Orbital selective directional conductor in the two-orbital Hubbard model

Employing a recently developed many-body technique that allows for the incorporation of thermal effects, the rich phase diagram of a two-dimensional two-orbital (degenerate $d_{xz}$ and $d_{yz}$) Hubbard model is presented varying temperature and the repulsion $U$. Our main result is the finding at intermediate $U$ of an antiferromagnetic orbital selective state where an effective dimensional reduction renders one direction insulating and the other metallic. Possible realizations of this state are discussed. In addition, we also study nematicity above the Néel temperature. After a careful finite-size scaling analysis, the nematicity temperature window appears to survive in the bulk limit, although it is very narrow.

Figure 1

Phase diagram of the two-orbital Hubbard model Eq. (1) in the MC-MF approximation and with hoppings from previous literature [23], at $J/U=0.25$ and $n=2$. Shown are results for a ${32}^2$ lattice. Besides the weak-coupling paramagnetic metal (PM-M) and the intermediate/large $U/W$ region with “preformed local moments” (gray), other states were identified: (1) A $(\pi ,0)$-spin-ordered metal, AF-M, where transport is anisotropic but metallic in both directions. (2) A novel $(\pi ,0)$-spin-ordered regime, the OSDC, with metallic (insulating) behavior along the $x$ ($y$) axis. (3) A $(\pi ,0)$-spin-ordered insulator, AF-I, with a full gap. (4) A very narrow spin nematic regime above $T_N$. The AF-M and OSDC states break the same symmetries and, thus, no sharp distinction between them is expected but a rapid crossover. $T_N$ has a nonmonotonic behavior, maximizing close to the OSDC/AF-I boundary.

Figure 2

(a) Spin structure factors and nematic order parameter at $U/W=1.16$ and using a ${32}^2$ lattice. (Statistical error bars are the size of the points; error bars due to trapping in metastable states are better represented by the fluctuations of the results with varying temperatures.) (b) Same as (a) but for the magnetic [for $S(\pi ,0)$ case] and nematic susceptibilities. Lorentzian fits of the data are shown. Vertical dashed lines (left to right) indicate the magnetic and nematic transition temperatures from susceptibility maximization.

Figure 3

(a) Finite-size scaling analysis for $T_N, T_{\mathrm{Nem}}$, and $T_{\mathrm{split}}$ at $U/W=1.16$, using $L=12,...,48$ ($L\times L$) lattices plotted vs $1/L$ (values at top). Data is fit with a scaling function $T_{\mathrm{ord}}\left(L\right)=T_{\mathrm{ord}}^{\mathrm{bulk}}+b{(1/L)}^{(1/\nu )}$, where $T_{\mathrm{ord}}^{\mathrm{bulk}}, b$, and $\nu$ are independent fitting parameters for the three data sets. From the fit we obtain a nematic temperature window of width approximately $0.002t$. (b) Density of states $N\left(\omega \right)$ at $U/W=0.5$ ($\mu$ denotes $E_F$). A pseudogap develops below $T_{\mathrm{Nem}}$ that deepens with reducing temperature but never becomes a full gap. On the other hand, in panel (c) at $U/W=1.16$ a clear gap develops upon cooling. (d) The orbital-resolved DOS at $T=0.005t$ and $U/W=0.5$. The $C_4$ spontaneous symmetry breaking makes the (nonzero) population of the two orbitals different at $E_F$ [33]. (e) Same as (d), but at $U/W=0.533$ in the OSDC regime where $N_{yz}(\omega =\mu )\ll N_{xz}(\omega =\mu )$. (f) The orbital-resolved total occupation $n_{xz}$ and $n_{yz}$ shown at low $T$ vs $U/W$. Dashed lines indicate the OSDC regime. Panels (b–f) were obtained using ${32}^2$ lattices.

Figure 4

(a) Resistivity along the $x$ spin-staggered (solid symbols) and $y$ spin-uniform (open symbols) directions at several $U/W$'s, illustrating the transport properties of the AF-M, OSDC, and AF-I regions of Fig. 1. Arrows indicate $T_N$ for each case. (b) Optical conductivity in the OSDC with electric fields along the $y$ and $x$ directions. In both panels a ${16}^2$ lattice is used.

Figure 5

(a,b,c) The $(\pi ,0)$ and $(0,\pi )$ magnetic structure factors, and the nematic order parameter (${\mathrm{\Psi }}_{\mathrm{Nem}}$) vs temperature for different lattice sizes, at $U/W=1.16$. (d,e,f) Magnetic and nematic susceptibilities at $U/W=1.16$ for the same lattice sizes as immediately above [i.e., (d) corresponds to (a), etc.]. The solid curves are Lorentzian fits. The dashed lines indicate the positions of the maxima in the susceptibilities, therefore indicating the estimated critical temperatures.

Figure 6

The Hubbard $U/t$ vs temperature $T/t$ phase diagram corresponding to the one-orbital Hubbard model in three dimensions using the Hartree approximation in the MC-MF technique. $T_N$ denotes the Néel critical temperature where antiferromagnetism with $(\pi ,\pi ,\pi )$ staggered magnetic order develops (region denoted as AF-I) according to measurements of the spin structure factor and the spin-spin correlation functions. The MC-MF results (red filled squares) are compared against data obtained using the determinant quantum Monte Carlo (white squares, reproduced from Staudt et al. [40]). The lattice size used in the MC-MF method is $4^3$. The expected $4t^2/U$ behavior at large $U/t$ is indicated by the black dashed line. The more crude mean-field Hartree-Fock approximation results are denoted by “H-F” (blue dashed line); it incorrectly predicts the growth of the critical temperature with increasing $U/t$. In the light-blue region, measurements of the spin square operator and the double occupancy indicate the presence of a local moment [10].

Figure 7

Optical conductivity of the 2D two-orbital model used in this study at the temperatures and couplings $U/W$ indicated. Panel (a) is in the AF-M regime; panel (b) in the OSDC regime; and panel (c) in the AF-I state. All data are using a ${16}^2$ system.