Time evolution with the density-matrix renormalization-group algorithm: A generic implementation for strongly correlated electronic systems

A detailed description of the time-step-targeting time evolution method within the density-matrix renormalization-group algorithm is presented. The focus of this publication is on the implementation of the algorithm and its generic application. The case of one-site excitations within a Hubbard model is analyzed as a test for the algorithm, using open chains and two-leg ladder geometries. The accuracy of the procedure in the case of the recently discussed holon-doublon photo excitations of Mott insulators is also analyzed. Performance and parallelization issues are discussed. In addition, the full open-source code is provided as supplementary material.

Figure 1

Example of a state $|\phi \rangle$ given by Eq. (4). The order in which the on-site operators are applied will depend on the order in which the sites appear as “central sites.” In the example above, $b=3$, and the operators are acting on sites 3, 4, and 7. In the sweeping procedure shown, the order will be ${\pi }^{\prime }\left(0\right)=4,{\pi }^{\prime }\left(1\right)=7,{\pi }^{\prime }\left(2\right)=3$, that is, $|\phi \rangle =B_3{B}_7{B}_4|\psi \rangle$.

Figure 2

Observable $X_{ij\uparrow }\left(t\right)$, Eq. (6), for Hamiltonian $H_0$, with $i=25$ and $j$ as indicated, on two geometries: (a) a $25\times 2$ ladder and (b) a 50-site chain. Circles and squares represent DMRG results, and solid lines show exact results. Inset: Labeling of sites on a two-leg ladder. The one-site excitation (creation of a “hole”) was applied on site $i=25$. Chain sites are labeled from left to right, starting at 0.

Figure 3

(a), (b) Density at site $p$ of the holon-doublon state Eq. (7) as a function of time for (a) $U=0$ and (b) $U=10$. A $2\times 4$ ladder with $t_x=1$ and $t_y=0.5$ was used, containing four up and four down electrons. The holon-doublon operator was applied at $i=2$, $j=4$. (c), (d) Same as for panels (a) and (b) but now on a $2\times 25$ ladder with 50 electrons. In this case, the holon-doublon operator was applied at $i=25$, $j=24$. In the vertical axes labels, hd${}_{ij}\left(t\right)$ denotes the state obtained by acting with a holon-doublon operator at sites $i$ and $j$ over the ground state and evolving that state to time $t$.

Figure 4

Comparison of $O_{j,i,p}\left(t\right)$ computed with DMRG (circles and squares) and with a $t\to 0$ expansion. For the latter $O_{j,i,p}\left(t\right)=O_{j,i,p}\left(0\right)+a_{j,i,p}{t}^2+O\left(t^4\right)$, where $a_{j,i,p}\equiv \langle \phi |H{n}_{p}H-H^2{n}_p|\phi \rangle$. Same parameters as in Fig. 3b.

Figure 5

(a), (b) Double occupation at site $p$ of the holon-doublon state Eq. (7) as a function of time for (a) $U=0$ and (b) $U=10$. A $2\times 4$ ladder with $t_x=1$ and $t_y=0.5$ was used, containing four up and four down electrons. The holon-doublon operator was applied at $i=2$, $j=4$. (c), (d) Same as in panels (a) and (b) but now on a $2\times 25$ ladder with 50 electrons. In this case, the holon-doublon operator was applied at $i=25$, $j=24$. In the vertical axes labels, hd${}_{ij}\left(t\right)$ denotes the state obtained by acting with a holon-doublon operator at sites $i$ and $j$ over the ground state and evolving that state to time $t$.