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arXiv:cond-mat/9707277 · Physical review. B, Condensed matter

Enhancement of the dx2−y2 pairing correlation in the two-dimensional Hubbard model: A quantum Monte Carlo study

Kazuhiko Kuroki, Hideo Aoki

Published Dec 1, 1997• 30 citations
arXivHTMLPDF

Abstract

Quantum Monte Carlo is used to investigate the possibility of ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductivity in the two-dimensional repulsive Hubbard model. A small energy scale relevant to possible pairing requires a care [i.e., sufficiently small level separation between the $k$ points $(\ensuremath{\delta}k,\ensuremath{\pi}\ensuremath{-}\ensuremath{\delta}{k}^{\ensuremath{'}})$ and $(\ensuremath{\pi}\ensuremath{-}\ensuremath{\delta}{k}^{\ensuremath{'}\ensuremath{'}},\ensuremath{\delta}{k}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}})$ with small $\ensuremath{\delta}k$'s] to detect enhanced correlations in finite-size studies, as motivated from a previous study on Hubbard ladders. Our calculation indeed detects long-tailed enhancements in the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing correlation when the system is near, but not exactly at, half-filling.

DOI
10.1103/physrevb.56.r14287

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Quantum Monte Carlo is used to investigate the possibility of dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ superconductivity in the two-dimensional repulsive Hubbard model. A small energy scale relevant to possible pairing requires a care [i.e., sufficiently small level separation between the kkk points (\ensuremathδk,\ensuremathπ\ensuremath−\ensuremathδk\ensuremath′)(\ensuremath{\delta}k,\ensuremath{\pi}\ensuremath{-}\ensuremath{\delta}{k}^{\ensuremath{'}})(\ensuremathδk,\ensuremathπ\ensuremath−\ensuremathδk\ensuremath′) and (\ensuremathπ\ensuremath−\ensuremathδk\ensuremath′\ensuremath′,\ensuremathδk\ensuremath′\ensuremath′\ensuremath′)(\ensuremath{\pi}\ensuremath{-}\ensuremath{\delta}{k}^{\ensuremath{'}\ensuremath{'}},\ensuremath{\delta}{k}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}})(\ensuremathπ\ensuremath−\ensuremathδk\ensuremath′\ensuremath′,\ensuremathδk\ensuremath′\ensuremath′\ensuremath′) with small \ensuremathδk\ensuremath{\delta}k\ensuremathδk's] to detect enhanced correlations in finite-size studies, as motivated from a previous study on Hubbard ladders. Our calculation indeed detects long-tailed enhancements in the dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ pairing correlation when the system is near, but not exactly at, half-filling.
In this study we have computed the pair correlation functions in the two-dimensional Hubbard model using a quantum Monte Carlo method. We employ a new diagonalization algorithm in the quantum Monte Carlo method which is free from the negative sign problem. We show that the d-wave pairing correlation function is indeed enhanced slightly for the positive on-site Coulomb interaction U when doping away from the half-filling. When the system size becomes large, the pair correlation function P d increases for U > 0 compared to the non-interacting case, while P d is suppressed for U > 0 when the system size is small. The enhancement ratio P d [ U ]/ P d [ U = 0] will give a criterion on the existence of superconductivity. The ratio P d [ U ]/ P d [ U = 0] increases almost linearly ∝ L when the system size L × L is increased. This increase is a good indication of the existence of a superconducting phase in the two-dimensional Hubbard model. There is, however, no enhancement of pair correlation functions in the half-filled case, which indicates the absence of superconductivity without hole doping.
The quantum Monte Carlo method is used to reexamine superconductivity in the single-band Hubbard model in two dimensions. Instead of the conventional pairing, we consider a ``correlated pairing,'' 〈c-${\mathrm{tilde}}_{\mathrm{i}\mathrm{\ensuremath{\uparrow}}}$c-${\mathrm{tilde}}_{{\mathrm{i}}^{|\mathrm{I}\mathrm{H}}\mathrm{\ensuremath{\downarrow}}}$〉 with c-${\mathrm{tilde}}_{\mathrm{i}\mathrm{\ensuremath{\sigma}}}$\ensuremath{\equiv}${\mathrm{c}}_{\mathrm{i}\mathrm{\ensuremath{\sigma}}}$(1-${\mathrm{n}}_{\mathrm{i}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$), which is inferred from the t-J model, the strong-coupling limit of the Hubbard model. The pairing in the d-wave channel is found to possess both a divergence like 1/T in the pairing susceptibility and a growth of the ground-state pairing correlation with sample size, indicating an off-diagonal long-range order near (but not exactly at) half-filling.
In the ground state of the doped two-leg Hubbard ladder there are power-law decaying ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-type pairing correlations. It is important to know the strength and the temperature scale of these correlations. For this purpose, we have performed determinantal quantum Monte Carlo (QMC) calculations of the reducible particle-particle interaction in the Hubbard ladder. In this paper, we report on these calculations and show that, at sufficiently low temperatures, resonant particle-particle scattering takes place in the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing channel for certain values of the model parameters. The QMC data presented here indicate that the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing correlations are strong in the Hubbard ladder.
An extensive quantum Monte Carlo calculation is performed for the two-leg Hubbard ladder model to clarify whether the singlet pairing correlation decays slowly, which is predicted from the weak-coupling theory but controversial from numerical studies. Our result suggests that the discreteness of energy levels in finite systems affects the correlation enormously, where the enhanced pairing correlation is indeed detected if we make the energy levels of the bonding and antibonding bands lie close to each other at the Fermi level to mimic the thermodynamic limit.
Superconducting correlation of the Hubbard model on a square lattice is reinvestigated by quantum Monte Carlo method. Results from more than one order of magnitude larger Monte Carlo steps than before are presented to discuss the convergence of the data to the ground state. By examining the dependence on the choice of initial trial states as well as on the Trotter number in detail, the superconducting correlation in the ground state is reliably obtained up to 12×12 lattice away from the half-filling and up to 16×16 lattice at the half-filling, which substantially improve former results near the half-filling. Various channels of pairing in the ground state show no system size dependence and remain quite short ranged at any filling including the region close to the half-filling. Numerical data show further evidence that neither extended s - nor d -wave superconducting state is possible in the two-dimensional Hubbard model.
To address the issue of electron correlation driven superconductivity in graphene, we perform a systematic quantum Monte Carlo study of the pairing correlation in the $t\ensuremath{-}U\ensuremath{-}V$ Hubbard model on a honeycomb lattice. For V=0V=0V=0 and close to half filling, we find that pairing with d+idd+idd+id (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}^{\ensuremath{'}}$ in its specific form) symmetry dominates pairing with extended-sss symmetry. However, as the system size or the on-site Coulomb interaction increases, the long-range part of the d+idd+idd+id pairing correlation decreases and tends to vanish in the thermodynamic limit. An inclusion of nearest-neighbor interaction VVV, either repulsive or attractive, has a small effect on the extended-sss pairing correlation, but strongly suppresses the d+idd+idd+id pairing correlation.