Ground-state phase diagram of the Hubbard model with hopping modulation in the y-direction

The Hubbard model serves as a testbed for exploring the 
essential physics of strongly correlated electron systems and has proven 
valuable in elucidating key aspects of unconventional superconductivity. 
Previously, the model with uniform and nearest-neighbor hopping has 
revealed the subtle competition between the charge density wave (CDW), 
spin density wave (SDW), and the superconducting order [1]. Notably, the 
ground state of the pure (with only uniform nearest-neighbor hopping) 
Hubbard model is found to be non-superconducting near optimal hole 
doping (i.e., 12.5%) [1]. However, the inclusion of 
next-nearest-neighbor hopping has been shown to induce superconductivity 
in the system [2,3]. Recently, it has been shown that periodic 
modulation of the hopping amplitude along the y-direction can enhance 
superconducting correlations [4]. In this work, we employ the state-of 
the-art density matrix renormalization group (DMRG) method to 
investigate the ground-state phase diagram of the modulated Hubbard 
model on a four-leg cylinder. We compute superconducting correlations 
and the pairing order parameter across a range of hole dopings and model 
parameters to probe different phases.

References:
[1] Mingpu et al., “Absence of Superconductivity in the Pure 
Two-Dimensional Hubbard Model,” Physical Review X 10, 031016 (2020).
[2] Xu et al., “Coexistence of superconductivity with partially filled 
stripes in the Hubbard model,” Science 384, 637 (2024).
[3] Jiang et al., “Superconductivity in the doped Hubbard model and its 
interplay with next-nearest hopping t′,” Science 365, 1424–1428 (2019).
[4] Jiang et al., “Stripe order enhanced superconductivity in the 
Hubbard model,” PNAS 119 (1), e2109406119 (2022).