Simulating Lattice Gauge Theories with Virtual Rishons

Abstract

Classical tensor network and hybrid quantum-classical algorithms are promising candidates for the investigation of real-time properties of lattice gauge theories. We develop here a novel framework which enforces gauge symmetry via a quantum-link virtual rishon representation applied at intermediate steps. Crucially, the gauge and matter degrees of freedom are dynamical variables encoded in terms of qubits, enabling analysis of gauge theories in $d+1$ spacetime dimensions. We benchmark this framework in a U(1) gauge theory with and without matter fields. For $d = 1$, the multi-flavor Schwinger model with $1\leq N_f\leq3$ flavors is analyzed for arbitrary boundary conditions and nonzero topological angle, capturing signatures of the underlying Wess-Zumino-Witten conformal field theory. For $d = 2$, we extract the confining string tension in close agreement with continuum expectations. These results establish the virtual rishon framework as a scalable and robust approach for the simulation of lattice gauge theories using both classical tensor networks as well as near-term quantum hardware.

Figures (11)

• Figure 1: Schematic of interactions in a U(1) lattice gauge theory. Fermion matter degrees of freedom (colored circles) sit on lattice sites connected by gauge links (yellow rectangles). The contributions to the Hamiltonian [Eq. \ref{['eq:Hamiltonian']}] and the support of the local Gauss operator $G_{\vec{n}}$ [Eq. \ref{['eq:gausslaw']}] are indicated.
• Figure 2: Virtual rishon formulation. Green arrows indicate the generic formulation, while purple arrows include the qubit encoding. 0) Hamiltonian with matter fields on the sites and U(1) rotors on the links; the Hilbert space is restricted due to the overlapping gauge constraints (dark blue boxes). 1) Each rotor is represented by two bosonic rishons (red and green boxes). 1*) Binary encoding of rishons using $N_r$-qubits. 2) In the rishon representation, Gauss law constraints do not overlap and the Hamiltonian block-diagonalizes into irreducible representations of the gauge group. 3a) The rishon link constraint $\bar{N}=N_a+N_b$ is enforced by projecting $H$ onto the relevant subspace via $P^{\bar{N}}$. 3b) In the qubit encoding, $P^{\bar{N}}$ factorizes into a product of two-qubit site projectors $\mathcal{P}^{(i)}$. The projector $P^{\bar{N}}$ preserves the block-diagonal structure of the Hamiltonian, yielding the qubit representation that explicitly conserves Gauss's law.
• Figure 3: Variation of entanglement entropy with subsystem-size $r$ for the Schwinger model with $N_f$ fermions on a ring. ($a$) The system consists of a ring of size $L \in \{16,24,32,64,96\}$ unit cells. Each cell has $N_f \leq 3$ and $N_r=4$ qubits to encode the rotor. At the quantum-critical point, the central charge is extracted by fitting the numerical data to Eq. \ref{['eq:CardyFormula']}. The expected central charges for the different quantum-critical points described by ${\rm SU}(N_f)_1$ WZW models are shown. Only bipartitions including full unit cells were considered in the fits. ($b$, $c$) For $N_f=2$ and $N_f=1$ with $\theta=\pi$ the obtained central charges are close to the expected results $c_{\rm SU(2)}=1$ and the $c_{\rm Ising}=1/2$ respectively. ($d$) For $N_f=3$ and $\theta=\pi$, the central charge is again obtained to be close to $c_{\rm SU(3)}=2$. ($e$, $f$) One or two of the masses are made much heavier than the rest to reach the $SU(2)_1$, and the Ising critical points, as a crosscheck of our results. The relatively strong deviation from $c_{\rm Ising}=1/2$ in panel $(f)$ is due to the $N_f$ dependence of the mass shift $\tilde{m}_\alpha = m_\alpha - ag^2N_f/8$Dempsey:2022nys; more details in the SM. While data are shown only for the renormalization group fixed points, results for the entire flow can be obtained similarly. Note that the quoted errors are from the fit only, see main text for further discussion.
• Figure 4: String tension in $d=2$ U(1) gauge theory without matter fields. $(a)$--$(c)$: Red and blue dots with separation $\Delta=5$ represent background charges. With increasing coupling constant $g$, the electric field profile narrows, resulting in a flux tube. $(d)$: Fit to linear scaling of ground-state energy with $\Delta$, allowing extraction of $\sigma$ for large coupling. The expected $g^2/2$ scaling of $\sigma$Kogut:1974ag is verified (blue fit). Gaussian decay is expected for small $g$hamerWeakcouplingExpansionsEffective1993 (red fit). $(e)$: Best extracted value for decay constant is $\nu_0=0.223$ (red fit), to be compared to expected $\nu_0=0.321$loanPathIntegralMonte2003. Finite size effects are estimated by $\sigma_\mathrm{min} = g^2/(2L_y)$hamerWeakcouplingExpansionsEffective1993 (green dashed line).
• Figure S1: Ladder operator of the qubit encoded rotor