Many-body quantum chaos in mixtures of multiple species

Abstract

We study spectral correlations in many-body quantum mixtures of fermions, bosons, and qubits with periodically kicked spreading and mixing of species. We take two types of mixing, namely, Jaynes-Cummings and Rabi, respectively, satisfying and breaking the conservation of a total number of species. We analytically derive the generating Hamiltonians whose spectral properties determine the spectral form factor in the leading order. We further analyze the system-size $(L)$ scaling of Thouless time $t^*$, beyond which the spectral form factor follows the prediction of random matrix theory. The $L$-dependence of $t^*$ crosses over from $\log L$ to $L^2$ with an increasing Jaynes-Cummings mixing between qubits and fermions or bosons in a finite-sized chain, and it finally settles to $t^* \propto \mathcal{O}(L^2)$ in the thermodynamic limit for any mixing strength. The Rabi mixing between qubits and fermions leads to $t^*\propto \mathcal{O}(\log L)$, previously predicted for single species of qubits or fermions without total number conservation.

Figures (8)

• Figure 1: Spectral form factor $K(t)$ using Eq. \ref{['series']} for different system sizes $L$ of the kicked chain with $JC$ mixing between fermions and qubits for $g=0.1,J=0.4$ in (a,b), and $g=0.4,J=0.1$ in (c,d). We take half-filling $N/L=1/2$. In (b) and (d), we show data collapse in scaled time $t/\log L$ and $t/L^{1.85}$, respectively.
• Figure S1: The largest eigenvalues $\lambda_i$ of $\mathcal{M}_{\text{JC}}^{\text{F}}$ with $i\in\{0,1,2,...,49\}$ for $g=0.1,J=0.4$ in $(a)$, and $i\in\{0,1,2,...,20\}$ for $g=0.4,J=0.1$ in $(b)$. The plots $(a)$ and $(b)$ show respectively $L$ and $3$ largest $\lambda_i$ being the same for different $N$.
• Figure S2: Comparison between the exact numerically computed SFF, $K(t)$ vs. $t$, with that obtained using the RPA for Jaynes-Cummings mixing between fermions and qubits. The red curve is exact SFF computed numerically using Eq. \ref{['S4']} and the blue (black dashed) curve is that calculated using the first-order and second-order term (only the first-order term) in time within the RPA. All, $\mathcal{N}_{\text{JC}}^{\text{F}}=15504$, eigenvalues of $\mathcal{M}_{\text{JC}}^{\text{F}}$ are used for the RPA result. For exact numerical computation, we fix $U_0=10,\alpha=1.4$, and $\omega_i,\Omega_i$ are chosen as Gaussian random variables with a mean $\langle\omega_i\rangle=\langle\Omega_i\rangle=1$ and a standard deviation $\sigma_{\omega_i}=\sigma_{\Omega_i}=0.3$. Averaging over $520$ realizations of disorder is performed for the direct SFF computation.
• Figure S3: The second-largest eigenvalue $\lambda_1$ of $\mathcal{M}_{\text{JC}}^{\text{B}}$ vs. $L$ for different total excitations $N$ when $g=0.4,J=0.1$. $\lambda_1$ for different $N$'s approach the same value with an increasing $L$ suggesting emergence of an approximate symmetry of $\mathcal{M}_{\text{JC}}^{\text{B}}$.
• Figure S4: Spectral form factor $K(t)$ using Eq. \ref{['S14']} for different system sizes $L$ of the kicked chain with $JC$ mixing between bosons and qubits for $g=0.1,J=0.4$ in $(a,b)$, and $g=0.4,J=0.1$ in $(c,d)$. We take half-filling $N/L=1/2$. In $(b)$ and $(d)$, we show data collapse in scaled time $t/\log L$ and $t/L^{1.86}$, respectively.