Statistics of Matrix Elements of Operators in a Disorder-Free SYK model

Abstract

Recently, studies have explored the statistics of matrix elements of local operators in the Lieb-Liniger model. It was found that the probability distribution function for off-diagonal matrix elements $\langle \boldsymbolμ|\mathcal{O}|\boldsymbolλ \rangle$ within the same macro-state is well described by the Fréchet distributions. This represents a significant development for the Eigenstate Thermalization Hypothesis (ETH). In this paper, we investigate a similar phenomenon in another solvable model: the disorder-free Sachdev-Ye-Kitaev (SYK) model. The Hamiltonian of this model consists of 4-body interactions of Majorana fermions. Unlike the conventional SYK model, the coupling strengths in this model are fixed to a constant, earning it the name ``disorder-free.'' We evaluate the matrix elements of operators constructed from products of $n$ Majorana fermions: $\mathcal{O} = χ_{a_1}χ_{a_2}\ldots χ_{a_n}$. For a general choice of indices and $n \geq 4$, we find that the statistics of the off-diagonal matrix elements are well-fitted by a generalized inverse Gaussian distribution rather than Fréchet distributions.

Figures (6)

• Figure 1: $\rho(\theta)$ under different temperatures; $\lambda=1/2$ here.
• Figure 2: Distribution of $\mathcal{M}_{\{a\}}$ ($\beta=1$, $\lambda=1/2$, $N=20000$; $10^7$ samples). Upper: Histograms for three randomly chosen index sets $\{a\}$ of sizes $|a|=4$, $6$, and $8$, overlaid by generalized inverse Gaussian (GIG, $p=1$) curves. Lower: Comparison between GIG and Fréchet fits for $|a|=6$: the (GIG, $p=1$) fit is accurate, while the Fréchet fit is worse. Meanwhile, two different $\{a\}$ sets (green and blue) yield indistinguishable distributions, with their respective histograms and fitted curves overlapping.
• Figure 3: Schematic illustrating why $|\langle \boldsymbol{n} |\chi_{\{a\}} |\boldsymbol{m} \rangle|$ is generally insensitive to temperature and the index set $\{a\}$. Here, $\beta=10$ (red) and $100$ (blue); the scales are illustrative. Different temperatures yield indistinguishable $\xi_n(\phi)$ as $n$ becomes large, where $\xi_n(\phi)\propto(1/\pi-\rho(|\frac{\phi}{n}|))\rho(|\frac{\phi}{n}|)$. Note that the state $|\boldsymbol{n}\rangle$ can be obtained from $|\boldsymbol{m}\rangle$ by exchanging the occupied states with unoccupied states (detailed at the end of Appendix. \ref{['state_of_eq']}); for random $|\boldsymbol{n}\rangle$ and $|\boldsymbol{m}\rangle$, the distribution of $\theta_k$ in $\langle \boldsymbol{n} |\chi_{\{a\}} |\boldsymbol{m} \rangle$ is thus characterized by $(1/\pi-\rho(\theta_k))\rho(\theta_k)$. For a given $|a|$, $\xi_n(\phi)$ is not sensitive to the temperature and exact $\{a\}$, except when $\xi_1(\phi)$ is sharply peaked ($\beta \rightarrow \infty$) or when the differences in indices are small.
• Figure 4: Distribution of $\widetilde{\mathcal{M}}_{\{a\}}=(\frac{N}{2|a|})^\frac{|a|}{2}\langle \boldsymbol{n} |\chi_{\{a\}} |\boldsymbol{m} \rangle$ in the complex plane for special $\{a\}$. $\beta=1$, $\lambda=1/2$, $N=20000$, and $10^7$ random samples for both cases. Left panel: $\{a\} = \{1,5001,10001,15001\}$; Right panel: $\{a\} = \{1,2501,5001,7501,10001,12501\}$. The color bar indicates the frequency of occurrence for each point.
• Figure 5: Distribution of $\mathcal{M}_{\{a\}}$ under different temperatures, $\lambda=1/2$, $N=20000$ and $10^7$ random samples for both cases. In the case $|a|=4$, we find that for $\delta\beta\sim100$, the temperature dependence become noticeable only when the index set $\{a\}$ includes separations $|\Delta a| \lesssim 10$. For a random choice of $\{a\}$, these curves are overlapped in general.