Revival Dynamics from Equilibrium States: Scars from Chords in SYK

TL;DR

The paper develops a general Krylov-space framework to generate quantum many-body scars by coupling two anticorrelated subsystems and embedding a grading operator to produce a tower of equally spaced scar energies, yielding finite-time revivals for purified equilibrium states. It then provides an explicit approximate realization in the two-sided double-scaled SYK model, where chord diagrams realize the Krylov subspace and the chord number operator acts as the grading, with H0 behaving as a harmonic-oscillator-like generator and revivals governed by a tunable μ. The authors analyze revival dynamics in DSSYK, including coherent-state trajectories and an approximate quantum-error-correcting description of the scar subspace, and discuss the λ → 0 continuum limit with holographic interpretations in terms of JT gravity and AdS2 isometries. Finite-size numerics corroborate the analytical predictions, showing robust non-ergodic dynamics within the Krylov sector even away from the strict double-scaling limit. The work highlights a universal mechanism for non-ergodic dynamics in bipartite quantum systems and opens pathways for experimental realizations and holographic explorations of quantum scars.

Abstract

We develop a novel framework to build quantum many-body scar states in bipartite systems characterized by perfect correlation between the Hamiltonians governing the two sides. By means of a Krylov construction, we build an interaction term which supports a tower of equally-spaced energy eigenstates. This gives rise to finite-time revivals whenever the system is initialized in a purification of a generic equilibrium state. The dynamics is universally characterized, and is largely independent of the specific details of the Hamiltonians defining the individual partitions. By considering the two-sided chord states of the double-scaled SYK model, we find an approximate realization of this framework. We analytically study the revival dynamics, finding rigid motion for wavepackets localized on the spectrum of a single SYK copy. These findings are tested numerically for systems of finite size, showing excellent agreement with the analytical predictions.

Figures (8)

• Figure 1: (a) The open chord state $\ket{2}$ from slicing open a chord diagram appearing in computation of $m_8$. (b) $\ket{3}$ contributing to $m_{10}$.
• Figure 2: $\abs{f_{\beta}(\theta, t)}$ for $\mathfrak{q}=0.8$, $\beta=2$ (left) and $\beta=10$ (right). For $\beta=2$, we sample $\abs{f_{\beta}(\theta, t)}$ for $\frac{\omega t}{2 \pi} = 0$ (blue), $\frac{\omega t}{2 \pi} = 0.1$ (orange), and $\frac{\omega t}{2 \pi} = 0.25$ (green). For $\beta=10$, we show $\frac{\omega t}{2 \pi} = 0$ (blue), $\frac{\omega t}{2 \pi} = 0.05$ (orange), and $\frac{\omega t}{2 \pi} = 0.25$ (green). The distribution gets reflected around $\theta=\frac{\pi}{2}$ at $\pi -\omega t$, we have plotted reflected distributions for the specified times with the same colors. The dashed lines are the result of fitting with the Gibbs form $\sim e^{-\frac{\beta_{\text{eff}}(t)}{2} E(\theta)}$ with $\beta_{\text{eff}}(t)$ given by Eq. \ref{['eq:beta_effective']}. The agreement is approximate in nature. The width of $\abs{f_{\beta}(\theta, t)}$ changes as a function of $t$ to satisfy Eq. \ref{['eq:survival_amplitude_as_f_convolution']}.
• Figure 3: The normalized wavepacket profile $\abs{f_{\beta}(\theta, t)}^2 \rho(\theta)$ for $\beta=2$ (left) $\beta=10$ (right) with $\mathfrak{q}=0.05$ (dashed lines) and $\mathfrak{q}=0.8$ (solid lines). We show the densities moving from the left at $t=0$ (blue) to a configuration symmetric about $\theta=\frac{\pi}{2}$ at $t=\frac{\pi}{2 \omega}$ (orange) to $t=\frac{\pi}{\omega}$ (green), when it gets reflected about $\theta=\frac{\pi}{2}$. For fixed $\beta$, smaller $\mathfrak{q}$ gives a broader distribution in $\ket{\theta}$ which can be traced back to how $\rho(\theta)$ depends on $\mathfrak{q}$. On increasing $\beta$, the wavepacket becomes more sharply peaked for the entire cycle.
• Figure 4: The entanglement entropy difference $S_{\infty}-S(t)$ for $\ket{\beta(t)}$ with $\beta=2$ (left) and $\beta=10$ (right). $S(t)$ is the von Neumann entropy of $\rho_{L/R}(t)$ as defined in Eq. \ref{['eq:one-sided']}. The entanglement entropy difference is correlated with $\langle \hat{H}_0(t) \rangle$ and $\langle \hat{k}_0(t) \rangle$. The entanglement entropy is well approximated by the thermal entropy evaluated at $\beta_{\text{eff}}(t)$ (dashed lines). For $\beta=2$, $E(\beta) \approx \beta$ across different values of $\mathfrak{q}$ and $\beta_{\text{eff}}(t) \approx \beta(0) \cos(\omega t)$. We used the Gaussian form $S_{\infty}-S(t) \propto \beta(0)^2 \cos^2(\omega t)$ in the left panel as $S(\beta_{\text{eff}}(t))$ and found good agreement. For $\beta=10$, we computed $\beta_{\text{eff}}(t)$ by directly numerically solving for $\beta_{\text{eff}}(t)$ in Eq. \ref{['eq:beta_effective']}
• Figure 5: The return probability $F(t)$ for $\mathfrak{q}=0.1$ (left) and $\mathfrak{q}=0.9$ (right), with $\beta=2$ (blue) and $\beta=10$ (orange). The dashed lines are fits with approximate analytical predictions: for the $\mathfrak{q}=0.1$ plot, we use the predictions coming from the $\mathfrak{q}=0$ case, computed analytically in Eq. \ref{['eq:survival_amplitde_q=0']}; while for the $\mathfrak{q}=0.9$ plot, we use as an analytical prediction the coherent state approximation computed in Eq. \ref{['eq:coherent_state_survival_probability']}. For this latter case, we used $z = -\frac{1}{2} E(\beta)$ and we see that the agreement with Eq. \ref{['eq:coherent_state_survival_probability']} worsens on increasing $\beta$.