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Real Time Dynamics and Confinement in the $\mathbb{Z}_{n}$ Schwinger-Weyl lattice model for 1+1 QED
Giuseppe Magnifico, Marcello Dalmonte, Paolo Facchi, Saverio Pascazio, Francesco V. Pepe, Elisa Ercolessi
TL;DR
The paper investigates out-of-equilibrium real-time dynamics in a Zn lattice version of 1+1-dimensional QED, focusing on pair production and string breaking. Using numerical simulations of the staggered-fermion Schwinger model, it finds that confinement strongly modifies these processes and produces stable or recurrent dynamics. The study also identifies finite-size and finite-n effects that limit extraction of asymptotic behavior and continuum results.
Problem
The study asks how confinement affects Dirac-vacuum stability, pair production, and string breaking in the out-of-equilibrium dynamics of discretized 1+1-dimensional QED.
Method
The authors simulate real-time evolution in a finite-group staggered-fermion Schwinger model using DMRG-based numerical methods across fermion masses, gauge couplings, and external-field conditions.
Results
Confinement suppresses spontaneous pair production and string breaking, enhances oscillatory behavior, and can leave string excitations localized rather than spreading.
Takeaways & Limitations
The results show that strong confinement and slow, recurrent dynamics occur in the discrete Zn model, not only in the U(1) Schwinger model.
Takeaways & Limitations
The simulations cannot extract the asymptotic density plateau or exact oscillation period, and finite-size and finite-n effects remain non-negligible.
Abstract
from arXiv · showhide
We study the out-of-equilibrium properties of $1+1$ dimensional quantum electrodynamics (QED), discretized via the staggered-fermion Schwinger model with an Abelian $\mathbb{Z}_{n}$ gauge group. We look at two relevant phenomena: first, we analyze the stability of the Dirac vacuum with respect to particle/antiparticle pair production, both spontaneous and induced by an external electric field; then, we examine the string breaking mechanism. We observe a strong effect of confinement, which acts by suppressing both spontaneous pair production and string breaking into quark/antiquark pairs, indicating that the system dynamics displays a number of out-of-equilibrium features.
1 Introduction
The paper studies out-of-equilibrium real-time dynamics in a finite-group lattice formulation of 1+1-dimensional QED, focusing on pair production and string breaking. DMRG simulations show that confinement strongly shapes these dynamics, producing stable or recurrent behavior and suppressing key processes.
- Model formulation: The Zn Schwinger model provides a finite-dimensional lattice implementation of QED, with systematic control of finite-n effects and a large-n connection to lattice U(1) theory.The formulation uses staggered fermions and finite-dimensional representations of the Weyl group.
- Research questions: The study examines Dirac-vacuum stability through spontaneous and externally induced particle/antiparticle pair production, alongside confinement-driven string breaking.These processes probe both vacuum stability and dynamical confinement in the model.
- Approach and dynamics: DMRG simulations vary the fermion mass m and gauge coupling g, revealing stable or recurrent evolution that deviates from the relaxation expected in generic non-integrable many-body systems.The authors study a wide parameter range and identify slow dynamics associated with confinement.
- Significance: Confinement is presented as a feature of the discrete Zn models as well as the U(1) Schwinger model, with implications for quantum simulations using platforms such as Rydberg atoms.The paper connects these results to proposed experiments with quantum simulators.
- Model phases: The model exhibits a quantum phase transition at m_c = −0.33, with a confined phase for m > m_c whose elementary excitations above the Dirac sea are mesonic.The transition belongs to the Ising universality class.
- Main findings: In an external electric field, simulations agree with Schwinger’s predicted pair-production rate, while string excitations are found to break into mesons across studied parameters.The external field also stimulates mesonic formation through a dynamical effect distinct from spontaneous production.
2 The model
The model discretizes 1+1-dimensional QED on a staggered-fermion lattice with a finite Z_n gauge group, enforcing Gauss’s law on physical states. Its mass-controlled phase transition separates confined and deconfined regimes, while gauge-mediated interactions make the model non-integrable and shape its real-time dynamics.
- Gauge constraints: Physical states satisfy Gauss’s law, G(x)|Ψ⟩ = 0, and in one dimension matter density and electric field determine each other up to a boundary-field constant.This permits equivalent effective descriptions retaining either matter or gauge variables.
- Discretization: The model combines spatial lattice discretization with staggered fermions and replaces U(1) by finite Z_n gauge variables, giving each gauge link finitely many local degrees of freedom.The discretization is designed to avoid fermion doubling and make the gauge-sector local Hilbert spaces finite.
- Phase structure: For m > m_c the ground state is confined with mesonic excitations, whereas for m < m_c it is deconfined with nonzero average electric field and broken symmetry.The transition occurs at m_c = −0.33 in the stated large-odd-n or continuum-related case.
- Gauge-invariant configurations: The Dirac sea fills odd sites and leaves even sites empty, while separated particle–antiparticle excitations form a string with constant nonzero electric field between them.These gauge-invariant configurations provide the initial states for studying string dynamics.
- Phase structure: The transition is in the Ising universality class for all n, with mean electric field or mean fermion density serving as the order parameter in the deconfined phase.The critical mass depends on n, and for large odd n it approaches the continuum value −0.33.
- Real-time dynamics: Gauge coupling makes the model non-integrable except at g = 0, so gauge-mediated interactions are studied through real-time t-DMRG evolution implemented with Runge–Kutta integration.The simulations target dynamical properties arising from interactions between matter and electric fields.
3 Spontaneous pair production
The study tests spontaneous pair production from the Dirac sea vacuum after quenches in the Z_n Schwinger model, using particle density, entanglement, and correlation dynamics. Confinement suppresses pair production and limits the spreading of particle–antiparticle pairs, especially at large |m|.
- 3.1 Mean particle density: For a four-site quench to m = 0.5 and g = 6/2π, the density approaches 1/2, indicating one meson, before recombination lowers it again.This reproduces the behavior observed in the referenced trapped-ion experiment.
- 3.1 Mean particle density: Pair production is strongest near the critical mass, while it is strongly suppressed for large |m| by confinement and recombination.The first density maximum is peaked around the phase-transition value; large positive or negative masses produce only small oscillations around low density.
- 3.2 Entanglement entropy: Entanglement grows faster and more monotonically near the critical mass, whereas confinement reduces its growth and produces oscillatory behavior at larger |m|.The entropy reaches values of order unity for small m, compared with approximately 0.5 for examples such as m = 2.0 or m = −3.0.
- 3.3 Correlation functions: Connected correlations spread in a light cone at small m but become localized and oscillatory in intermediate and strong confinement regions.This indicates that pairs are cyclically created and recombined without spreading through the chain.
4 Pair production in an external field
External electric fields induce pair production from a stable Dirac vacuum, with rates showing a linear regime and saturation, while entanglement remains oscillatory and does not grow with production.
- External-field-induced pair production: The vacuum remains stable with small oscillations at weak external fields, while stronger fields produce a linear-density regime followed by saturation.The simulations use ϵ = E0/Ec with Ec = m^2/g.
- External-field-induced pair production: The simulated pair-production rate agrees with the continuous prediction, although finite-size and small-n effects remain non-negligible.Agreement improves as the chain size N and gauge-group parameter n increase.
- Entanglement dynamics: Increasing the external field rapidly increases pair production but does not increase or spread half-chain entanglement, which remains oscillatory with modest maxima.This contrasts with spontaneous production, where entangled pairs spread along the chain.
- Entanglement dynamics: The external field changes pair-production dynamics in a manner distinct from spontaneous production, stimulating mesonic excitations without producing the same entanglement growth.The paper identifies different mechanisms for pair production in the absence and presence of the external field.
5 The string breaking mechanism
String dynamics separates into weak, intermediate, and strong confinement regimes, ranging from deconfined meson production to a stable, localized string.
- Setup: The simulations initialize a central string between a particle and antiparticle and track link electric fields for different mass and coupling values.The reported Z3 setup uses a chain of length N = 80 and an initial string spanning 19 central links.
- String dynamics: Weak confinement lets the string spread and break into two deconfined mesons localized near its edges.The mesons stabilize after an initial period of pair production and recombination.
- String dynamics: Intermediate confinement prevents string spreading but allows breakup into two confined mesons that remain at a fixed separation.The resulting edge-localized mesons are confined rather than freely separating.
- String dynamics: Strong confinement leaves the string stable, without spreading or breaking into mesons.The electric field remains localized, apart from small oscillations in the corresponding entanglement dynamics.
- Confinement regimes: The parameter-space contour plot identifies weak, intermediate, and strong confinement using the asymptotic central electric field relative to its initial value.The 10% and 50% level curves separate the three dynamical regimes.
6 Conclusions
The conclusions attribute oscillatory pair-production observables and suppressed string spreading or breaking to confinement in the Zn model, with slower dynamics and reduced entanglement paralleling other constrained systems.
- Main conclusions: Confinement strongly affects Zn real-time dynamics, enhancing oscillations during pair production and suppressing string breaking and spreading.The conclusions emphasize that confinement affects the Zn model as well as the U(1) Schwinger model.
- Relation to other systems: The constrained physical subspace imposed by Gauss’s law connects this model to systems showing reduced entanglement and slowed dynamics.Related examples include long-range-interaction models, Ising and Potts models, quantum scars, and Abelian quantum-link chains.
A Finite size scaling and large-n limit
Finite-size and finite-n analyses assess how the lattice Zn model approaches continuum behavior, with larger chains and larger n improving the approximation.
- Large-n limit: The Zn model recovers the 1+1-dimensional Schwinger model accurately already near N = 50 sites for n = 3.The continuum and finite-n limits were studied theoretically and checked numerically.
- Finite-size scaling: Finite-size analysis tracks particle-density maxima, density scaling at t0 = 0.52, and long-time density evolution as the chain size changes.The analysis is illustrated for m = 2.0.
- Large-n limit: Contour plots extend the density analysis to Z5 and Z7 models, with a reduced mass range for Z7 because simulations are computationally demanding.The Z5 range is m ∈ [−5.0, +5.0], while the Z7 range is m ∈ [−2.0, +1.0].
- Large-n limit: Pair-production-rate comparisons across chain sizes and Zn models show that continuum-limit agreement improves as N and n increase.The comparison includes Z3, Z5, and Z7 models against the continuous result.
A.1 Numerical precision
The simulations use time-dependent DMRG with controlled time integration, truncation, and initial-state preparation for pair-production and string-breaking analyses.
- A.1 Numerical precision: Time-dependent DMRG evolves the system with fourth-order Runge–Kutta integration using δ = 0.01, reaching tmax ∼4–5 while maintaining at least 93% norm precision.Up to 1200 DMRG states keep the truncation error below 10^-6 at each time step.
A.2 Spontaneous pair production
The simulations extrapolate spontaneous pair-production dynamics to the infinite-size limit and find similar behavior for Z5 and Z7 across the quenched-mass range.
- A.2 Spontaneous pair production: The infinite-size limit ρ∞(t) is extracted from simulations with chain sizes N = 16, 20, 24, 28, 32, 36, 40.The extrapolation is performed at each time t0 using a finite-size fit.
- A.2 Spontaneous pair production: At m = 2.0 and t0 = 0.52, the extrapolation gives ρ∞(t0) = 0.2175± 0.0001 and β(t0) = 0.1703 ± 0.0001.
- A.2 Spontaneous pair production: Z5 and Z7 show very similar ρ∞(t) behavior to Z3 across the quenched-mass range m ∈[−5, +5].The comparison is presented through contours of ρ∞(t).
A.3 Pair production in an external field
The pair-production-rate analysis tests finite-size effects and the large-n approach toward the U(1) limit by comparing Z5 and Z7 simulations with the Schwinger formula.
- A.3 Pair production in an external field: The pair-production-rate simulations use chain sizes N = 50, 60, 70, 80, 90 to evaluate finite-size effects and assess the large-n limit.
- A.3 Pair production in an external field: Even at the largest numerically accessible system sizes, the results still exhibit significant finite-size effects in comparison with the Schwinger formula.