Quantum chaos dynamics in long-range power law interaction systems

We use an out-of-time-order commutator (OTOC) to diagnose the propagation of chaos in one-dimensional long-range power law interaction system. We map the evolution of OTOC to a classical stochastic dynamics problem and use a Brownian quantum circuit to exactly derive the master equation. We vary two parameters: The number of qubits $N$ on each site (the on-site Hilbert space dimension) and the power law exponent $\alpha$. Three light cone structures of OTOC appear at $N=1$: (1) logarithmic when $0.5<\alpha \lesssim 0.8$, (2) sublinear power law when $0.8\lesssim \alpha \lesssim 1.5$, and (3) linear when $\alpha \gtrsim 1.5$. The OTOC scales as $exp\left(\lambda t\right)/x^{2\alpha }$ and $t^{2\alpha /\zeta }/x^{2\alpha }$, respectively, beyond the light cones in the first two cases. When $\alpha \ge 2$, the OTOC has essentially the same diffusive broadening as systems with short-range interactions, suggesting a complete recovery of locality. In the large $N$ limit, it is always a logarithmic light cone asymptotically, although a linear light cone can appear before the transition time for $\alpha \gtrsim 1.5$. This implies the locality is never fully recovered for finite $\alpha$. Our result provides a unified physical picture for the chaos dynamics in a long-range power law interaction system.

Figure 1

Upper panel: The light cone structures and the scaling forms of OTOC when varying $\alpha$ at $N=1$. In the power law light cone regime between 0.8 and 1.5, the OTOC grows algebraically in time. The linear light cone regime with $\alpha \gtrsim 1.5$ can be further divided into two subregimes at $\alpha =2$ according to the different scaling forms of OTOC. Lower panel: The light cone structure and scaling form of OTOC when varying $\alpha$ in the large $N$ limit. The values of the critical $\alpha$ are estimated from the numerics.

Figure 2

Transition rate induced by an interaction term between spins from dot $j$ and dot $i$. The flipping rate of $N=1$ case (shown in the table) for spin at dot $i$ is only nonzero when spin $j$ is up; the rate for flipping up is higher, giving rise to the operator spreading.

Figure 3

End point of an operator. Three panels show typical height configurations at $\alpha =\infty$, where the green block represents $h=1$ and is empty for $h=0$. The red block marks the rightmost site with height 1. The flip occurs at the nearest-neighbor site. Arrows label the transition rates from the middle state to the top and bottom ones. There are also flipping processes to the left of the end point, but they quickly equilibrate.

Figure 4

The spreading of OTOC at various exponent $\alpha$. OTOC $=h_{\mathrm{sat}}/2$ on the black line [light cone $t_{\mathrm{LC}}\left(x\right)]$ in panels (b), (c), and (d). In (a), the locality is fully lost.

Figure 5

Light cone scalings. (a) Linear behavior of $t_{\mathrm{LC}}$ on semilog scale indicates a logarithmic light cone when $\alpha \lesssim 0.8$. The solid line ($L=10000$) stretches linearly longer than the dashed line ($L=2000$), showing that logarithmic light cone will persist in the infinite system. (b) The power law light cone $t=x^{\zeta }$ on the log-log scale for $\alpha \gtrsim 0.8$. We extract the exponent $\zeta$ at various $\alpha$ in the inset.

Figure 6

Data collapse of $\overline{h\left(x\right)}$ curves at various $t$ in the linear light cone regime of $\alpha$. Insets show curves before collapsing. (a) The dashed line is the complementary error function $\frac{3}{8}\text{erfc}(x/3.2)$. It matches well with the collapsed curve at $\alpha =2.5$. The curves in the inset have the same time differences. This applies to all the data collapses below. (b) The data collapse of $\overline{h\left(x\right)}$ at $\alpha =1.75$. Notice that it has a tail which is close to a power law function.

Figure 7

Right end point distribution $\rho \left(x\right)$. (a) When $\alpha =2.5, \rho \left(x\right)$ at various $t$ (solid curves) fit well with the Gaussian packets which broaden diffusively with time (dashed curves). (b) $\rho \left(x\right)$ at $\alpha =1.75$ on the semilog scale (solid curve) compared with a Gaussian packet (dashed curve). The deviation is obvious, especially for large $x$.

Figure 8

Data collapse of $\overline{h\left(x\right)}$ curves for $\alpha =1.25$ (power law light cone) and $\alpha =0.75$ (logarithmic light cone) respectively. In both insets, $\overline{h\left(x\right)}$ decays as $x^{-2\alpha }$ (dashed line). In panel (b) we take $q=\infty$ to access larger system size ($L=100000$).

Figure 9

Light cone structures in large $N$ limit. (a) The curve $t_{\mathrm{LC}}\left(x\right)$ switches from linear to logarithmic at $\alpha =5$. (b) The transitions from linear to logarithmic light cone at various $\alpha$.

Figure 10

The scaling of the front in the linear-to-logarithmic transition in the large $N$ limit at $\alpha =10$. (a) Scaling collapse for the exponential tail on the semilog scale at early time in the linear light cone regime. The front propagates linearly in time. (b) Scaling collapse on the log-log scale at late time in the logarithmic light cone regime. The front propagates exponentially in time. The dashed linear scales as $1/x^{20}$.

Figure 11

(a) The front shape of $\overline{h(x,t)}$ vs $x$ at different times on the semilog scale for $\alpha =10$. (b) Scaling collapse of $\overline{h(x,t)}$ on the log-log scale at $\alpha =1$. The dashed linear scales as $1/x^2$.