Compact U(1) Lattice Gauge Theory in Superconducting Circuits with Infinite-Dimensional Local Hilbert Spaces

TL;DR

This work proposes a continuous-variable superconducting-circuit realization of a compact $U(1)$ lattice gauge theory in which Gauss's law arises from Kirchhoff's laws rather than external constraints and rotor-based d.o.f. avoid Hilbert-space truncation. The minimal gauge–matter coupling emerges from a Josephson nonlinearity, while a magnetic plaquette term is generated perturbatively at fourth order as $J_{\square}=-\frac{5\lambda^4}{16m^3}$ in the static-matter regime, reproducing Kogut–Susskind dynamics. Numerical analysis on a single plaquette reveals vortex excitations and highlights the need for large local Hilbert spaces to access the continuum regime. The scheme operates with experimentally feasible circuit parameters and scales toward multi-plaquette lattices, offering a scalable platform for analog quantum simulation of nonperturbative gauge dynamics with continuous-variable encodings.

Abstract

We propose a superconducting-circuit architecture that realizes a compact U(1) lattice gauge theory using the intrinsic infinite-dimensional Hilbert space of phase and charge variables. The gauge and matter fields are encoded directly in the degrees of freedom of the rotor variables associated with the circuit nodes, and Gauss's law emerges exactly from the conservation of local charge, without auxiliary stabilizers, penalty terms, or Hilbert-space truncation. A minimal gauge-matter coupling arises microscopically from Josephson nonlinearities, whereas the magnetic plaquette interaction is generated perturbatively via virtual matter excitations. Numerical diagonalization confirms the emergence of compact electrodynamics and coherent vortex excitations, underscoring the need for large local Hilbert spaces in the continuum regime. The required circuit parameters are within the current experimental capabilities. Our results establish superconducting circuits as a scalable, continuous-variable platform for analog quantum simulation of non-perturbative gauge dynamics.

Figures (12)

• Figure 1: Lumped-element superconducting-circuit realization of a compact $\mathrm{U(1)}$ plaquette. Matter d.o.f are encoded in site-node phases $\hat{\phi}_i$ and conjugate charges $\hat{n}_i$, while gauge fields reside on link-node phases $\hat{\theta}_{ij}$ and charges $\hat{n}_{ij}$. Josephson junctions generate the minimal gauge–matter coupling $\cos{( \hat{\phi}_i+\hat{\theta}_{ij} - \hat{\phi}_j )}$. Gauss’s law emerges exactly from charge conservation at each node and requires no auxiliary constraints.
• Figure 2: Ground-state properties of the plaquette Hamiltonian. (a) Expectation value of the plaquette operator $\langle \hat{\square}\rangle$ as a function of the couplings $\lambda / g$ and $\lambda / m$. (b) Electric-field fluctuations $\langle \hat{n}_{12}^2 \rangle$ on a representative link. Large values of $\lambda / g$, corresponding to the weak-coupling (continuum) limit of compact electrodynamics, require large local Hilbert spaces, highlighting the necessity of untruncated rotor d.o.f. Insets show cuts at fixed values of $\lambda /m$ and share a common legend. Data are obtained for the local Hilbert space of dimension $N = 13$. Numerical convergence is discussed in Appendix \ref{['app_sec:Numerical_results']}.
• Figure 3: Coherent vortex dynamics on a single plaquette. Time evolution of (a) the plaquette operator $\langle \hat{\square}\rangle$ in \ref{['eq:plaq_operator_circuit']}, (b) the electric-field variance $\langle \hat{n}^2_{12} \rangle$, and (c) the fidelity $\mathcal{F}_{\Theta} = |\langle \Theta | \Psi_{t} \rangle|^2$. As an initial state, we have considered a vortex state prepared by applying $\hat{U}_{\text{vortex}}$ in \ref{['eq:vortex_op']} to the ground state of the plaquette Hamiltonian \ref{['eq:plaquette_hamiltonian']}. Persistent oscillations indicate coherent vortex dynamics accessible with experimentally realistic parameters. Results are shown for $\lambda / m = \lambda / g = 50$ and local Hilbert space of dimension $N = 13$.
• Figure 4: Lumped-element representation of the branch circuit. (a) Node-flux formulation realizing the minimal gauge-matter coupling $\cos{( \hat{\phi}_i+\hat{\theta}_{ij} - \hat{\phi}_j )}$. The circuit consists of three capacitors-two with capacitance $C_{\rm m}$ and one with capacitance $C_{\rm g}$-and a Josephson junction with Josephson energy $E_{\rm J}$. (b) Inclusion of classical control parameters to account for charge and flux offsets. Gate voltages $V_{\rm s}$ and gate capacitors with capacitance $C_{\rm s}$ introduce offset charges, while an external magnetic flux $\varphi_{\rm ext}$ threading a SQUID renders the Josephson energy tunable. In both panels, the green box indicates a small shunt capacitance $C_{\rm J}$ associated with the Josephson element.
• Figure 5: Lumped-element representation of the plaquette circuit, indicating the independent nodes and loops as determined by Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL). The system has four physical degrees of freedom.