A Path to Quantum Simulations of Topological Phases: (2+1)D Quantum Electrodynamics with Wilson Fermions

Abstract

Quantum simulation offers a powerful approach to studying quantum field theories, particularly (2+1)D quantum electrodynamics (QED$_3$) with Wilson fermions, which hosts a rich landscape of physical phenomena. A key challenge in lattice formulations is the proper realization of topological phases and the Chern-Simons terms, where fermion discretization plays a crucial role. In this work, we highlight the differences between staggered and Wilson fermions coupled to $\text{U}(1)$ gauge fields in the Hamiltonian formulation. We analyze why staggered fermions fail to induce (2+1)D topological phases, while Wilson fermions admit a variety of topological phases including Chern insulator and quantum spin Hall phases. Additionally, we uncover a rich phase diagram for the two-flavor Wilson fermion model in the presence of a chemical potential. Our findings resolve existing ambiguities in Hamiltonian formulations and provide a theoretical foundation for future quantum simulations of lattice field theories with topological phases. We further outline connections to experimental platforms, offering guidance for implementations on near-term quantum computing architectures. A complementary presentation of the analytical calculations, the identification of robust topological structure and response, and extensive numerical results is contained in a joint submission [1].

Figures (3)

• Figure 1: The analytically computed Chern number $C$ (left) vs. the shifted mass $M=m+2R$ for a ${\text{U}}(1)$ gauge group and arbitrary system size $L$, and the numerically obtained many-body Chern number (right) on a $2\times 2$ lattice with ${\amsmathbb Z} _2$ gauge fields for $e^2=0.01$. The pink regions are trivial insulator phases, the orange one has $C = -1$, while the yellow one has $C = +1$. This figure demonstrates the robustness of topological phases even under severe gauge-group and lattice truncations, making them ideal targets for quantum simulations. Derivations of the analytical and many-body Chern numbers are relegated to Ref PRD.
• Figure 2: 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 PRD, which allows access to large system sizes. Details on deriving the phase diagram may be found in Ref. PRD.
• Figure 3: The level crossings obtained by exact diagonalization on a $2\times2$ lattice for a single Wilson fermion with shifted mass $M=m+2R$ coupled to ${\amsmathbb Z} _2$ gauge fields at $e^2 = 0.01$. The blue line denotes the ground state and the red line denotes the first excited state, both in the trivial-flux sector with $({\cal W}_x,{\cal W}_y) = (1, 1)$. Details on the projection to the trivial-flux sector and the unprojected plot may be found in Ref. PRD.