Tuning between Continuous Time Crystals and Many-Body Scars in Long-Range $XYZ$ Spin Chains

Persistent oscillatory dynamics in nonequilibrium many-body systems is a tantalizing manifestation of ergodicity breakdown that continues to attract much attention. Recent works have focused on two classes of such systems: discrete time crystals and quantum many-body scars (QMBS). While both systems host oscillatory dynamics, its origin is expected to be fundamentally different: discrete time crystal is a phase of matter which spontaneously breaks the ${\mathbb{Z}}_2$ symmetry of the external periodic drive, while QMBS span a subspace of nonthermalizing eigenstates forming an su(2) algebra representation. Here, we ask a basic question: is there a physical system that allows us to tune between these two dynamical phenomena? In contrast to much previous work, we investigate the possibility of a continuous time crystal (CTC) in undriven, energy-conserving systems exhibiting prethermalization. We introduce a long-range $XYZ$ spin model and show that it encompasses both a CTC phase as well as QMBS. We map out the dynamical phase diagram using numerical simulations based on exact diagonalization and time-dependent variational principle in the thermodynamic limit. We identify a regime where QMBS and CTC order coexist, and we discuss experimental protocols that reveal their similarities as well as key differences.

Figure 1

Schematic summary of the dynamical phase diagram of the model in Eq. (1) as a function of U(1) symmetry breaking $J_{\mathrm{U}(1)}$, SU(2) symmetry breaking $J_{\mathrm{SU}(2)}$ and interaction range $\alpha$. Within the prethermal regime (yellow), CTC phase emerges for small $\alpha$ and small $J_{\mathrm{U}(1)}$ (green). QMBS (red) are independent of $\alpha$ but require small $J_{\mathrm{SU}(2)}$. The solvable Lipkin-Meshkov-Glick (LMG) limit ($\alpha =0$) is shown in gray.

Figure 2

Signatures of the continuous time crystal. (a) Expectation value of the prethermal Hamiltonian, Eq. (2), in the time evolved state. The value is normalized by its value at time $t=0$. (b) Order parameter $⟨{\sigma }^{+}(t)⟩$ defined in the text. (c) Entanglement entropy $S_E(t)$. All plots are for the infinite long-range $XYZ$ model in Eq. (1) with $\alpha =1.13$, $J_x=-0.4$, $J_y=-2.0$, $J_z=-1$. The value of the field $h_z$ is indicated in panels (a)–(c), while $h_z=1$ in panels (d)–(f). The initial state is given by Eq. (3) with $\varphi =0$ in (a)–(c), and $\varphi =\{0,\pi /6,\pi /4,\pi /3,\pi /2\}$ in panels (d)–(f). Dashed lines in (f) denote the average magnetization $\cos (\varphi )/2$ after the rapid initial relaxation.

Figure 3

Quantum many-body scars near the isotropic limit of the model in Eq. (1), with $J_x=-0.8$, $J_y=-1$, $J_z=-0.95$, $h_z=3$ and system size $N=16$. The initial state $|\psi (0)⟩$ is $x$-polarized [$\varphi =0$ in Eq. (3)]. Top row: Eigenstate overlap with $|\psi (0)⟩$ for both long-range (a) and short-range (b) models. In both cases, the top band of eigenstates are the QMBS eigenstates, which are well approximated by maximal-spin SU(2) basis states in the $z$-direction, denoted by diamonds. (c) Quantum fidelity revivals from the initial state $||\psi (0)⟩$, for both long- and short-range models. (d) Finite-size scaling of the fidelity density $-\ln (f_0)/N$, where $f_0$ is the height of the first fidelity revival. The fidelity density, was obtained using finite TDVP with, bond dimension $\chi =300$ and time step $\delta t=0.02$.