Interplay of localization and topology in disordered dimerized array of Rydberg atoms

This paper numerically demonstrates that in a disordered, dimerized one-dimensional array of Rydberg atoms, both positional disorder and dimerization induce a unique localized phase characterized by Hilbert space fragmentation and an extensive fraction of Symmetry Protected Topological (SPT) states across the energy spectrum, distinct from standard many-body localization.

The Gist

The Big Picture: A Dance Floor with a Twist

Imagine a long line of dancers (the Rydberg atoms) on a stage. In a perfect world, they stand in a neat, evenly spaced line, holding hands and dancing in perfect sync. This is a "clean" quantum system.

But in this experiment, the scientists did two things to mess things up:

1. They shuffled the floor (Disorder): They randomly moved the dancers slightly left or right, so the spacing between them is uneven.
2. They paired them up (Dimerization): They forced the dancers to stand very close to their neighbors in specific pairs, creating "strong bonds" between some and "weak bonds" between others.

The question the paper asks is: When you mix random shuffling with forced pairing, what happens to the dance? Does the whole line eventually forget the original choreography and just jiggle randomly (thermalize), or does it get stuck in a frozen, weird state?

The Surprise: It's Not the Usual "Freeze"

Usually, when physicists mess up a quantum system with too much disorder, they expect Many-Body Localization (MBL). Think of this like a traffic jam where every car is stuck in its own spot, completely ignoring the cars next to it. The system freezes, and information about the start is preserved forever.

But this paper found something stranger.

Instead of a total traffic jam, the system forms a "Hilbert Space Fragmentation."

• The Analogy: Imagine the dance floor is a giant puzzle. In a normal jam, every piece is stuck. In this new state, the puzzle breaks into separate, isolated islands.
• Some dancers are stuck in a tiny 2-person huddle. Others are in a 4-person huddle.
• Because of the specific way they are paired and shuffled, the dancers in one huddle can't "talk" to the dancers in another huddle. They are trapped in their own little bubbles.
• The system isn't just frozen; it's shattered into many tiny, isolated rooms where different rules apply.

The Ghost in the Machine: Spin Glass vs. Topology

The scientists noticed two competing forces fighting for control of this shattered dance floor:

1. The "Glassy" Mess (Spin Glass):
   Because the dancers are randomly shuffled, some pairs get stuck in awkward, rigid positions. They form a "spin glass"—a state that is disordered and frozen, like a glass of water that has been supercooled but hasn't turned to ice. It's messy and chaotic.

2. The "Topological" Secret (SPT Order):
   Despite the mess, the scientists found that a huge number of the dancers are actually holding a secret, special handshake. This is called Symmetry Protected Topological (SPT) order.

   • The Analogy: Imagine that even though the dance floor is a mess, the dancers at the very ends of the line (the edges) are still holding hands in a special, unbreakable way that the dancers in the middle can't replicate.
   • Usually, if you have a "glassy" mess, you lose these special edge handshakes. But here, the "shattered islands" allow the special edge handshakes to survive even in the middle of the chaos.

The Big Discovery: The system manages to host both the messy, frozen glass state and the special, protected topological state at the same time. It's like having a chaotic party where, surprisingly, a secret club still exists in the corner.

How They Proved It (The "Disconnect" Test)

How do you know if the dancers at the ends are really holding hands secretly? You can't just look at the middle.

The scientists used a clever math trick called "Disconnected Entropy."

• The Analogy: Imagine you want to check if the person at the far left of the line is connected to the person at the far right. Instead of looking at the whole line, you cut the line into four pieces: Left, Middle-Left, Middle-Right, and Right.
• You measure the "connection" between the Left and Right pieces ignoring the middle.
• If the connection is strong and "quantized" (comes out as a perfect whole number), it proves the dancers at the ends are sharing a secret, topological bond that survives the chaos.

They found that for many of the "shattered islands," this secret bond was indeed there.

The Future: Watching the Dance

Finally, the paper suggests how to see this in real life (using Rydberg atoms in optical tweezers).

• The Experiment: If you start the dancers in a specific pattern and watch them over time, you should see the dancers at the ends of the line oscillating (swinging back and forth) in a very specific rhythm.
• The Result: The time it takes for them to swing back depends on the length of the line in a very specific way (scaling with the cube or sixth power of the size). This proves that the "secret handshake" is real and that the system is governed by these special topological rules, not just random noise.

Summary in One Sentence

By shuffling a line of quantum dancers and forcing them into pairs, the scientists discovered that the system doesn't just freeze; it shatters into isolated islands where a special, secret "topological" connection between the ends survives, even amidst a chaotic, glassy mess.

Technical Summary

1. Problem Statement

The paper investigates the interplay between positional disorder and dimerization in a one-dimensional array of Rydberg atoms. Specifically, it addresses how these factors affect the localization properties and topological order of excited states in a long-range interacting spin-1/2 system.

• Context: While Many-Body Localization (MBL) is a well-known mechanism for ergodicity breaking driven by strong disorder, recent research has identified other non-ergodic phases, such as those arising from Hilbert Space Fragmentation (HSF).
• Gap: It is unclear how topological phases, specifically Symmetry Protected Topological (SPT) orders, survive or coexist with localization mechanisms that are not standard MBL, particularly in systems with long-range interactions and positional disorder.
• Goal: To characterize the phase diagram of a disordered, dimerized XY model realized in Rydberg tweezer arrays, determining whether the system exhibits standard MBL, HSF, and if non-trivial SPT states persist across the energy spectrum.

2. Methodology

Model System

The authors study a 1D spin-1/2 chain described by a long-range XY Hamiltonian:
$$H = J \sum_{i>j} \frac{1}{r_{ij}^3} (s_i^x s_j^x + s_i^y s_j^y)$$
where $r_{ij}$ is the distance between spins. The system possesses $U(1)$ symmetry (conservation of total magnetization) and an antiunitary symmetry $S = (\prod s_i^x) \circ K$, protecting SPT order.

Disorder and Dimerization:
Disorder is introduced via positional randomness. The spin positions $r_i$ are defined as:
$$r_i = \left( i + \delta(-1)^{i+1} - \frac{1}{2} \right)a + i D_{rb} + W_i$$

• $\delta$ (Dimerization): Controls the pairing pattern. $\delta < 0$ creates a non-trivial topological phase (SSH-like), while $\delta > 0$ creates a trivial phase.
• $W$ (Disorder Strength): Random shifts $W_i$ drawn from a uniform distribution $[-W, W]$.
• Constraints: The geometry imposes a triangular parameter space constraint on $W$ and $\delta$ to prevent spin overlap.

Numerical Techniques

• Exact Diagonalization (ED): Used for system sizes up to $L=16$ (and up to $L=18$ for specific statistical averages) to compute full spectra.
• Polynomially Filtered Exact Diagonalization (POLFED): Used for larger system sizes ($L > 16$) to access specific spectral regions.
• Real-Space Renormalization Group for Excited States (RSRG-X): A theoretical framework used to analytically understand the structure of eigenstates in the localized regime, predicting the formation of effective dimers and edge modes.

Observables

• Statistical Properties: Mean gap ratio ($\langle r_n \rangle$) and half-chain entanglement entropy ($S_{ent}$) to distinguish ergodic (GOE) from localized (Poissonian/Sub-Poissonian) phases.
• Spin-Glass Order: Edwards-Anderson order parameter ($m_{EA}$) to detect long-range Ising correlations.
• Topological Order:
  • Standard Entanglement Spectrum (failed due to "cat states").
  • Disconnected Entropy ($S_D$): Specifically $S_D^2$, defined via a specific partition of the chain to detect long-range entanglement between edge states.
  • Time Dynamics: Quench dynamics of specific initial states to observe edge oscillations.

3. Key Contributions and Results

A. Localization Phase Diagram (Non-Standard MBL)

The system does not exhibit standard Many-Body Localization. Instead, it enters a localized phase driven by Hilbert Space Fragmentation (HSF).

• Gap Ratio Behavior:
  • In the region of large disorder ($W$) and small dimerization ($\delta \approx 0$), the gap ratio is slightly above the Poissonian value ($r_{Poi} \approx 0.389$), indicating residual level repulsion.
  • In the region of strong dimerization ($|\delta| \to 1$), the gap ratio becomes strongly sub-Poissonian. This is a signature of HSF, where the Hilbert space breaks into disconnected sectors, leading to "level attraction" and closely spaced energy pairs.
• Entanglement: The localized phase exhibits a multimodal distribution of entanglement entropy, with peaks at integer values ($S_{ent} \approx 1, 2$), corresponding to frozen domains or "cat states" rather than the zero-entropy typical of standard MBL.

B. Spin-Glass Order

Despite the absence of explicit $ZZ$ interactions, the localized regime exhibits a partial spin-glass order.

• The Edwards-Anderson parameter ($m_{EA}$) extrapolates to a non-zero value in the thermodynamic limit for the localized phase.
• This glassy order is driven by the formation of domains of parallel spins (chemical potential-like terms generated by the RSRG-X procedure).

C. Coexistence of Topology and Localization

A major finding is that non-trivial SPT states coexist with the spin-glass order across a significant fraction of the energy spectrum.

• SPT Identification: Using the disconnected entropy $S_D^2$, the authors identify states with quantized values ($S_D^2 = 1, 2$).
  • $S_D^2 = 2$: Corresponds to maximally correlated edge states (topologically non-trivial).
  • $S_D^2 = 1$: Corresponds to domain walls in the bulk.
• Thermodynamic Limit: Extrapolation suggests that a finite fraction of these topological states survives in the thermodynamic limit, even in the presence of disorder and glassy order.
• Distinction: The topological states exhibit significantly weaker $ZZ$ correlations than the rest of the spectrum, meaning they contribute negligibly to the global spin-glass order parameter. Thus, the system hosts a "glassy" background with embedded "topological" islands.

D. Time Dynamics Verification

The authors propose experimental verification via quench dynamics:

• Edge Oscillations: Preparing a state with frozen edge spins and dimerized bulk leads to coherent oscillations between edge states.
• Scaling: The oscillation period $T$ scales as $L^3$ for simple edge modes and $L^6$ for more complex exchange processes, matching the predictions of the long-range tunneling Hamiltonian and RSRG-X theory.

4. Significance

1. Beyond Standard MBL: The paper provides a concrete example of a localized phase that is distinct from standard MBL. It demonstrates that Hilbert Space Fragmentation induced by positional disorder and dimerization can lead to sub-Poissonian statistics and multimodal entanglement, challenging the universality of the MBL paradigm.
2. Robustness of Topology: It resolves a theoretical tension regarding whether SPT order can exist in glassy, disordered systems. The results show that topological order can survive in excited states even when the system exhibits spin-glass characteristics, provided the symmetry is not spontaneously broken by the specific eigenstate.
3. Experimental Relevance: The model is directly realizable with current Rydberg atom tweezer arrays. The predicted scaling of oscillation periods ($L^3$ and $L^6$) offers a clear experimental signature to distinguish between different topological and localization mechanisms.
4. Methodological Advance: The successful application of RSRG-X to long-range interacting systems and the use of disconnected entropy ($S_D^2$) to detect topology in the presence of "cat states" provide new tools for analyzing complex quantum phases.

Conclusion

The study reveals a rich phase diagram where positional disorder and dimerization drive a system into a non-ergodic, Hilbert-space fragmented phase. This phase hosts a partial spin-glass order while simultaneously supporting a robust fraction of Symmetry Protected Topological states across the entire energy spectrum. The findings suggest that topological protection can be remarkably resilient, persisting even in the presence of glassy dynamics and strong disorder, offering a new platform for studying the interplay of localization and topology in quantum matter.