Long-time soliton dynamics via a coarse-grained space-time method

TL;DR

The paper develops a spacetime-coarse-graining method (DEC-QED) that discretizes 1+1D field theories on a Minkowski grid to achieve stable, long-time simulations of perturbed Sine-Gordon dynamics and the bosonized Schwinger model. It validates the approach on Josephson transmission line fluxons and breathers, then applies it to the massive Schwinger model to reveal long-lived bound states (Schwinger atoms) and bound pairs (Schwinger positronium) that radiatively relax toward stationary or quasi-stationary configurations. A key finding is that radiative boundary conditions and a dynamical mass term stabilize complex, nonperturbative structures, with energy conserved and local charge continuity enforced to machine precision by DEC. The results suggest avenues for analog quantum simulation of relativistic QED phenomena using JTLs, and establish a robust framework for exploring non-perturbative structure formation in nonlinear field theories with long-time dynamics.

Abstract

We investigate the long-time dynamics of the Sine-Gordon (SG) model under a class of perturbations whose quantum field theoretic analog - via bosonization - corresponds to the massive Schwinger model describing 1+1D relativistic QED of Dirac fermions. Classical SG solutions offer critical insight into non-perturbative effects in this quantum theory, but capturing their long-time behavior poses significant numerical challenges. To address this, we extend a coarse-graining method to spacetime using a dual-mesh construction based on the Minkowski-metric. We first validate the approach against the well-studied variant of the SG model describing magnetic fluxon dynamics in Josephson transmission lines (JTLs), where analytical and numerical benchmarks exist. We then apply the method to the Schwinger-inspired SG model and uncover long-lived bound states - "Schwinger atoms" - in which a soliton is trapped by a fixed central charge. In certain regimes, the system exhibits limit cycles that give rise to positronium-like states of oppositely charged solitons, while in others such formation is suppressed. Accessing such long-time solutions requires a rigorous implementation of outgoing boundary conditions on a finite computational domain that provide radiative dissipation to allow relaxation toward states that exist only in an infinite domain. Here we provide such a construction. Our results also suggest the possibility of analog quantum simulation of relativistic quantum field theories with JTLs. These results demonstrate the utility of spatio-temporal coarse-graining methodology for probing non-perturbative structure formation in non-linear field theories.

Figures (26)

• Figure 1: (a) 3D Schematic of a long junction in which the generalized flux variable $\varphi(x,t)$ varies along the length of the junction. (b) A view at an $xz$-slice of the junction. $\lambda_L$ is the London penetration depth and $\lambda_J$ is the Josephson penetration depth. (c) $\varphi(x)$, $\partial_x\varphi$ and $\partial_t\varphi$ of a $2\pi-$kink solution of the pure SG equation.
• Figure 2: (a) A schematic of the 1+1D spacetime grid used for the DEC algorithm. The vertices and edges on the primal mesh are colored orange, while the dual vertices and dual edges are in green. An example cell $v^\dagger$ that is the dual of a primal vertex $v$ is also shaded in green. The primal edges connected to $v$ are bolded in red. (b) A closed-up view of an internal primal vertex $v(i,j)$ along with its single dual face $v^\dagger$, the neighboring vertices, and the primal edges associated with this vertex. Dual faces of neighboring vertices are not shown here. (c) A closed-up view of a boundary vertex $v(i,j)$ with the boundary primal edges associated with it.
• Figure 3: Long-time dynamics of a single fluxon trapped inside a Josephson junction. The junction length is $L=100$, the total time of the dynamics is $T_m=50000$, and the initial velocity of the fluxon is $u=0.55$, with no external bias on the boundary ($\eta=\xi=0$). (a), (b) and (c) show the value of $\varphi$, $\phi_t$, and $\phi_t$ respectively.
• Figure 4: Comparisons on the final positions of the fluxon in the junction at the last time step ($t=T_m=50000$) obtained with different discretization steps $\Delta x$ and $\Delta t$. (a) shows the results from using Euler method, and (b) shows the results from DEC-QED. In all the plots from both figures, $\Delta t=0.8\Delta x$, $u=0.55$, and $L=100$. The inset in (b) shows the difference $\Delta D$ in the distance traveled by the soliton calculated with different values of $\Delta x$ to the distance $D$ covered by the soliton simulated with $\Delta x = 0.025$.
• Figure 5: (a) Simulated dynamics of the field $\phi_x(x,t)$ that corresponds to a vortex-antivortex pair in a long Josephson junction. The initial distance between the pair is $d=66$, with each soliton having an initial speed of $u=0.55$ and set to travel towards each other. (b) The field $\phi_x(x,t)$ that corresponds to a breather traveling inside a long JJ also with speed $u=0.55$.