Meson dynamics from locally exciting a particle-conserving $Z_2$ lattice gauge theory

TL;DR

The paper investigates real-time meson dynamics in a particle-conserving 1D $Z_2$ lattice gauge theory by locally exciting a central region. By focusing on the two-particle sector and mapping to center-of-mass and relative coordinates, the authors reveal linear confinement as the origin of propagating mesons whose sizes oscillate in time. They show that the average meson size and its oscillation frequency are controlled by the confinement strength $h/J$, and that higher initial energy yields longer, slower-moving mesons, leading to size-based separation in the evolving state. Analytically, a Wannier-Stark description explains the large-$h/J$ regime with breathing dynamics $r_s \propto (J/h)|\sin(ht)|$, and the approach is validated by exact numerics and a proposed spin-model implementation for near-term quantum simulators.

Abstract

Quantum simulation of lattice gauge theories is an important avenue to gain insights into both particle physics phenomena and constrained quantum many-body dynamics. There is a growing interest in probing analogs of high energy collision phenomena in lattice gauge theories that can be implemented on current quantum simulators. Motivated by this, we characterize the confined mesons that originate from a local high energy excitation in a particle-conserving 1D $Z_2$ lattice gauge theory. We focus on a simple, experimentally accessible setting that does not require preparation of colliding wavepackets and isolates the effects of gauge field confinement strength and initial state energy on the nature of propagating excitations. We find that the dynamics is characterized by the propagation of a superposition of differently sized mesons. The linear confinement leads to meson size oscillations in time. The average meson size and oscillation frequency are controlled by the strength of the gauge field confinement. At a constant confinement field, the average meson length is controlled by the initial excitation's energy. Higher energies produce longer mesons and their effective mass depends strongly on their size: longer mesons propagate more slowly out of the central excitation. Mesons of different sizes get spatially filtered with time due to different speeds. We show that this phenomenology is a consequence of linear confinement and remains valid in both the strong and weak confinement limit. We present simple explanations of these phenomena supported by exact numerics.

Figures (11)

• Figure 1: (a) Meson dynamics: A meson propagates out of the localized energy excitation in both directions in a 1D chain governed by a $Z_2$ lattice gauge theory. The state is described by a superposition of different meson sizes. The confinement field and the initial excitation energy determine the meson size distribution. The propagation speed varies with the meson size. (b) A basis state of the 1D $Z_2$ lattice gauge theory model. Open and filled circles represent absence and presence of a boson on sites labeled by index $i$. The red and blue arrows are link gauge spins representing $\sigma^z = \pm 1$ respectively. The quantity, $G_i = \sigma^z_{i-1,i}(-1)^{n_i}\sigma^z_{i,i+1} = 1$ is conserved on each site. The red ellipse denotes a meson where the two bosons are next to each other, called an $r=1$ meson.
• Figure 2: (i),(ii): Particle density, $n$ as a function of site index (x-axis) and time, $Jt$ (y-axis) for (i) $h/J = 1.1$ and (ii) $h/J = 0.3$. With time, the meson delocalizes in light-cones like structures. The outermost cones represent faster moving excitations. The slower moving inner cones show substructures with diamond shaped particle tracks denoting temporal oscillation of meson size. (iii),(iv): Snapshot of occupation probability, $p$ as a function of meson size ($r$) and its center-of-mass position ($c$) at (iii) $Jt = 50$ for $h=1.1J$ and (iv) $Jt = 30$ for $h=0.3J$. The red dashed lines in (i),(ii) denotes these two chosen time steps. In both cases, smaller sized mesons delocalize the farthest, suggesting larger mesons propagate slower. The different speeds lead to a spatial filtering of meson sizes as such that it is more probable to detect a longer meson closer to the center during time evolution. Note that we have removed zero density pixels to correct for a parity effect in (iii),(iv) because when $r$ is even, the center-of-mass $c$ only takes integer values while for odd $r$, it only takes half-integer values.
• Figure 3: Dynamics after initializing an $r=1$ meson at the center of a chain of $L=100$ and $L=120$ sites for $h=1.1J$and $h=0.3J$ respectively. (a,c) Average size of propagating mesons, $r_{\text{avg}}$, vs time, $t$. The meson size oscillates in time about a value of $r'_{\text{avg}} = 1.46$ and $r'_{\text{avg}} = 3.28$ for $h/J=1.1J$ and $h=0.3J$ respectively. (b,d) Width, $c_s$, of the center-of-mass distribution vs time, $t$. Due to the propagating meson, the width of the center-of-mass distribution increases monotonically with time. The width grows faster for lower value of $h/J$.
• Figure 4: Long-time average size of the propagating meson, $r'_{\text{avg}}$, vs confinement field, $h/J$ shown by open markers. The solid blue line is a fit to a function, $r'_{\text{avg}} - 1 \propto 1/h$. The error bars show the long time average of the standard deviation of the meson size, $r'_{\text{sd}}$. It shows the range of meson size fluctuations. Frequency of oscillations, $\omega$ of average meson size, $r_{\text{avg}}$ vs field $h/J$ shown by solid green triangles. The solid red line is a fit of $\omega$ as a linear function of $h$.
• Figure 5: A central excitation in a superposition of an $r=1$ and $r=2$ meson, tuned by the angle $\theta$ of one of the gauge spins. The gray circles show a finite probability between $0$ and $1$ of having a particle controlled by $\theta$. The excitation energy is, $E(\theta) = \sin \theta + 2h \sin^2 (\theta/2) + 2h$.