[Paper Review] Hybrid rank-1 and rank-2 U(1) lattice gauge theory, the F3 model, and its effective field theory

Jintae Kim, Yun-Tak Oh+1 more|arXiv (Cornell University)|Mar 28, 2022

Physics of Superconductivity and Magnetism4 citations

TL;DR

This paper identifies the F3 model—a three-dimensional spin model combining features of the 3D toric code and X-cube model—as arising from a hybrid rank-1 and rank-2 U(1) lattice gauge theory (LGT) via Higgsing. It constructs the effective field theory (EFT) of the F3 model, showing that fracton dynamics are tied to fluxon degrees of freedom through a modified BF-type Lagrangian, and derives hydrodynamic equations of motion that reflect constrained quasiparticle dynamics.

ABSTRACT

A number of exactly solvable spin models, including the Kitaev toric code in two and three dimensions and the X-cube model in three dimensions, can be related to their respective parent lattice gauge theories (LGT) through the mathematical process of 'Higgsing'. Field theories of the low-energy excitations of these spin models can be developed subsequently, building upon the symmetry of the parent LGTs. Recently, two of the present authors proposed a variant of the three-dimensional toric code which we now call the F3 model, whose elementary excitations consist of freeon and fluxon excitations of the three-dimensional toric code and fracton excitations of the X-cube model. In this work, we identify the parent LGT of the F3 model as the hybrid rank-1 and rank-2 U(1) LGT, and develop the corresponding field theory. The resulting Lagrangian of the F3 model is that of a three-dimensional toric code with an extra term, which ties the dynamics of fractons to that of fluxons. The matter part of the effective action for the F3 model can be derived as well, by carefully keeping track of the gauge invariance of the F3 model. Hydrodynamic equations of motion of the quasiparticles are derived, which properly reflect their constrained dynamics. Finally, we present a tight-binding model for the quasiparticle motion in the F3 model.

Motivation & Objective

• To identify the parent lattice gauge theory (LGT) of the F3 model, which exhibits combined excitations of the 3D toric code and X-cube model.
• To construct the effective field theory (EFT) of the F3 model by systematically deriving the gauge-invariant Lagrangian from the hybrid LGT.
• To derive hydrodynamic equations of motion for the quasiparticles (freeons, fluxons, fractons) that correctly encode their constrained dynamics.
• To establish a tight-binding model for quasiparticle motion in the F3 model, grounded in the derived EFT.
• To clarify the gauge transformation properties of the hybrid rank-1 and rank-2 gauge fields and their role in ensuring consistency of the EFT.

Proposed method

• Identify the parent LGT as a hybrid rank-1 and rank-2 U(1) gauge theory, with vector gauge fields $A_i^a$ (rank-1) and antisymmetric tensor fields $A_i^{ab}$ (rank-2), defined on a 3D cubic lattice.
• Apply the Higgsing procedure to the parent LGT to derive the stabilizer Hamiltonian of the F3 model, confirming that the F3 model arises from this hybrid gauge structure.
• Construct the effective Lagrangian by promoting the discrete gauge symmetries to continuous ones and introducing Lagrange multipliers to enforce Gauss’s laws and Bianchi identities.
• Derive the hydrodynamic equations of motion for matter and gauge fields by taking the continuum limit and identifying conserved currents from the Noether procedure.
• Re-express the EFT in terms of BF-type field theory language, with the Lagrangian including terms coupling electric fields $E^a_i$ to matter currents $j^e_i$, and additional terms linking fracton density to the double curl of $E^a_i$.
• Derive a tight-binding model for quasiparticle motion by mapping the EFT dynamics onto a discrete lattice model with constrained hopping terms reflecting the gauge-invariant structure.

Experimental results

Research questions

• RQ1 What is the parent lattice gauge theory of the F3 model, which hosts freeons, fluxons, and fractons?
• RQ2 How does the effective field theory of the F3 model encode the constrained dynamics of fractons and fluxons?
• RQ3 What is the role of the hybrid rank-1 and rank-2 U(1) gauge structure in mediating the coupling between fracton and fluxon degrees of freedom?
• RQ4 How do the hydrodynamic equations of motion for the quasiparticles reflect their mobility constraints?
• RQ5 Can a consistent tight-binding model be derived from the EFT that captures the quasiparticle dynamics of the F3 model?

Key findings

• The F3 model is shown to arise from a hybrid rank-1 and rank-2 U(1) lattice gauge theory, with the parent theory providing the gauge structure underlying the model’s stabilizer Hamiltonian.
• The effective field theory of the F3 model is derived as a modified BF-type Lagrangian that includes a term coupling the double curl of the electric field to the fracton density, enforcing fracton confinement via fluxon mediation.
• Hydrodynamic equations of motion are derived that correctly reflect the immobility of fractons and the one-dimensional mobility of lineon-like excitations, with the constraints encoded via Bianchi identities and Gauss’s laws.
• The matter part of the effective action is constructed while preserving gauge invariance, with the fracton and fluxon current operators derived from the Noether procedure applied to the gauge symmetry.
• The Lagrangian is re-expressed in continuum form via field redefinitions, yielding a BF-like action that unifies the dynamics of electric and magnetic excitations with fractonic degrees of freedom.
• A tight-binding model for quasiparticle motion is constructed, showing that the hopping terms are constrained by the gauge structure, with fractons requiring collective motion and fluxons obeying directional mobility rules.