Toward the continuum limit of a $(1+1)\mathrm{D}$ quantum link Schwinger model

The solution of gauge theories is one of the most promising applications of quantum technologies. Here, we discuss the approach to the continuum limit for $U(1)$ gauge theories regularized via finite-dimensional Hilbert spaces of quantum spin-$S$ operators, known as quantum link models. For quantum electrodynamics (QED) in one spatial dimension, we numerically demonstrate the continuum limit by extrapolating the ground state energy, the scalar, and the vector meson masses to large spin lengths $S$, large volume $N$, and vanishing lattice spacing $a$. By exactly solving Gauss’s law for arbitrary $S$, we obtain a generalized PXP spin model and count the physical Hilbert space dimension analytically. This allows us to quantify the required resources for reliable extrapolations to the continuum limit on quantum devices. We use a functional integral approach to relate the model with large values of half-integer spins to the physics at topological angle $\mathrm{\Theta }=\pi$. Our findings indicate that quantum devices will in the foreseeable future be able to quantitatively probe the QED regime with quantum link models.

Figure 1

(a) In a $U(1)$ QLM, matter fields (blue dots) reside on the sites of a lattice, while gauge fields, represented by spins (red arrows), live on the links connecting two neighboring lattice sites. Gauss’s law ties consecutive gauge fields to the matter in between as indicated by the shaded ellipses. The ground state energy, shown in (c), and the first two excited states (with vector and scalar quantum numbers), shown in (b), in the zero momentum sector of QED obtained from the $U(1)$ QLM using ED and iMPS show excellent agreement with the analytical prediction for $m/e=0$. In (b), the gray solid lines indicate the leading order (LO) analytical expansions [65] for small $m/e$ and $e/m$, respectively. The error bars indicate an estimated systematic uncertainty (see Sec. 4 in the Supplemental Material [60]).

Figure 2

We illustrate the sequence of extrapolations from left to right, required to reach the continuum limit for the ED data with $m/e=0$. The energies of the vacuum (bottom row), vector particle (middle row), and scalar particle (top row) are extrapolated to $S\to \infty$ (left column), $N\to \infty$ (middle column), and $a\to 0$ (right column), as discussed in the main text. The circles in the middle row indicate the values obtained from the corresponding $S$ extrapolations shown in the left column. Similarly, the ticks in the right column indicate the values corresponding to the $N$ extrapolations in the middle column. For clarity, we only show selected values of $ae$ and $N$ for the first two extrapolations. For comparison, the green crosses indicate the exact analytical results. All fits employed for the extrapolations are polynomials and ${\chi }^2$ denotes the resulting normalized square error of the fit (see Sec. 4 of the Supplemental Material [60] for details, where we also provide a table with our quantitative results).

Figure 3

Final result for vector (lower) and scalar (upper) masses, with the leading dependence $2m/e$ subtracted. Our results reproduce the analytic prediction of the massless limit ($m/e=0$) and are quantitatively consistent with the perturbative expectation (gray solid line). For the vector mass, both IMPS (red dots) and ED (blue diamonds) results agree with each other and previously obtained results (black crosses) at infinite spin length.

Figure 4

(a) Closed path traced by a quantum spin $S$ on the Bloch sphere. (b) With increasing $S$, the path of the spin is forced along the equator to minimize fluctuations of the electric energy term.