Origin of strong dispersion in Hubbard insulators

Using cluster perturbation theory, we explain the origin of the strongly dispersive feature found at high binding energy in the spectral function of the Hubbard model. By comparing the Hubbard and $t-J-3s$ model spectra, we show that this dispersion does not originate from either coupling to spin fluctuations $\left(\propto J\right)$ or the free hopping $\left(\propto t\right)$. Instead, it should be attributed to a long-range, correlated hopping $\propto t^2/U$, which allows an effectively free motion of the hole within the same antiferromagnetic sublattice. This origin explains both the formation of the high-energy anomaly in the single-particle spectrum and the sensitivity of the high-binding-energy dispersion to the next-nearest-neighbor hopping $t^{\prime }$.

Figure 1

(left) Spectral function $A(\mathbf{k},\omega )$ of the undoped Hubbard model with $U=8t$ and $t^{\prime }=0$ calculated using CPT on a $4\times 4$ cluster for the high-symmetry directions of the 2D Brillouin zone. The blue dotted (green dashed) box guides the eye for the LBE (HBE) features. (right) The high-symmetry directions of the 2D Brillouin zone.

Figure 2

Spectral function $A(\mathbf{k},\omega )$ for the undoped Hubbard model with $U=8t$ and (a) $t^{\prime }=-0.3t$ and (b) $t^{\prime }=0.3t$ calculated using CPT on a $4\times 4$ cluster.

Figure 3

(a) Top: schematic cartoon showing the three-site correlated hopping elements $\propto t^2/U$ in an exchange process via virtual double occupancy $\propto U$; bottom: various three-site paths in the AF background. Arrows (circles) denote electron spins (holes). Spectral function $A(\mathbf{k},\omega )$ for the undoped (b) $t-J$ and (c) $t-J-3s$ model with $J=0.5t$ calculated using CPT on a $4\times 4$ cluster. (d) The same as (c), but with the spectral weight normalized in the Hubbard-like way; see text.

Figure 4

(a) Idealized propagation via IASH (dashed lines) triggered by, e.g., the three-site terms: the hole motion does not disturb the AF background. (b)–(f) Spectral functions $A(\mathbf{k},\omega )$ for various three-site term values in $t-J-3s$ models using SCBA on a $40\times 40$ cluster ($J=0.5t$ and broadening $\delta =0.15t$), with $J_{3s}=0,J/2,J,3J/2$, and $4J$, respectively. [The canonical parameter $J_{3s}=J$ is highlighted by the red box around (d)]. Dotted lines indicate fitted SP dispersion, while dashed lines indicate IASH dispersion [Eq. (3)].

Figure 5

(a) The electron density $n_{\mathbf{k}}$ used for normalization in Fig. 3. (b) The momentum-integrated correlation function $\xi \left(\omega \right)$ [Eq. (A5)] calculated for the CPT $A(\mathbf{k},\omega )$ results for the $t-J$ and Hubbard models (dot-dashed line), $t-J-3s$ and Hubbard models (dashed lines), and $t-J-3s$ models normalized in the Hubbard-like way [or $(t-J-3s$)$/N$] and Hubbard models (solid line).

Figure 6

Comparison of $A(\mathbf{k},\omega )$ for the $t-J_z-3s$ model calculated using SCBA for various strengths of the three-site term $J_{3s}$: (a) $J_{3s}=0$, (b) $J_{3s}=J/2$, (c) $J_{3s}=J$, (d) $J_{3s}=2J$, and (e) $J_{3s}=4J$; see text for further details. The dashed line indicates the IASH dispersion following from the inclusion of the three-site terms in the model.