Symmetry-induced fragmentation and dissipative time crystal

TL;DR

This paper addresses whether time-crystalline order can be stabilized in open quantum many-body systems beyond traditional localization or Floquet prethermalization. It develops a scheme where a $U(1)$ weak symmetry fragments the Liouville space, creating a spectrum with purely imaginary eigenvalues that support persistent oscillations and time-crystal behavior in dissipative settings. Under symmetry-breaking perturbations, a prethermal regime emerges with stage-wise dynamics, protected by the Liouvillian skin effect and a permutation-group representation of excitations above a real-space Fermi sea, leading to algebraic suppression of dissipation. The findings reveal a new mechanism—Hilbert-space fragmentation coupled to non-Hermitian effects—for robust time crystals with potential experimental realization in atom–cavity platforms, highlighting the role of non-Hermitian dynamics in many-body open systems.

Abstract

Time crystals are a peculiar state of matter. Their emergence hinges on ergodicity breaking, which typically originates from many-body localization or Floquet prethermalization. Here we propose a novel scheme for devising robust dissipative time crystals where the ergodicity is broken through symmetry-induced fragmentation. Building upon a U(1)-symmetry-induced Liouville-space fragmentation, we first propose a generic Liouvillian with long-time oscillations typical of time crystals. We then show that, even when the U(1) symmetry is broken, a prethermal time-crystal behavior survives, with distinct oscillation frequencies at different times of the steady-state approaching dynamics. Intriguingly, the stage-wise prethermal dynamics derive from Fermi statistics and the Liouvillian skin effect of our model -- as the excitations above the boundary-localized dark states can be mapped to the irreducible representations of the permutation group, the branching rules of the permutation group ensure the robustness of the prethermal time crystal. Our work paves the way for devising time crystals through Hilbert-space fragmentation. It also sheds light on the dynamic effects of non-Hermitian physics in many-body quantum open systems.

Figures (9)

• Figure 1: Symmetry-induced fragmentation and time crystal. (a) Block-diagonalized Liouvillian matrix. The red boxes indicate the diagonal symmetry sectors with $f_i=f_j$, the inset shows a zoomed-in view of the matrix elements (tinged according to the color bar) within a symmetry sector. (b) The Liouvillian spectrum on the complex plane. The inset shows the non-dissipative eigenmodes on the imaginary axis. (c) The long-time dynamics of $\langle \hat{c}_5^\dagger \hat{c}_2+ \hat{c}_2^\dagger \hat{c}_5\rangle$ (black: $L=10, N=4$; purple: $L=8, N=4$; orange: $L=6, N=3$). (d) Two periods of the oscillatory dynamics starting at $t_0\gamma=10^5$ (upper panel) and $t_0\gamma=10^7$ (lower panel), respectively marked by the blue and green vertical lines in (c). Unless otherwise specified, all panels are obtained with $s=0$, $\Delta/\gamma=0.2$, $L=8$, and $N=4$.
• Figure 2: Prethermal time crystal. (a) Tri-diagonal Liouvillian matrix under symmetry-breaking perturbations. The red and green boxes respectively indicate the diagonal ($f_i=f_j$) and sub-diagonal ($|f_i-f_j|=1$) blocks. (b) The Liouvillian spectrum and its zoomed-in view (inset) near the imaginary axis. (c)(d) The real and imaginary eigenvalue components of the weakly-dissipative eigenmodes [in the inset of (b)], with respect to their corresponding unperturbed values ($-i\Delta f$). The dashed lines are numerical fits with the power-law scaling $\lambda_f+i\Delta f\sim s^p$, where $p=2,4,6,8$ from top to bottom. For (a)(b), we take $L=8$, $N=4$, and $s/\gamma=0.2$; while for (c)(d), $L=10$ and $N=5$. For all calculations, we set $\Delta/\gamma=0.2$.
• Figure 3: Mapping to the permutation group. (a) Excited states, in the excitation basis, generated from the dark state $|0\rangle$ by $\hat{K}^\dag$. The expansion coefficients (red) coincide with the dimensions of the corresponding irreducible representation (of the permutation group). An example of the Young-diagram correspondence of the excitation basis is given in the upper right corner, with which the structure in (a) corresponds to the Young's lattice. (b) Relation of the exponent $p=2l^{|D^{B}\rangle}+2$ with $B$ for $L=10$ and $N=5$, obtained analytically from the group analysis. (c) Maximum exponent $p_m$ for eigenmodes characterized by $f$, numerically calculated for different $N$ and $L=10$. For (b)(c), we take $s/\gamma=0.2$, and $\Delta/\gamma=0.2$.
• Figure 4: Stage-wise dynamics of the prethermal time crystal. (a) Time evolution of $P_{|f|}$. (b) Numerically simulated evolution of $\langle\hat{K}_2+\hat{K}^\dagger_2\rangle$, where $\hat{K}_2=\sum_{j=1}^{L-2} \hat{c}_j^\dagger \hat{c}_{j+2}$ (black: $L=10, N=4$; purple: $L=8, N=4$; orange: $L=6, N=3$). In (a)(b), stages with different dominant eigenmodes are separated by gray vertical lines. (c) Oscillations zoomed in, starting at different times $t_0\gamma=10^3$ (upper), $t_0\gamma=10^5$ (middle) and $t_0\gamma=10^7$ (lower), which are labeled with vertical lines in (a)(b) with the corresponding colors. The oscillation period is given by $T_f=2\pi/f\Delta$, with $f=1,3,2$, respectively, for upper, middle and lower panels, consistent with (a). All panels are obtained with $s/\gamma=0.2$, and $\Delta/\gamma=0.2$, and we take $L=10$, $N=4$ in (a) and (c).
• Figure 5: (a) Liouvillian spectrum for Eq. (\ref{['Lioudoublon']}) with $N=6$. (b) The long-time dynamics of $\langle\sigma_1^x\rangle$. (c) A zoomed-in view of (b) over four periods, with different starting times of $t_0\gamma=10^5$ (blue) and $t_0\gamma=10^7$ (green), respectively, as marked by the vertical dashed lines in (b). For all simulations, we take $U=\gamma$.