopping regimes.
  • $U < 2V$ (CDW Dominance): In this regime, nearest-neighbor interactions favor charge-density-wave (CDW) ordering. The localized states shift from the edges to the center of the chain, forming domain-wall bound states. These states correspond to a CDW domain wall where the charge ordering phase flips (e.g., high-low-high to low-high-low). The spin density in this regime is vanishingly small in the bulk, consistent with suppressed SDW order.

• Topological vs. Interaction-Driven States:

  • In the topological phase with weak interactions, standard SSH edge modes exist. However, as $U$ increases (with $V=0$), these modes eventually merge into the bulk continuum and disappear around $U \approx 1.7$.
  • Crucially, the introduction of $V$ can restore edge localization even in regimes where pure $U$ would destroy it, provided $U > 2V$.
  • In the critical and trivial phases, where non-interacting edge modes are absent, localized states emerge purely due to the interplay of $U$ and $V$. For $U > 2V$, these are interaction-driven edge SDW modes; for $U < 2V$, they are interaction-driven mid-chain CDW domain walls.

• Spin-Split Edge Configurations:

  • In the topological phase with $U < 2V$, a unique spin-split configuration emerges where the spin-up and spin-down sectors localize at opposite ends of the chain, resulting in opposite spin densities at the two edges.

• Critical Ratio:

  • The edge probability peaks sharply when $2V \approx U$, coinciding with the boundary between SDW and CDW dominance. As the system moves away from this ratio into the CDW-dominated regime ($2V > U$), edge localization decays rapidly, giving way to mid-chain domain walls.

Significance and Claims
The paper establishes that localized states in finite one-dimensional chains with open boundaries are intrinsic, correlation-driven excitations of the extended Hubbard model. The authors claim that the spatial distribution of these states (edge vs. domain wall) is controlled by the interaction ratio $2V/U$ and is largely independent of the SSH band topology.

This finding challenges the notion that edge localization is solely a topological property protected by bulk invariants. Instead, the results demonstrate that strong electron correlations can drive localization phenomena that persist across topological, critical, and trivial phases, provided the interaction parameters satisfy specific conditions. The work highlights the nontrivial role of interactions in driving localization beyond the non-interacting topological picture, providing a systematic understanding of how on-site and extended interactions compete to determine the real-space properties of eigenstates.