[Paper Review] A Chern-Simons theory for dipole symmetry

Xiaoyang Huang|arXiv (Cornell University)|May 4, 2023

Black Holes and Theoretical Physics80 references4 citations

TL;DR

This paper introduces a novel Chern-Simons theory unified with both U(1) and dipole gauge fields to describe topological phases with dipole symmetry, enabling consistent coupling to curved spacetime and revealing that only the highest multipole symmetry supports the 't Hooft anomaly. The framework establishes a boundary-dependent bulk-edge correspondence and generalizes particle-vortex duality, while also deriving a fracton-elasticity duality via non-Abelian Chern-Simons theory in 3D, with level quantization derived from dipole flux threading.

ABSTRACT

We present effective field theories for dipole symmetric topological matters that can be described by the Chern-Simons theory. Unlike most studies using higher-rank gauge theory, we develop a framework with both U(1) and dipole gauge fields. As a result, only the highest multipole symmetry can support the 't Hooft anomaly. We show that with appropriate point group symmetries, the dipolar Chern-Simons theory can exist in any dimension and, moreover, the bulk-edge correspondence can depend on the boundary. As two applications, we draw an analogy between the dipole anomaly and the torsional anomaly and generalize particle-vortex duality to dipole phase transitions. All of the above are in the flat spacetime limit, but our framework is able to systematically couple dipole symmetry to curved spacetime. Based on that, we give a proposal about anomalous dipole hydrodynamics. Moreover, we show that the fracton-elasticity duality arises naturally from a non-abelian Chern-Simons theory in 3D.

Motivation & Objective

• To develop a systematic effective field theory for dipole symmetric topological matter using a unified U(1) and dipole gauge field framework.
• To resolve limitations of higher-rank gauge theories by enabling consistent coupling to curved spacetime and preserving distinct scalar/vector charge degrees of freedom.
• To clarify the role of point group and rotational symmetries in determining which multipole symmetries can support anomalies and topological order.
• To generalize particle-vortex duality to dipole phase transitions and establish a link to torsional anomalies in quantum Hall systems.
• To derive a non-Abelian Chern-Simons theory that naturally gives rise to fracton-elasticity duality in 3D.

Proposed method

• Formulates a gauge theory with two independent gauge fields: $A_\mu$ (U(1) gauge field) and $A^a_\mu$ (dipole gauge field), transforming nontrivially under combined U(1) and dipole shifts.
• Derives the action for the dipolar Chern-Simons theory in flat spacetime, with coupling to both $A_\mu$ and $A^a_\mu$, and ensures gauge invariance under the full symmetry group.
• Applies large gauge transformations to compute the phase acquired by a dipole moment moving in a loop, leading to flux quantization conditions.
• Uses the partition function invariance under large dipole gauge transformations to derive level quantization: $C_2 = k/(4\pi)$, $k \in \mathbb{Z}$, for the 2+1D case.
• Extends the formalism to curved spacetime by coupling to dipole Goldstone modes and constructing a non-Abelian Chern-Simons theory in 3D to realize fracton-elasticity duality.
• Applies differential forms and flux integrals over spheres and 3-spheres to generalize the quantization condition to higher dimensions.

Experimental results

Research questions

• RQ1 Which multipole symmetries can support an 't Hooft anomaly in a Chern-Simons theory, and why is only the highest multipole symmetry sufficient?
• RQ2 How does the bulk-edge correspondence in dipole symmetric systems depend on the boundary conditions and point group symmetries?
• RQ3 Can the dipole anomaly be analogized to the torsional anomaly in U(1) quantum Hall states, and what are the implications for topological order?
• RQ4 How can particle-vortex duality be generalized to include dipole symmetry in phase transitions?
• RQ5 What is the role of non-Abelian Chern-Simons theory in realizing the fracton-elasticity duality in 3D?

Key findings

• Only the highest multipole symmetry can support the 't Hooft anomaly in the dipolar Chern-Simons theory, as lower symmetries do not close the anomaly under gauge transformations.
• The bulk-edge correspondence in the dipole Chern-Simons theory depends on the boundary's point group symmetry, leading to non-universal edge behavior across different boundary types.
• The dipole anomaly is shown to be analogous to the torsional anomaly in U(1) quantum Hall states, with both arising from flux quantization under large gauge transformations.
• A generalized particle-vortex duality is established for systems with dipole symmetry, extending the standard duality to include multipole-conserving phases.
• The fracton-elasticity duality emerges naturally from a non-Abelian Chern-Simons theory in 3D, with the theory coupling to both dipole and translational symmetries.
• Level quantization is derived as $C_2 = k/(4\pi)$ with $k \in \mathbb{Z}$, based on the requirement that the partition function remains invariant under large dipole gauge transformations.