In a Fermi system near or at half-filling, a specific superconducting pairing channel, if not explicitly included in the Hamiltonian, can be boosted by suppressing a competing pairing channel; this is exemplified by the enhancement of extended
s-wave correlations upon suppressing
s-wave Cooper pairing. This phenomenon, originally found by the use of generalized uncertainty relations is referred to as order by projection. The case of zero on-site Coulomb interaction in the thermodynamic limit, confirms this mechanism through the analytical solution. In this study, we go further and systematically investigate this mechanism for a strongly correlated fermionic Hubbard model, now with finite on-site interaction, on a square lattice with an extended set of hopping parameters. We explore the behaviors of different pairing channels when one of them is suppressed, utilizing density matrix renormalization group calculations. Our findings provide numerical evidence supporting the existence of order by projection in the strongly correlated system we studied. We also investigate the effect of the strength of Hubbard
U, next-nearest-neighbor ${t}^{\ensuremath{'}}$, hole doping, as well as finite-size scaling approaching the thermodynamic limit.
Series for the reduced susceptibility $\overline{\ensuremath{\chi}}$, the reduced specific heat
CH, and second moment ${\ensuremath{\mu}}_{2}$ of the static correlation function for the three-dimensional
S=21 Ising model on both the simple cubic (sc) and face-centered cubic (fcc) lattices with different coupling strengths in different lattice directions have been analyzed to determine the crossover exponent $\ensuremath{\phi}$ describing the behavior of the critical temperature as a function of the anisotropy parameter
R in the Hamiltonian $\mathcal{H}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j}\ensuremath{\equiv}\ensuremath{-}{J}_{\mathrm{xy}}(\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}+R\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j})$. Here
si=+1, the first sum is over all nearest-neighbor pairs in the
xy plane, and the second sum is over all pairs coupled in the
z direction. The constant gap exponent we obtain for successive derivatives of $\overline{\ensuremath{\chi}}$ and
CH with respect to
R confirms the exponent predictions of scaling in the parameter
R for thermodynamic functions, while the results of the ${\ensuremath{\mu}}_{2}$ series confirm the exponent predictions of scaling with respect to
R for the two-spin correlation function. Our results agree with the predictions for $\ensuremath{\phi}$ of Abe and Suzuki, and also with rigorous relations satisfied by the exponents describing the derivatives of the various functions. Our results do not agree with previously published results, which are based on an analysis of only the susceptibility on only the sc lattice.
An ideal relativistic Bose-gas model with pair production is solved exactly in
d dimensions. The critical behavior is shown to be the same as the usual nonrelativistic Bose gas. Scaling functions (including nonuniversal parameters) for the equation of state and the correlation functions are obtained in a closed form. The scale-factor universality is verified by the formation of the expected universal ratios. Various limiting cases of the model depending on the relative magnitudes of the three basic length scales, i.e., Compton wave-length, thermal wavelength, and interparticle distance are discussed. In these limiting cases, previously known results are recovered, wherever these are available.