Grand Canonical vs Canonical Krylov Complexity in Double-Scaled Complex SYK Model

Abstract

We consider the complex SYK model in the double-scaling limit. We obtain the transfer matrix for the grand canonical ensemble and symmetrize it. In the (n,Q)- basis of chord states, the grand canonical transfer matrix is block diagonal, where each block is the canonical transfer matrix for the respective charge sector. We therefore conclude that the Krylov complexity for the grand canonical ensemble is given by the sum of the complexities in the charge sectors weighted by a probability function that depends on the chemical potential. Finally, we compute the Krylov complexity analytically in the limit of early and late time in the charge sector and numerically for both canonical and grand canonical ensemble.

Figures (6)

• Figure 2.1: Part of a cut-open moment diagram expressed in terms of string of $Xs$ and $O$s
• Figure 2.2: States in the physical Hilbert space that contribute to the $k$th moment. The operators $a$ and $b$ are defined in \ref{['sec:bosonic_sector']}.
• Figure 3.1: Numerical results for $C_{\mathcal{K}}^{(\mathcal{Q})}(t)$ for different values of $\mathcal{Q}$ at $p = 10$, $\lambda=0.001$ and $\mathcal{J}=1$.
• Figure 3.2: We compare the linear and quadratic asymptotes (dashed lines) for $C_{\mathcal{K}}^{(\mathcal{Q})}(t)$ given in \ref{['eq:kryltwo']} to the numerical results (solid lines). The parameters for the plot are $p = 50$, $\lambda=0.001$ and $\mathcal{J}=1$. The two figures show the same data. However, the bottom plot has a log scale on the $x$-axis.
• Figure 3.3: Numerical results for $C_{\mathcal{K}}(t)$ calculated via the $\mu$ dependent Lanczos coefficients \ref{['eq:Lanczos_coeff_GC']} for different values of $\mu$ at $p = 10$, $\lambda=0.001$ and $\mathcal{J}=1$.