Long-living prethermalization in nearly integrable spin ladders

Relaxation rates in nearly integrable systems usually increase quadratically with the strength of the perturbation that breaks integrability. We show that the relaxation rates can be significantly smaller in systems that are integrable along two intersecting lines in the parameter space. In the vicinity of the intersection point, the relaxation rates of certain observables increase with the fourth power of the distance from this point, whereas for other observables one observes standard quadratic dependence on the perturbation. As a result, one obtains exceedingly long-living prethermalization but with a reduced number of the nearly conserved operators. We show also that such a scenario can be realized in spin ladders.

Figure 1

(a) Sketch of the ladder in Eq. (1) with marked $S^z\text{−}{S}^z$ interactions ($\mathrm{\Delta },U$) and nearly conserved operators ($Q_n^{\pm }$). (b,c) Correlation functions $C_n^{\pm }\left(t\right)$ defined in Eq. (3) obtained for a ladder with $L=14$ rungs. (b) Dashed and continuous curves show $C_3^{+}\left(t\right)$ and $C_3^{-}\left(t\right)$, respectively. (c) The same as in panel (b) but for $C_4^{\pm }\left(t\right)$. Horizontal dotted lines in panels (b) and (c) define times, $t_{n,\gamma }^{\pm }$, when $C_n^{\pm }\left(t_{n,\gamma }^{\pm }\right)=\gamma =0.3$. The legends in panels (b) and (c) are common for both panels. Panels (d) and (e) show $1/t_{3,\gamma }^{\pm }$ and $1/t_{4,\gamma }^{\pm }$ for NI systems with $U=\mathrm{\Delta }$ and $\gamma =0.3$ (see text for details).

Figure 2

Ratio $R_n\left(\gamma \right)$ calculated for $L=12$. Blank regions correspond to parameters for which $C_n^{+}\left(t\right)>\gamma$ for the numerically accessible times $t\sim {10}^4$, e.g., see $C_n^{+}\left(t\right)$ for $U=0.1$ in Fig. 1.

Figure 3

Overlaps $\langle Q_n^{+}{I}_n\rangle$ calculated analytically for $L\to \infty$, see Eq. (4) and Ref. [48] for more details. Black solid curves are isolines. Note that $Q_n^{+}$ and $I_n$ are strictly conserved for $U=0$ and $\mathrm{\Delta }=0$, respectively.

Figure 4

(a)–(d) Relaxation rates $1/t_{n,\gamma }^{\pm }$ estimated from the correlation functions in Eq. (3) via relation $C_n^{\pm }\left(t_{n,\gamma }^{\pm }\right)=\gamma$ for $\gamma =0.3$ and $L=12$. Continuous curves represent isolines. Dashed lines in panels (a) and (b) mark parameters for which relaxation rates are shown in Figs. 1 and 1. Panels (e) and (f) show relaxation rates $1/t_{n,\gamma }^{-}$ along the dashed lines marked in panel (c).