发表机构
Tata Institute of Fundamental Research（塔塔基础科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究利用归一化的Krylov - Wigner负性率来诊断边界态二阶矩捕捉波包扩散情况。从种子归一化Krylov - Wigner分布出发，在宏观极限下得到解析贝塞尔形式并计算总负性，发现负性与时间的关系，且负性率在特定维度有特殊性质。
AI 中文摘要
在扩散复杂性中，算符沿其Krylov链的平均位置能恢复AdS中下落粒子的右径向动量，但它是一阶矩的量度，与波包偏离经典轨迹的扩散无关。可提出归一化Krylov - Wigner负性率作为捕捉这种扩散的边界态二阶矩的诊断方法。从‘种子归一化’Krylov - Wigner分布出发，在宏观极限下得到解析贝塞尔形式并明确计算其总负性。一直保留贝塞尔变量，我们发现负性随$\sinh^{4\Delta}(\pi t/\beta)$变化，而原始的、种子归一化态负性饱和，这由$O(\sqrt{D})$界决定。利用Krylov链的精确负二项统计和Caputa等人的动量字典，当且仅当$\Delta = 1$（$AdS_3$中的Breitenlohner - Freedman饱和维度）时，负性率在晚期渐近中与Krylov方差的增长率成比例。这个维度很特殊，因为负性率是固有径向位置和动量的乘积，即$\mathcal{R} \propto \mathcal{C} P_\rho$，也就是附近测地线落入视界的潮汐拉伸率。我们评论了未来研究的方向，特别是通过共同的$SU(1,1)$离散系列用Krylov数算符解释横向弦尺寸算符。
英文摘要
In spread complexity, the growth rate of the average position of a time-evolved state along its Krylov chain recovers the proper radial momentum of an infalling particle in AdS. It is, however, a measure of the first moment only, irrespective of the spread of the wavepacket away from its classical trajectory. The rate of a normalized Krylov-Wigner negativity can be proposed as a diagnostic of the second moment of the boundary state that captures this spreading. Starting with the seed-normalized Krylov-Wigner distribution, that is, the Wigner transform of the descendant cloud with the decaying return amplitude divided out, we obtain an analytic Bessel form in the macroscopic limit and compute its total negativity explicitly. Retaining the Bessel variable all the way through, we find that the negativity goes as $\sinh^{4Δ}(πt/β)$. But the raw normalized-state negativity saturates to an $O(1)$ constant, well below the $O(\sqrt{D})$ bound. Using the exact negative binomial statistics of the Krylov chain, the normalized negativity is at late times a fixed power of the second moment of the Krylov wavepacket, $\mathcal{N}(t)\propto[\mathrm{Var}(\hat{N}_{\mathrm{Krylov}})]^Δ$, for every $Δ>1/2$. The relation linearizes precisely at $Δ=1$. In this regime via the momentum dictionary of Caputa et al. (arXiv:2410.23334), the rate becomes the product of the proper radial position and momentum, $\mathcal{R}\propto\mathcal{C}\,P_ρ$, which we interpret as the rate of the tidal stretch of nearby geodesics falling into the horizon. We also comment briefly on a speculative direction for future research, in particular the interpretation of the transverse string size operator in terms of the Krylov number operator through the common $\mathrm{SU}(1,1)$ discrete series.
Comments41 pages , 5 figures
