Hamiltonian Lattice QED$_3$ with One and Two Flavors of Wilson Fermions: Topological Structure and Response

Abstract

The quantum simulation of topological phases in (2+1)D quantum electrodynamics with Wilson fermions provides a promising route toward realizing topological phenomena in near-term lattice experiments. We show that the commonly used staggered-fermion discretizations in Hamiltonian gauge theories possesses an exact time-reversal symmetry, which forbids the emergence of nontrivial topological phases and has led to confusion in the existing literature. In this work, we resolve this obstacle by systematically analyzing fermion discretization effects in (2+1)D lattice Hamiltonians of fermions coupled to U(1) gauge fields that satisfy Gauss' law. We show that Wilson fermions, already in the minimal one-flavor theory, naturally enable topological regimes with nonzero Chern numbers, and that the two-flavor extension at finite chemical potential further enriches the accessible topological structure. We develop gauge-invariant diagnostics of topological response, including many-body Chern numbers and current correlators that remain robust probes at weak coupling. Finally, through extensive exact diagonalization calculations across both flavor settings, we characterize the spectrum, correlators, and topological invariants, providing a concrete foundation for near-term quantum simulations of topological phases in lattice field theories. The implications of this work for quantum simulations of lattice field theory are analyzed in a joint submission [1].

Figures (21)

• Figure 1: Chern number (solid black line) vs. the shifted mass $M$ for $R=1$. The pink regions are trivial insulator phases, the orange one has $c_1[b] = -1$, while the yellow one has $c_1[b] = +1$.
• Figure 2: A plot of the average occupation fraction $f_a$ vs. $\mu$, where $a=1$ corresponds to $\uparrow$ (red) and $a=2$ corresponds to $\downarrow$ (blue). The purple segment in between corresponds to the regime where they red and blue coincide. Clearly, as we tune the chemical potential, the theory is driven to different phases. This plot was produced for the $N_f=2$ theory with equal masses $M=-1.2$ on a $32\times 32$ spatial lattice.
• Figure 3: The phase diagram of $N_f=2$ Wilson fermions coupled to a ${\text{U}}(1)$ gauge field for singlet (left) and triplet (right) mass configurations at weak coupling. In the left figure, the pairs are labeled as $(c_1[b], \expval{s})$. In the right figure, the triplets of Chern numbers are labeled as $(c_\uparrow, c_\downarrow, c_\text{tot})$. Here, "$\uparrow\downarrow$" denotes zero average spin. The diagram highlights the existence of IQH phases for singlet masses and QSH phases for triplet masses. This plot was produced on a $16\times 16$ lattice using the analytical solution, which allows access to large system sizes.
• Figure 4: The analytical current one-point functions for both the singlet and triplet cases, which are non-trivial in the IQH and QSH phases respectively.
• Figure 5: The lowest five energies versus $M$ for the $N_f=1$ theory before projecting onto the trivial-flux sector (left), and the level crossing after projecting onto the trivial-flux sector (right) with $({\cal W}_x, {\cal W}_y) = (1,1)$ via $\prod_k\frac{1+{\cal W}_k}{2}$. The results were obtained via ED on a $2\times 2$ lattice with ${\amsmathbb Z} _2$ gauge fields and $e^2 = 0.01$. We emphasize that the left figure did not require a projection into the approximate $({\cal W}_x, {\cal W}_y)$ superselection sectors, but rather were found to naturally lie in the four sectors labeled as $(\pm1, \pm1)$