Statistical structural properties of many-body chaotic eigenfunctions and applications

Abstract

In this paper, we employ a semiperturbative theory to study the statistical structural properties of energy eigenfunctions (EFs) in many-body quantum chaotic systems consisting of a central system coupled to an environment. Under certain assumptions, we derive both the average shape and the statistical fluctuations of EFs on the basis formed by the direct product of the energy eigenbases of the system and the environment. Furthermore, we apply our results to two fundamental questions: (i) the properties of the reduced density matrix of the central system in an eigenstate, and (ii) the structure of the off-diagonal smooth function within the framework of the eigenstate thermalization hypothesis. Numerical results are also presented in support of our main findings.

Figures (3)

• Figure 1: (a) Averaged width of NPT region $\overline{\omega_{\text{NPT}}}$ and (b) averaged weight of the NPT region $\overline{W^{\text{NPT}}}$ versus coupling strength $\lambda$. The dashed line in (a) indicates predicted the scaling $\propto \lambda^2$ given in Eq. \ref{['w-NPT']}. The results are obtained by averaging over $10$ eigenstates in the middle of the spectrum, and over $10$ different realizations of $H^{I\mathcal{E}}$ and $H^{\mathcal{E}}$.
• Figure 2: Averaged shape of eigenfunctions $\overline{|C_{n}^{r}|^{2}}$ for (a) $\lambda = 0.1$ and (b) $\lambda = 0.2$. The dashed line indicate the predicted scaling $\propto(E_{n}-E_{r}^{0})^{-2}$ given in Eq. \ref{['nr-PT']}. The results are obtained using a single eigenstate in the middle of spectrum and averaging over $100$ independent realizations of $H^{I\mathcal{E}}$ and $H^{\mathcal{E}}$.
• Figure 3: Averaged off-diagonal elements of operator $O=\sigma_z$ for (a) $\lambda = 0.1$ and (b) $\lambda = 0.2$. The dashed line indicates the scaling $\propto \omega^{-2}$ given in Eq. \ref{['feomega-sim']}.