Matrix-product-state method with a dynamical local basis optimization for bosonic systems out of equilibrium

We present a method for simulating the time evolution of one-dimensional correlated electron-phonon systems which combines the time-evolving block decimation algorithm with a dynamical optimization of the local basis. This approach can reduce the computational cost by orders of magnitude when boson fluctuations are large. The method is demonstrated on the nonequilibrium Holstein polaron by comparison with exact simulations in a limited functional space and on the scattering of an electronic wave packet by local phonon modes. Our study of the scattering problem reveals a rich physics including transient self-trapping and dissipation.

Figure 1

Graphical representation of the MPS (a) in the original TEBD algorithm and (b) in the TEBD-LBO algorithm.

Figure 2

Comparison of LFS and TEBD-LBO for various $d_O$. (a) Time evolution of the phonon number calculated for $L=L_H=6, \gamma =2$, and ${\omega }_0=1$. (b) Relative deviations of the TEBD-LBO data from the LFS results. Deviations for TEBD with a bare basis of dimension $d=86$ are also shown.

Figure 3

(a) Electronic density $\langle n_j\rangle$ as a function of site $j$ and time $t$ for $L_H=6, \gamma =1$, and ${\omega }_0=2.25$. The Holstein chain corresponds to sites $j=280,\cdots ,285$. (b) Total electronic density $n_H$ in a Holstein chain of length $L_H=6$ (red solid line), 3 (blue dotted line), and 1 (black dashed line) as a function of time $t$. The inset shows the same data on a logarithmic scale.

Figure 4

(a) Electronic density $\langle n_j\rangle$ as a function of site $j$ and time $t$ for $L_H=1, \gamma =2.5$, and ${\omega }_0=1.65$. The position of the EP site is $j=90$. (b) Phonon energy $E_{\text{ph}}={\omega }_0{N}_{\text{ph}}$ as a function of time for $L_H=1, {\omega }_0=1.5$, and several values of $\gamma$.