Ergodicity breaking meets criticality in a gauge-theory quantum simulator

TL;DR

This work tackles how gauge constraints and quantum criticality influence real-time dynamics and thermalization in lattice gauge theories. It implements a spin-$1/2$ U(1) quantum link model in $(1+1)$ dimensions using a programmable Rydberg-atom array, employing a ramp-then-quench protocol to induce quantum many-body scars and probe Kibble-Zurek effects. The study uncovers a broad ergodicity-breaking regime with long-lived scar revivals that persists near the Coleman transition, and shows that the density and internal structure of electron-positron pairs generated during ramps critically shape post-quench dynamics; slower ramps promote domain-wall configurations that stabilize scars. Overall, the results establish Rydberg-atom quantum simulators as a powerful platform for exploring the interplay between gauge constraints, quantum criticality, and nonthermal dynamics in lattice gauge theories.

Abstract

Recent advances in quantum simulations have opened access to the real-time dynamics of lattice gauge theories, providing a new setting to explore how quantum criticality influences thermalization and ergodicity far from equilibrium. Using QuEra's programmable Rydberg atom array, we map out the dynamical phase diagram of the spin-1/2 U(1) quantum link model in one spatial dimension by quenching the fermion mass. We reveal a tunable regime of ergodicity breaking due to quantum many-body scars, manifested as long-lived coherent oscillations that persist across a much broader range of parameters than previously observed, including at the equilibrium phase transition point. We further analyze the electron-positron pairs generated during state preparation via the Kibble-Zurek mechanism, which strongly affect the post-quench dynamics. Our results provide new insights into nonthermal dynamics in lattice gauge theories and establish Rydberg atom arrays as a powerful platform for probing the interplay between ergodicity breaking and quantum criticality.

Figures (18)

• Figure 1: Simulating the dynamics of a spin-1/2 U(1) QLM in $(1+1)\mathrm{D}$ with Rydberg atoms. (a) ${}^{87}\mathrm{Rb}$ atoms are trapped in a 1D tweezer array. The two pseudo-spin states, the ground state $\ket{g}$ and the Rydberg state $\ket{r}$, are coupled by a two-photon process at the Rabi frequency $\Omega$ and detuning $\Delta$. Nearest-neighbor excitations are suppressed by van der Waals interactions within the Rydberg blockade radius $R_b$, giving rise to the gauge-invariant constraint Surace2020. (b) The Rydberg spin states are associated with electric fields in the QLM, with each spin-flip representing a fermion pair production or annihilation process, accompanied by the corresponding change in the direction of electric field (shaded boxes). This mapping, the details of which are given in Appendix \ref{['a:mapping']}, fixes the parameters of the QLM to $J=\Omega/2$ and $m=\Delta/2$, while the average number of electron-positron pairs in the QLM becomes the average density of Rydberg excitations. (c) The phase diagram as a function of $m/J$. The representative ground states in the two phases are indicated, in both the Rydberg and QLM representations. The two phases are separated by an Ising phase transition at the critical mass $m_c\approx 0.65J$. (d) Schematic of our two-stage dynamical protocol. The initial state is prepared by ramping from deep in the disordered phase ($m_0 \to -\infty$) into the ordered phase at some $m_i$, potentially crossing the critical point $m_c$. The ramp function $f(x)=x(1-k)/(k-2k\lvert x\rvert +1)$ is a sigmoid with the curvature parameter $k$, see Appendix \ref{['a:experiment']}. For present purposes, we only need to distinguish between 'fast' (piecewise linear) ramps ($k\to 0$) and 'slow' ($k\to 1$) ramps, whose dependencies are sketched in the figure. The second stage of the protocol is the quench where we rapidly change $m_i \to m_f$ and monitor the subsequent dynamics. (e)-(f) Measurements of the average number of electron-positron pairs after the quench to $m_f=0$. A slow ramp, $k=0.8$ in panel (e), prepares a state close to one of the flux states in panel (c). The subsequent dynamics give rise to persistent oscillations, first observed in Ref. Bernien2017. By contrast, a fast ramp, $k=0$ in panel (f), prepares a state with many electron-positron pairs, which causes a much faster decay of the oscillations in the bulk. The measurements in (e)-(f) were taken at a fixed $J=7.715\, \mathrm{kHz}$.
• Figure 2: Dynamical phase diagram of ergodicity breaking in the spin-1/2 U(1) QLM in $(1+1)\mathrm{D}$. (a) Deviation from thermal value, $D_\mathrm{th}$ in Eq. (\ref{['eq:msd']}), shown on the color bar, as a function of fermion masses, $m_i$ and $m_f$. Data obtained by exact diagonalization of the Rydberg model [Eq. (\ref{['eq:ryd']})] for $L=15$ atoms. Red regions correspond to ergodicity breaking. Dashed lines ($C_1$-$C_3$) are the three cuts that have been measured experimentally, with $C_2$ close to the critical value $m_i\approx J$. The measured $D_\mathrm{th}$ values at these cuts are plotted in panels (e)-(g), while the dynamics of charge density and PCA entropy at representative points $P_1$ and $P_2$ are shown in panels (b)-(e). (b)-(c) Time series of QLM charge density $Q(t)$, Eq. (\ref{['eq:Q']}), for quenches at $P_1$ ($m_i=J$ and $m_f=J/2$) and $P_2$ ($m_i=J$ and $m_f=0$), respectively. (d)-(e) Time series of the normalized PCA entropy, $S_\mathrm{PCA}/\log(L)$, for the same quenches as in (b)-(c). In panels (b)-(e), points are experimental data for $L=61$ atoms, while lines are theoretical predictions for $L=21$. (e)-(g) Experimental data along the line cuts $C_1$-$C_3$. The data for a smaller system with $L=21$ (blue dots) are seen to be in excellent agreement with the much larger system $L=61$ (red pluses), where we have collected fewer points due to limited resources. The error bars in all the plots represent statistical errors. The dashed and dot-dashed black lines show numerical results for $L=21$, without and with exponential damping, respectively.
• Figure 3: Impact of electron-positron pairs on the nonergodic dynamics in the spin-1/2 U(1) QLM in $(1+1)\mathrm{D}$. (a)-(b) The post-quench dynamics of total matter field density, $\tilde{Q}$ in Eq. (\ref{['eq:n_d']}), for a fast ramp with $k=0$ (a) versus slow ramp with $k=0.8$ (b). Dots are experimental results, solid line is their interpolation, dashed line is the numerical simulation of the Rydberg model [Eq. (\ref{['eq:ryd']})]. (c)-(d) $\tilde{Q}$ in the initial state (red crosses, left $y$-axis) compared against the first peak of the Fourier transform of $\tilde{Q}(t)$ (blue circles, right $y$-axis), as a function of ramp curvature $k$. Close agreement between the two for $k\lesssim 0.6$ demonstrates that the density of electron-positron pairs in the initial state governs the decay of QMBS oscillations. Panel (c) is numerics, (d) is experimental data. (e)-(f) Beyond the sheer number of electron-positron pairs, different types of particle-string domains can have a strong impact on QMBS dynamics. This is illustrated by the numerical simulations of the dynamics for 'even' (e) vs. 'odd' (f) type of particle-string defects in the initial state, schematically shown on top of each plot. (g) The measured number of paired (e.g., $\textcolor{red}{\bullet}\textcolor{blue}{\bullet}\textcolor{red}{\bullet}\textcolor{blue}{\bullet}$) and unpaired (e.g., $\textcolor{red}{\bullet}\textcolor{blue}{\bullet}\textcolor{red}{\bullet}$) particle strings, which serve as domain walls between vacuum domains with uniform electric field. Dashed lines are theoretical predictions. (h) Density of gauge violations $\varepsilon$ after the ramp and before the quench. In all the panels, the parameters $(m_i=2J, m_f=0)$ correspond to the point with the most robust QMBS signatures along the $C_3$ line in Fig. \ref{['fig:phase_diagram']}. The system size used in all panels except (e) and (f) for both the experiment and the numerics is $L=21$. Panels (e) and (f) are numerics for $L=61$ atoms using time-dependent variational principle algorithm in TenPy tenpy, with a maximum bond dimension $1000$.
• Figure 4: Ramp protocol. (a) Plot of the sigmoid part of the ramp from Fig. \ref{['fig:schematic']}(d), for different values of $k$. (b) The complete protocol, with the simultaneous modulation of both $\Delta$ and $\Omega$, according to Eqs. (\ref{['eq:delta']})-(\ref{['eq:omega']}).
• Figure 5: Spatial arrangement of atoms. Left panel shows six parallel chains of 21 atoms, right panel is one long chain of 61 atoms looping around the available physical space.