Real-Time Evolution Using the Density Matrix Renormalization Group

We describe an extension to the density matrix renormalization group method incorporating real-time evolution. Its application to transport problems in systems out of equilibrium and frequency dependent correlation functions is discussed and illustrated in several examples. We simulate a scattering process in a spin chain which generates a spatially nonlocal entangled wave function.

Figure 1

(a) Tunneling current through a noninteracting quantum dot and a junction as defined in Eqs. (2) and (5) in Ref. 5, respectively. The full lines correspond to exact results. (b) Tunneling current through an interacting junction, with $V=0.5$ and $V=1.1$. All the DMRG results where obtained with $M=128$ and a time step $\tau =0.2$.

Figure 2

Time evolution of the local magnetization $\langle S^z(x)\rangle$ of a 200 site spin-1 Heisenberg chain after $S^{+}(100)$ is applied.

Figure 3

The single-magnon line of the spin-1 Heisenberg antiferromagnetic chain. The entire spectrum is obtained from one DMRG run, by Fourier transforming the time and position dependent correlation function $\langle S_l^{-}(t)S_0^{+}(0)\rangle$. The broad solid curve shows the location of the maximum in the spectra for a particular $q$, in units of the Haldane gap, 0.41050(2), for a system of $L=600$ sites, using a time step $\tau =0.02$, running for $T=27.3$, and keeping $m=200$ states. For comparison, results from two other runs are shown: $L=400$, $\tau =0.1$, $T=60$, and $m=150$ (dashed curve); and. $L=400$, $\tau =0.4$, $T=72$, and $m=200$ (dotted curve). The solid curve peaked at $q=\pi$, shown only for the first run, is the weight $A_0$ in this quasiparticle peak, i.e., $S(\omega )\approx A_0\delta (\omega -{\omega }_0)$.

Figure 4

A Gaussian magnon wave packet with momentum $k=0.8\pi$, $S^z=1$, and half-width 16 scattering off the left end of a 200 site spin-1 chain. The chain end states initially have $S^z=-1/2$. We measure $\langle S^z(x)\rangle$ at time $t$, and $\tau =0.2$. In (a) we show $t=0$ and $t=20$. In (b), we show the leftmost 50 sites for a number of times. In (c), we show the whole chain for $t=0$ and $t=110$. After the scattering, the system is in a nonlocal superposition of spin-flip and non-spin-flip states.