Statistics of Schmidt coefficients and the simulability of complex quantum systems

We show that the transition from regular to chaotic spectral statistics in interacting many-body quantum systems has an unambiguous signature in the distribution of Schmidt coefficients dynamically generated from a generic initial state, and thus limits the efficiency of the time-dependent density-matrix renormalization-group algorithm. We investigate this mechanism on the tilted Bose-Hubbard model; however the emergence of universal spectral properties allows translation of our conclusions to generic many-body quantum systems.

Figure 1

Distribution of average Schmidt coefficients ${\lambda }_{\alpha }$ obtained from averaging over ten initial states of eight particles initially distributed over eight lattice sites. The total lattice size was 64 to eliminate boundary effects. Shown are the $\chi =100$ largest average Schmidt coefficients, sorted in descending order, for $U/J=1$ (thick lines) and $U/J=10$ (thin lines), and for different values of the static tilt $F/J=1$ (solid), 1.5 (dashed), and 2 (dotted) at $t=4.8\times T_B$.

Figure 2

Mean square deviation ${\Delta }^2$, Eq. (6), of the Floquet-Bloch operator’s [Eq. (4)] nearest-neighbor distribution from Poissonian (dashed lines) and Wigner-Dyson statistics (solid lines), for eight particles on nine lattice sites, as a function of $F/J$. $U/J=1$ (thick lines) and 10 (thin lines).

Figure 3

Many-particle entanglement vs $F/J$, measured by the average von Neumann entropy $S$, extracted from $t\text{-DMRG}$ simulations of eight particles for ten different initial states at $t=4.8\times T_B$. $U/J=1$ (thick) and 10 (thin line). Inset: $S$ vs $t/T_B$ for various tilt strengths $F/J=1$ (solid), 1.5 (dashed), and 2 (dotted lines), and the same values of $U/J$.

Figure 4

Average number of Schmidt coefficients larger than $ϵ=0.01$ vs $F/J$, after a simulation time $t=4.8\times T_B$. $U/J=1$ (thick) and 10 (thin lines).