Charge dynamics in magnetically disordered Mott insulators

With the aid of both a semianalytical and a numerically exact method, we investigate the charge dynamics in the vicinity of half-filling in the one- and two-dimensional $t\text{−}J$ model derived from a Fermi-Hubbard model in the limit of large interaction $U$ and hence small exchange coupling $J$. The spin degrees of freedom are taken to be disordered. So we consider the limit $0<J\ll T\ll W$, where $W$ is the bandwidth. We focus on evaluating the local spectral density of a single hole excitation and the charge gap that separates the upper and the lower Hubbard band. We find indications that no band edges exist if the magnetic exchange is taken into account; instead of band edges, Gaussian tails seem to appear. A discussion of the underlying physics is provided.

Figure 1

For a large enough $U>0$, the local density of states $\rho \left(E\right)$ splits into a lower and an upper Hubbard band at half-filling. Each band has the effective bandwidth $W_{\mathrm{eff}}$. Charge excitations take the form of DOs of electron or hole character with a minimum excitation energy of the gap $\mathrm{\Delta }$. Decreasing the on-site interaction reduces the gap until it closes at the critical interaction $U_c$.

Figure 2

Retarded Green's function of the chain for the complete generalized $t-J$ model with parameters (A) for various chain lengths $N$ (CET) and loop orders $m$ (iEoM). The parabola (blue triangles) from the expansion, cf. Appendix pp1, in powers of $t$ up to $O\left(t^3\right)$ is depicted for orientation. The iEoM result for $m=6$ first starts to deviate from the CET results at about $t\ge 9/t_0$.

Figure 3

Minimum eigenvalues ${\omega }_{\min }$ (symbols) for a chain and parameters (A) as a function of the loop order $m$ of the iEoM. The three different Hamiltonians [NN hopping only ($T_0$), generalized hopping in second order ($H_{0,\text{eff}}$), and the inclusion of magnetic exchange ($H_{\text{eff}}$)] are shown by different colors. The solid lines mark fits of the form (38) where applicable. The fit parameter $c=-{\omega }_{\min }(m\to \infty )$ is displayed as short horizontal bars at the right boundary of the graph.

Figure 4

Analysis of the lower and the upper tail of the spectral density as obtained by iEoM via the logarithm of the primitives defined in Eq. (39) for the parameter set (A); the fits are defined in Eq. (40). The optimum fit parameters read $W_{-}=0.28, {\sigma }_{-}=0.24t_0, x_{-}=-1.88t_0$ for the lower tail, and $W_{+}=0.80, {\sigma }_{+}=0.26t_0, x_{+}=1.62t_0$ for the upper tail.

Figure 5

Same as Fig. 4, but for parameters (B) and the fit parameters $W_{-}=0.12, {\sigma }_{-}=0.69t_0, x_{-}=-2.26t_0$ for the lower tail, and $W_{+}=0.37, {\sigma }_{+}=0.70t_0, x_{+}=1.81t_0$ for the upper tail.

Figure 6

Spectral density $A\left(\omega \right)$ vs $\omega$ for a chain and the parameter set (A) in (14a), broadened by $\sigma =0.15t_0$. Solid lines represent CET results, dashed lines iEoM results. The band edges ${\omega }_{\min }$ determined from (37) are indicated by vertical dashed lines.

Figure 7

Same as Fig. 6, but for the parameters (B) in (14b).

Figure 8

Spectral density $A\left(\omega \right)$ for hole motion on the square lattice and parameter set (A). See the caption of Fig. 6 for further explanations.

Figure 9

Spectral density $A\left(\omega \right)$ for hole motion on the square lattice and parameter set (B). See the caption of Fig. 6 for further explanations.

Figure 10

Spectral densities $A\left(\omega \right)$ for the square lattice, parameter set (A), and the various contributions, calculated with CET for $N=18$ (solid) and iEoM for $m=4$ for $T_0$ and $H_{0,\text{eff}}$ and $m=3$ for $H_{\text{eff}}$ (dashed). All data are convolved with $\sigma =0.45t_0$. The band edges ${\omega }_{\min }$ calculated according to (37) are indicated by vertical dashed lines.

Figure 11

Same as in Fig. 10, but for parameter set (B).