  The recent discovery of superconductivity in infinite-layer nickelates has
drawn considerable attention; however, a consensus on the fundamental building
blocks and common ingredients necessary to understand and describe their ground
states and emergent properties is lacking. A series of experimental and
theoretical studies have suggested that an effective two-band Hubbard model
with Ni 3$d_{x^2-y^2}$ and rare-earth ($R$) 5$d$ character may describe the
low-energy physics. Here, we study the ground state properties of this two-band
model on four-leg cylinders using the density-matrix renormalization group
(DMRG) technique to better grasp whether such a simple model can embody the
essential physics. A key difference compared to single-band Hubbard materials
is that the system is self-doped: even at overall half-filling, the $R$-band
acts as an electron reservoir, hole-doping the Ni-layer and fundamentally
altering the physics expected from an undoped antiferromagnet. On the four-leg
cylinder, the ground state is consistent with a Luttinger liquid, with
anti-phase modulations of the charge density in the Ni- and $R$-layers having
corresponding wavevectors that lock together. Light hole doping away from 1/2
filling releases the locking between the Ni and the $R$ charge modulations, as
the electron density in the $R$-band decreases and eventually becomes exhausted
at a hole doping concentration that depends sensitively on the effective
splitting between the Ni and the $R$ orbitals. The ground state of the doped
system is consistent with a Luther-Emery liquid, possessing quasi-long-range
superconducting correlations in the Ni layer, similar to the single-band
Hubbard model. Our results are consistent with experimental observations and
may help to reveal the microscopic mechanism for pairing and other emergent
properties.
