[Paper Review] Towards the continuum limit of a $(1+1)$d quantum link Schwinger model

Torsten V. Zache, Maarten Van Damme+3 more|arXiv (Cornell University)|Mar 31, 2021

Quantum and electron transport phenomena4 citations

TL;DR

This paper demonstrates the approach to the continuum limit in a (1+1)d U(1) quantum link Schwinger model by numerically extrapolating ground state energy and meson masses to large spin length $S$, large system size $N$, and vanishing lattice spacing $a$. It shows that half-integer spin representations naturally realize the topological angle $\Theta = \pi$, and analytically counts the physical Hilbert space dimension via a generalized PXP model, enabling resource estimation for quantum simulations of QED.

ABSTRACT

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' 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 $Θ=π$. Our findings indicate that quantum devices will in the foreseeable future be able to quantitatively probe the QED regime with quantum link models.

Motivation & Objective

• To demonstrate the approach to the continuum limit in a (1+1)d U(1) quantum link model (QLM) for quantum electrodynamics (QED).
• To numerically extrapolate physical observables—ground state energy and scalar/vector meson masses—toward the continuum limit using large spin length $S$, large volume $N$, and vanishing lattice spacing $a$.
• To analytically count the physical Hilbert space dimension of the QLM by solving Gauss’ law exactly, leading to a generalized PXP spin model.
• To establish a connection between large half-integer spin representations and the topological angle $\Theta = \pi$ via the Berry phase in the large-$S$ limit.
• To quantify the quantum resources required for reliable extrapolations to the continuum limit on near-term quantum devices.

Proposed method

• Exact solution of Gauss’ law for arbitrary spin $S$ yields a generalized PXP spin model, enabling analytical counting of the physical Hilbert space dimension.
• Numerical diagonalization and infinite matrix product state (iMPS) techniques are used to compute ground state energy and meson masses in the zero momentum sector.
• Functional integral formulation maps the large-$S$ limit of the QLM to the Wilson lattice gauge theory action, with the Maxwell term emerging in the continuum limit.
• Berry phase contributions from coherent states of spin-$S$ operators are shown to generate a topological term sensitive to the integer vs. half-integer nature of $S$.
• The topological term contributes $2\pi S$ in the large-$S$ limit, yielding $\pi$ phase for half-integer $S$, equivalent to $\Theta = \pi$ in the Wilson formulation.
• Systematic extrapolation of observables in $S$, $N$, and $a$ is performed to approach the continuum limit, with error estimates provided.

Experimental results

Research questions

• RQ1 Can the continuum limit of the (1+1)d quantum link Schwinger model be approached through systematic extrapolation in spin length $S$, system size $N$, and lattice spacing $a$?
• RQ2 How does the choice of spin $S$—specifically integer vs. half-integer—affect the emergence of topological terms in the effective action?
• RQ3 What is the exact dimension of the physical Hilbert space in the quantum link model for arbitrary $S$, and how does it scale?
• RQ4 Can the large-$S$ limit of the QLM reproduce the Wilson lattice gauge theory action, including the Maxwell and topological terms?
• RQ5 What are the resource requirements for simulating the QED regime of the Schwinger model on near-term quantum devices using QLMs?

Key findings

• The ground state energy and scalar/vector meson masses in the (1+1)d quantum link Schwinger model converge to the analytical predictions in the continuum limit as $S$, $N$, and $1/a$ are increased.
• The physical Hilbert space dimension of the QLM is analytically determined via exact Gauss’ law solution, yielding a generalized PXP model with $\mathcal{O}(2^N)$ states for $S=1/2$.
• Half-integer spin representations ($S = 1/2, 3/2, \dots$) generate a Berry phase of $\pi$ in the large-$S$ limit, corresponding to the topological angle $\Theta = \pi$ in the Wilson formulation.
• Integer spin representations ($S = 1, 2, \dots$) yield a $2\pi$-periodic phase, which is topologically trivial and does not contribute to the effective action beyond the plaquette term.
• The functional integral approach confirms that the large-$S$ limit reproduces the Wilson lattice gauge theory action, with the Maxwell term emerging independently of $S$, while the topological term depends on the spin parity.
• The study provides a quantitative roadmap for resource estimation, showing that reliable extrapolations to the continuum limit are feasible on near-term quantum devices.