Occupation-selective topological pumping from Floquet gauge fields

This paper demonstrates that periodically driving a one-dimensional superlattice with density-dependent hopping induces occupation-selective topological pumping, where two-body bound states acquire distinct Chern numbers and exhibit quantized transport driven by a dynamical gauge field mechanism, even when single-particle transport is trivial.

The Gist

Imagine a crowded dance floor where people are trying to move from one side of the room to the other. In the world of quantum physics, this "dance floor" is a crystal lattice, and the "people" are particles like electrons or atoms.

Usually, scientists study how single particles move. They found that if you wiggle the floor in a specific, rhythmic way (a process called "topological pumping"), you can force these single dancers to march in a perfectly organized line, moving exactly one step forward every time the music completes a cycle. This is a very predictable, "one-size-fits-all" rule.

The Big Twist in This Paper
The authors of this paper asked a fascinating question: What happens if the rules of the dance floor change depending on how many people are standing on a single spot?

In the real world, if you are alone, you can walk through a door easily. If you are with a friend, you might have to squeeze through. If you are with a whole group, you might get stuck. This paper proposes a quantum system where the "door" (the tunneling mechanism) literally changes its size and shape based on how many particles are trying to pass through it.

Here is the breakdown of their discovery using simple analogies:

1. The "Smart Door" (Dynamical Gauge Field)

Imagine the dance floor has doors between rooms.

• Normal Physics: The doors are fixed. Whether one person or ten people are there, the door opens the same amount.
• This Paper's Physics: The doors are "smart." They are connected to a sensor.
  • If one person approaches, the door opens normally.
  • If two people (a "doublon") are huddled together, the door senses them and changes its behavior completely. It might open wider, or even swing in the opposite direction!

This "smart door" is what the scientists call a Dynamical Gauge Field. It's a rule that changes in real-time based on the crowd.

2. The "Ghost Walk" vs. The "Real Walk"

The most surprising result is what happens when you run the "pumping" cycle (the rhythmic wiggling of the floor):

• The Single Dancer: In some settings, the single dancer just stands still. The floor wiggles, but they end up exactly where they started. In physics terms, this is a "trivial" state (boring).
• The Double Dancer (The Doublon): Even though the single dancer is standing still, the pair of dancers (the doublon) suddenly starts marching! They move forward exactly one step, perfectly quantized.

The Analogy: Imagine a conveyor belt that is broken. A single box placed on it doesn't move. But if you tape two boxes together, the friction changes, and suddenly the pair of boxes starts moving perfectly down the line. The "broken" belt works perfectly for the pair, but not for the single box.

3. The "Reverse March"

Even cooler, the paper shows that you can tune the system so that:

• The single dancer moves Left.
• The double dancer moves Right.

They are moving in opposite directions at the same time, on the same floor, with the same music playing. This is called Counter-propagating Pumping. It's like a two-way street where cars and trucks are forced to drive in opposite lanes, even though the road itself hasn't changed.

4. Why Does This Matter?

For decades, physicists thought topological transport (this perfect, quantized movement) was a property of the material itself, like the texture of the floor. Everyone assumed that if the floor was "topological," everything on it would move the same way.

This paper proves that topology can be "occupation-selective."

• It's not just about the floor; it's about who is walking on it.
• By making the movement rules depend on the number of particles, we can create a system where different groups of particles have completely different "personalities" and destinies.

The "How-To" (The Recipe)

The authors suggest how to build this in a lab using ultracold atoms (atoms cooled to near absolute zero).

1. The Trap: Put the atoms in a laser grid (the dance floor).
2. The Wiggle: Shake the lasers rhythmically to create the pump.
3. The Magic Sauce: Use a second, faster vibration to make the "doors" between atoms sensitive to how many atoms are crowded together. This creates the "smart door" effect.

The Takeaway

This research opens the door to a new kind of quantum control. Instead of just building a machine that moves everything the same way, we can build machines that sort particles by their "group size."

• Want to move pairs but leave singles alone? Done.
• Want to move singles left and pairs right? Done.

It's a fundamental shift from thinking of particles as independent travelers to realizing that in the quantum world, who you are with changes where you can go.

Technical Summary

1. Problem Statement

Conventional topological pumping (e.g., the Thouless pump) is governed by single-particle band topology. In interacting systems, while correlations can modify dynamics or induce breakdown, the fundamental tunneling rules typically remain occupation-independent. Consequently, the topological invariants (Chern numbers) and resulting quantized transport are usually shared across different particle number sectors (e.g., single particles and bound states move in the same direction or with the same quantization).

The authors address a fundamental question: Can topological pumping become "occupation-selective"? That is, can different occupancy sectors (e.g., single particles vs. two-body bound states) exhibit distinct, independently tunable quantized transport responses under the same driving cycle?

2. Methodology

The authors propose a theoretical framework utilizing periodically driven (Floquet) systems with density-dependent hopping.

• Model System: A one-dimensional interacting bosonic superlattice with a time-dependent Hamiltonian $\hat{H}(t)$.
  • The system includes standard inter-cell hopping modulated by a slow phase $\phi(t)$.
  • Crucially, it features intra-cell hopping that depends on the local particle density ($\hat{n}$). This density-dependent term acts as a microscopic dynamical gauge field.
  • The Hamiltonian includes on-site interaction $U$, staggered potential $\Delta_0$, and modulated hopping parameters.
• Regime: The study focuses on the strong-interaction regime ($U \gg J, \Delta_0$), where two-body bound states (doublons) are isolated from the scattering continuum.
• Theoretical Tools:
  • Degenerate Perturbation Theory: Used to derive an effective single-particle Hamiltonian for the doublon sector, treating the bound state as a composite particle.
  • Topological Characterization: Calculation of Chern numbers ($C_d$) via Berry curvature integration and many-body polarization ($P_d$) using the projected position operator (Wilson loop).
  • Floquet Engineering: A proposal to realize the required density-dependent hopping experimentally using ultracold atoms in optical lattices with fast Raman-assisted tunneling and Feshbach resonance modulation.
• Dynamics: Adiabatic pumping simulations using a gap-adapted driving protocol ($\omega \propto \Delta_{min}^2$) to ensure high-fidelity transport despite transient gap closings.

3. Key Contributions

1. Occupation-Selective Topology: The paper establishes that promoting tunneling to a dynamical, occupation-conditioned variable fundamentally reshapes the topological paradigm. The Chern number of a bound state ($C_d$) can differ from that of a single particle ($C_s$) under identical driving conditions.
2. Dynamical Gauge Field Mechanism: The authors identify that the density-dependent hopping acts as a dynamical gauge field. This field renormalizes the effective intra-cell tunneling for bound states differently than for single particles, inducing topological phase transitions unique to the bound-state sector.
3. Geometric Compression: The dynamical gauge field concentrates Berry curvature into sharply localized resonant regions in parameter space. This "geometric compression" allows for sharp topological switching without breaking adiabaticity.
4. Experimental Proposal: A concrete Floquet engineering scheme is outlined for ultracold atoms, demonstrating how to synthesize the required occupation-dependent link synchronized with the pumping cycle.

4. Key Results

• Divergent Chern Numbers:
  • In the absence of the dynamical gauge field ($\gamma_0 = 0$), the doublon topology follows the single-particle topology.
  • With a non-zero dynamical gauge field ($\gamma_0 \neq 0$), the system exhibits occupation-selective phases. For example, the single-particle pump can be topologically trivial ($C_s = 0$) while the doublon pump is non-trivial ($C_d = \pm 1$).
• Counter-Propagating Pumping:
  • The most striking result is the ability to tune the pumping direction independently. The authors demonstrate regimes where single particles pump in one direction (e.g., left, $\Delta X = -1$) while doublons pump in the opposite direction (e.g., right, $\Delta X = +1$) under the same adiabatic cycle.
• Robust Quantized Transport:
  • Simulations confirm that doublons undergo quantized transport (shifting by exactly one unit cell per cycle) even when the single-particle sector is trivial.
  • The use of a gap-adapted driving protocol successfully suppresses non-adiabatic Landau-Zener transitions near the sharp Berry curvature peaks, ensuring the state tracks the Wannier center accurately.
• Extension to Higher Occupancy:
  • The Supplemental Material extends the analysis to three-particle bound states ("triolons"). The effective Hamiltonian for triolons shows that the dynamical gauge field contribution scales with occupancy (e.g., $\gamma + 2\gamma_0 \sin \phi$), leading to distinct phase boundaries and topological invariants ($C_t$) compared to doublons and single particles.

5. Significance

• Beyond Single-Particle Paradigm: This work breaks the conventional view that topological transport in driven systems is a universal property inherited from single-particle bands. It proves that topology can be an intrinsic property of specific many-body sectors.
• Control of Many-Body Topology: It introduces a powerful mechanism (dynamical gauge fields via density-dependent hopping) to engineer and control geometric responses in interacting systems, allowing for the selective manipulation of bound states.
• Experimental Feasibility: By providing a realistic Floquet engineering protocol for ultracold atoms, the paper bridges the gap between abstract topological concepts and experimental realization. The predicted "counter-propagating" pumping offers a clear, observable signature to distinguish occupation-selective effects from standard interaction-induced breakdown.
• New Regime of Quantum Simulation: The results open a pathway to studying hierarchical topological transport across different particle-number sectors, potentially leading to new forms of quantum matter where transport properties are encoded in the particle number itself.