Dynamical correlations and domain wall relocalization in transverse field Ising chains

Abstract

We study conventional and out-of-time-ordered correlators (OTOCs) for a wide variety of transverse field Ising chains: classical and quantum, clean and disordered, and integrable and generic. The setting we consider is that of a quantum quench. We find a remarkably rich phenomenology, ranging from stable periodic signals to ones decaying with varying rates. This variety is due to a complex interplay of constraints on thermalization imposed by integrability and symmetry. A process we term dynamical domain wall relocalization provides a long-lived signal in the clean, integrable case, which can be degraded by the addition of disorder even without interactions. Our results shed light on a proposal to use an OTOC as a dynamical diagnostic of a quantum phase more powerful than a standard observable.

Figures (6)

• Figure 1: Magnetization (top) and OTOC (bottom) for the quantum (left) and classical (right) Ising chain given in Eqs. \ref{['eq:tfi_hamiltonian']} and \ref{['eq:classical_model_chain']}. We use $N=500$ spins, coupling strength $g=0.9$, open boundary conditions, and $O_\mathrm{m}=O_\mathrm{p}=\sigma^z_{0}$. Static spatial disorder is added to the transverse field (Gaussian with standard deviation $W=0.04$). (d) shows results averaged over a time window of $\sim 0.03t$ and over the nine sites closest to the perturbation to reduce fluctuations. For the classical simulations beyond $t \gtrsim 50$, the time evolution of the OTOC appears chaotic.
• Figure 2: The schematic creation of $d_{0'}(t)=\sum_{l'} A_{0'l'} c_{l'} + B_{0'l'} c_{l'}^\dagger$, Eq. \ref{['eq:final_d_opertors']}, with relocalization (red, green) and delocalization (orange) in three steps. The blue lines indicate the envelopes of the prefactors $A_{0'l'},B_{0'l'}$ in space when expressing the intermediate stages $c_{0'}$, $c_{0'}(-t)$, $\sigma^z_0 c_{0'}(-t) \sigma^z_{0}$ as sum $A_{0'l'} c_{l'} + B_{0'l'}c_{l'}^\dagger$, respectively. First step: The operator splits into two wavepackets with positive (green) and negative (red) front velocity $v$ creating $c_{0'}(-t)$ . Second step: Applying $\sigma^z_0$ yields a global perturbation by shifting the phase of the left half by $\pi$, indicated by the dashed lines. This decomposes in a phase shift of the left moving wavepacket and a local perturbation shown in orange. Third step: The two wavepackets (green, red) are relocalized, and the orange perturbation delocalizes. Their sum is $d_{0'}(t)$.
• Figure 3: Time-averaged correlators in the nonintegrable ANNNI model at $\Delta=1/2$: OTOC (circles, from Ref. heylpollmann and our own numerics), compared to domain wall density observables (triangles) evaluated in $|\!\!\uparrow^N\rangle$ and $\sigma^z_j(t)|\!\!\uparrow^N\rangle$. The window for time averaging was chosen after initial transients (by eye), up to the time where boundary effects become visible, see Supplemental Material \ref{['sm:annni_finite']}. The data is most reliable for intermediate values of $g \approx 1$: for small $g$ no clear plateau in the OTOC appears up to $t=120$, while for large $g$, boundary effects set in early.
• Figure 4: Central row of $|X_{j'l'}|^2$, with $J = 1, g = 0.3, N=4000$, and times $700<t<5300$. The wide gray line indicates the exponential decay due to the relocalized domain wall pairs, $X^r$ [Eq. \ref{['eq:result_x_rel']}]. The inset shows its deviation $|X_{j'l'}^d|^2=|X_{j'l'}-X^r_{j'l'}|^2$; these curves collapse upon rescaling the axes with $x_r=v_\mathrm{wf}t\approx 0.3t$ and $y_r=0.04/t^2$, respectively. The data are averaged over $10$ adjacent $j'$ and two $l'$.
• Figure 5: Numerical simulation results for the ANNNI model for $N=22$ (orange) and $N=24$ (blue) for three different exemplary values of $g$, as indicated in green, while $J=1$ and $\Delta=0.5$. The results were obtained using Krylov time evolution. The red lines indicate the manually selected time window over which the OTOC has been averaged, yielding the results shown in Fig. \ref{['fig:compare_annni_dw_density']}.