Spin, charge, and orbital correlations in the one-dimensional $t_{2\mathrm{g}}$-orbital Hubbard model

We present the zero-temperature phase diagram of the one-dimensional $t_{2\mathrm{g}}$-orbital Hubbard model, obtained using the density-matrix renormalization group and Lanczos techniques. Emphasis is given to the case of the electron density $n=5$ corresponding to five electrons per site, while several other cases for electron densities between $n=3$ and 6 are also studied. At $n=5$, our results indicate a first-order transition between a paramagnetic (PM) insulator phase, with power-law slowly decaying correlations, and a fully polarized ferromagnetic (FM) state by tuning the Hund’s coupling. The results also suggest a transition from the $n=5$ PM insulator phase to a metallic regime by changing the electron density, either via hole or electron doping. The behavior of the spin, charge, and orbital correlation functions in the FM and PM states are also described in the text and discussed. The robustness of these two states against varying parameters suggests that they may be of relevance in quasi-one-dimensional Co-oxide materials, or even in higher dimensional cobaltite systems as well.

Figure 1

Ground-state phase diagram for the one-dimensional three-orbital Hubbard model for the electron density $n=5$, using a six-site chain. FM and PMI denote the regions with ferromagnetism and paramagnetism (insulator), the latter with robust short-range correlations, respectively. We also present a schematic picture of the electron configurations. AFO indicates the staggered population of orbitals in the FM state. The reader should consult the text for more details, as well as Fig. 2.

Figure 2

States with the largest weight in the ground state of a four-site chain solved exactly. Note that each state has eightfold degeneracy. At $J=0$, these three states have the same weight. On the other hand, for nonzero $J$, the state (a) (and its eightfold degenerate states) has the largest weight, with a spin (orbital) structure factor peaked at $\pi ∕2$ $\left(\pi \right)$. The states (b) and (c) (each one also with degeneracy 8) have the second- and third-largest weights, respectively, for nonzero $J$.

Figure 3

(Color online) (a) The spin-spin correlation function $C_{\text{spin}}\left(j\right)$ vs $j$ for $L=48$ and density $n=5$. The dashed curve indicates a fit using Eq. (5). The inset shows the spin structure factor $S\left(q\right)$ for $L=16$ and $L=48$. (b) The linear-log plot of the module of the spin-spin correlation function $\mid C_{\text{spin}}\left(l\right)\mid$ for densities $n=3$, 4, and 5 with $L=48$, as well as the fit used in (a). For details, see the main text. (c) The spin structure factor $S\left(q\right)$ for several densities $n$, and using $L=16$. The arrows indicate the peak positions. In all plots $U_{\mathrm{eff}}=10$ and $J=1$, as indicated. Inset shows $S\left(q\right)$ for $n=4$ and 5.

Figure 4

(Color online) (a) The charge gap $\Delta$ vs $1∕L$ at particular values of $U_{\mathrm{eff}}$ and $J$, and densities $n=4$ and $n=5$. (b) Same as (a) but for noninteger densities, and $U_{\mathrm{eff}}=10$ and $J=1$. (c) and (d) denote the charge gap for density $n=5$ and $L=12$. (c) contains $\Delta$ vs $U^{\prime }$ at $J=1$, while (d) shows $\Delta$ as a function of $J$ at $U^{\prime }=11$.

Figure 5

(Color online) The charge structure factor $N^{\gamma ,{\gamma }^{\prime }}\left(q\right)$ and the charge-charge correlation function $C\left(j\right)$, at density $n=5$. (a) and (b) are for $U_{\mathrm{eff}}=10$, $J=10$, and $L=64$. This corresponds to the FM regime of Fig. 1. The dashed curve is a fit using the function $a\cos \left(\pi j\right)∕j$ with an appropriate fitting parameter $a$. (c) and (d) are for $U_{\mathrm{eff}}=10$, $J=1$, and $L=32$. This is in the PM regime of Fig. 1. (e) and (f) are the same as (c) and (d), respectively, but for $J=5$.

Figure 6

(Color online) The charge structure factor $N^{zx,zx}\left(q\right)$ for several densities and using $U_{\mathrm{eff}}=10$, $J=1$, and $L=16$. The arrows indicate the cusp positions.

Figure 7

(Color online) (a) The orbital structure factor $T\left(q\right)$ versus momentum for $U_{\mathrm{eff}}=10$, $J=10$, and $L=64$ with ${\theta }_i=\theta =0$. (b) The orbital correlation $C_{\text{orbital}}\left(j\right)$ for the same parameters as used in (a). The dashed curve is a fit using the function $a\cos \left(\pi j\right)∕j$. (c) and (d) are the same as (a) and (b), but for $U_{\mathrm{eff}}=10$, $J=1$, and $L=32$. (e) and (f) contain the correlations $C_{\text{spin}}\left(j\right)$ and $C_{\text{orbital}}\left(j\right)$ for $U_{\mathrm{eff}}=10$, $J=0$, and $L=32$. All the results are for the density $n=5$.

Figure 8

(Color online) (a) The size dependence of the orbital structure factor $T\left(q\right)$ at $q=\pi$ with ${\theta }_i=\theta =0$, at density $n=5$. (b) $T\left(q\right)$ vs $\theta$ for particular values of $q$.