Filling-dependent doublon dynamics in the one-dimensional Hubbard model

The fate of a local two-hole doublon excitation in the one-dimensional Fermi-Hubbard model is systematically studied for strong Hubbard interaction $U$ in the entire filling range using the density-matrix renormalization group (DMRG) and the Bethe ansatz. For strong $U$, two holes at the same site form a compound object whose decay is impeded by the lack of phase space. Still, a partial decay is possible on an extremely short time scale where phase-space arguments do not yet apply. We argue that the initial decay and the resulting intermediate state are relevant for experiments performed with ultracold atoms loaded into an optical lattice as well as for (time-resolved) CVV Auger-electron spectroscopy. The detailed discussion comprises the mixed ballistic-diffusive real-time propagation of the doublon through the lattice, its partial decay on the short time scale as a function of filling and interaction strength, as well as the analysis of the decay products, which are metastable on the intermediate time scale that is numerically accessible and which show up in the two-hole excitation (Auger) spectrum. The ambivalent role of singly occupied sites is key to understanding the doublon physics; for high fillings, ground-state configurations with single occupancies are recognized to strongly relax the kinematic constraints and to open up decay channels. For fillings close to half-filling, however, their presence actually blocks the doublon decay. Finally, the analysis of the continua in the two-hole spectrum excludes a picture where the doublon decays into unbound electron holes for generic fillings, different from the limiting case of the completely filled band. We demonstrate that the decay products as well as the doublon propagation should rather be understood in terms of Bethe ansatz eigenstates.

Figure 1

Eigenvalues of the one-dimensional Hubbard model on $L=100$ lattice sites (periodic boundary conditions) in the subspace of two electrons spanned by the states given in Eq. (2) and classified according to the center-of-mass momentum $K$. Calculations for different $U$ as indicated. For $\left|U\right|>0$, only the bound-state solutions are shown.

Figure 2

DMRG results (color code) for the bandlike parts of the two-hole spectral function (9) for $L=60$ lattice sites, $U=6$ and the fillings $n=1.8$ (left), 1.5 (middle), and 1.1 (right) with overlaid borders of the Bethe ansatz continua as coloured lines (see explanation in the text). The spectrum at $n=1.1$ has been multiplied by a factor of 10 to maintain a common scale.

Figure 3

Sketches of the doublon decay processes, Eqs. (11) and (12), as determined from the band-like part in Fig. 2. The horizontal axis shows the magnitude and direction of the involved momenta, the vertical axis the magnitude of the involved energies (such that a flat line corresponds to the Fermi energy). Purple circles: doublons. Red circles: holons. Blue circles: spinons.

Figure 4

Total hole density $n_{\text{tot}}^h\left(t\right)$, normalized to $n_{\text{tot}}^h\left(0\right)$, for various fillings as indicated. Time-dependent DMRG calculation at $U=6$ for $L=40$. (Left inset) Characteristic decay time $t_{\min }$, defined as the location of the first minimum, as a function of filling. (Right inset) Decayed fraction of doublons, defined as the time average $1-\overline{n_{\text{tot}}^h\left(t\right)/n_{\text{tot}}^h\left(0\right)}$ between $t=4$ and the maximal propagation time, as a function of filling.

Figure 5

Inverse square of the decay time $t_{\min }$, where $n_{\text{tot}}^h\left(t\right)$ exhibits its first pronounced minimum, as a function of $U^2$ for different fillings. Lines: linear fits of the DMRG data based on the result of the two-site model, Eq. (14), see text.

Figure 6

Comparison of the unnormalized hole density $n_{\text{tot}}^h\left(t\right)$ with the expected relaxed ground state expectation value (indicated by arrows of the same color) for various values of $U$ at a filling of $n=1.5$ and $L=40$.

Figure 7

Space-time plot of $〈{\tilde{n}}_j^h〉\left(t\right)n=1.8$. DMRG calculation at $U=6$ for $L=40$. The inset shows $R^2\left(t\right)$ (after subtracting the initial value) with the fit according to Eq. (21). The red line in the main plot indicates $vt$ with the velocity extracted from the fit.

Figure 8

Space-time plot of $〈{\tilde{n}}_j^h〉\left(t\right)$ for $n=1.2$ and other parameters as in Fig. 7. The fit is limited to $t=3$.

Figure 9

Symbols: velocity $v$ of the holon wavefront extracted from the fit of $R^2\left(t\right)$ via Eq. (21) (red circles) and directly from the DMRG data (red triangles). Yellow circles show the diffusion constant $\mathcal{D}$ from the same fit. White circles indicate where the fitting procedure for $v$ and $\mathcal{D}$ was restricted to times $t\le 3$ (see text). Lines: maximal velocities obtained via Eq. (25) from different Bethe ansatz eigenstates as functions of the filling.