Fractional Fermi liquid in a generalized $t\text{−}J$ model

Inspired by the recent discovery of superconductivity in the nickelate ${\mathrm{Nd}}_{1-x}{\mathrm{Sr}}_x\mathrm{Ni}{\mathrm{O}}_2$, we study a generalized $t\text{−}J$ model to investigate the correlated phases induced by doping spin-1 ${\mathrm{Ni}}^{2+}$ into a spin-$1/2$ Mott insulator formed by ${\mathrm{Ni}}^{1+}$. Based on a three-fermion parton mean-field analysis, we identify a robust fractional Fermi liquid (FL*) phase for almost every doping level. The FL* state is characterized by a small Fermi pocket on top of a spin liquid, which violates the Luttinger theorem of a conventional Fermi liquid and is an example of a symmetric pseudogap metal. Furthermore, we verify our theory in one dimension through density matrix renormalization group simulations on both the generalized $t\text{−}J$ model and a two-orbital Hubbard model. The fractional Fermi liquid reduces to a fractional Luttinger liquid (LL*) in one dimension, which is connected to the conventional Luttinger liquid through a continuous quantum phase transition by tuning interaction strength. Our findings offer insights into correlated electron phenomena in nickelate superconductors and other multiorbital transition-metal oxides with a spin-triplet $d^8$ state.

Figure 1

Mean-field ansatz at zero temperature with doping $x$ for $N=2$ and $N=10$ on square lattice. We used the parameters $t=1$, $J=J_d=J^{\prime }=0.5$. $t_{\mathrm{eff}}=t_{\psi ;22}$ is the hopping of ${\psi }_2$ and is in units of $t$. $Z=|\frac{1}{2}\langle {\sum }_{\alpha }{f}_{i;\alpha }^{†}{\psi }_{i;1\alpha }\rangle {|}^2$. The hoppings of the spinons $f,{\psi }_1$ are listed in Fig. 5.

Figure 2

Results for $x=\frac{1}{3}$ from iDMRG. We used the $t\text{−}J$ model as defined in Eq. (15) with parameter $t=2J=1$ and $J_d=J^{\prime }=J$. The momentum is in units of $2\pi$. (a) We vary the bond dimension $D$ from 1000 to 4000 to fit the central charge from $S=\frac{c}{6}log\xi$, where $\xi$ is the correlation length. The obtained central charge is $c=2.99$. (b) Momentum distribution function $n\left(\mathbf{k}\right)=\langle c^{†}\left(\mathbf{k}\right)c\left(\mathbf{k}\right)\rangle$. The dashed line is at $k_F^{*}=\frac{x}{4}2\pi$. (c) Spin structure factor with peaks at $\mathbf{q}=2k_F^{*}$ and $\mathbf{q}=\pi$. (d) Density-density correlation function with weak discontinuities at $2k_F^{*}$ and $4k_F^{*}$.

Figure 3

Results of the two-orbital Hubbard model defined in Eq. (1) at $x=0.2$. We tune $U$ and Hund's coupling $J_H$ together with fixed ratio $U=4J_H,U^{\prime }=2J_H$. For the hopping defined in Eq. (2), we use $t_1=t_2=V=1$, ${\epsilon }_{dd}=2$, and $t_{12}=0$. In the electron picture, the filling is fixed to ${\nu }_T=3-x$, or in the hole picture ${\nu }_T=1+x$. Panel (a) shows the Fermi momentum in the conventional LL phase at smaller-$U$ side and the fractional LL phase at large-$U$ side. The Fermi momentum in these two phases is identified from the consistent evidence, including the sudden jump in momentum distribution in (b), the kinks in the spin structure factor in (c), and the second-order derivative of the charge density structure factor in (d).

Figure 4

$\frac{\partial E}{\partial U}$ from iDMRG with bond dimension $D=5000$ for doping $x=\frac{1}{3}$. The parameters are the same as in Fig. 3; we choose the doping $x=\frac{1}{3}$ because it is easier for iDMRG. $\frac{\partial E}{\partial U}$ is continuous, implying a continuous phase transition. For $U<U_c$, $\frac{\partial E}{\partial U}$ can be fitted with $A{(U_c-U)}^{\alpha }+C$ with $\alpha \approx 0.64$.

Figure 5

Mean-field ansatz obtained from the self-consistent equations at zero temperature with doping $x$ for $N=2$ and $N=10$ on square lattice for the proposed $\text{SU}\left(N\right)$ $t\text{−}J$ model. We used the parameter $t=1$, $J=J_d=J^{\prime }=0.5$.

Figure 6

Results from finite DMRG with $L=100$ and $D=2000$. The parameters are still $t=2J$ and $J_d=J^{\prime }=J$ as defined in Eq. (15). Doping is varied from 0 to 1 with $\delta x=0.02$. (a) and (b) show a first-derivative jump of energy at $x=0.5$. In (c) and (d) we show that the system is in a LL* phase from $x\le 0.7$ except at $x=0.5$. At $x=0.5$ we find that $n\left(\mathbf{k}\right)$ does not have sharp $k_F^{*}$, indicating that a single electron is gapped. The Fourier transformation is done using the region in $[L/4,3L/4]$ to avoid the boundary effects which still causes some wiggles. Here, the momentum is in units of $2\pi$.

Figure 7

Zoom-in plots of $n\left(\mathbf{k}\right)$ in Fig. 2 of the main text for the iDMRG result at $x=\frac{1}{3}$. We focus on $k=k_F^{*}=\frac{x}{4}$ in panel (a) and $k=\frac{1}{2}-k_F^{*}\approx 0.416$ in panel (b). Here, the momentum is in units of $2\pi$.