Real-time collisions of fractional charges in a trapped-ion Jackiw-Rebbi field theory

Abstract

We propose and analyze a trapped-ion quantum simulator of the Jackiw-Rebbi model, a paradigmatic quantum field theory in (1+1) dimensions where solitonic excitations of a scalar field can bind fermionic zero modes leading to fractionally charged excitations. In our approach, the scalar field is a coarse-grained description of the planar zigzag ion displacements in the vicinity of a structural phase transition. The internal electronic states of the ions encode spins with interactions mediated by the transverse phonons and in-plane spin-phonon couplings with a zigzag pattern, which together correspond to a Yukawa-coupled Dirac field. Instead of assuming a fixed soliton background, we study the effect of back-reaction and quantum fluctuations on the coupled dynamics of the full fermion-boson system. We start by applying a Born-Oppenheimer approximation to obtain an effective Peierls-Nabarro potential for the topological kink, unveiling how the fermionic back-reaction can lead to localization of the kink. Beyond this limit, a truncated Wigner approximation combined with fermionic Gaussian states captures the quantum spreading and localization of a kink and kink-antikink scattering. Our results reveal how back-reaction and quantum fluctuations modify the stability and real-time evolution of fractionalized fermions, predicting experimentally accessible signatures in current trapped-ion architectures.

Figures (16)

• Figure 1: Trapped-ion Jackiw-Rebbi model: A linear chain of ions is tuned across the linear (a) to zigzag (b) structural phase transition, so that transverse displacements along the $z$ axis act as a coarse-grained scalar field $\phi(x)$ supporting topological kink configurations that interpolate between the "zig-zag" and "zag-zig" configurations $\langle \phi \rangle = \pm \Phi_0$, as depicted in (b). Two internal electronic states of each ion encode a staggered fermionic field via a Jordan--Wigner transformation, and phonon-mediated spin-spin interactions induced by a Mø lmer-Sø rensen laser scheme acting alomg $\Delta\boldsymbol{k}_L$ yield an effective Dirac kinetic term (see inset of (b)). A state-dependent optical dipole force along the crystal plane $\Delta\tilde{\boldsymbol{k}}_L$ (inset) couples spin and motion, implementing the Yukawa-type interaction. Together, these elements realize an analog trapped-ion Q$\ell$S of the Jackiw-Rebbi model that would alow to explore real-time dynamics of scalar solitons and fractional fermion bound states, particularly looking into characteristic effects due to back-reaction and quantum fluctuations that go beyond typical approximations.
• Figure 2: Classical solitons of a lattice scalar field: The dotted white line shows the classical kink solution \ref{['kink_cont']} with $q_t=1$, plotted on top of a histogram of initial conditions for the truncated Wigner approach. This histogram is drawn from the Wigner distribution of the ground state of the model of coupled harmonic oscillators resulting from the linearization of the $\lambda\phi^4$ QFT around the kink solution, as defined in Eq. \ref{['harmonic_chain_ground_state']}. Parameters used are $N=100$ lattice sites, lengthscale $\xi_0/a=3.0$, and amplitude $\Phi_0=5$, which uniquely identify the bare mass and quartic coupling.
• Figure 3: Peierls-Nabarro potential: Potential energy felt by the soliton as a function of its center. Parameters used are $N=160$, $\Phi_0=3$ and $\xi_0/a=1$.
• Figure 4: Low-lying excitations around the classical soliton: a) Frequencies of the first two lowest lying modes of the equations of motion linearized around the classical soliton (eqs. \ref{['ec_dispersion_kink_lattice']}) for an inter-site kink (light-blue) and an on-site kink (dark-blue). This evidences the on-site kink as linearly unstable in discrete lattices. Parameters used are: $N=160$, $\Phi_0=3$. b) Amplitudes of these modes for the inter-site solution at $\xi_0/a=6$.
• Figure 5: Fermionic spectrum and zero mode: a) Spectrum as a function of the solution width. (Eq. \ref{['eq:matrix_fermions']}) for an on-site kink. b) Fermionic zero-mode eigenfunctions . The circles denote numerical values obtained through Eq. \ref{['fermionic_ground_state']}, while the solid lines show the fermionic zero-mode of the continuous model in Eq. \ref{['continuous_zero_mode']}. Parameters used are $N=160$, $\Phi_0=3$, $Ja=3$, $ga=2$.