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Journal of Physics Condensed Matter

Quantum Monte Carlo study of the Hubbard model with next-nearest-neighbor hopping t ′: pairing and magnetism

Shuhui Yang, Tao Ying, Weiqi Li, Jianqun Yang, Xiudong Sun, Xingji Li

Published Dec 14, 2020• 9 citations
Publisher

Abstract

Using the finite-temperature determinant quantum Monte Carlo (DQMC) algorithm, we study the pairing symmetries of the Hubbard Hamiltonian with next-nearest-neighbor (NNN) hopping t' on square lattices. By varying the value of t', we find that the d-wave pairing is suppressed by the onset of t', while the p + ip-wave pairing tends to emerge for low electron density and t' around -0.7. Together with the calculation of the anti-ferromagnetic and ferromagnetic spin correlation function, we explore the relationship between anti-ferromagnetic order and the d-wave pairing symmetry, and the relationship between ferromagnetic order and the p + ip-wave pairing symmetry. Our results may be useful for the exploration of the mechanism of the electron pairing symmetries, and for the realization of the exotic p + ip-wave superconductivity.

DOI
10.1088/1361-648x/abd33a

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Using the finite-temperature determinant quantum Monte Carlo (DQMC) algorithm, we study the pairing symmetries of the Hubbard Hamiltonian with next-nearest-neighbor (NNN) hopping t' on square lattices. By varying the value of t', we find that the d-wave pairing is suppressed by the onset of t', while the p + ip-wave pairing tends to emerge for low electron density and t' around -0.7. Together with the calculation of the anti-ferromagnetic and ferromagnetic spin correlation function, we explore the relationship between anti-ferromagnetic order and the d-wave pairing symmetry, and the relationship between ferromagnetic order and the p + ip-wave pairing symmetry. Our results may be useful for the exploration of the mechanism of the electron pairing symmetries, and for the realization of the exotic p + ip-wave superconductivity.
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, ferromagnetism, 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.
We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (t2{t}_{2}t2​) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if t2{t}_{2}t2​\ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.
Determinant quantum Monte Carlo (DQMC) is used to determine the pairing and magnetic response for a Hubbard model built up from four-site clusters---a two-dimensional square lattice consisting of elemental $2\ifmmode\times\else\texttimes\fi{}2$ plaquettes with hopping ttt and on-site repulsion UUU coupled by an interplaquette hopping ${t}^{\ensuremath{'}}\ensuremath{\le}t$. Superconductivity in this geometry has previously been studied by a variety of analytic and numeric methods, with differing conclusions concerning whether the pairing correlations and transition temperature are raised near half filling by the inhomogeneous hopping or not. For U/t=4U/t=4U/t=4, DQMC indicates an optimal ${t}^{\ensuremath{'}}/t\ensuremath{\approx}0.4$ at which the pairing vertex is most attractive. The optimal ${t}^{\ensuremath{'}}/t$ increases with U/tU/tU/t. We then contrast our results for this plaquette model with a hamiltonian which instead involves a regular pattern of site energies whose large site energy limit is the three-band CuO2{\mathrm{CuO}}_{2}CuO2​ model; we show that there the inhomogeneity rapidly, and monotonically, suppresses pairing.
To clarify the effects of next-nearest-neighbor electron hopping ${t}^{\ensuremath{'}}$ on the spin, charge, and pairing correlations, we studied the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ Hubbard model using the constrained-path Monte Carlo technique. We found that a negative ${t}^{\ensuremath{'}}$ suppresses the spin fluctuations near the Brillouin zone point $M=(\ensuremath{\pi},\ensuremath{\pi})$ and the charge fluctuations near the point $X=(\ensuremath{\pi},0),$ while a positive ${t}^{\ensuremath{'}}$ has the opposite effect. For a negative ${t}^{\ensuremath{'}},$ we also found that the long-range vertex contributions to the pairing correlations are heavily suppressed, but the local contributions are enhanced. Over the entire hole doping range studied, the d-wave long-range vertex contributions to pairing dominated those of the extended s-wave vertex contributions. In general, we did not find any tendency for these correlations to grow with increasing system size. We also did a limited number of simulations with an attractive nearest neighbor density-density interaction added to the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ model. It enhanced the long-range vertex contributions to the pairing correlations only for weak values of U.
A combination of analytical approaches and quantum Monte Carlo simulations is used to study both magnetic and pairing correlations for a version of the Hubbard model that includes second-neighbor hopping ${\mathit{t}}^{\ensuremath{'}}$=-0.35t as a model for high-temperature superconductors. Magnetic properties are analyzed using the two-particle self-consistent approach. The maximum in magnetic susceptibility as a function of doping appears both at finite ${\mathit{t}}^{\ensuremath{'}}$ and at ${\mathit{t}}^{\ensuremath{'}}$=0 but for two totally different physical reasons. When ${\mathit{t}}^{\ensuremath{'}}$=0, it is induced by antiferromagnetic correlations while at ${\mathit{t}}^{\ensuremath{'}}$=-0.35t it is a band structure effect amplified by interactions. Finally, pairing fluctuations are compared with T-matrix results to disentangle the effects of van Hove singularity and of nesting on superconducting correlations. The addition of antiferromagnetic fluctuations increases slightly the d-wave superconducting correlations despite the presence of a van Hove singularity which tends to decrease them in the repulsive model. Some aspects of the phase diagram and some subtleties of finite-size scaling in Monte Carlo simulations, such as inverted finite-size dependence, are also discussed.
The search for superconductivity with unconventional pairing symmetry has been a central focus in the study of strongly correlated electron systems. In this work, we report a numerically exact study of the spin-triplet pairing in a two-dimensional Hubbard model with repulsive interactions, employing the determinant quantum Monte Carlo method. The model includes next-nearest-neighbor and third-nearest-neighbor hopping terms and maintains spin balance. By tuning the fermion filling close to a type-II Van Hove singularity (VHS) in the model, we numerically investigate the ordering tendencies of several possible pairing channels with different symmetries. Our numerical results provide clear evidence of the spin-triplet ppp-wave pairing instability approaching low temperatures, as revealed by the vertex contribution to the pairing susceptibility. This signature becomes increasingly pronounced as the interaction strength increases in the weak to intermediate regime. We further find that, near the type-II VHS, the dominant spin-spin correlations in the system are ferromagnetic, suggesting its close relation to the spin-triplet pairing instability. Our findings offer a reliable approach to realize spin-triplet ppp-wave superfluidity in the repulsive Hubbard model from an unbiased numerical perspective.