Discrete non-Abelian gauge theories in two-dimensional lattices and their realizations in Josephson-junction arrays

TL;DR

The work tackles the challenge of realizing discrete non-Abelian gauge theories in two-dimensional lattices, proposing a Hamiltonian framework suitable for solid-state platforms and analyzing two distinct constraint realizations: a gauge-invariant model and a kinematical Gauss-law model. It identifies and characterizes fluxon excitations and their non-Abelian Aharonov-Bohm interactions, showing how topological degeneracy can be protected by geometry (holes) and by energy gaps. The authors propose concrete Josephson-junction array designs, particularly for the non-Abelian groups $D_3$ and $D_4$, and derive the hierarchy of energy scales necessary to realize and control these topological states. The work points to potential applications in universal quantum computation with fault-tolerant, adiabatically controlled operations, and highlights open questions about broader classes of topological phases and their experimental verification.

Abstract

We discuss real-space lattice models equivalent to gauge theories with a discrete non-Abelian gauge group. We construct the Hamiltonian formalism which is appropriate for their solid-state physics implementation and outline their basic properties. The unusual physics of these systems is due to local constraints on the degrees of freedom which are variables localized on the links of the lattice. We discuss two types of constraints that become equivalent after a duality transformation for Abelian theories but are qualitatively different for non-Abelian ones. We emphasize highly nontrivial topological properties of the excitations (fluxons and charges) in these non-Abelian discrete lattice gauge theories. We show that an implementation of these models may provide one with the realization of an ideal quantum computer, that is the computer that is protected from the noise and allows a full set of precise manipulations required for quantum computations. We suggest a few designs of the Josephson-junction arrays that provide the experimental implementations of these models and discuss the physical restrictions on the parameters of their junctions.

Figures (12)

• Figure 1: Schematics of the Josephson-junction array equivalent to the Abelian theory. Physical variables (superconducting phases) are defined on the hexagonal lattice $\Lambda$: $\phi_a$ are phases of the islands while $u_{ab}$ describe the phase differences. Each bond represents a superconducting element which is $2\pi/n$ periodic in the phase difference $\phi_a-\phi_b$. The state of each bond is described by the discrete variable $u_{ab}$ according to Eq. (\ref{['Delta_phi_ab']}). Alternatively, the state of the bond can be described by the variable $u_{ij}$ defined on the links of the dual (triangular) lattice $\Lambda_d$.
• Figure 2: The global topological invariant characterizing the ground state of the system is defined as a product of operators along contour which joins inner and outer boundaries (left bold line). Excitations (fluxons) correspond to the string of operators that are illustrated by the right bold line.
• Figure 3: The closed paths $\gamma_1$ and $\gamma_2$ encircle only one fluxon, whereas $\gamma$ encircles both. The conjugacy classes of $\Phi({\gamma_1})$, $\Phi({\gamma_2})$, and $\Phi({\gamma})$ are gauge-invariant and are also invariant under "smooth" deformation of the paths. However, $\Phi({\gamma})$ is not conjugated to $\Phi({\gamma_1}) \Phi({\gamma_2})$, except if all the paths have the same origin.
• Figure 4: The path $\gamma'_{2}$ is equivalent to $\gamma_{1}^{-1}\gamma_{2}\gamma_{1}$ which is not in the same class as $\gamma_{2}$ as this figure illustrates. Consequently, the fluxes $\Phi(\gamma_2)$ and $\Phi(\gamma'_{2})$ belong to the same conjugacy class, but they are in general different, in spite of the fact that $\gamma_{2}$ and $\gamma'_{2}$ have the same origin $O$, that they both wind exactly once counterclockwise around fluxon 2, and do not wind around fluxon 1.
• Figure 5: Fluxon moving from plaquette (123) to plaquette (234) involves a modification of $g_{23}$. In the initial state (left), $g_{23}^{init.}=g_{24}g_{43}$, and in the final state (right) $g_{23}^{fin.}=g_{21}g_{13}$. The operator which transforms the former into the latter state is ${\cal M}_{23}$.