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Towards the continuum limit of a $(1+1)$d quantum link Schwinger model
Torsten V. Zache, Maarten Van Damme, Jad C. Halimeh, Philipp Hauke, Debasish Banerjee
TL;DR
Finite-dimensional quantum link models require a sequence of extrapolations to recover continuum QFT information. This paper numerically extrapolates observables across spin length, volume, and lattice spacing, demonstrating the continuum limit with small spin values.
Problem
Extracting continuum-QFT information from quantum link models, particularly in the low-dimensional Hilbert spaces available on current quantum devices, requires a sequence of extrapolations.
Source: p. 1Method
The authors numerically extrapolate QLM observables through large spin length and volume, then vanishing lattice spacing, to approach the QFT limit.
Source: p. 3Results
The authors numerically demonstrate the continuum limit of (1+1)-d QED for several observables, with small spin values S ≲3−5 sufficing.
Source: p. 5Takeaways & Limitations
Systematic finite-size scaling enables quantitative estimates of quantum-device resources needed to realize the continuum limit.
Source: p. 5The mass-spectrum calculation is a stepping stone toward more complex tasks, and the situation changes for real-time dynamics.
Source: p. 6
Abstract
from arXiv · showhide
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 $\Theta=\pi$. Our findings indicate that quantum devices will in the foreseeable future be able to quantitatively probe the QED regime with quantum link models.
I. SPIN LENGTH AND LATTICE UNITS IN THE QLM
The section establishes the unit conventions and dimensionless lattice scaling, then describes how quantum link models represent gauge fields with fixed-length spin operators. The spin operators obey angular-momentum algebra consistent with the continuum canonical commutation relations.
I. SPIN LENGTH AND LATTICE UNITS IN THE QLM
- In standard lattice QED, quantities are rescaled with the lattice spacing a to become dimensionless, with g2 = e2a^(3−d) and µ = am.The main text uses units c = ℏ = 1.Source: §I · p. 9
- Quantum link models replace dimensionless gauge fields, including the electric field and compact U(1) link operator, with spin operators.Source: §I · p. 9
- The fixed-length spin operators obey angular-momentum algebra, and their identifications are consistent with the continuum canonical commutation relations.Source: §I · p. 9
II. EQUIVALENT SPIN MODEL AND HILBERT SPACE DIMENSION
The section reduces the gauge-constrained theory to an equivalent spin model and counts its physical Hilbert space. The physical dimension grows exponentially with system size but only linearly with spin length.
Equivalent spin model
- Gauss' law uniquely fixes the fermionic configurations from the spin configurations, allowing physical-subspace operators to be written in terms of spin operators alone.The projector onto allowed spin configurations is essential because it induces interactions between neighboring spins.Source: §II · p. 10
Hilbert space dimension
- The physical dimension is counted using a (2S + 1)-dimensional matrix whose entries mark allowed neighboring-spin pairs, with Gauss' law alternating between even and odd sites.For periodic boundary conditions, the total count is expressed as a trace over allowed configurations.Source: §II · p. 10
- The physical Hilbert space dimension scales exponentially with system size N but only linearly with spin length S.This scaling is used in the paper's exact-diagonalization numerics and resource estimates.Source: Figure 5 · p. 10, §II · p. 10
III. ANALYTICAL EXPRESSIONS FOR THE VECTOR AND SCALAR MASSES
This section presents exact vector and scalar masses in the strong-coupling limit and quotes perturbative strong-coupling expansions used for the main-text plots.
III. ANALYTICAL EXPRESSIONS FOR THE VECTOR AND SCALAR MASSES
- The cited passage states that, in the strong-coupling limit, the exact masses are known and satisfy m/e = μ/g = 0.Source: §III · p. 11
- Perturbative strong-coupling expansions are quoted as predictions used for the main-text plots.The section refers to and references therein for details.Source: §III · p. 11
IV. EXTRAPOLATIONS
The section describes stepwise extrapolations to the continuum Schwinger model’s ground-state energy and excited-state energies, with convergence checks for the data used in the fits.
- The extrapolations proceed stepwise in S, N, and g; the large-S fit uses a 1/S dependence, large-N fits account for leading excited-state finite-size corrections, and the continuum fit includes a second-order term for the ground state.The 1/S form is motivated as the simplest polynomial dependence and is supported a posteriori by the data; the second-order continuum term improves ground-state accuracy, indicating the lattice data remain far from the continuum limit. The iMPS extrapolations omit the N fit.Source: §IV · p. 11
- As g approaches zero, fixed-S,N data diverge, so data are discarded when extrapolated values differ by more than 10% from the largest-S value; ED data receive an analogous N-convergence check.For the reported results, the S-convergence filter is applied at fixed g and j, and the N-convergence check is applied to ED data.Source: §IV · p. 11
- After convergence filtering, the final continuum extrapolation dominates uncertainty; plotted error bars represent systematic fit-range errors, while individual-fit errors are negligible and χ2 has no statistical interpretation.The normalized square error illustrates the negligible individual-fit errors; all calculations are deterministic.Source: §IV · pp. 11–12
- The final extrapolations give the ground-state energy density ω0(m/e) and first two energies ω1,2(m/e) above the ground state; Figure 6 summarizes ground-state energy and vector and scalar masses.The figure caption identifies the plotted masses as MV (S) for the m/e values corresponding to the main-text data.Source: §IV · p. 11, Figure 6 · p. 12
V. PATH INTEGRAL REPRESENTATION
The section derives a coherent-state path integral for the constrained pure-gauge U(1) quantum link model and shows how its large-S limit yields Wilson lattice gauge theory. It also connects the spin Berry phase to a topological angle Θ=π for half-integer spins.
V. PATH INTEGRAL REPRESENTATION
- The path integral uses coherent states and Trotter time-slicing of the locally constrained Hamiltonian, with periodic imaginary-time gauge fields and projection onto allowed configurations.Taking β→∞ gives ground-state results; the transfer-matrix expansion is valid up to O(ϵ^2).Source: §V · pp. 12–13
- At large S, spin directions localize near the equator, yielding the Villain plaquette/Maxwell action and the Wilson lattice gauge theory in Euclidean space.The effective action follows from Gaussian fluctuations around the equator.Source: §V · p. 13
- In the large-S limit, half-integer spins produce the Wilson theory with background field Θ=π, while integer spins contribute only an irrelevant phase of multiples of 2π.The Berry phase arises from overlaps of coherent states; the topological contribution is 2πS, distinguishing half-integer from integer representations.Source: §V · p. 14