Hole doping of a Mott insulator with orbital degrees of freedom

We study the effects of hole doping on one-dimensional Mott insulators with orbital degrees of freedom. We describe the system in terms of a generalized $t\text{−}J$ model. At a specific point in parameter space the model becomes integrable in analogy to the one-band supersymmetric $t\text{−}J$ model. We use the Bethe ansatz to derive a set of nonlinear integral equations which allow us to study the thermodynamics exactly. Moving away from this special point in parameter space we use the density-matrix renormalization group applied to transfer matrices to study the evolution of various phases of the undoped system with doping and temperature. Finally, we study a one-dimensional version of a realistic model for cubic titanates which includes the anisotropy of the orbital sector due to Hund’s coupling. We find a transition from a phase with antiferromagnetically correlated spins to a phase where the spins are fully ferromagnetically polarized, a strong tendency toward phase separation at large Hund’s coupling, as well as the possibility of instability toward triplet superconductivity.

Figure 1

(Color online) Specific heat for integrable model (2) with $J/t=2$ and $x=y=1/4$ calculated by BA. The left inset shows the spin-orbital $\left(v_{\text{so}}\right)$ and the charge velocity $\left(v_c\right)$ as a function of particle density $n$. The right inset compares the BA results (solid lines) with the low-temperature asymptotics obtained from CFT (dashed lines) [see Eq. (4)].

Figure 2

(Color online) Magnetic susceptibility ${\chi }_s$ for the integrable model. The dashed lines denote the zero-temperature limit according to CFT.

Figure 3

(Color online) Compressibility ${\chi }_c$ for the integrable model. The lines denote the zero-temperature limit according to CFT. The inset shows the dressed charge as a function of density.

Figure 4

(Color online) Drude weight $D$ for the supersymmetric point $J=2t$ (black solid line) and for $J=0$ (red dashed line), respectively. In the insets the regions of small and large densities are shown in detail.

Figure 5

(Color online) (a) Free energy $f$ calculated by TMRG (symbols) with $N=360$ states kept compared to the exact solution (lines). The black circles (black solid lines) denote the result for $\mu =-1.7$, the red squares (red dashed lines) for $\mu =-1.9$, and the blue diamonds (blue dot-dashed lines) for $\mu =-2.5$, respectively. (b) Absolute error $\left|{\Delta }_f\right|$ of the TMRG results presented in (a). (c) TMRG results for the density compared to the exact solution with symbols and lines denoting the same chemical potentials as in (a). (d) Absolute error $\left|{\Delta }_n\right|$ of the TMRG results in (c).

Figure 6

(Color online) The density $n$ as a function of chemical potential $\mu$ for different temperatures. Here $J/t=3$ and $x=y=1/4$. The lines are guides for the eyes.

Figure 7

(Color online) Spin (orbital) susceptibility for $J/t=0.5$ and $x=y=1/2$ and different chemical potentials. Insets: (a) corresponding densities as function of temperature. (b) Leading spin (orbital) dimer correlation length ${\xi }_D$.

Figure 8

(Color online) TMRG results for $J/t=0.5$ with $x=1/2$ and $y=-1/2$. Main figure: spin susceptibility in the undoped case and for $\mu =-0.1$ as function of temperature. Insets: (a) for $\mu =-0.1$ the density $n\sim 0.84$ is almost independent of temperature. (b) Nearest-neighbor spin-spin and orbital-orbital expectation values for the undoped model and for $\mu =-0.1$ $\left(n\sim 0.84\right)$. The circles on the $T=0$ axis denote 1/4 and $-\text{ln}2+1/4$, respectively. (c) Charge compressibility for $\mu =-0.1$.

Figure 9

(Color online) Leading correlation lengths for $J/t=0.5$, $x=1/2$, and $y=-1/2$. The leading spin-spin correlation length is nonoscillating $\left(k=0\right)$, whereas the orbital-orbital and the pair correlations show incommensurate oscillations at low temperatures as depicted in the inset.

Figure 10

(Color online) (a) Parameters $J_{\text{eff}}$ for $J=0.5$ and $z$ as defined in Eq. (12) as a function of $\eta =J_H/U$. (b) Parameters $x$ and $y$ as functions of $\eta$. (c) Parameters $\delta$ and $\gamma$ related to the $xxz$ anisotropy of the orbital sector as a function of $\eta$. The dashed blue lines in (a) and (b) indicate the approximate values for the phase transitions described in the text.

Figure 11

(Color online) Density as a function of the chemical potential for $J/t=0.5$ and $\eta =0.3$. The inset shows the same for $\eta =0.25$. The lines are guides for the eyes.

Figure 12

(Color online) TMRG results for model (11) with $J=0.5$ and $\eta =0.1,0.2,0.25$. The chemical potential is set to $\mu =0.6$ for $\eta =0.1$, $\mu =0.0$ for $\eta =0.2$, and $\mu =-0.8$ for $\eta =0.25$, respectively. (a) Nearest-neighbor spin-spin correlation $⟨{\mathit{S}}_j{\mathit{S}}_{j+1}⟩$, (b) nearest-neighbor orbital-orbital correlation $⟨{\mathit{\tau }}_j{\mathit{\tau }}_{j+1}⟩$, and (c) the density as a function of temperature. Note that for the chemical potentials chosen here the densities depend only weakly on temperature, $n\sim 0.8–0.85$.

Figure 13

(Color online) TMRG results for the nearest-neighbor correlation function $⟨{\mathit{S}}_j{\mathit{S}}_{j+1}⟩$ in the undoped case for $J=0.5$ and different $\eta$. The red dashed curve denotes the result for $\eta =0.2$. The spins are fully polarized in the ground state, $⟨{\mathit{S}}_j{\mathit{S}}_{j+1}⟩=1/4$ (black dot), if $\eta \gtrsim 0.2$.

Figure 14

(Color online) Leading correlation lengths for $J=0.5$, $\eta =0.25$ and a chemical potential $\mu =-0.8$ corresponding to a density $n\sim 0.84$ at low temperatures [see Fig. 12c]. The orbital-orbital and the density-density correlations show incommensurate oscillations at low temperatures as depicted in the inset, whereas the spin-spin and the spin triplet/orbital singlet correlations are both nonoscillating.