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Effective Gauge Fields and Topological Band Structures in Pilot-Wave Hydrodynamics

Ethan Andersson, Valeri Frumkin

TL;DR

Pilot-wave hydrodynamics provides a macroscopic, wave-particle platform to realize topological band structure, edge states, and synthetic gauge fields. By engineering bath topography, the work demonstrates Bloch-band gaps in a submerged lattice, and, with a honeycomb lattice, valley-Hall–like edge transport along a 1D interface. A chiral annular geometry induces an emergent vector potential that yields an Aharonov-Bohm–like phase difference between clockwise and counter-clockwise droplet orbits, evidencing a gauge-field effect without fluid rotation. This table-top system offers direct access to wave-particle manifestations of topological transport and pioneer exploration of synthetic gauge fields in a macroscopic setting.

Abstract

We demonstrate that pilot-wave hydrodynamics provides a macroscopic platform for realizing band-structure physics, topological edge states, and gauge-field-induced phase shifts. We show that a submerged square lattice produces frequency-dependent transmission governed by Bloch bands. An inversion-asymmetric honeycomb lattice confines the droplet to a domain wall, revealing a hydrodynamic analog of a valley-Hall edge state. And a chiral annular structure generates an effective gauge field that produces an Aharonov-Bohm-like phase difference between clockwise and counter-clockwise orbits. Unlike conventional wave analogs, pilot-wave hydrodynamics couples a localized particle to its self-generated wave field, providing direct access to topological wave-particle behavior normally associated with quantum systems.

Effective Gauge Fields and Topological Band Structures in Pilot-Wave Hydrodynamics

TL;DR

Pilot-wave hydrodynamics provides a macroscopic, wave-particle platform to realize topological band structure, edge states, and synthetic gauge fields. By engineering bath topography, the work demonstrates Bloch-band gaps in a submerged lattice, and, with a honeycomb lattice, valley-Hall–like edge transport along a 1D interface. A chiral annular geometry induces an emergent vector potential that yields an Aharonov-Bohm–like phase difference between clockwise and counter-clockwise droplet orbits, evidencing a gauge-field effect without fluid rotation. This table-top system offers direct access to wave-particle manifestations of topological transport and pioneer exploration of synthetic gauge fields in a macroscopic setting.

Abstract

We demonstrate that pilot-wave hydrodynamics provides a macroscopic platform for realizing band-structure physics, topological edge states, and gauge-field-induced phase shifts. We show that a submerged square lattice produces frequency-dependent transmission governed by Bloch bands. An inversion-asymmetric honeycomb lattice confines the droplet to a domain wall, revealing a hydrodynamic analog of a valley-Hall edge state. And a chiral annular structure generates an effective gauge field that produces an Aharonov-Bohm-like phase difference between clockwise and counter-clockwise orbits. Unlike conventional wave analogs, pilot-wave hydrodynamics couples a localized particle to its self-generated wave field, providing direct access to topological wave-particle behavior normally associated with quantum systems.
Paper Structure (1 section, 8 equations, 4 figures)

This paper contains 1 section, 8 equations, 4 figures.

Table of Contents

  1. Introduction.

Figures (4)

  • Figure 1: A schematic illustration of the experimental setup: (a) A loudspeaker provides harmonic forcing, $f=\gamma \sin \omega t$ to a circular base plate that holds a small bath of silicon oil with 3D printed topography. (b) A rhombus-shaped bath containing a square lattice of circular pillars of radius $r=0.5$ mm, depth $h=10$ mm, and lattice constant $a=6$ mm, used as a barrier between its two sides. (c) A circular channel $2$ mm deep, with inner radius of $18.5$ mm and outer diameter of $24.5$ mm, enclosing a strip with chiral topography. The well at the center is $8$ mm deep and $10$ mm in radius.
  • Figure 2: Band-gap behavior in pilot-wave hydrodynamics: (a) Transmission probability as a function of driving frequency. At $71$ Hz the droplet is transmitted through the lattice; transmission probability drops with an increase in frequency, and starting at $82$ Hz, a hydrodynamic band-gap is opened and the droplet is reflected at all times. (b) As the droplet enters a square lattice of circular pillars (radius $r=0.5$ mm, lattice constant $a=6$ mm) in the transmission regime ($f=71$ Hz), its pilot wave exhibits a spatially modulated envelope in accordance with Bloch’s theorem for waves in periodic media. (c) Outside the lattice, the pilot-wave returns to its unperturbed horseshoe profile.
  • Figure 3: Topological edge states in pilot-wave hydrodynamics: (a) A honeycomb lattice of circular pillars of radius $r=0.8$ mm, with two pillar heights $a=12$ mm and $b=13$ mm, corresponding to a controlled modulation of the local depth on the $A$ and $B$ sublattices (lattice constant $c=12$ mm), producing a 1D interface running along the center of the bath. (b) When driven at frequencies outside the band gap of the bulk honeycomb crystal (here $79$ Hz), the droplet is not confined to the interface, exploring the entire lattice. (c) Inside the band gap (here $83$ HZ), the droplet does not penetrate into either domain, locking onto the boundary and traveling along it.
  • Figure 4: Geometric phase accumulation in pilot-wave hydrodynamics: (a) A top view of the experiment showing the skewing of the droplet's pilot-wave by the underlying chiral topography. (b) Accumulative phase difference per cycle between clockwise and counter-clockwise rotations of a droplet circling submerged chiral topography, demonstrating Aharonov-Bohm-like level splitting. An extended measurement of the accumulative phase, showing an excellent linear fit over $40$ cycles, is provided in the SM.