Su-Schrieffer-Heeger model driven by sequences of two unitaries: periodic, quasiperiodic, aperiodic, and random protocols

This paper investigates the topological and dynamical properties of the Su-Schrieffer-Heeger model driven by sequences of two unitaries under periodic, quasiperiodic, aperiodic, and random protocols, revealing discrepancies between end mode counts and winding numbers in periodic drives, and characterizing the distinct Loschmidt echo behaviors—ranging from long-lived oscillations to rapid decay—across different driving sequences.

The Gist

Imagine a long, narrow chain of atoms, like a string of beads. In this specific chain, called the Su-Schrieffer-Heeger (SSH) model, the beads are connected by springs of two different strengths. Sometimes the springs between beads in a pair are tight, and the springs connecting pairs are loose. Sometimes it's the other way around.

When the "loose" springs are weaker than the "tight" ones, something magical happens at the very ends of the chain: a special, invisible "ghost" particle appears. It's stuck to the end and doesn't want to move into the middle of the chain. This is called a topological end mode.

The scientists in this paper asked a big question: What happens if we shake this chain?

Instead of leaving the springs alone, they decided to rhythmically switch the spring strengths back and forth. They used two different "shaking patterns" (let's call them Shake A and Shake B) and applied them in different orders to see how the ghost particle at the end would react.

Here is what they found, broken down by how they shook the chain:

1. The Rhythmic Shaker (Periodic Driving)

Imagine shaking the chain in a perfect, repeating pattern: Shake A, Shake B, Shake A, Shake B...

• The Surprise: Sometimes, this rhythm creates ghost particles at the ends. But here's the catch: the number of ghosts doesn't always match the "mathematical rule" (called the winding number) that physicists usually use to predict them. It's like having a recipe that says "add 2 eggs," but sometimes you end up with 3, and sometimes 1, depending on exactly how you mix them.
• The Echo: When they started with a ghost particle and watched it dance, it didn't just sit still. It bounced back and forth with a very specific rhythm. If you listened to this bounce, you could hear a clear "note" (a frequency) that told them exactly how much energy the ghost particle had.

2. The Fibonacci Shaker (Quasiperiodic Driving)

Now, imagine a more complex pattern based on the Fibonacci sequence (1, 1, 2, 3, 5, 8...). You shake the chain using a pattern that grows like this: A, AB, ABA, ABAAB, ABAABABA...

• The Magic of Stability: If the difference between Shake A and Shake B is tiny, and the shakes are fast, the ghost particle at the end is incredibly stubborn. It refuses to leave. Even after millions of shakes, it stays right where it started, vibrating slightly but never fading away.
• The "Almost" Perfect: The scientists found that the longer they shook it, the more the ghost particle held on. It was as if the chaotic-looking Fibonacci pattern actually created a "shield" that protected the particle.
• The Breaking Point: However, if they shook it too long (billions of times) or if the difference between the two shakes was too big, the shield eventually cracked, and the ghost particle finally faded away.

3. The Thue-Morse Shaker (Aperiodic Driving)

This is another complex pattern, but it's generated differently (like flipping a coin but with strict rules: A, AB, ABBA, ABBABAAB...).

• The Result: This behaved very similarly to the Fibonacci shaker. The ghost particle stayed safe for a very long time. The complex, non-repeating pattern still managed to protect the particle, much like the Fibonacci pattern did.

4. The Random Shaker (Random Driving)

Finally, they tried shaking the chain with no pattern at all. Just pure chaos: A, B, A, A, B, B, A...

• The Disaster: The ghost particle didn't stand a chance. It faded away almost immediately. The lack of order meant there was no "shield" to protect it. The randomness scrambled the particle's memory of where it started, and it disappeared into the middle of the chain very quickly.

The "Why" Behind the Magic

The scientists explained this using a concept called the commutator (a fancy math way of saying "order matters").

• In the ordered patterns (Fibonacci/Thue-Morse): The specific way the shakes are arranged causes the "mistakes" or "jitters" to cancel each other out. It's like walking in a zig-zag pattern where every step to the left is perfectly balanced by a step to the right, keeping you in the same spot.
• In the random pattern: The mistakes pile up. It's like taking random steps in a crowd; eventually, you wander far away from where you started.

Summary

The paper shows that order matters. Even if the pattern isn't a simple repetition (like a metronome), as long as it follows a specific, structured rule (like Fibonacci), it can protect special particles at the edge of a material. But if you introduce pure randomness, that protection vanishes instantly.

This helps us understand how to keep delicate quantum states alive in future technologies by carefully designing how we "shake" or drive them, rather than just shaking them randomly.

Technical Summary

Technical Summary: Su-Schrieffer-Heeger Model Driven by Sequences of Two Unitaries

Problem Statement
The paper investigates the dynamical behavior of end modes in a one-dimensional topological system, specifically the Su-Schrieffer-Heeger (SSH) model, when subjected to various driving protocols. While periodic driving is well-studied for engineering topological phases and generating Floquet time crystals, the effects of quasiperiodic, aperiodic, and random driving on the stability of topological end modes remain less explored. The authors aim to determine how different sequences of two distinct unitary operators ($U_1$ and $U_2$), which differ slightly in their inter-cell hopping amplitudes, affect the existence, topological nature, and long-time stability of these boundary states.

Methodology
The study utilizes the SSH model Hamiltonian with alternating intra-cell ($J_1$) and inter-cell ($J_2$) hopping amplitudes. The system is driven by two unitary operators, $U_1 = e^{-iH_1T/2}$ and $U_2 = e^{-iH_2T/2}$ (or $e^{-iH_{1,2}T}$ in later sections), where $H_1$ and $H_2$ correspond to SSH Hamiltonians with slightly different inter-cell hopping parameters ($J_2$ and $J_2 + \epsilon$).

The authors analyze four distinct driving protocols:

1. Periodic: Alternating application of $U_1$ and $U_2$ ($U_2U_1$).
2. Quasiperiodic: Application following a Fibonacci sequence.
3. Aperiodic: Application following a Thue-Morse sequence.
4. Random: Application of $U_1$ and $U_2$ in a random order.

Key observables include:

• Floquet Spectrum: Calculation of eigenvalues and eigenvectors of the time-evolution operator to identify end modes and their quasienergies.
• Topological Invariants: Computation of the winding number for the periodic case using a symmetrized Floquet operator to address ambiguities in defining invariants for driven systems.
• Loschmidt Amplitude (LA) and Echo (LE): The LA is defined as $\langle \phi(t) | \phi(0) \rangle$, and the LE as $|\langle \phi(t) | \phi(0) \rangle|^2$, where $|\phi(0)\rangle$ is an initial end mode of $H_1$. These quantities measure the memory of the initial state over time.

Key Contributions and Results

• Periodic Driving:

  • When $U_1$ and $U_2$ are applied alternately, end modes can appear. However, the number of these modes does not always agree with the winding number (a $\mathbb{Z}$-valued topological invariant).
  • The authors distinguish between topological end modes (with Floquet eigenvalues exactly at $\pm 1$) and non-topological localized modes (with eigenvalues not equal to $\pm 1$).
  • The Loschmidt amplitude (LA) exhibits pronounced oscillations. The Fourier transform of the LA reveals peaks corresponding to the quasienergies of the end modes, specifically linking the dominant peak to the gap in the quasienergy spectrum.

• Quasiperiodic (Fibonacci) and Aperiodic (Thue-Morse) Driving:

  • When $U_1$ and $U_2$ are applied in Fibonacci or Thue-Morse sequences, and the difference in hopping parameters ($\epsilon$) and the time period ($T$) are small, the Loschmidt Echo (LE) remains remarkably stable.
  • The LE oscillates around a mean value very close to 1 for a very long duration (up to $n \sim 10^7$ drives) before eventually decaying to zero.
  • The deviation of the saturation value of the LE from 1 scales as $\epsilon^2$ for a fixed $T$.
  • The stability time $T_p$ (before decay begins) is found to be independent of $T$ (for small $\epsilon T$) but depends significantly on the intra-cell hopping $J_1$. As $J_1$ decreases (making end modes more localized), $T_p$ increases.
  • The authors attribute this long-time stability to the recursive nature of Fibonacci and Thue-Morse sequences, which leads to significant cancellations of first-order commutators in the Baker-Campbell-Hausdorff expansion of the product of unitaries. The effective Hamiltonian remains close to a topological phase for long times.

• Random Driving:

  • In contrast to the ordered sequences, random driving causes the LE to decay rapidly to zero, even for small $\epsilon$ and $T$.
  • The decay time $T_p$ (defined as the time for LE to drop to 0.9) scales approximately as $1/\epsilon^2$.
  • This rapid decay is attributed to the unbounded growth of fluctuations in the effective Hamiltonian due to the random walk nature of the sequence, where commutator terms do not cancel out.

Significance and Claims
The paper claims to provide a qualitative and quantitative understanding of how different driving protocols affect the stability of topological end modes. The primary finding is that quasiperiodic and aperiodic driving (specifically Fibonacci and Thue-Morse sequences) can protect end modes from decoherence and decay for exponentially long times compared to random driving, provided the driving parameters ($\epsilon$ and $T$) are sufficiently small.

The authors highlight that the stability in ordered sequences arises from the suppression of heating and decoherence mechanisms (commutator terms) due to the specific recursive structure of the sequences. They note that while the LE remains close to 1 for long times, it eventually decays for extremely large $n$ (order $10^8$), likely due to higher-order commutators.

The work suggests that the phase diagram of the SSH model under quasiperiodic driving may exhibit self-similar structures, analogous to findings in the transverse field Ising model, and points to the existence of emergent conservation laws in these driven integrable systems as a direction for future study. The paper does not propose new experimental setups but rather analyzes the theoretical dynamics of existing models under these specific driving conditions.