Quantum Critical Dynamics Induced by Topological Zero Modes

This paper demonstrates that in a disordered Su-Schrieffer-Heeger chain near the topological delocalization transition, the hybridization of topological domain wall zero modes with stretched-exponential spatial decay drives an anomalous logarithmic scaling of ac conductivity at criticality.

The Gist

Imagine a long, crowded hallway where people (electrons) are trying to walk from one end to the other. In a normal hallway, if the floor is messy or full of obstacles (disorder), people get stuck, bump into walls, and can't move far. This is what happens in most "insulators"—materials that don't conduct electricity.

However, this paper explores a very special, weird kind of hallway called the Su-Schrieffer-Heeger (SSH) chain. In this hallway, the obstacles are arranged in a specific, "topological" way. It's like the hallway has a hidden rule: even if the floor is messy, there are special "ghost paths" that allow people to move, but only under very specific conditions.

Here is the breakdown of what the scientists found, using simple analogies:

1. The Setup: The "Ghost" Walkers

In this special hallway, there are two types of floor tiles: Type A and Type B.

• Normal Hallway: If you mess up the tiles randomly, everyone gets stuck.
• Topological Hallway: Because of the specific pattern of the tiles, two special "ghost walkers" (called Zero Modes) appear at the ends of any isolated section of the hallway. These ghosts don't like to stay put; they want to meet up.

2. The Problem: How Do They Move?

The scientists wanted to know: If we shake the hallway with a gentle rhythm (low-frequency electricity), how fast can these ghosts move?

Usually, in a messy hallway, the more you shake it, the harder it is for people to move. But here, something strange happens.

3. The Discovery: The "Hybrid" Dance

The key finding is that when two "ghost walkers" from opposite sides of a broken section of the hallway get close enough, they don't just bump into each other. They dance together.

• They form a pair: one is a "happy" dance (bonding), and one is a "sad" dance (anti-bonding).
• The distance between them determines how fast they can switch partners.
• The Twist: In a normal messy hallway, the chance of them meeting drops off very quickly as they get farther apart (like a light fading). But in this topological hallway at a critical point, the chance of them meeting drops off much, much slower. It's like the ghosts have a "super-stretchy" connection.

4. The Result: The "Logarithmic" Magic

The scientists measured how well electricity flows (conductivity) as they changed the shaking speed (frequency).

• Normal Insulators: If you shake them slowly, almost nothing happens. The flow drops off sharply.
• This Special Hallway: Even when you shake it very slowly, the flow doesn't drop off sharply. Instead, it follows a weird, slow curve called logarithmic scaling.
  • Analogy: Imagine a bucket with a hole. In a normal bucket, the water drains fast at first, then slows down quickly. In this special bucket, the water drains at a steady, slow, "creeping" pace that seems to defy the usual rules.

5. Why Does This Happen? (The "Stretched Exponential")

The paper explains that this happens because the "ghost walkers" have a unique shape.

• Normal Ghosts: Their presence fades away like a standard exponential decay (think of a candle flame going out: bright, then dim, then gone).
• Critical Ghosts: Their presence fades away like a stretched exponential. Imagine a rubber band being pulled. It resists snapping back for a long time. This "stretchy" nature means the ghosts can "feel" each other across much larger distances than usual.

Because they can feel each other from so far away, they can still "dance" (conduct electricity) even when the hallway is very long and messy.

6. The "Glassy" Metal

The authors call this a "Glassy Metal."

• Metal: Usually, metals conduct electricity easily and fast.
• Glass: Usually, glass is a solid that doesn't conduct.
• Glassy Metal: This material is in between. It conducts, but it does so very slowly and "lazily," like a glass that is slowly melting. It's a state where the material is technically an insulator, but the topological rules force it to act like a very sluggish metal.

Summary in One Sentence

The paper shows that in a specific type of messy quantum material, the unique "shape" of the electrons allows them to form long-distance partnerships, creating a slow, steady flow of electricity that defies the usual rules of how messy materials should behave.

Why does this matter?
It helps us understand how electricity moves in complex, messy materials (like certain polymers or disordered atomic wires). It suggests that if we can engineer materials with these "topological" rules, we might be able to create new types of electronic devices that are robust against damage and disorder.

Technical Summary

1. Problem Statement

The paper investigates the low-frequency alternating current (ac) transport properties of the Su-Schrieffer-Heeger (SSH) chain in the presence of chiral disorder (disorder affecting only hopping elements, preserving chiral symmetry).

• Context: In disordered topological insulators, transport differs significantly from trivial insulators. While the quantum Hall plateau transition is a famous example of delocalization in low dimensions, it remains poorly understood. The SSH chain serves as a tractable 1D model where a single zero-energy mode becomes delocalized at a topological phase transition.
• The Gap: While the static properties (density of states, critical exponents) of this model are known, the dynamic response (ac conductivity, $\sigma(\omega)$) near the topological delocalization transition was not fully understood. Specifically, the interplay between the Dyson singularity (divergent density of states at zero energy) and the spatial decay of wavefunctions in determining transport was unclear.
• Objective: To determine the scaling behavior of ac conductivity at the critical point and away from it, and to identify the microscopic states responsible for transport.

2. Methodology

The authors employ a combination of analytical frameworks and numerical simulations:

• Mott Hybridization Argument: They adapt the framework introduced by Mott for Anderson insulators, which posits that ac transport is dominated by rare, resonant pairs of localized states. In this topological context, these are identified as hybridized pairs of topological domain wall zero modes (forming bonding $\psi_+$ and anti-bonding $\psi_-$ orbitals).
• Real-Space Renormalization Group (RSRG): They utilize RSRG techniques to analyze the distribution of effective hopping parameters and the evolution of the system's energy cutoff. This allows them to derive the probability distributions of hopping elements and the resulting density of states.
• Wavefunction Analysis: A critical component is the analysis of the spatial decay of zero-mode wavefunctions. The authors distinguish between:
  • Off-critical: Exponential decay ($\psi \sim e^{-x/\xi}$).
  • Critical: Stretched-exponential decay ($\psi \sim e^{-s\sqrt{x}}$) due to the random walk nature of the wavefunction exponent.
• Kubo-Greenwood Formula: Numerical simulations are performed using the Kubo-Greenwood formula to calculate $\sigma(\omega)$ for various disorder strengths and dimerization parameters, validating the analytical predictions.

3. Key Contributions and Results

A. Mechanism of Transport

The authors demonstrate that transport is dominated by optical transitions between hybridized pairs of zero modes located at the ends of rare topological segments.

• The transition amplitude (dipole matrix element) $\langle \psi_- | \hat{x} | \psi_+ \rangle$ is proportional to the spatial separation $d$ of the zero modes.
• The energy splitting $\omega$ between these modes depends on the overlap of the wavefunctions.

B. Scaling Away from Criticality ($\delta \neq 0$)

When the system is away from the critical point (parameterized by distance $\delta$):

• Zero modes decay exponentially ($\psi \sim e^{-x/\xi}$).
• The energy splitting scales as $\omega \sim e^{-d/\xi}$.
• The density of states (DOS) scales as $\nu(E) \sim |E|^{2|\delta|-1}$.
• Result: The ac conductivity follows the law:
  $$\sigma(\omega) \sim \omega^{2|\delta|} \log^2 \omega$$
  Here, the DC conductivity vanishes because the DOS divergence is insufficient to compensate for the reduction in low-energy excitations.

C. Scaling at Criticality ($\delta = 0$)

At the topological phase transition:

• The system exhibits a Dyson singularity in the DOS: $\nu(E) \sim |E|^{-1} \log^{-3} |E|$.
• Crucially, the zero-mode wavefunctions exhibit stretched-exponential decay: $\psi(x) \sim e^{-s\sqrt{x}}$.
• This leads to an energy splitting scaling as $\omega \sim e^{-s\sqrt{d}}$, implying the spatial separation scales as $d \sim \log^2 \omega$.
• Result: The dipole matrix element scales as $\log^2 \omega$. Combined with the DOS, the ac conductivity exhibits anomalous logarithmic scaling:
  $$\sigma_{crit}(\omega) \sim \log(\Omega/\omega)$$
  This represents a "glassy" metallic behavior where the DC conductivity diverges logarithmically, contrasting with the insulating behavior of standard Anderson models ($\sigma \sim \omega^2 \log^2 \omega$).

D. Topological Enhancement

The paper highlights a topological enhancement of conductivity. In the topological phase ($\delta > 0$), the characteristic length of impurity resonances increases, enhancing the dipole matrix element. This enhancement persists over long length scales, not just at the lattice scale.

E. Crossover Behavior

The authors identify a crossover regime. Even away from criticality, at sufficiently high frequencies, the system can exhibit the critical $\log \omega$ scaling. This occurs because finite critical topological segments (which are short enough to be unaffected by the finite localization length) dominate the transport at those frequencies.

4. Significance and Implications

• Resolution of Transport Anomaly: The paper resolves why the SSH chain with chiral disorder exhibits a divergent DC conductivity at criticality despite being a 1D disordered system. The topological protection (chiral symmetry) prevents the localization that typically occurs in 1D, leading to a sub-diffusive metallic phase.
• Generalization of Mott's Theory: It successfully extends Mott's theory of hopping conductivity to topological systems, showing that the specific form of wavefunction decay (stretched-exponential vs. exponential) dictates the transport scaling.
• Universality: The findings suggest that similar "glassy" metallic behaviors and activated dynamical scaling may occur in other topological critical points, such as the Quantum Hall plateau transition or disordered Chern insulators, where single delocalized states exist at criticality.
• Experimental Relevance: The predicted scaling laws ($\sigma \sim \log \omega$ at criticality and $\sigma \sim \omega^{2|\delta|} \log^2 \omega$ off-critical) provide specific signatures for experimental verification in disordered atomic wires or engineered photonic/electronic systems.
• Variable-Range Hopping: The work connects the critical dynamics to variable-range hopping, predicting a transition from thermally activated transport to polynomial temperature scaling ($\sigma(T) \sim \exp[-(T_0/T)^{1/(1+z)}]$) as the system approaches criticality, where the dynamical exponent $z$ diverges.

In summary, the paper establishes that the stretched-exponential decay of topological zero modes is the fundamental mechanism driving the anomalous logarithmic scaling of ac conductivity at the topological delocalization transition, distinguishing it sharply from conventional disordered insulators.