Energy diffusion in frustrated quantum spin chains exhibiting Gaussian orthogonal ensemble level statistics

Frustrated quantum $XXZ$ spin chains with the next-nearest-neighbor couplings are typically deterministic many-body systems exhibiting Gaussian orthogonal ensemble spectral statistics. We investigate energy diffusion for these spin chains in the presence of a periodically oscillating magnetic field. Diffusion coefficients are found to obey the power law with respect to both the field strength and driving frequency with its power varying depending on the linear response and nonperturbative regimes. The widths of the linear response and the nonperturbative regimes depend on the strength of frustrations. We have also elucidated a mechanism for oscillation of energy diffusion in the case of weakened frustrations.

Figure 1

(Color online) Time evolution of energy diffusion for (a) $L=10$ and (b) $L=14$. The parameters are the following: $J_1=J_2=1.0$, $\Delta =0.3$, $B_0=1.0$.

Figure 2

Driving frequency dependence of the diffusion coefficients. The chained line and the solid line are just eye guides for $D\propto {\omega }^{\beta }$ with $\beta =1$ and 2, respectively. The symbols (◇) are the average of the diffusion coefficients calculated for several values of $\Delta$ $(0.3⩽\Delta ⩽0.8)$. The parameters are the following: $L=10$, $J_1=1.0$; (a) $J_2=1.0$, (b) $J_2=0.2$.

Figure 3

Dependence of the diffusion coefficients on the product of field strength $B_0$ and driving frequency $\omega$ for (a) $L=10$ and (b) $L=14$. The symbols (◇) are the average of the diffusion coefficient calculated for several values of $\Delta$ $(0.3⩽\Delta ⩽0.8)$. The parameters are $J_1=J_2=1.0$; for the inset, $J_1=1.0$ and $J_2=0.2$. The chained line and the solid line are just eye guides for $D\propto {\left(B_0\omega \right)}^{\beta }$ with $\beta =1$ and 2, respectively. Some error bars are too short to see.

Figure 4

Examples for time evolution of energy variances: (a) $\delta E{\left(t\right)}^2$ and (b) $\delta \tilde{E}{\left(t\right)}^2$ (see the text). Solid lines are for $J_2=1.0$; broken lines, $J_2=0.2$. The parameters are the following: $L=10$, $J_1=1.0$, $\Delta =0.3$, $B_0=1.5$, $\omega =0.5$.

Figure 5

Parts of energy spectra depending on adiabatically fixed time $t$ with $0⩽t⩽T∕4$. Effective period is $\omega T^{\prime }=2\pi ∕10$. The parameters are the following: $L=10$, $J_1=1.0$, $\Delta =0.3$, $B_0=0.8$; (a) $J_2=1.0$, (b) $J_2=0.2$.

Figure 6

(Color online) Level-spacing distributions at $t=\pi ∕4$ for lowest 300 levels from the ground state (about 10% of all 3003 levels). Blue histogram is for $J_2=1.0$; red bars, $J_2=0.2$; solid curve, GOE spectral statistics. The other parameters are the following: $L=14$, $S_{\mathrm{tot}}^z=1$, $J_1=1.0$, $\Delta =0.3$, $B_0=0.8$. The inset is for all levels when $J_2=1.0$. The numerical methods to obtain the level-spacing distributions are referred to in Refs. 17, 18.