Interference of critical dynamics associated with zero modes

This paper investigates interference patterns of critical dynamics associated with zero modes (ICDZM) in generalized Creutz ladders, demonstrating how closed quench paths through critical points generate distinct oscillations and period doubling that can be detected via boundary particle number deviations and serve as probes for topological zero-mode dynamics.

The Gist

Imagine a quantum system as a vast, intricate landscape of hills and valleys. In this landscape, there are special "zero modes"—think of them as tiny, invisible marbles that like to sit right on the very edge of the cliff, never falling into the middle. These marbles are special because they are protected by the shape of the landscape itself (topology).

This paper is about what happens when we shake this landscape rapidly, forcing the marbles to move, and then watch how they behave when we stop. Specifically, the researchers are interested in a phenomenon called Interference of Critical Dynamics associated with Zero Modes (ICDZM).

Here is a simple breakdown of their journey and discoveries:

The Setup: The "Creutz Ladder"

The researchers used a model called the "generalized Creutz ladder." You can imagine this as a two-track train track. The "marbles" (particles) can hop between the tracks and along the length of the ladder. By changing the speed of the wind or the angle of the tracks (parameters called $\theta$ and $\mu$), they can change the shape of the landscape, creating different "phases" of matter. Some phases are "trivial" (boring, flat ground), and some are "topologically nontrivial" (complex, winding paths that protect the edge marbles).

The Experiment: The "Closed Loop" Drive

Usually, scientists study what happens when they push a system through a critical point once (like driving a car over a single speed bump). But here, the researchers did something more complex: they drove the system through two critical points in a closed loop.

Imagine driving a car:

1. Protocol 1: You drive from Point A, cross a speed bump, go through a winding, complex valley, cross a second speed bump, and end up back at a point that looks exactly like where you started.
2. Protocol 2: You drive from Point A, cross a speed bump, turn around immediately, and cross that same speed bump again to return home.
3. Protocol 3: You drive from Point A, cross a speed bump, go through a flat, boring plain, cross the speed bump again, and return home.

The Discovery: The "Interference Pattern"

When you drive through these loops, the "edge marbles" (zero modes) don't just stay put or move randomly. They create an interference pattern, much like ripples in a pond when two stones are dropped in. The researchers measured how likely it was for a marble to jump from one edge state to its partner state (the "transfer probability").

They found three distinct outcomes based on the path taken:

1. The "Period Doubling" Surprise (Protocol 1):
   When the car drove through the complex, winding valley (the topologically nontrivial phase) between the two bumps, the marbles created a special pattern. The rhythm of their movement was twice as slow as the rhythm seen in the middle of the system (the bulk).

   • Analogy: Imagine the bulk of the system is a drum beating at a fast pace. But the edge marbles, having traveled through the complex valley, decided to beat at half that speed. The researchers call this "period doubling."

2. The "Silent" Return (Protocol 2):
   When the car crossed the same speed bump twice (returning immediately), the edge marbles barely moved. The interference pattern was so weak it almost vanished.

   • Analogy: It's like trying to create a ripple by splashing water in the exact same spot twice in a row; the waves cancel each other out or fail to build up. The bulk of the system still showed ripples, but the special edge marbles went quiet.

3. The "Standard" Rhythm (Protocol 3):
   When the car drove through the flat, boring plain (the topologically trivial phase), the edge marbles behaved normally. Their rhythm matched the rhythm of the bulk system exactly.

   • Analogy: The edge marbles and the bulk marbles are now dancing to the exact same beat.

The "Why": The WKB Map

The researchers used a mathematical tool called "WKB analysis" to explain this. Think of this as a map that calculates the "phase" (or timing) the marbles accumulate as they travel.

• In the complex valley, the "energy gap" (the distance between the marbles' energy levels) is effectively halved because of the special edge states. This halving causes the rhythm to slow down (period doubling).
• In the flat plain, there is no such halving, so the rhythm stays standard.

How to See It: The "Edge Defect"

You might ask, "How do we actually see these invisible marbles?"
The researchers showed that you don't need to see the marbles directly. You can just count the number of particles on the very first rung of the ladder.

• Initially, the edge has a "fractional" charge (like having 1.5 particles on average).
• After the drive, if the number of particles on that edge changes, it tells you exactly how the marbles interfered.
• Analogy: It's like checking the water level at the edge of a pool. Even if you can't see the waves in the middle, the water level rising and falling at the edge tells you exactly what kind of waves are happening.

The Bottom Line

This paper shows that by driving a quantum system in a closed loop and watching the edge particles, we can detect the "topological memory" of the path taken.

• If the path went through a complex, topological region, the edge particles show a slowed-down, doubled rhythm.
• If the path went through a simple region, they show a standard rhythm.
• If the path retraced its steps, the edge particles go silent.

This provides a new way to "listen" to the critical dynamics of topological systems using simple edge measurements, revealing hidden information about the journey the system took.

Technical Summary

Technical Summary: Interference of Critical Dynamics Associated with Zero Modes

Problem Statement
The paper addresses the intersection of nonadiabatic critical dynamics and topological zero modes in low-dimensional systems. While the Kibble-Zurek (KZ) mechanism is well-established for describing defect production during continuous phase transitions, and topological zero modes (such as Majorana modes) are known to exhibit unique behaviors, the specific interplay between critical dynamics and zero modes remains under-explored. Previous studies using "one-way" quench protocols (driving a system across a single critical point) primarily reveal generated nonadiabatic excitations but fail to fully expose the phase information embedded in the critical dynamics. The authors investigate whether a "closed" quench path, which passes through two critical points and returns to the starting point, can reveal interference patterns specific to zero modes (Interference of Critical Dynamics associated with Zero Modes, or ICDZM).

Methodology
The study employs the generalized Creutz ladder model, a two-leg fermionic ladder with complex hopping amplitudes. The Hamiltonian is parameterized by hopping strengths $J_X, J_Y, J_D$ and phases $\theta, \phi$. The authors restrict the analysis to the parameter plane where $\phi=0$ and $J_X=J_D=K$, introducing a dimensionless parameter $\mu = J_Y/2K$.

Three distinct closed quench protocols are simulated numerically (with system size $L=500$) and analyzed theoretically:

1. Protocol 1: Varying $\theta$ from $\pi/2$ to $5\pi/2$ at fixed $\mu=0$. This path crosses two distinct critical points ($\theta=\pi$ and $\theta=2\pi$) and traverses a topologically nontrivial phase with winding number $\nu=-1$.
2. Protocol 2: Varying $\theta$ from $\pi/2$ to $3\pi/2$ and back to $\pi/2$ at fixed $\mu=0$. This path crosses the same critical point ($\theta=\pi$) twice.
3. Protocol 3: Varying $\mu$ from $0$ to $2.5$ and back to $0$ at fixed $\theta=\pi/2$. This path enters the topologically trivial region ($\nu=0$) and crosses the critical point $\mu=1$ twice.

The dynamics are governed by the time-evolution operator $U(t, t_i)$. The authors calculate the zero-mode transfer probability ($p_Z$), defined as the probability of transferring an occupation from an initially occupied zero mode to its particle-hole partner. This is compared against the bulk excitation density ($p_B$). Theoretical analysis utilizes the WKB approximation to derive the interference phase accumulated during the quench.

Key Results
The study reveals that the ICDZM pattern is highly dependent on the specific quench path and the topological nature of the region traversed:

• Protocol 1 (Nontrivial Phase, Two Critical Points): The zero-mode transfer probability exhibits a smooth background following an anomalous power-law decay ($p_Z^0 \sim \tau_Q^{-1.33}$). Crucially, the oscillatory part ($p_Z^{osc}$) displays period doubling relative to the bulk interference period ($T_Z \simeq 2T_B$).
• Protocol 2 (Nontrivial Phase, One Critical Point Crossed Twice): While the bulk excitation density shows standard interference oscillations, the zero-mode transfer probability's oscillatory component is strongly suppressed. No stable ICDZM period is extracted, despite the system visiting the same topological sector as Protocol 1.
• Protocol 3 (Trivial Phase): The zero-mode transfer probability oscillates around a value of $1/2$, indicating an approximately balanced occupation of the zero-mode pair. The oscillation period matches the bulk interference period ($T_Z \simeq T_B$).

Mechanism and Theoretical Explanation
The authors attribute these phenomena to the interference phase accumulated during the quench, analyzed via the WKB framework:

• Period Doubling: In Protocol 1, the interference phase is accumulated within a topologically nontrivial region. Here, the energy gap relevant to zero-mode transitions ($\Delta_Z$) is approximately half the bulk energy gap ($\Delta_B$) due to the presence of exponentially small zero-mode energies ($E_{1,\pm} \approx 0$). Since the oscillation period is inversely proportional to the integrated energy gap, the halved gap results in a doubled period ($T_Z \simeq 2T_B$).
• Suppression: In Protocol 2, although the path is similar, crossing the same critical point twice leads to a cancellation or strong suppression of the specific zero-mode interference term, preventing the observation of the period-doubled pattern.
• Standard Period: In Protocol 3, the phase is accumulated in a topologically trivial region where no zero modes exist to reduce the energy gap. Consequently, the zero-mode dynamics follow the bulk gap, resulting in $T_Z \simeq T_B$.

Observability and Edge Defects
The paper demonstrates that the zero-mode transfer probability can be experimentally accessed without direct measurement of zero-mode operators. The authors show that the left edge defect ($d_{left}$), defined as the deviation of the boundary particle number from its initial fractional value ($N_1^i - N_1^f$), faithfully reproduces the ICDZM oscillation patterns. This connects the interference phenomenon to the concept of boundary charge and fractionalization, showing that the edge defect captures both the ICDZM oscillation and the anomalous scaling of one-way quench dynamics.

Significance
The paper claims to identify ICDZM and the corresponding edge defect as effective probes for critical dynamics associated with topological zero modes. The primary contribution is demonstrating that the interference pattern (specifically the period and visibility) encodes information about the topological sector visited during the quench. Specifically, the period doubling serves as a signature of phase accumulation in a topologically nontrivial region involving zero modes, distinguishing it from bulk dynamics or dynamics in trivial regions. This provides a method to distinguish topological features in non-equilibrium processes that are not accessible via standard one-way quench protocols or bulk observables alone.