Scattering and induced false vacuum decay in the two-dimensional quantum Ising model

TL;DR

This work demonstrates real-time scattering and non-perturbative phenomena in the two-dimensional quantum Ising spin lattice using tree tensor networks on a $24 \times 24$ lattice. It uncovers a ladder of bound-state excitations and interwoven resonances that mediate elastic and inelastic scattering, and shows that high-energy collisions in a symmetry-broken regime can nucleate a true-vacuum bubble that expands ballistically. The authors develop wave-packet preparation and an efficient TTN-based summation method to access large 2D systems, highlighting the potential of TTN simulations for lattice gauge theories and strongly interacting condensed-matter settings beyond perturbation theory. These results provide a computational pathway to probe non-perturbative dynamics and collective excitations in higher dimensions with quantum-simulation relevance.

Abstract

We study scattering in the quantum Ising model in two dimensions. In the ordered phase, the spectrum contains a ladder of bound states and intertwined scattering resonances, which enable various scattering channels. By preparing wave packets on a $24 \times 24$ lattice and evolving the state with tensor networks, we explore and characterize these regimes, ranging from elastic scattering in the perturbative regime, to non-perturbative processes closer to the critical point. Then, we break the spin inversion symmetry and study the stability of the metastable false vacuum state on the collision of its excitations. We find that a highly-energetic scattering process can induce a violent decay of the false vacuum, and investigate the spread of the resulting true vacuum bubble.

Figures (5)

• Figure 1: The spectrum of the 2D Ising model. (a) Low-energy excitations above the polarized state are domains of flipped spins. Here we show the excitations up to three, and the lowest-energy four-spin excitation. (b) A sketch of the single-excitation spectrum. The bands correspond to excitations shown in panel (a). We prepare wave packets in the magnon band (red). The scattering energy is resonant with the 3-spin and 4-spin bands. (c) The many-body spectrum of a $5 \times 5$ lattice for $g/J = 1$ for momentum $\mathbf k = (k,k)$. The color indicates the change of magnetization with respect to the fully polarized state. Red crosses denote twice the energy of the one-spin excitation; this corresponds to the energy available in scattering.
• Figure 2: Scattering of magnon wave packets. (a) The evolution of energy density for varying $g/J$. The wave packets have momenta $\mathbf{k} = \pm\left(\frac{\pi}{2}, \frac{\pi}{2}\right)$. (b) The evolution of energy density at $g/J=1.5$ for two values of $\mathbf{k} = \pm(k,k)$. (c) The evolution of the quasiparticle correlators for representative values of $g/J$. The value at $tJ=10$ is subtracted to disentangle the scattering from the background effects. (d) The $g$-dependence of the peak value of the correlators. (e) The long-range correlations. The points where the correlations are measured are sketched in the top right panel of (a) using the same colors scheme. As they are smaller than the two-point measurements, the three-point correlators are multiplied by a factor of 3 for visibility. (f) The $g$-dependence of the peak value of the long-range correlations. The curves drawn with different shades of the same color correspond to simulations with different bond dimension, the darkest being the largest. We use 100 and 150 in the left, 100, 150, 200 in the left central, and 100, 150, 200, 250 in the two panels on the right.
• Figure 3: Scattering in the false vacuum. (a) A sketch of the energy landscape of a true vacuum bubble with radius $r$. (b) The evolution of the energy density during scattering with $h/J = 0.1$ and $h/J = 0.3$, both with $g/J=1.5$ and $\mathbf{k} = \left(\frac{\pi}{2}, \frac{\pi}{2}\right)$. (c) The change in magnetization per site with varying $h$ for $g/J=1.5$ and $\mathbf{k} = \left(\frac{\pi}{2}, \frac{\pi}{2}\right)$. (d) The dependence of the threshold value $h^*$ on $g/J$. The gray dashed line corresponds to the threshold $h^*$ expected for $g\rightarrow 0$. See text for details. (e) The evolution of the radius of the true vacuum bubble in log-log scale for varying $g$, with some longitudinal field larger than the threshold value. We subtract the time $t_0$ when the radius reaches $r_0 = 3$ on the horizontal, and $r_0$ on the vertical axis. The black dashed line corresponds to linear growth and acts as a guide to the eye. (Inset) The evolution of the radius in lin-lin scale. In (c) and (e), the curves with different shades of the same color correspond to simulations with different bond dimension, darker is bigger. Here we use 150 and 200.
• Figure 4: Preparation of a wave packet. (a) The dispersion obtained with the adiabatic quench to two values of $g$ (colors) with different $\tau$ (shades). The wave packet has width $\sigma=2$ and momentum $(k,k)$. (b) The dependence of the bandwidth, $\delta E = E(k=\pi)-E(k=0)$, on $g$. The dashed lines are perturbative results, black for second (Eq. \ref{['eq:dispersion']}) and red for up to fourth order. (c) The evolution of the energy density for a wave packet generated with $\tau J=10$, $\mathbf{k} = (\frac{\pi}{2}, \frac{\pi}{2})$ and $g/J=1.5$.
• Figure 5: Summation of TTN states. (a) A schematic representation of a tensor network corresponding to an overlap between two TTN states; $\langle \psi \vert \mathds{1} \vert \phi_i \rangle$. (b) One step of the iterative calculation of the effective operators. The two effective operators and the tensors from $\psi$ and $\phi_i$ are contracted into a new effective operator, which lives one layer higher. (c) The operator representing the effective projection of $\phi_i$ onto $\psi$.