Density matrix renormalization group study of a three-orbital Hubbard model with spin-orbit coupling in one dimension

Using the density matrix renormalization group technique we study the effect of spin-orbit coupling on a three-orbital Hubbard model in the ${\left(t_{2g}\right)}^4$ sector and in one dimension. Fixing the Hund coupling to a robust value compatible with some multiorbital materials, we present the phase diagram varying the Hubbard $U$ and spin-orbit coupling $\lambda$, at zero temperature. Our results are shown to be qualitatively similar to those recently reported using the dynamical mean-field theory in higher dimensions, providing a robust basis to approximate many-body techniques. Among many results, we observe an interesting transition from an orbital-selective Mott phase to an excitonic insulator with increasing $\lambda$ at intermediate $U$. In the strong $U$ coupling limit, we find a nonmagnetic insulator with an effective angular momentum $\langle {\left({\mathbf{J}}^{\mathrm{eff}}\right)}^2\rangle \ne 0$ near the excitonic phase, smoothly connected to the $\langle {\left({\mathbf{J}}^{\mathrm{eff}}\right)}^2\rangle =0$ regime. We also provide a list of quasi-one-dimensional materials where the physics discussed in this paper could be realized.

Figure 1

In (a) we show the connections between $t_{2g}$ orbitals using dashed lines, while in (b) the dashed lines represent the nonzero connections present in the ($j^{\mathrm{eff}},m$) basis if we use the proper hopping and crystal-field parameters satisfying the constraints described in Appendix pp1.

Figure 2

$\lambda \text{−}U$ phase diagram (note the log scale in $U/W$ axis). RBI, PM-M, B, FM, OO, IC, EXI, AFM, and NMI stands for relativistic band insulator, paramagnetic metal, block phase, ferromagnetic, orbital ordering, incommensurate, excitonic insulator, antiferromagnetic, and nonmagnetic insulator, respectively. Lines separating phases are guides to the eyes. The actual small circles indicate specific values of data points that were investigated with DMRG. Their high density indicates that this effort has been computationally demanding.

Figure 3

(a) shows the noninteracting bands of our model at $\lambda /W=0.0$. As explained in the text, the almost fully populated bands are degenerate and superimposed. In (b) and (c), we show the bands at $\lambda /W=0.2$ and 1.0, respectively. (c) displays a clear opening of a gap, i.e., the system becomes a band insulator. Colors are decided depending on the relative contributions from the three $(j^{\mathrm{eff}},m)$ bands, with the pure cases shown in the legend of (b). (d) contains the occupation numbers in the $(j^{\mathrm{eff}},m)$ basis, while (e) has the local magnetic moments strengths (see legend) as well as $\langle \mathbf{S}·\mathbf{L}\rangle$, all at $U/W=0.02$. Calculations for (d) and (e) were performed with DMRG using a $L=16$ chain, while (a)–(c) are from exact analytical formulas.

Figure 4

DMRG results obtained at $U/W=1$ (intermediate coupling) and using a $L=16$ system. (a) shows occupation number in the $(j^{\mathrm{eff}},m)$ bands while (b) shows the excitonic parameter ${\mathrm{\Delta }}_m$ defined in Eq. (8) varying $\lambda /W$. (c) shows the three local moment strengths as well as $\langle \mathbf{S}.\mathbf{L}\rangle$.

Figure 5

DMRG resuls obtained at $U/W=1$. (a) and (b) contain the spin structure factor in the block phase and in the EXI phase, respectively, for various number of sites $L=8$ (black), 16 (red), 24 (green), and 32 (blue). (c) and (d) display the real-space spin correlations at $L=32$ corresponding to the block and EXI phases, respectively, for the $\lambda /W$'s indicated.

Figure 6

DMRG results obtained at $U/W=10$ (strong coupling regime). (a) shows the occupation number in the $(j^{\mathrm{eff}},m)$ bands, while (b) displays the excitonic order parameter dependence on $\lambda /W$. (c) displays the local moment strengths and also $\langle \mathbf{S}.\mathbf{L}\rangle$. (d) contains the spin structure factor for a number of sites $L$ equal to 8 (black), 16 (red), 24 (green), and 32 (blue). (e) shows $\langle {\tau }_z\left(i\right){\tau }_z\left(j\right)\rangle$ for $L=32$. Both (d) and (e) are in the ferromagnetic and orbitally ordered phase at $\lambda /W=0.046$.

Figure 7

DMRG results obtained at $U/W=10$. (a) depicts the spin structure factor in the EXI phase at $\lambda /W=0.115$, for a number of sites $L$ equal to 8 (black), 16 (red), 24 (green), and 32 (blue). (b) shows the real-space averaged spin-spin correlations for $\lambda /W=0.115$. In (c) and (d), we present the pair-pair correlation in momentum and real space, respectively, for a $L=16$ system. In (d) $j=1/2$, $\tilde{j}=3/2$, and $m=\pm 1/2$ were used.

Figure 8

Averaged local charge fluctuations of the $(j^{\mathrm{eff}},m)$ states (as indicated in the top panel legend) corresponding to (a) $U/W=0.02$ (weak coupling), (b) $U/W=1$ (intermediate coupling), and (c) $U/W=10$ (strong coupling).

Figure 9

Occupations of the $t_{2g}$ orbital states corresponding to (a) $U/W=1$ (intermediate coupling) and (b) $U/W=10$ (strong coupling). To a good approximation, the excitonic condensate phase behaves similarly as the block (OSMP) phase, namely with one orbital having occupation of approximately one electron.