Anyons in the $π$-flux phase of fermionic matter coupled to a $\mathbb{Z}_2$-gauge field

TL;DR

The work rigorously demonstrates that spinful lattice fermions coupled to a dynamical $\mathbb{Z}_2$ gauge field realize a $\pi$-flux topological phase with massive monopoles and a fourfold ground-state subspace on large tori. By leveraging reflection positivity and chessboard estimates, the authors prove the $\pi$-flux sector is energetically favored and that a small staggered mass opens a full bulk gap, yielding a topologically ordered phase robust to weak interactions. They construct loop operators and adiabatic flux threading to reveal toric-code–like anyon braiding: monopoles are bosons with trivial self-braiding, while their braiding with fermions yields the correct $-1$ phase, and the composite $e=m\times\epsilon$ is a boson with a nontrivial mutual braiding with $m$. The results rigorously connect a natural lattice model to the toric-code topological order, persisting under weak coupling and twisted boundary conditions, and they quantify ground-state degeneracy, spectral gaps, and local topological order. This bridge between a dynamical gauge theory with itinerant fermions and a fault-tolerant topological quantum memory highlights a robust, experimentally relevant route to Abelian anyon physics in lattice systems.

Abstract

We consider a system of weakly interacting spinful lattice fermions coupled to a dynamical $\mathbb{Z}_2$ gauge field. Using reflection positivity, we prove that the ground state lies in the sector of a uniform $π$-flux per plaquette and that the monopoles are massive. In the presence of a staggered mass for the fermions, this yields a fully gapped, four-dimensional ground state space on large tori. It is topologically ordered. By considering adiabatic $π$-flux insertion, we construct dressed monopole excitations, show that their self-braiding is proportional to the Hall conductance and hence vanishes, and prove that their braiding with the fermionic excitations is that of the toric code.

Figures (21)

• Figure 1: (a) Effect of mass term on $\pi$-flux dispersion relations: the gap opened by the pertubation is proportional to $\frac{m}{t}$. Solid and dotted lines represent respectively lattice and continuum dispersion relations. (b) The spectral structure of the full Hamiltonian: $\Delta$ is the energy necessary to create a monopole, while $m$ is the minimal energy needed to create a fermion (for $U=0$). The ground state splitting is exponentially small. The picture is stable for $|U|$ small enough.
• Figure 2: Picture of the flux threading procedure in a portion of the torus: the hoppings on the edges cut by the blue line acquire an extra $e^{\pm i \phi}$ (depending on the orientation).
• Figure 3: Braiding of a monopole around a fermion, its annihilation into the vacuum, and the resulting anyonic phase $-1$.
• Figure 4: Oriented lattice $\Gamma_L$ and its dual $\Gamma^*_L$
• Figure 5: The operator $A_i$ acts on all edges corresponding to the lattice site $i\in\Gamma_L$.

Theorems & Definitions (36)

• Remark 2.1
• Definition 2.2: $\mathbb{Z}_2$-charges
• Definition 2.3: Physical observables
• Remark 2.4
• Definition 2.5: Gauss' law
• Remark 2.6
• Example 2.7
• Theorem 3.1: Spectral structure of the model
• Theorem 3.2: Ground state topological order
• Remark 3.3