Fate of a Fractional Chern Insulator under Nonlocal Interactions in Synthetic Dimensions

This paper demonstrates that infinite-range nonlocal interactions in synthetic dimensions can adiabatically transform a Laughlin-type fractional Chern insulator into a charge-ordered Tao-Thouless state without closing the many-body gap, revealing that conventional topological markers remain unchanged despite the loss of locality-protected robustness.

The Gist

Imagine you are trying to build a very special, intricate sandcastle on a beach. In the world of quantum physics, this "sandcastle" is a state of matter called a Fractional Chern Insulator (FCI). It's a magical, topological state where particles dance together in a way that is incredibly robust; if you poke it or push it locally, the whole structure holds its shape. This is usually only possible in a flat, two-dimensional world where particles only talk to their immediate neighbors.

Now, imagine you want to build this sandcastle using a new, futuristic tool: Synthetic Dimensions.

The Setup: The "Magic Elevator"

In a normal lab, atoms sit on a grid. But in this experiment, scientists use a trick called a "synthetic dimension." Think of it like a magic elevator built into the side of your grid.

• The Real World (x-axis): The atoms sit in a line on the floor.
• The Synthetic World (y-axis): Instead of moving up a real staircase, the atoms can "hop" between different internal energy levels (like different floors of the elevator) using laser beams.

The problem? In the real world, if you are on the 1st floor and your friend is on the 10th floor, you can't touch them. But in this synthetic elevator, because all these "floors" are actually just different states of the same atom sitting in the same spot on the floor, everyone is physically co-located.

This creates a weird, nonlocal interaction: The atom on the "1st floor" of the elevator feels a strong force from the atom on the "10th floor," even though they are technically the same particle. It's like if you could instantly feel a hug from your future self.

The Experiment: Tuning the "Hug"

The researchers asked: What happens to our magical sandcastle if we turn up the strength of these "elevator hugs" (the nonlocal interactions)?

They started with a perfect, topological sandcastle (the Laughlin state). In this state:

1. The Pattern: The particles are arranged in a complex, entangled dance that is hard to break.
2. The Markers: They checked the "ID cards" of the state (mathematical markers like the Chern number and entanglement entropy). These cards said, "Yes, this is a topological state!"

Then, they slowly increased the strength of the nonlocal interactions (the "hugs").

The Surprise: The "Fake-Out"

As they turned up the interaction, something bizarre happened.

1. The ID Cards Didn't Change:
The mathematical "ID cards" (Chern number, entanglement entropy) still said, "This is a topological state!" It looked like nothing had changed. The energy gap (the safety buffer) stayed open. It seemed like the sandcastle was perfectly safe.

2. The Reality Check:
But when they looked closer, they realized the sandcastle had actually collapsed into a different shape.

• The Robustness Test: In a true topological state, if you poke it with a tiny stick (a local perturbation), it doesn't care. But in this new state, a tiny poke shattered the structure. The "magic" was gone.
• The New Shape: The particles had rearranged themselves into a simple, rigid line of alternating density (a Charge Density Wave). Think of it like the sandcastle turning into a straight, rigid fence. It looks orderly, but it's not "topological" anymore. It's just a regular, boring fence.

The Metaphor:
Imagine a jigsaw puzzle that is glued together in a way that if you try to pull one piece out, the whole puzzle resists (Topological Order).
Now, imagine you have a magnetic glue that connects every piece to every other piece, regardless of where they are.

• At first, the puzzle looks like the glued jigsaw.
• As you turn up the magnetic glue, the pieces snap into a rigid, straight line (the fence).
• If you look at the "shape" of the puzzle, it still looks like it has a complex structure (the ID cards say "Topological").
• But if you try to pull one piece, the whole thing falls apart because the "glue" is now just holding a rigid line, not a complex web. The "topological protection" is gone, even though the math says it's still there.

The Big Discovery: A Secret Shortcut

The most exciting part of this paper is what this means for the future.

Usually, to turn a topological state (the sandcastle) into a normal state (the fence), you have to break the energy gap. It's like having to melt the sandcastle completely before you can build a fence. This takes time and energy.

But here, because of the synthetic dimension, the researchers found a secret tunnel.

• They can smoothly slide from the "Sandcastle" to the "Fence" without ever breaking the energy gap.
• It's like a magic morph in a video game where the character changes form without losing health.

Why Does This Matter?

1. New Way to Build States: Scientists can now prepare these complex quantum states much faster. They can start with a simple, easy-to-make "fence" (a charge-ordered state) and smoothly morph it into the complex "sandcastle" (the topological state) without the system crashing.
2. Testing the Rules: This experiment shows that our old rules for identifying "topological order" might be too simple. If you have these weird, long-range "elevator hugs," the usual math tricks can lie to you. It tells us we need new ways to understand quantum matter when we break the rule of "locality" (things only talking to neighbors).

In a nutshell: The researchers found a way to trick a quantum system into changing its fundamental nature without breaking the rules of energy, using a "synthetic elevator" to create long-range connections that act like a secret shortcut between two very different worlds.

Technical Summary

1. Problem Statement

The paper addresses a fundamental tension in the study of Synthetic Dimensions (SDs) in ultracold atomic systems. While SDs offer a powerful route to engineer topological lattice models (such as Fractional Chern Insulators, or FCIs) by coupling internal atomic degrees of freedom, they introduce an intrinsic nonlocality.

• The Conflict: In conventional 2D topological phases, topological order (TO) relies on locality and is robust against local perturbations. However, in SDs, atoms at different positions along the synthetic dimension physically occupy the same real-space site. This leads to infinite-range, anisotropic interactions ($U_i$) along the synthetic direction, regardless of their synthetic separation.
• The Question: Can a genuine topological phase (specifically a bosonic Laughlin state) survive in the presence of these strong, nonlocal interactions? Do standard topological markers remain valid, or does the system undergo a transition to a trivial phase without closing the energy gap?

2. Methodology

The authors investigate an extended Harper-Hofstadter model for bosons on a 2D lattice ($L_x \times L_y$) with a synthetic gauge field.

• Hamiltonian: The model includes kinetic hopping ($t$), a Peierls phase ($\alpha$) mimicking a magnetic field, onsite interactions ($U_0$), and the key infinite-range column interaction ($U_i$) coupling all particles within the same physical column ($x$).
  $$\hat{H} = -t \sum \dots + \frac{U_0}{2} \sum \hat{n}(\hat{n}-1) + U_i \sum_{x, y<y'} \hat{n}_{x,y}\hat{n}_{x,y'}$$
• Parameters: The study focuses on the hard-core limit ($U_0 \to \infty$) and a specific filling factor where the 1D physical density is non-integer ($\rho_{1D} = N/L_x = 1/2$). This regime is chosen to avoid Mott insulating transitions that occur at integer densities.
• Numerical Techniques:
  • Exact Diagonalization (ED): Used for small lattices (up to $L_x=10$) to calculate energy spectra and entanglement properties.
  • Tree Tensor Networks (TTN): Employed for larger systems ($16 \times 8$) to compute the topological entanglement entropy and entanglement spectra efficiently.
• Diagnostic Tools: The authors monitor several markers as $U_i$ is tuned from 0 to large values:
  • Many-body energy spectrum (gap and degeneracy).
  • Many-body Chern number ($C$) via twisted boundary conditions.
  • Topological Entanglement Entropy (TEE, $\gamma$).
  • Robustness to local pinning potentials.
  • Particle Entanglement Spectrum (PES) counting.
  • Momentum-space four-point correlators.

3. Key Results

A. Persistence of "Standard" Topological Markers

As $U_i$ increases, the system undergoes a smooth evolution. Surprisingly, several conventional topological indicators remain unchanged:

• Gap and Degeneracy: The many-body energy gap remains open, and the ground state remains two-fold degenerate (characteristic of the Laughlin state) across the entire range of $U_i$.
• Chern Number: The many-body Chern number remains quantized at $C = 1/2$.
• Topological Entanglement Entropy: The TEE ($\gamma$) remains approximately $1/2$ (consistent with a Laughlin state at filling $\nu=1/2$).
• Conclusion from these markers: Naively, these results suggest the system remains in the same topological universality class as the $U_i=0$ Laughlin state.

B. Breakdown of Topological Order

Despite the persistence of the above markers, the authors demonstrate that the system loses genuine topological order at large $U_i$:

1. Loss of Robustness: When a weak local pinning potential is introduced, the ground-state degeneracy is lifted for large $U_i$. Unlike the $U_i=0$ case where degeneracy is robust to local disorder, the large-$U_i$ states become locally distinguishable. This indicates the phase is topologically trivial in the thermodynamic limit.
2. Particle Entanglement Spectrum (PES) Restructuring: The PES counting, which is a robust fingerprint of the phase, changes drastically.
   • At $U_i=0$, the counting matches the Laughlin state ($N=50$ for the specific sector).
   • At large $U_i$, the counting shifts to match a Charge Density Wave (CDW) state ($N=20$). This signals a transition from a topological fluid to a trivial, ordered state.
3. Momentum-Space Ordering: While real-space correlations remain featureless (no symmetry breaking in real space), the momentum-space four-point correlator $C^{(4)}(k_y, k'_y)$ develops a pronounced periodic structure. This identifies the new phase as a $k$-DW (momentum-space density wave), analogous to a Tao-Thouless state in the thin-torus limit.

C. Adiabatic Connectivity

A critical finding is that the transition from the topological Laughlin state to the trivial $k$-DW state occurs without closing the many-body gap.

• This implies the existence of an adiabatic path connecting a topologically ordered phase to a trivial phase.
• This challenges the conventional wisdom that a phase transition between topological and trivial phases must involve a gap closing (a critical point).

4. Significance and Implications

• Redefining Topological Order in Synthetic Dimensions: The paper highlights that in systems with engineered nonlocality, standard topological markers (Chern number, TEE, gap) can be "decoupled" from the actual topological nature of the state. A state can possess non-zero Chern numbers and TEE yet be topologically trivial due to the loss of locality protection.
• New State Preparation Protocol: The adiabatic connection without gap closure offers a novel route for state preparation. One could theoretically prepare a complex Fractional Quantum Hall (FQH) state by:
  1. Initializing a simple, trivial Charge Density Wave (CDW) state (easy to prepare in optical lattices).
  2. Adiabatically tuning the nonlocal interaction strength $U_i$ to zero.
  3. This avoids the need to cross a critical point, potentially allowing for faster and more robust preparation of FQH states.
• Theoretical Challenge: The results challenge the foundational assumptions of topological order theory, which relies heavily on locality. It suggests that "topological" invariants in synthetic dimensions may not always imply topological protection against local perturbations.
• Role of Density: The study clarifies that this phenomenon is specific to incommensurate 1D densities ($\rho_{1D} = 1/2$). At commensurate densities ($\rho_{1D} = 1$), the topological degeneracy breaks down immediately with non-zero $U_i$, leading to a trivial CDW without an adiabatic bridge.

Conclusion

Geraghty et al. demonstrate that strong nonlocal interactions in synthetic dimensions can drive a system from a Laughlin-type Fractional Chern Insulator to a trivial momentum-space density wave. This transition preserves the energy gap, Chern number, and topological entanglement entropy but destroys the robustness to local perturbations and reshapes the entanglement spectrum. This reveals a "hidden" transition where topological markers fail to diagnose the loss of true topological order, while simultaneously opening a new pathway for the adiabatic preparation of topological states.