Typical Relaxation of Isolated Many-Body Systems Which Do Not Thermalize

We consider isolated many-body quantum systems which do not thermalize; i.e., expectation values approach an (approximately) steady longtime limit which disagrees with the microcanonical prediction of equilibrium statistical mechanics. A general analytical theory is worked out for the typical temporal relaxation behavior in such cases. The main prerequisites are initial conditions which appreciably populate many energy levels and do not give rise to significant spatial inhomogeneities on macroscopic scales. The theory explains very well the experimental and numerical findings in a trapped-ion quantum simulator exhibiting many-body localization, in ultracold atomic gases, and in integrable hard-core boson and $XXZ$ models.

Figure 1

(Symbols) The experimentally measured Hamming distance $\tilde{D}(t)$ from Fig. 3(a) of Ref. [16] for $W=4J_{\max }$, averaged over 30 realizations of the disorder in Eq. (9). (Line) Corresponding theory from Eq. (7). (Insets) Theory (red curves) and numerical solutions (blue curves) for two representative realizations of the disorder in Eq. (9).

Figure 2

(Symbols) Experimental mean integrated squared contrast from Fig. 3 in Ref. [14] for integration lengths $L=18$, 40, 60, and $100\mu \mathrm{m}$ (from top to bottom) and vertically shifted by 0.3, 0.2, 0.1, and 0.0, respectively, for better visibility. (Lines) Theoretical approximation (7), (8) with $T=3\mathrm{nK}$.

Figure 3

(Symbols) Numerical results from Fig. 1(e) of Ref. [9]. (Line) Theoretical approximation (7), (8). For further details, see the text.

Figure 4

(Symbols) Numerical results for the spin-spin correlation $C^z(t)$ from Fig. 8 of Ref. [29] for a spin-$1/2$ $XXZ$ model with 16 spins, coupling $J$, anisotropy $\mathrm{\Delta }=1/2$, $\hslash =1$, and a so-called pairs of parallel spins initial condition. (Line) Theoretical approximation (7), as specified in the text.