Floquet thermalization by power-law induced permutation symmetry breaking

TL;DR

Problem: Understanding how power-law induced permutation symmetry breaking affects Floquet thermalization in a long-range spin system. Approach: Analyze a periodically kicked, Kac-normalized Hamiltonian with $1/r^{α}$ couplings as $α$ varies from $0$ to large, using dynamical probes $J^2$, $S_{N/2}$, and Floquet-eigenstate metrics such as $D_{eff}$ and level statistics; large-$α$ limits are treated analytically via Jordan-Wigner mapping. Findings: Dynamics stay near the permutation-symmetric subspace for small $α$, thermalize to full Hilbert space statistics at intermediate $α$ with $⟨J^2⟩$ approaching $3N/4$ and $⟨S_{N/2}⟩$ near the Page value, and revert to integrable kicked-Ising behavior at large $α$; drive period $τ$ shifts the onset and width of the thermalization window. Significance: The work demonstrates controllable Floquet heating through symmetry-breaking range and provides a framework for tuning thermalization in experimental platforms with long-range interactions.

Abstract

Permutation symmetry plays a central role in the understanding of collective quantum dynamics. On the other hand, interactions are rarely uniform in real systems. By introducing power law couplings that algebraically decay with the distance between the spins $r$ as $1/r^α$, we break this symmetry with a non-zero $α$, and probe the emergence of new dynamical behaviors, including thermalization. As we increase $α$, the system interpolates from an infinite range spin system at $α=0$ exhibiting permutation symmetry, to a short range integrable model as $α\rightarrow \infty$ where this permutation symmetry is absent. We focus on the change in the behavior of the system as $α$ is tuned, using dynamical quantities like total angular momentum operator $J^2$ and the von Neumann entropy $S_{N/2}$. Starting from the chaotic limit of the permutation symmetric Hamiltonian at $α=0$, we find that for small $α$, the steady state values of these quantities remain close to the permutation symmetric subspace values corresponding to $α=0$. At intermediate $α$ values, these show signatures of thermalization exhibiting values corresponding to that of random states in full Hilbert space. On the other hand, the large $α$ limit approaches the values corresponding to integrable kicked Ising model. In addition, we also study the dependence of thermalization on the driving period $τ$, with results indicating the onset of thermalization for smaller values of $α$ when $τ$ is large, thereby extending the intermediate range of $α$. We further confirm these results using effective dimension and spectral statistics.

Figures (6)

• Figure 1: (a): The plot of $\left \langle J^2 \right \rangle$ with respect to time $n$ for $N=16$ at $\tau=1.0$ for different $\alpha$ values. (b): The plot of time averaged $J^2$ with respect to $\alpha$ for the same $\tau$ and $N$. The inset shows the plot of $J^2$ with respect to $\alpha$ for $\tau=0.1$ and $\tau=5.0$.
• Figure 2: (a): The plot of $\left \langle S_{N/2} \right \rangle$ with respect to time $n$ for $N=14$ at $\tau=1.0$ for different $\alpha$ values. (b): The plot of time averaged $S_{N/2}$ with respect to $\alpha$ for the same $\tau$ and $N$. The inset shows the same for $\tau=0.1$ and $\tau=5.0$.
• Figure 3: (a). The plot of $\left \langle J^2 \right \rangle$ with respect to time $n$ for $N=14$ at $\tau=1.0$ for $\alpha=15$. The dashed line represents the analytical results and dots correspond to the numerical results. (b). The plot of $S_{N/2}$ with respect to $n$ for $N=14$ at $\tau=1.0$ and $\alpha=15$. The dashed line represents the analytical results and the dots represents the numerical values. The initial state is $\ket{0}^{\otimes N}$.
• Figure 4: The plot of $\overline{S_{N/2}}$ with $N=800$ for $\alpha=0$ (corresponding to random permutation symmetric subspace in $N+1$ dimensions) and $1\leq \alpha \leq 2$ (Page value in $2^N-$dimensional Hilbert space). For large $\alpha$, $\overline{S_{N_{A}}}$ for different subsystem size $N_A$ is computed using semi-analytics to obtain $\overline{S_{N/2}}$ from the fitting.
• Figure 5: The plot of $D_{\rm eff}$ with respect to $\alpha$ for $N=12$ at $\tau=1.0$. The inset shows $D_{\rm eff}$ for different $\tau$ values at same parameter values.