Time-step targeting methods for real-time dynamics using the density matrix renormalization group

We present a time-step targeting scheme to simulate real-time dynamics efficiently using the density matrix renormalization group. The algorithm works on ladders and systems with interactions beyond nearest neighbors, in contrast to existing Suzuki-Trotter-based approaches.

Figure 1

Error $E(t=8)$ for the Haldane chain $(L=32)$, according to Eq. (6), as a function of the number of states kept $m$. We show results of simulations using the Runge-Kutta algorithm, for different time steps and distributions of weights.

Figure 2

Same as in Fig. 1, using first-order Suzuki-Trotter breakup (gray symbols), fourth-order Suzuki-Trotter (empty symbols), and fourth-order Runge-Kutta (filled symbols).

Figure 3

Error $E(t=8)$ for the Haldane chain for different time steps $\tau$: (a) First-, second-, and fourth-order Suzuki-Trotter breakups and $m=160$; (b) Runge-Kutta and $m=100$. Time is in units of the Heisenberg exchange $J$.

Figure 4

Number of states required to keep a truncation error of ${10}^{-8}$, as a function of time. The results correspond to a R-K simulation of a Haldane chain with $L=32$.

Figure 5

Structure factor $A(\mathbf{k},\omega )$ for the Heisenberg ladder using fourth-order Runge-Kutta, $m=256$ states, and time step $\tau =0.1$. The solid line is centered at the quasiparticle peak. The tones of gray are proportional to the quasiparticle weight (dashed curve). The symbols are Lanczos results for $L=12$ from Ref. 14.