Doublon dynamics in the extended Fermi-Hubbard model

Two fermions occupying the same site of a lattice model with strongly repulsive Hubbard-type interaction $U$ form a doublon, a long-living excitation, the decay of which is suppressed because of energy conservation. By means of an exact-diagonalization approach based on the Krylov-space technique, we study the dynamics of a single doublon, of two doublons, and of a doublon in the presence of two additional fermions prepared locally in the initial state of the extended Hubbard model. The time dependence of the expectation value of the double occupancy at the different sites of a large one-dimensional lattice is analyzed by perturbative arguments. In this way the spatiotemporal evolution of the doublon can be understood. The initial decay takes place on a short time scale $1/U$, and the long-time average of the decayed fraction of the total double occupancy scales as $1/U^2$. We demonstrate how the dynamics of a doublon in the initial state is related to the spectrum of two-fermion excitations obtained from linear-response theory, we work out the difference between doublons composed of fermions vs doublons composed of bosons, and we show that despite the increase of phase space for inelastic decay processes, the stability of a doublon is enhanced by the presence of additional fermions on an intermediate time scale.

Figure 1

Temporal evolution of the expectation value (see color code) of the double occupancy at sites $i=1,...,L$. Left: Numerical results for the one-dimensional Hubbard model ($V=0$) with $L=100$ sites and periodic boundary conditions at $U=8J$. Initially, at $t=0$, the two-fermion state has been prepared as a doublon at $i_0=50$. Top: Time evolution of the expectation value of the total double occupancy. Right: Corresponding analytical results of the effective model; see Eq. (3). Note that most of the color range is used to display expectation values less than $0.1$, as shown in the color bar below. The time $t$ is measured in units of the inverse hopping $1/J$.

Figure 2

Time-dependent expectation value of the local (main panels) and total double occupancy (small top panels) for two fermions initially prepared at the same site $i_0$ of a one-dimensional lattice with $L=100$ sites and periodic boundary conditions. Results for on-site interaction $U=0,5,10$ and nearest-neighbor coupling $V=0,\pm 5,\pm 10$, as indicated. Note that for the total double occupancy the scale of the $y$ axis differs from case to case. The color code is the same as in Fig. 1 and the same for all plots.

Figure 3

(a) Short-time behavior of the total double occupancy for $U=10,12,15,20,25,30,40$ (from bottom to top) and $V=0$. (b) First local minimum point (“decay time”) of the time-dependent total double occupancy plotted against $1/U$. The line is a guide to the eye.

Figure 4

Long-time average (top) and relative standard deviation (bottom) of the total double occupancy $\langle \mathcal{D}\left(t\right)\rangle$ for interaction strengths $-10<U<10$ and $-10<V<10$. $\overline{\mathcal{D}}$ and $(\overline{{\mathcal{D}}^2}-{\overline{\mathcal{D}}}^2){}^{1/2}/\overline{\mathcal{D}}$ are calculated for the time interval $50<t<100$. The color or grey-scale code is given on the right.

Figure 5

Sectional views of the stability map Fig. 4 for $U,V=0,5,10$. Blue lines show $\overline{\mathcal{D}}$, error bars indicate the absolute standard deviation. Green lines with error bars: time average and standard deviation of the nearest-neighbor occupancy, $\langle \frac{1}{2}{\sum }_{\langle ij\rangle }{\sum }_{\sigma {\sigma }^{\prime }}{\widehat{n}}_{i\sigma }^c\left(t\right){\widehat{n}}_{j{\sigma }^{\prime }}^c\left(t\right)\rangle$. Red: sum of double and nearest-neighbor occupancy.

Figure 6

Scheme of dominant first-order [$\left(1\right)$, right] and second-order [$\left(2\right)$, left] hopping processes from an initial state with two doublons on nearest-neighbor sites to the possible final states. The involved interactions $U$ and $V$ are depicted by wiggly lines in red and blue, respectively.

Figure 7

Time dependence of the expectation value of the local (main panels) and the total double occupancy (small top panels) for different $U$ and $V$ as indicated. Axis and color code as in Fig. 1. Calculations are performed for different initial states as indicated by the bracketed symbols: two pairs fermions (two doublons) initially prepared as nearest neighbors ($x$${}_{\mathrm{nn}}$), next-nearest neighbors ($x$${}_{\mathrm{nnn}}$), or further separated with $|i-j|=10$ ($x$${}_{\mathrm{sep}}$), respectively. Results for one-dimensional lattice with periodic boundary conditions and $L=50,51$, and 49 sites, respectively.

Figure 8

Time average of the total double occupancy $\langle \mathcal{D}\left(t\right)\rangle$ for interaction strengths $-10<U<10$ and $-10<V<10$. Calculations for an initial state with two doublons placed at sites $i$ and $j$. (a) $i$ and $j$ nearest neighbors, (b) next-nearest neighbors, and (c) $|i-j|=10$. Average over the time interval $50<t<100$. The color code is given on the right.

Figure 9

Scheme of dominant first-order [$\left(1^{\prime }\right)$, right] and second-order [$\left(2^{\prime }\right)$, left] hopping processes from an initial state with two doublons on next-nearest-neighbor sites to the possible final states. The inverse of process $\left(2_d\right)$ (see Fig. 6) is not shown again. $U$ and $V$ are depicted by wiggly lines in red and blue, respectively.

Figure 10

Coefficient $m$ obtained from a fit of the time-averaged expectation value for the total double occupancy $\overline{\mathcal{D}\left(t\right)}$ to the observed $U$ dependence: $\overline{\mathcal{D}\left(t\right)}\simeq \langle \mathcal{D}\left(0\right)\rangle (1-m/U^2)$. $m$ values are obtained from linear regression of our data in the range $8\le U\le 45$. Results are shown for different initial states as indicated and discussed in the text. $n=1$ refers to the results of a time-dependent DMRG calculation by Al-Hassanieh et al.; see Ref. 16 and text for discussion.