A Monte Carlo world-line algorithm for a Cu
O2 plane based on a breakup of the plane into five-site cells, Cu
O4 has been suggested. The algorithm has fast convergency and gives the possibility of investigating the superconducting state symmetry. The calculation of pairing correlation functions was made with an advent of additional imaginary-time slices to the Monte Carlo scheme. The characteristics of a two-dimensional Cu-O cluster with number of sites
Na=768 (16\ifmmode\times\else\texttimes\fi{}16 Cu
O2 cells) were calculated. It has been shown within the chosen interval of parameters of the Emery Hamiltonian (
Ud=6t, $\ensuremath{\varepsilon}=1\ensuremath{-}3t$,
Up=V=0), temperatures (
T> 0.125t), and carrier concentrations (
0.7< x< 1.5) that (i) the long-range off-diagonal order associated with the superconducting state was not observed in the thermodynamic limit in any pairing channels and (ii) a tendency to divergence in
s∗ channels and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ channels, which is noted as the temperature decreases, is due to the antiferromagnetic ordering rather than the superconducting pairing. If the carrier concentration
x is equal to 1.0 (undoped dielectric state) the pairing correlations have maximum amplitude and demonstrate clear antiferromagnetic ordering in the copper sublattice, and the characteristic correlation length is close to the earlier-reported antiferromagnetic length.
A novel minimal-size pairing operator ${\mathrm{\ensuremath{\Delta}}}_{D0}^{\ifmmode\dagger\else\textdagger\fi{}}$ with
d-wave symmetry in
CuO2 planes is introduced. This pairing operator creates on-site Cooper pairs at the four oxygens that surround a copper atom. Via the time evolution of ${\mathrm{\ensuremath{\Delta}}}_{D0}^{\ifmmode\dagger\else\textdagger\fi{}}$, an additional interorbital pairing operator ${\mathrm{\ensuremath{\Delta}}}_{Dpd}^{\ifmmode\dagger\else\textdagger\fi{}}$ with
d-wave symmetry is generated that pairs fermions located in a Cu and its four surrounding O atoms. The subsequent time evolution of ${\mathrm{\ensuremath{\Delta}}}_{Dpd}^{\ifmmode\dagger\else\textdagger\fi{}}$ generates an intraorbital
d-wave pairing operator ${\mathrm{\ensuremath{\Delta}}}_{Dpp}^{\ifmmode\dagger\else\textdagger\fi{}}$ involving the four O atoms that surround a Cu atom, as well as the
d-wave operator ${\mathrm{\ensuremath{\Delta}}}_{D}^{\ifmmode\dagger\else\textdagger\fi{}}$ traditionally used in single-band models for cuprates. Because we recover the larger size operators extensively used in the three-orbital Hubbard model, we suggest that long-range order using the canonical extended operators occurs together with long-range order in the new minimal operators. However, our minimal
d-wave operators could be more practical to study
d-wave superconductivity because in the finite-size relatively small systems accessible to computational techniques it is easier to observe long-range order using local operators. Moreover, an effective model with the usual tight-binding hopping of the
CuO2 planes supplemented by an attractive potential
V in the
d-wave channel is introduced. Using mean-field techniques, we show that a paired ground state is stabilized for any finite value of
V. We observed that the values of
V that lead to gap sizes similar to those in the cuprates are smaller for
d-wave pairing operators that include Cu
d orbitals than those that include only
p orbitals. In all cases the gap that opens in the spectrum has standard
d-wave symmetry. Finally, a simpler effective model is introduced to study the phenomenology of multiorbital
d-wave superconductors, similar to how the negative-
U Hubbard model is used for properties of
s-wave superconductors.
The Quantum Monte Carlo simulation of the two-dimensional Emery model of the CuO2 plane of hight Tc superconductors were performed. The method based on the direct-space proposed by Suzuki and Hirsch was used. Contrary to the method based on the Hubbard-Stratonovich transformation, the states generated by this method are basis states in occupation number representation, i. e. configurations of fermions can be observed on real two-dimensional array.Energy and specific heat were computed for different dopings. Specific heat curves show peaks at low temperature which could be assigned to electronic transitions. Quantity similar to current-current correlation function were computed. The static electric conductivity curves obtained by this way show metal-insulator transitions and two different metallic behaviours. On the direct-space states generated at low temperature and zero doping, the fermions form antiferromagnetic loops while they form antiferromagnetic chains for other dopings. The loops seem to appear when the conductivity becomes zero while yhe conductivity increases with the numbers of chains but superconductivity is not unambiguously evident.
In a recent Letter, Kuroki and Aoki [Phys. Rev. Lett. 76, 440 (1996)] presented quantum Monte-Carlo (QMC) results for pairing correlations in the three-band Hubbard model, which describes the Cu-d_{x^2-y^2} and O-p_{x,y} orbitals present in the CuO_2 planes of high-T_c materials. In this comment we argue that (i) the used parameter set is not appropriate for the description of high-T_c materials since it does not satisfy the minimal requirement of a charge-transfer gap at half-filling, and (ii) the observed increase in the d_{x^2-y^2} channel is dominantly produced by the pair-field correlations without the vertex part. Hence, the claim of evidence of ODLRO is not justified.
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.
We utilize numerical linked-cluster expansions (NLCEs) and the determinantal quantum Monte Carlo algorithm to study pairing correlations in the square-lattice Hubbard model. To benchmark the NLCE, we first locate the finite-temperature phase transition of the attractive model to a superconducting state away from half filling. We then explore the superconducting properties of the repulsive model for the
d-wave and extended
s-wave pairing symmetries. The pairing structure factor shows a strong tendency to
d-wave pairing and peaks at an interaction strength comparable to the bandwidth. The extended
s-wave structure factor and correlation length are larger at higher temperatures but clearly saturate as temperature is lowered, whereas the
d-wave counterparts, which start off lower at high temperatures, continue to rise near half filling. This rise is even more dramatic in the
d-wave susceptibility. The convergence of NLCEs breaks down as the susceptibilities and correlation lengths become large, so we are unable to determine the onset of long-range order. However, our results extend the conclusion, previously restricted to only magnetic and charge correlations, that NLCEs offer a unique window into pairing in the Hubbard model at strong coupling.