Vortices as Fractons

Fracton phases of matter are characterized by local excitations with restricted mobility, stemming from conservation laws of multipole moments of charge density. While theoretical advancements have significantly progressed the understanding of fracton phases, conclusive experimental evidence remains elusive. This study proposes a simple and experimentally accessible system to address this gap: superfluid vortices. Superfluids, such as superfluid Helium, Bose-Einstein condensates, polariton superfluids, and nonlinear media, exhibit quantized vortices whose dynamics are well-understood and experimentally manipulatable. The authors hypothesize that the behavior of these vortices aligns with the characteristics of fractons, specifically their constrained mobility, and will allow for the direct experimental observation and study of fracton physics. The importance lies in the potential to provide the first conclusive experimental verification of theoretical predictions within the field of fracton physics, opening new avenues of research and technological applications.

The paper reviews existing literature on fracton phases, highlighting their theoretical description using tensor and multipole gauge theories. It acknowledges the extensive theoretical work exploring fracton excitations and their potential applications, while emphasizing the lack of experimental confirmation. The paper also briefly touches on crystalline defects in quantum crystals and liquid crystals as examples of excitations with restricted mobility, providing a broader context for understanding the concept of constrained mobility. Existing research on vortex dynamics in superfluids and their Hamiltonian formulation is reviewed as the foundation for the analysis. The authors particularly draw on previous studies of the classical w-algebra and its relation to vortex hydrodynamics.

The authors begin by analyzing the dynamics of point vortices in a two-dimensional incompressible ideal fluid, using the Euler equations and deriving the Helmholtz equation for vorticity. The complex velocity field is expressed in terms of the positions and circulations of individual vortices. The Hamiltonian formulation of vortex dynamics is reviewed, showing that it conserves the dipole and quadrupole moments of vorticity. The quantization of vorticity in superfluids is crucial for the equivalence with a scalar charge. The authors then discuss the traceless scalar charge theory (TSCT), showing how the conservation laws for dipole and quadrupole moments restrict the mobility of vortices (similar to fractons): isolated vortices are immobile, while dipoles move perpendicularly to their dipole moment. The hydrodynamic limit, where the number of vortices is large, is explored using the Wiegmann-Abanov Hamiltonian and Poisson brackets. The authors demonstrate the conservation of dipole and quadrupole moments in this limit, obtaining a continuity equation consistent with the Helmholtz equation. A collective field theory approach is then presented to derive the traceless symmetric tensor current for an arbitrary number of vortices, further reinforcing the connection to TSCT. Finally, the behavior of vortices on curved surfaces is analyzed using the Helmholtz equation generalized to curved spaces, demonstrating how the dynamics of vortices and fractons align, particularly on surfaces with constant curvature. The authors highlight the breaking of dipole conservation and the loss of fractonic properties on surfaces with varying curvature.

The core finding is the established equivalence between vortex dynamics in two-dimensional superfluids and the traceless scalar charge theory (TSCT). This equivalence is shown for both finite numbers of vortices and in the hydrodynamic limit where a large number of vortices are present. Specifically, the authors demonstrate that: 1. **Conservation Laws:** The Hamiltonian system governing vortex motion conserves the dipole and quadrupole moments of vorticity, mirroring the conservation laws of TSCT. This is true for both a finite number of vortices and in the hydrodynamic limit. 2. **Fracton-like Behavior:** Isolated vortices behave like immobile fractons, while vortex dipoles move perpendicular to their dipole moment, consistent with the predicted behavior of fractons in TSCT. 3. **Hydrodynamic Limit:** Emergent vortex hydrodynamics, derived from the Wiegmann-Abanov Hamiltonian, also conserves the relevant dipole and quadrupole moments, further supporting the connection to fracton physics. 4. **Collective Field Theory:** A collective field theory derivation yields an expression for the traceless symmetric tensor current, providing a microscopic description consistent with the macroscopic conservation laws. 5. **Curved Space:** The behavior of vortices on curved surfaces, which can be experimentally realized, mirrors that of fractons. On surfaces with constant curvature, isolated vortices remain immobile, and dipoles move along geodesics perpendicular to their moment, consistent with fracton behavior. On surfaces with variable curvature, however, the fractonic properties are lost due to the breaking of dipole conservation. These findings demonstrate that superfluid vortices provide an experimentally accessible platform to study fracton physics, bridging the gap between theoretical predictions and experimental verification.

The equivalence between superfluid vortex dynamics and TSCT provides a direct experimental route to observe and study fracton quasiparticles. The ability to create and manipulate individual vortices and vortex dipoles using current technology offers a significant advantage. The results not only confirm theoretical predictions but also expand the understanding of fracton physics to include curved space considerations. The conservation laws derived, particularly the conservation of dipole and quadrupole moments, are not a consequence of internal symmetries as in previous studies, but rather arise from spatial symmetries and non-commutativity in configuration space. This suggests a deeper relationship between non-commutative field theories and fracton physics. Furthermore, the study's findings are extendable to systems of charged particles in a strong magnetic field, opening possibilities for exploring fracton phenomena in different physical contexts. The limitations of the study regarding the integrability of vortex dynamics, and ergodicity are acknowledged, suggesting that further investigation is warranted to address these issues and the broader implications on classical and quantum turbulence.

This paper establishes a clear connection between superfluid vortices and fracton physics, demonstrating that vortices offer a practical system for experimental verification of fracton behavior. The conservation of dipole and quadrupole moments, observed in both finite and hydrodynamic vortex systems, strengthens the equivalence with the traceless scalar charge theory. Furthermore, the study's extension to curved spaces highlights the interplay between fracton physics and geometry. Future work could focus on exploring higher-dimensional vortex systems, refining experimental probes to study fracton dynamics in superfluids, and investigating applications to other systems like electrons in the lowest Landau level.

The many-vortex dynamics are chaotic, which might affect the experimental observation of some of the predicted behaviours. For instance, the theoretical analysis focuses on idealized vortex configurations; real-world systems are influenced by factors such as thermal effects, the presence of impurities, and the boundaries of the superfluid. Additionally, while the equivalence to TSCT is established, the study acknowledges open questions around the ergodicity of the vortex system, which could have implications for the statistical mechanics of vortices. The authors acknowledge several avenues for future research which can improve the understanding of limitations associated with this study.