Isolated Heisenberg magnet as a quantum time crystal

We demonstrate analytically and numerically that the paradigmatic model of quantum magnetism, the Heisenberg XXZ spin chain, does not equilibrate. It constitutes an example of persistent nonstationarity in a quantum many-body system that does not rely on external driving or coupling to an environment. We trace this phenomenon to the existence of extensive dynamical symmetries. We discuss how the ensuing persistent oscillations that seemingly violate one of the most fundamental laws of physics could be observed experimentally.

Figure 1

(a) Nonstationary behavior of the numerically computed dynamical susceptibility of $O=O_3={\sum }_j{s}_j^x{s}_{j+1}^x{s}_{j+2}^x$ for $\mathrm{\Delta }=-1/2,h=J$ (dots) and compare it to the exact asymptotic result (red curve). As expected, the single-body observable $O=O_1={\sum }_j{s}_j^x$ relaxes to stationarity (blue curve). In (b) we plot the data for the quench from the ferromagnetic state, showing nonstationarity of the three-point function $o=o_3=s_j^{+}{s}_{j+1}^{+}{s}_{j+2}^{+}+s_j^{-}{s}_{j+1}^{-}{s}_{j+2}^{-}$ (blue) and relaxation of a single-point function $o=o_1=s_j^{+}+s_j^{-}$ (red), for $\mathrm{\Delta }=-0.5,h=2J$. In both (a) and (b) we choose four different observables, though qualitatively the same two kinds of behavior (persistent oscillations or relaxation) would be seen for any choice of observables. In (c) we present the effects of perturbing $\mathrm{\Delta }$ at $h=J$ on the dynamical susceptibility for $O=O_3$. In (d) we present the effects that the integrability breaking next-to-nearest neighbor interaction $\alpha$ has on the dynamical susceptibility for $O=O_3$ at $\mathrm{\Delta }=-1/2,h=J$. All simulations were performed using DMRG with the system size $N=100$.

Figure 2

The period $T=2\pi /\omega$ of the persistent oscillations of dynamical symmetries depending on $\mathrm{\Delta }$. Inset shows a close-up illustrating the nowhere continuous (fractal) nature of the curve. Crucially, as the families of the $Y_{\mathrm{\Delta }}$ operators are different at each $\mathrm{\Delta }$ the observables that have overlap with them and thus persistently oscillate also change via (4) and (5). The arrows show examples of operators that have overlap with the $Y_{\mathrm{\Delta }}$ at a given $\mathrm{\Delta }$ and thus persistently oscillate.