Entanglement entropy and computational complexity of the periodically driven Anderson impurity model

We study the growth of entanglement entropy and bond dimension with time in density matrix renormalization group simulations of the periodically driven single-impurity Anderson model. The growth of entanglement entropy is found to be related to the ordering of the bath orbitals and to the relation of the driving period $T$ to the convergence radius of the Floquet-Magnus expansion. Ordering the bath orbitals by their Floquet quasienergy is found to reduce the exponential growth rate of the computation time at intermediate driving periods, suggesting new ways to optimize matrix product state calculations of driven systems.

Figure 1

The density of states of the bath orbitals ${\varepsilon }_k$. We consider a semicircle DOS with a half bandwidth $E$. The bath is initially half-filled, and the $d$-orbital energy ${\varepsilon }_d=\pm \left|{\varepsilon }_d\right|$ oscillates between the two values shown every half driving period $T/2$ across the Fermi level.

Figure 2

Comparison of time dependence of entanglement entropy of quenched [lower (blue) curve] and driven [upper (red) curve] Anderson impurity model. The driving period $ET=10$, the Hubbard $U=0$, the bath is half-filled, and the impurity-bath coupling $V/E=0.25$. Initially the impurity is empty. The entanglement entropy is computed between the 500 bath orbitals below the Fermi level and the rest of the system.

Figure 3

Linear growth rate of entropy over a cycle of the drive in steady state plotted against the driving period for various amplitudes $|{\varepsilon }_d|$. Hubbard $U=0$ and the impurity-bath coupling $V/E=0.25$.

Figure 4

The bath orbital energies ${\varepsilon }_k$ of the Floquet Hamiltonian $H_F$ for $ET=3$ (blue) and $ET=4$ (red). The orbital energies are unaffected by the periodic driving if $ET<\pi$ but get aliased to quasienergies within $[-\pi /T,\pi /T]$ modulo $2\pi /T$ if $ET>\pi$.

Figure 5

The growth of entropy $S_{N/2}$ for the driven and quenched models with energy-ordered and quasienergy-ordered bath orbitals. Hubbard $U=0$ and impurity-bath coupling $V/E=0.25$. Period $ET=10$.

Figure 6

(a) The upshift $\mathrm{\Delta }{S}_{N/2}$ of entropy in the quenched SIAM at $|{\varepsilon }_d|=0$ when bath orbitals are ordered by quasienergy of driving period $T$. (b) The slope of $S_{N/2}$ vs $lnt$ in the driven SIAM at $|{\varepsilon }_d|/E=0.1$. Bath sizes for long periods need to reach $N=3000$ to obtain accurate data.

Figure 7

The dependence of the coefficient $c$ in Eq. (15) on the driving amplitude $|{\varepsilon }_d|$. Hubbard $U=0$ and impurity-bath coupling $V/E=0.25$.

Figure 8

The $d$ occupancy $n_d=\langle n_{d\uparrow }\rangle +\langle n_{d\downarrow }\rangle$ and double occupancy $D=\langle n_{d\uparrow }{n}_{d\downarrow }\rangle$ of the quenched and driven SIAMs vs time at Hubbard $U/E=1$, impurity-bath coupling $V/E=0.25$, driving amplitude $|{\varepsilon }_d|/E=0.1$, and period in (a) $ET=10$ and (b) $ET=20$. The dashed gray line is $n_d^2/4$ of the quenched $n_d$.

Figure 9

(a) The $d$ occupancy $n_d$ and double occupancy $D$ of the SIAM at driving amplitudes $|{\varepsilon }_d|/E=0,0.1,...,0.8$ and period $ET=10$. Other parameters are the same as Fig. 8. The gray dashed line is $n_d^2/4$ of the quenched $n_d$. (b) Amplitude $\mathrm{\Delta }{n}_d$ ($1/2$ of peak-to-peak value) of $n_d$ and the time-averaged double occupancy $\overline{D}$ over a full period.

Figure 10

The maximum entanglement entropy $S_{\max }$ reached in (a) and CPU time $t_{\mathrm{CPU}}$ spent in (b) to run to different simulation times $Et$. Parameter values $U/E=1$, $V/E=0.25$, $|{\varepsilon }_d|/E=0.1$, and $ET=6$. The red curve in (a) is slightly concave upward as $Et$ approaches 100 when the period-$ET$ oscillations are eliminated by moving average.

Figure 11

Crossing time of maximum entanglement entropies of the energy-ordered and quasienergy-ordered simulations at various driving periods $T$. Fixed parameter values $U/E=1$, $V/E=0.25$, $|{\varepsilon }_d|/E=0.1$. The red line is a smooth guideline of the data points in blue dots.