Magnetically bound nature of a holon-doublon pair in two-dimensional photoexcited Mott insulators

We revisit the holon-doublon binding problem in two-dimensional (2D) photoexcited Mott insulators. Low-energy photoexcited states in Mott insulators are described as a pair state of a doublon and a holon. The most basic question is whether its bound state is formed in the lowest-energy state, and negative and positive responses have been discussed in the past. In this study we begin with the 2D Hubbard model, and transform it into the first effective model, which is based on the $t/U$ expansion, with $U$ and $t$ being the Hubbard $U$ and the electron hopping energy, respectively. We find that quantitative reliability is assured for $U/t\gtrsim 10$. Furthermore, we transform it into a second effective model that selects essential states in the low-energy region. In both effective models we distinguish two magnetic terms, namely, the spin-exchange interaction and the three-site transfer, and parametrize the two terms with the parameters $J_{\mathrm{ex}}$ and $J_{3\mathrm{site}}$. By changing the parameters apart from the restriction given by the Hubbard model, any positive $J_{\mathrm{ex}}$ value with $J_{3\mathrm{site}}=0$ yields a finite amount of binding, whereas a finite value of $J_{3\mathrm{site}}$ suppresses the binding significantly, still leaving the Hubbard case of $U=10t$ in the vicinity of the bound-unbound boundary.

Figure 1

Illustrations of the interactions included in the effective model I. The case of a holon is presented as an example. The two-way arrow, the solid one-way arrow, and the dashed one-way arrow represent the spin-exchange interactions, the direct transfer, and the three-site term, respectively. Site $e$ is an empty site, and the $k$ and $l$ sites are singly occupied sites.

Figure 2

(a) Cluster of 18 sites and (b)–(d) optical conductivities calculated using the Hubbard model and the effective model I. The broadening factor is commonly $0.1t$.

Figure 3

(a) Cluster used in the present calculation and (b) comparison of the present result with those from tDMRG calculations. The broadening factor is commonly $0.2t$. The periodic boundary condition is adopted in our calculation, whereas in the tDMRG calculation, the open boundary condition is adopted. The tDMRG results are reprinted from Ref. [20].

Figure 4

Optical conductivities and charge patterns for $U=10t$. In the latter, the hole resides at the origin and the relative density distribution of the doublon is plotted with interpolation. In (a) and (b), $J_{\mathrm{ex}}=J_{3\mathrm{site}}=J$, whereas in (c) and (d), $J_{\mathrm{ex}}=J$, but $J_{3\mathrm{site}}=0$. In (b) and (d), the summed density inside the dashed area (cluster size) is normalized to unity.

Figure 5

Results in the absence of $J_{3\mathrm{site}}$. (a) Optical conductivity for $J_{\mathrm{ex}}=3.0t$, (b) $J_{\mathrm{ex}}$ dependence of the energy intervals appearing in (a), (c) charge-distribution maps, and (d) a schematic showing A–D states.

Figure 6

Schematics showing the two processes corresponding to $\mathrm{A}\to \mathrm{B}$ transitions.

Figure 7

Spin correlation function $S\left(l\right)$ for $J_{\mathrm{ex}}/t=3.0$ and $J_{3\mathrm{site}}=0$. In (a), $S\left(l\right)$ is plotted for the lowest four states in Fig. 5. For comparison, $S\left(l\right)$ in the ground state is also shown in (b).

Figure 8

In (a), $J_{\mathrm{ex}}$-dependent low-lying energy levels obtained without $J_{3\mathrm{site}}$, for $N=32$. The squares (circles) show the energies calculated using the effective model I (II). In (b), a schematic for the effective model II is illustrated, and, in (c), the density of states (DOS) for $J_{\mathrm{ex}}/t=3.0$, and $N=20000$ is plotted with the energy levels for the case with the same $J_{\text{ex}}/t$ and $N=32$ (vertical bars). Note that both are calculated by the effective model II. The single bound state for $N=20000$ is located very close to $e\left(1\right)$. The energies in (a) and (c) are measured from $e\left(2\right)$ for $N=32$.

Figure 9

Relative energies of the bound state measured from the lower edge of the continuum for various $J_{\mathrm{ex}}$ values and corresponding charge patterns. Note that $J_{3\mathrm{site}}=0$.

Figure 10

Results for $J_{3\mathrm{site}}>0$. (a) Energy levels determined by the effective model II for $N=32$, with $J_{\mathrm{ex}}=J_{3\mathrm{site}}=J$ (triangles), and $J_{\mathrm{ex}}=J$ and $J_{3\mathrm{site}}=0$ (circles). (b) Optical conductivity spectrum from the effective model I (curve) and the energy levels from the effective model II (vertical lines), both obtained with $N=32$ and $J_{\mathrm{ex}}=J_{3\mathrm{site}}=3.0t$. (c) Phase diagram, where the two variables $J_{3\mathrm{site}}$ and $J_{\mathrm{ex}}$ are selected independently. (d) DOS and local DOS (LDOS), and (e) a charge pattern, for $J_{\mathrm{ex}}/t=0.4$ and $J_{3\mathrm{site}}/t=0.07$. In (a), the triangles and circles correspond to the cases with and without $J_{3\mathrm{site}}$, respectively. The energy in (a) is measured from $e\left(2\right)$ for the case of $N=32$ and $J_{3\mathrm{site}}=0$, and that in (d) is measured from the lower edge of the continuum. In (b) the energies of the levels are shifted by $(-2.3)t$ to make them match the peaks of $\sigma \left(\omega \right)$.

Figure 11

Charge correlations for the cases corresponding to the points specified on the left side. $J_{\mathrm{ex}}$ is fixed at $0.4t$. The case of $J_{3\mathrm{site}}=0.07t$ (square) corresponds to the case of the Hubbard model, with $U=10t$.

Figure 12

$J_{\mathrm{ex}}$ dependence of the relative energies. Energy intervals of the first and third peaks obtained by the effective model II with $J_{3\mathrm{site}}>0$ are plotted as circles and squares. Here the circles show the data without any adjustment for $J_{3\mathrm{site}}$, that is, for $J_{3\mathrm{site}}=J_{\mathrm{ex}}$, while the squares are obtained with adjustment. The results from the effective model I are specified by triangles.