Topological Boundary Time Crystal Oscillations

Abstract

Boundary time crystals (BTCs) break time-translation symmetry and exhibit long-lived, robust oscillations insensitive to initial conditions. We show that collective spin BTCs can admit emergent topological winding numbers in operator space. Expanding the density operator in a spherical tensor basis, we map the Lindblad dynamics onto an effective local hopping problem, where collective degrees of freedom label sites of an emergent two-dimensional operator space lattice and identify topological obstructions that enforce the delocalization of operator modes on the lattice. The resulting spectral delocalization provides a natural explanation for the robust oscillatory dynamics observed in BTCs. When combined with non-reciprocal transport of operator weight across operator space, this mechanism moreover also leads to the universality of long-time dynamics across a broad class of initial states. Our results frame BTC dynamics as a form of topologically constrained operator space transport and suggest a close connection to non-Hermitian skin-effects.

Figures (7)

• Figure 1: Collective Spin System and Spherical Tensor Operators. a) Schematic diagram of the BTC model in Eq. (\ref{['eq:BTC_master_Eq']}), showing the collective spin configuration for the maximally polarized subsector $j=N/2$. b) Spherical tensor basis [Eq. (\ref{['eq:spherical_tensor_basis']})] shown as operator multiplets, where the $k^{\mathrm{th}}$ multiplet contains $(2k+1)$ operators. Here, $T^0_0 \propto \mathbb{I}$ and the triplet $(T^1_{-1}, T^1_0, T^1_1)$ is equivalent to $(J_-, J_z, J_+)$ up to normalization factors. The multiplets form a wedge-shaped lattice, whose sites are the spherical tensor operators.
• Figure 2: Non-Hermitian Hopping Model and One-Dimensional Spectral Localizer Probe. a) Effective one-dimensional hopping chain along the probe coordinate $k$, constructed in the spherical tensor basis [Eq. (\ref{['eq:non_hermitian_hopping_mapping']})]. b) As $p$ is varied along $S^1$, the map $p \mapsto \mathcal{L}_U(p)-\lambda_0$ traces a non-contractible loop in the complex plane, signaling a non-trivial winding around the point-gap at $\lambda_0$. c) Local topological index $\nu^L$ as a function of the sweep coordinate $x_0$, probing the rank chain shown in a). Results are shown for fixed $\lambda_0=0$ and $\tilde{\Gamma}\equiv\Gamma/\Omega=0,\,1.0,\,2.0$ with $N=20$ spins. Data are truncated to $x_0\leq 5$ for clarity; $\nu^L$ remains zero beyond this range.
• Figure 3: Local Topological Islands. For fixed operator space coordinate $x_0$, we show the complex-frequency–resolved local index $\nu^L(\lambda_0)$ for four representative cases. Panels (a,b) correspond to $x_0=1$ with $\tilde{\Gamma}\equiv \Gamma/N = 0.75$ and $1.50$, respectively, for $N=10$. Panels (c,d) show the same parameter sets evaluated at $x_0=2$. Hollow circular markers denote the eigenvalues of the Liouvillian $\mathcal{L}$. Colour shading indicates the local value of $\nu^L$, revealing spatially resolved regions of non-trivial topology in the complex-frequency plane.
• Figure 4: Eigenmode Delocalization. Rank-resolved mode weights $w_k$ are overlaid on the local topological domain structure $\nu^L(x_0)$. Six representative cases are shown, each labeled by the reference frequency $\lambda_0$ corresponding to a specific Liouvillian eigenmode. The first and second rows correspond to $\tilde{\Gamma}\equiv \Gamma/\Omega = 0.5$ and $1.5$, respectively, with $N=10$ throughout.
• Figure S1: Topological Domains as a Function of $\kappa$. We show the dependence of the localizer index $\nu^L$ and gap $\mu$ on the resolution parameter $\kappa$ for $\kappa\in\{0.01,0.1,0.5,1,2,5\}$ and fixed $\lambda_0=0$. A stable $\kappa$-window is observed in which the gap remains open (inside topologically non-trivial regions) and the topological classification is unchanged. Results are shown for fixed $\tilde{\Gamma}\equiv \Gamma/\Omega = 1$, system size $N=10$ and are truncated to $x_0 \leq3$ (beyond which $\nu^L$ remains zero.).