Effects of the Next-Nearest-Neighbor Hopping on the Low-Dimensional Hubbard Model: Ferromagnetism, Antiferromagnetism, and Superconductivity

TL;DR

This work surveys numerical studies of the Hubbard model with next-nearest-neighbor hopping ($t_2$) in one and two dimensions, focusing on how $t_2$ reshapes magnetism and superconductivity. By formulating the $t_1-t_2-U$ and related $t_1-t_2-J_1-J_2$ models and employing methods such as DMRG, QMC, tensor networks, and exact diagonalization, the authors map out phase diagrams featuring AFM, FM, SC, CDW, and PDW across dopings. A key finding is that NNN hopping can stabilize superconductivity on the electron-doped side in 2D and induce FM and PDW phenomena in 1D and near van Hove regions, though the presence of a SC dome on the hole-doped side remains contested. Altogether, the study highlights the critical role of band topology and longer-range hopping in shaping correlated phases and informs ongoing debates about cuprate-like superconductivity and related materials.

Abstract

The Hubbard model has attracted considerable interest due to its prototypical role in describing strongly interacting electronic systems, such as high-critical-temperature superconductors as well as many novel quantum materials. By introducing next-nearest-neighbor (NNN) hoppings to the Hubbard model, the phase diagram becomes richer, and fascinating phenomena arise in both, one-dimensional chains and square lattices, such as: antiferromagnetism (AFM), ferromagnetism (FM), superconductivity (SC), as well as charge orders, among others. Moreover, NNN hoppings play a fundamental role in understanding effects of doping on magnetism and pairing orders in strongly interacting regimes. In this article, we review the recent progress in understanding the different competing phases of this model in one and two dimensions from a computational perspective. We comment on the pressing technical challenges, illustrate the controversial results concerning the emergence of the SC phase, and conclude with our perspectives on future explorations.

Figures (6)

• Figure 1: Lattice structures and non-interacting band dispersions. (a) one-dimensional lattice; (b) two-dimensional square lattice; (c) non-interacting bands for one-dimensional chains; (d)-(h): two-dimensional energy dispersion colormaps; (i)-(m): energy dispersion along the dashed-white line cuts in (d)-(h).
• Figure 2: Left: Phase diagram for $t_1-t_2-J$ model with $t_2=-0.5$; Right: Phase diagram for $t_1-t_2-J$ model with $t_2=0.5$. AFM: antiferromagnetic; CDW: charge density wave; SDW: spin density wave; FM: ferromagnetic; PDW: pair density wave; PS: phase separation; SC: superconducting; SG: spin-gapped. my_t1_t2_J
• Figure 3: Top: Spin-spin correlations (left and middle) and spin structure factors (right) for the one-dimensional $t_1-t_2-U$ model with various values of $t_2$ and $U=8$. Results are obtained by DMRG for the chains with length $L=64$ and $1/16$ hole doping. Bottom: Magnetic phase diagram as a function of $t_2$ for $1/16$ hole doping, where the green region represents spin density wave, the yellow area depicts spin-gapped phase, the blue zone stands for AFM, and the red sector shows the AFM with anti-phase domain walls.
• Figure 4: Left: Ground state correlations for the $t_1-t_2-J$ chain with $t_2=-0.5$, $J=3$ and density $n=20/64$, where the system is in the PDW phase. Middle: Correlations for the $t_1-t_2-J$ chain with $t_2=0.5$, $J=5.3$ and density $n=24/64$, where the system is in the SC phase. Insets: Pairing orders shown in a linear scale. my_t1_t2_J Right: Photoemission spectrum for the $t_1-t_2-J$ chain with $t_2=-0.5$, $J=2.4$ and $n=6/48$. my_pdw
• Figure 5: Spin-spin and triplet pairing correlations in the fully polarized phase of $t_1-t_2-J$ model. The correlations are measured for the highest spin sectors, and are shown in a log-log scale. Insets: triplet-SC order shown in a linear scale. my_t1_t2_J