Spectral properties of orbital polarons in Mott insulators

We address the spectral properties of Mott insulators with orbital degrees of freedom, and investigate cases where the orbital symmetry leads to Ising-type superexchange in the orbital sector. The paradigm of a hole propagating by its coupling to quantum fluctuations, known from the spin $t\text{−}J$ model, then no longer applies. We find instead that when one of the two orbital flavors is immobile, as in the Falicov-Kimball model, trapped orbital polarons coexist with free hole propagation emerging from the effective three-site hopping in the regime of large on-site Coulomb interaction $U$. The spectral functions are found analytically in this case within the retraceable path approximation in one and two dimensions. On the contrary, when both of the orbitals are active, as in the model for $t_{2g}$ electrons in two dimensions, we find propagating polarons with incoherent scattering dressing the moving hole and renormalizing the quasiparticle dispersion. Here, the spectral functions, calculated using the self-consistent Born approximation, are anisotropic and depend on the orbital flavor. Unbiased conclusions concerning the spectral properties are established by comparing the above results for the orbital $t\text{−}J$ models with those obtained using the variational cluster approximation or exact diagonalization for the corresponding Hubbard models. The present work makes predictions concerning the essential features of photoemission spectra of certain fluorides and vanadates.

Figure 1

(Color online) Hole propagation in 1D strong-coupling model (2.2). Two top panels show a hole doped into (a) mobile $a$ orbitals (empty boxes), and (b) immobile $b$ orbitals (filled boxes). Solid (dashed) arrows indicate possible hopping processes with hopping elements $t$ and $\tau$, respectively. In case of a hole added to the $b$ orbital the latter process occurs only after the initial hopping by $t$; see panel (c). Panel (d) shows the exact spectral functions $A_a\left(k,\omega \right)$ and $A_b\left(k,\omega \right)$ of a hole added into the $a$ orbital (middle dispersive feature between $\omega =-0.4t$ and $\omega =0$) and the $b$ orbital (two side dispersionless maxima) as obtained from 1D strong-coupling model (2.2). Parameters: $J=0.4t$, $\tau =0.1t$, and peak broadening $\delta =0.01t$. The spectral functions obtained using the VCA for 1D Hubbard model (2.1) with $U=10t$ (not shown) are identical to the exact result.

Figure 2

(Color online) Spectral function $A_b\left(\omega \right)$ of a hole doped into the $b$ orbital in the 1D model with (a) $\tau =0$, (b) $\tau =0.5t$, (c) $\tau =t$, and (d) $\tau =2t$. Dotted (solid) lines for $J=0$ $\left(J=0.4t\right)$, respectively, with broadening $\delta =0.01t$.

Figure 3

(Color online) The spectral functions for the 2D FK model, as obtained for mobile $a$ orbitals (middle dispersive feature between $\omega =-1.6$ and $\omega =0$) and immobile $b$ orbitals (two side dispersionless maxima): (a) with the RPA of Ref. 43 for 2D FK strong-coupling model (3.2) with $J=0.4t$ and $\tau =0.1t$, and (b) by a numerical diagonalization of a $20\times 20$ cluster for 2D FK Hubbard model (3.1) with $U=10t$. Peak broadening $\delta =0.01t$.

Figure 4

(Color online) Schematic view of a five-site polaron which occurs after the removal of one $b$ electron in 2D FK model (3.2). (a) Initial state with a hole (empty circle) on $B$ sublattice—crosses represent immobile $b$ electrons and filled circles mobile $a$ electrons. Five-site polaron is indicated by the dotted line. (b) The hole may delocalize within it to the nearest neighbors by hopping $t$ (solid line), generating three broken bonds which cost energy $3J/4$ for each hopping process. The hole can hop directly between the external sites (dashed line) by second-order three-site hopping $\tau =J/4$ via a doubly occupied site (outside the polaron).

Figure 5

(Color online) Propagation of a hole added into the $a$ orbital in centipede strong-coupling model (4.3): (a) schematic picture of a hole doped at site $a$ and its possible delocalization via hopping $t$ (solid lines) and three-site effective $\tau$ term (dashed lines); (b) spectral function $A_a\left(k,\omega \right)$. Parameters: $J=0.4t$, $\tau =0.1t$, peak broadening $\delta =0.01t$. The chain is oriented along the $b$ axis, and nonequivalent positions of the orbitals which do not permit hopping along this direction are labeled $b$, $u$, and $d$ in panel (a).

Figure 6

(Color online) Characteristic features in the spectra obtained for the 1D centipede model (Fig. 5) for increasing $J$: (a) the bandwidth $W_{1,2}$, (b) the spectral weight $a_{\text{QP}}$, and (c) the distance △ between the two peaks. The solid (dotted) line in (a) corresponds to the first (second) dispersive peak in $A_a\left(k,\omega \right)$, whereas the solid (dashed) lines in the lower panels show results for $k=0$ $\left(k=\pi /2\right)$. The light solid line in (a) is merely a guide for the eyes to show that the bandwidth of the first peak is a function with a positive second derivative. Parameter: $\tau =J/4$.

Figure 7

(Color online) Schematic view of the hole motion in strong-coupling $t_{2g}$ orbital model (5.1) with AO order. Circles depict holes while horizontal (vertical) rectangles depict occupied orbitals with electrons that can move only horizontally (vertically). The hole inserted in the AO state (a) can move via NN hopping $t$, but it has to turn by $90°$ in each step along its path and leaves behind broken bonds, leading to string excitations with ever increasing energy (b) and (c). After moving by $270°$ around a plaquette (d), the hole cannot return to its initial position as would be necessary to complete the Trugman path (Ref. 35).

Figure 8

Diagrammatic representation of the perturbative procedure used within the SCBA: top—the Dyson equation for the $G_{BB}\left(\mathbf{k},\omega \right)$ and $G_{AA}\left(\mathbf{k},\omega \right)$ Green’s functions; bottom—the summation of diagrams for the self-energy ${\Sigma }_{BB}\left(\mathbf{k},\omega \right)$. The densely dotted and the dashed-dotted rainbow lines in the self-energy (lower part) connect the two vertices $N\left(\mathbf{k},\mathbf{q}\right)$ and $M\left(\mathbf{k},\mathbf{q}\right)$, respectively.

Figure 9

Spectral function as obtained in the SCBA for effective $t_{2g}$ model (5.8) for a hole doped into: (a) $a$ orbital and (b) $b$ orbital. Parameters: $J=0.4t$, $\tau =0.1t$, and peak broadening $\delta =0.01t$.

Figure 10

(Color online) Schematic representation of two three-site terms in $t_{2g}$ orbital model (5.2). Circles depict holes while horizontal (vertical) rectangles depict occupied orbitals with electrons that can move only horizontally (vertically), respectively. Processes shown in panels (a)–(c) result from forward propagation (5.5), while the ones shown in panels (d)–(f) and given by Eq. (5.6) create a defect in the AO order with the energy cost indicated by the lines (broken bonds) in (f).

Figure 11

(Color online) Staggered magnetization $m_{\text{stagg}}$ for increasing $J/t$ as obtained for the $t_{2g}$ orbital Hubbard model, spin Hubbard model (called Hubbard on the figure), and for the $e_g$ orbital Hubbard model, respectively.

Figure 12

Spectral function $A\left(\mathbf{k},\omega \right)$ obtained with VCA for 2D $t_{2g}$ Hubbard model (5.1) for (a) $a$ orbitals, and (b) $b$ orbitals. Parameter: $U=10t$.

Figure 13

(Color online) Dependence of total weight found in the photoemission spectrum on momentum $k$ for (a) a hole inserted into the $b$ orbital of $t_{2g}$ model (5.1) and (b) for a hole with spin up in the SU(2) symmetric spin Hubbard model. All data were obtained by VCA. For illustration, we added a line at $n\left(k\right)=0.5$, corresponding to the constant $n\left(k\right)$ for $t\text{−}J$-like models. The insets show the first quadrant of the first BZ: The arrows indicate the path taken for the main panel; the shaded area gives momenta with $n_b>0.5$ and $n_{\uparrow }>0.5$.

Figure 14

(Color online) Quasiparticle properties obtained for the 2D $t_{2g}$ model within the SCBA for increasing superexchange $J$ (with $\tau =J/4$): (a) the bandwidth of the QP $W_1$ (solid line) and the second dispersive feature $W_2$ (dotted line), (b) the spectral weight $a_{\text{QP}}$, and (c) the distance between the first two peaks in the spectra (pseudogap) △. The solid (dashed) lines in (b) and (c) give the results for $\mathbf{k}=\left(0,0\right)$ $\left[\mathbf{k}=\left(\pi /2,\pi /2\right)\right]$. The light solid line in (c) indicates the $t{\left(J/t\right)}^{2/3}$ law (see text).

Figure 15

Spectral density $A_b\left(\mathbf{k},\omega \right)$ obtained within the VCA method for a hole inserted into $b$ orbitals of $t_{2g}$ model (5.1), supplemented by finite NNN hopping (6.1). Parameters: $U=10t$, and $t_2=0.15t$.

Figure 16

Photoemission $\left[\left(\omega -\mu \right)<0\right]$ and inverse photoemission $\left[\left(\omega -\mu \right)>0\right]$ part of the spectral density $A_b\left(\mathbf{k},\omega \right)$ for a hole inserted into $b$ orbitals, obtained within VCA for $t_{2g}$ model (5.1) with an additional longer-range third-neighbor hopping $t_3$ (6.2). The value $t_3$ was selected to suppress dispersion arising from three-site effective hopping (5.5) (a) in the hole (photoemission) sector with $t_3=0.1t=J/4$, and (b) in the inverse photoemission sector with $t_3=-0.1t=-J/4$. Parameter: $U=10t$.