Dynamical Behaviour of Density Correlations Across the Chaotic Phase for Interacting Bosons

Abstract

We investigate the propagation of two-point density correlations in the one-dimensional Bose-Hubbard Hamiltonian in the thermodynamic limit in terms of the correlation transport distance (CTD), an experimentally measurable magnitude that characterizes the spatial spreading of correlations in time. We confirm that the integrable limits of the model exhibit CTD ballistic growth, while the onset of the chaotic phase leads to the emergence of a pronounced sub-ballistic regime, in agreement with previous results for finite systems. By a meticulous analysis of the spatio-temporal correlation profiles, we show that the correlation front nonetheless propagates ballistically for all interaction strengths, and that the chaos-induced slowdown of the CTD originates from the emergence of long-time distance-dependent correlation tails, together with an enhanced decay of the correlation front amplitude. Our results thus provide a detailed characterization of correlation transport that goes beyond a simple light-cone picture.

Figures (8)

• Figure 1: Optimal parameters used in the iTEBD simulations, (a) maximum local occupation number $n_{\rm max}$, (b) time step $\delta$, (c) cutoff value $\varepsilon$, and (d) maximum accessible simulation time $\tau_{\rm max}$ as functions of $\gamma$.
• Figure 2: (a) Time evolution of the CTD for $L = \infty$ and varying $\gamma \in \bqty{0.00316, 100}$, as indicated by the color scale. The grey dashed lines highlight the initial quadratic growth [Eq. \ref{['eq:Univ_Quad_Growth']}] and the asymptotic ballistic behaviours. The values of the CTD computed analytically in the integrable limits, following Eqs. \ref{['eq:Limit_Beh_gamma_Inf_full']} and \ref{['eq:Limit_Beh_gamma_0_full']}, are shown by two thicker black lines. (b) Parameters $\beta$ (main panel) and $\alpha$ (inset) of the CTD power-law fit $\ell\pqty{\tau} = \alpha \tau^\beta$, as functions of $\gamma$ for the indicated fitting intervals. Horizontal dotted grey lines highlight ballistic ($\beta = 1$) and diffusive ($\beta = 1/2$) behaviour. The vertical dashed grey line marks the onset of the chaotic phase at $\gamma = 0.11$. Horizontal dashed black lines follow from the fit of the analytical signals in the integrable limits, given by Eqs. \ref{['eq:Limit_Beh_gamma_Inf_full']} and \ref{['eq:Limit_Beh_gamma_0_full']}, in the intervals $\tau \in \bqty{2.2, 3.9}$ and $\tau \in \bqty{2.2, 5.5}$, respectively. In the lower inset, all data for different $L$ collapse and the thick grey lines correspond to the coefficients of the dominant term in Eqs. \ref{['eq:Limit_Beh_gamma_Inf']} and \ref{['eq:Limit_Beh_gamma_0']}. When not visible, error bars are contained within symbol size.
• Figure 3: Pseudo-distribution $G_d\pqty{\tau}$ [Eq. \ref{['eq:PseudoDistrib']}] in log scale as a function of the distance $d$ for $L = \infty$ at five equispaced instants in time, highlighted by different colours, and different $\gamma$ values, shown in independent panels, as indicated, and accompanied by the corresponding power-law fit exponents $\beta$ [see Fig. \ref{['fig:CTD+PowerlawFits']}(b)]. Crosses in the panels for the lowest and highest $\gamma$ display the analytical results in the integrable limits, Eqs. \ref{['eq:PseudoDistrib_gamma_inf_full']} and \ref{['eq:PseudoDistrib_gamma_0_full']}. Vertical dashed lines indicate the normalized CTD $\ell_{\mathcal{N}}(\tau)$ [Eq. \ref{['eq:normalizedCTD']}] for each $\tau$ and the two vertical arrowheads mark the correlation front (see main text).
• Figure 4: Spatio-temporal correlation profiles for $L=\infty$ and different $\gamma$ as indicated. For each $\gamma$, the upper panel displays a density plot of the normalized correlation $\mathcal{G}_d(\tau)$ [Eq. \ref{['eq:normalizedGd']}] versus distance $d$ and time $\tau$. The corresponding lower panel shows $G_d(\tau)$ as a function of time for different $d\in[3,25]$ as indicated by the color scale. Black dots mark the correlation front [maximum of $G_d(\tau)$ over time for each $d$].
• Figure 5: (a) Time evolution of $G_d(\tau)$ for $L=\infty$, different distances $d$ and values of $\gamma$ (as indicated). Dashed lines represent the saturation values $G^{\rm (sat)}_d$ as defined in the main text. (b) Saturation values $G^{\rm (sat)}_d$ as a function of $\gamma$ for different $d$ (colour coded). The vertical dashed grey line marks the onset of the chaotic phase, $\gamma = 0.11$.