Overview
The 3-4-5-0 model is a 1+1D theory of four chiral fermions carrying U(1) charges \((3,4,5,0)\). It is an example of a fermionic system that admits symmetric mass generation (SMG): a fully symmetric, non-perturbative interaction can gap every flavor without breaking the U(1) symmetry, even though no fermion bilinear mass exists. After bosonization the four fermions become four compact scalar fields propagating as a four-component Luttinger liquid, and the SMG interaction becomes two cosine perturbations built from Haldane null vectors. The physical question this page is built around is: how does a chiral mode scatter off a region where the SMG interaction is turned on? It is the continuum, real-time counterpart of the Wang–Wen mirror-decoupling and Tong et al. monopole-scattering setups.
Theory
Bosonized Luttinger liquid
The four fermion flavors \(\psi_I\), \(I=1,2,3,4\), are bosonized by four compact scalar fields
\[\phi_I \sim \phi_I+2\pi, \qquad \psi_I\sim e^{i\phi_I}.\]In the standard 3-4-5-0 ordering, the first two flavors are left-movers and the last two are right-movers. The free Luttinger Lagrangian is
\[\mathcal{L}_0= \frac{1}{4\pi} \left( K_{IJ}\,\partial_t\phi_I\,\partial_x\phi_J -V_{IJ}\,\partial_x\phi_I\,\partial_x\phi_J \right),\]with the chirality matrix and Luttinger metric
\[K=\operatorname{diag}(1,1,-1,-1), \qquad V=\operatorname{diag}(v_1,v_2,v_3,v_4).\]The eigenvalues of \(K\) make \(\phi_1,\phi_2\) left-movers and \(\phi_3,\phi_4\) right-movers in the gapless region, with per-flavor velocities \(v_I\).
SMG interface
The SMG interaction is built from two integer Haldane null vectors
\[\ell_1=(1,-2,1,2)^\intercal, \qquad \ell_2=(2,1,-2,1)^\intercal,\]which satisfy \(\ell_a^\intercal K^{-1}\ell_b=0\) and which preserve the full \(U(1)\times U(1)\times \mathbb{Z}_{10}\times\mathbb{Z}_{10}\) symmetry of the 3-4-5-0 model. The interacting Lagrangian is
\[\mathcal{L}_{\mathrm{int}} = g_1(x)\cos(\ell_1^\intercal\phi) + g_2(x)\cos(\ell_2^\intercal\phi).\]To model a region in which the SMG interaction is turned on smoothly, we work at the \(\mathbb{Z}_2\)-symmetric point \(g_1(x)=g_2(x)=g(x)\) with a sigmoid profile
\[g(x)=\frac{g_0}{1+\exp[-(x-c)/w]},\]so that \(g(x)\to 0\) on one side of \(x=c\) and \(g(x)\to g_0\) on the other, with \(w\) setting the crossover width.
Wave equation
Varying \(\mathcal{L}_0+\mathcal{L}_{\mathrm{int}}\) with respect to \(\phi_I\) gives the nonlinear wave equation
\[K_{IJ}\,\partial_t\partial_x\phi_J -V_{IJ}\,\partial_x^2\phi_J +2\pi g(x)\,\ell_{1I}\sin(\ell_1^\intercal\phi) +2\pi g(x)\,\ell_{2I}\sin(\ell_2^\intercal\phi) =0.\]In the gapless region (\(g\to 0\)) this reduces to the free first-order chiral wave equation \(K\,\partial_t\phi - V\,\partial_x\phi = 0\). In the deep SMG region (\(g\to g_0\)) the cosines pin the two null-vector combinations \(\ell_1^\intercal\phi\) and \(\ell_2^\intercal\phi\), opening a fully symmetric gap. The interface mediates the scattering between these two phases.
Numerics
The simulator integrates the wave equation above in real time on a one-dimensional interval. The interval is split into a gapless region and an SMG region by the sigmoid profile \(g(x)\), and a Gaussian wave packet is launched in one of the four channels \(\phi_I\). The four field profiles \(\phi_I(x,t)\) are displayed in stacked panels, reduced modulo \(2\pi\) into the window \([-\pi,\pi)\); the PDE itself is integrated on the universal cover. The shaded background under each panel tracks \(g(x)\), so the location and width of the interface are visible at a glance.
How to use it
- Launch channel: which of \(\phi_1,\dots,\phi_4\) carries the initial packet. \(\phi_{1,2}\) are left-movers and \(\phi_{3,4}\) are right-movers.
- Amplitude / Center / Width: shape of the Gaussian initial profile.
- SMG Region: which part of the line hosts the SMG interaction. Left and Right create a single interface. Both creates two interfaces at \(x=\pm c\), sets the packet center to \(0\), and uses \(g(x)=g_{\mathrm{left}}(x)+g_{\mathrm{right}}(x)\).
- SMG interface: strength \(g_0\), crossover width \(w\), and center \(c\) of the sigmoid interface.
- Discretization: spatial grid extent and resolution, time step \(dt\), total number of steps, sampling stride, and outer boundary condition. The default Absorbing boundary mimics outgoing radiation; Dirichlet, Neumann, and Periodic are useful as sanity checks.
- Solver: Automatic chooses the best browser-side solver for the current boundary condition, so most users should leave it there. The manual choices are simplified labels for the leading research branches: Stable default, Stiff interface, Plateau pinning, Conservative transport, and Legacy baseline.
- Velocity matrix: per-flavor velocities \(v_I\) on the diagonal of \(V\).
A useful starting experiment is to set \(g_0=0\) to recover free chiral propagation as a baseline, then gradually turn the SMG interaction on and watch how each channel begins to mix near the interface.
Caveats
This is an exploratory visualizer rather than a controlled scattering calculation:
- The solver menu is synced to the current research frontrunners, but these are still lightweight browser implementations for interactivity, not exact reproductions of the Python benchmark harness.
- The nonlinear region can become numerically stiff for large \(g_0\) or small \(w\); if the trace develops obvious artifacts, reduce \(dt\) or widen the interface.
- The compact identification \(\phi\sim\phi+2\pi\) is applied only for display; the PDE itself is integrated on the universal cover, so apparent jumps at \(\pm\pi\) are display wraps rather than dynamics.
- Extracting reflected, transmitted, and twisted-sector amplitudes from the outgoing wave — the comparison most relevant to the Wang–Wen and Tong et al. analyses — is not done in this demo.
References
- J. Wang and X.-G. Wen, A Solution to the 1+1D Gauged Chiral Fermion Problem, arXiv:1807.05998.
- T. Onogi, T. Wada, and S. Yamaoka, Discrete Symmetry and ‘t Hooft Anomalies for 3450 Model, arXiv:2501.18156.
- M. van Beest, P. Boyle Smith, D. Delmastro, Z. Komargodski, and D. Tong, Monopoles, Scattering, and Generalized Symmetries, arXiv:2306.07318.