Finite Size Scaling in the tt'-Hubbard Model and in BCS-Reduced Hubbard Models

Abstract

With the projector quantum Monte Carlo algorithm and the stochastic diagonalization it is possible to calculate the ground state of the Hubbard model for small finite clusters. Nevertheless the usual finite size scaling of the Hubbard model has problems of deducing the behavior of the infinite system correctly from the numerical data of small system sizes. Therefore we study the finite size scaling of the superconducting correlation functions in superconducting BCS-reduced Hubbard models to analyze the finite size behavior in small finite clusters. The ground state of the BCS-reduced Hubbard models is calculated with the stochastic diagonalization without any approximations. As result of these analyses we propose a new finite size scaling ansatz for the Hubbard model, which is able describe the finite size effects in a consistent way taking the corrections to scaling into account, which are dominant for weak interaction strength and small clusters. With this new finite size scaling ansatz it is possible to give evidence for superconductivity for all interaction strengths for both the attractive tt'-Hubbard model (with s-wave symmetry) and the repulsive tt'-Hubbard model (with d x 2 −y 2 -wave symmetry). Int. J. Mod. Phys. C 1998.09:943-985. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/03/15. For personal use only.

Quantum Monte Carlo study of Spin, Charge, and Pairing correlations in thet−t′−UHubbard model

“…Except for the N e ϭ30 case, decreasing tЈ suppresses the d-wave pairing. On the 4ϫ4 lattice, our past 20 and present results plus previous exact-diagonalization data 14 show a similar behavior to the N e ϭ30 case. For weak coupling (Uϭ1.0Ϫ2.0) Husslein et al 18 and Kuroki and Aoki 23 argue that their QMC results support a nonzero tЈ enhancing d-wave pairing.…”

“…Since the work of Lin and Hirsch a variety of mean-field calculations, 7-11 exact diagonalization studies, [12][13][14] and quantum Monte Carlo simulations were performed. 14 -24 The mean-field calculations suggested a number of phases as a function of doping, the ratio tЈ/t, and the relative strength of the on-site Coulomb repulsion U/t.…”

Glass Model, Hubbard Model and High-Temperature Superconductivity

“…Given the observation of plateaus supporting the existence of d-wave pairing in finite clusters, what are the implications for bulk superconductivity? Up to now it was thought necessary to scale the finite cluster calculations to infinite size in order to answer this question, which unfortunately presents difficulties as corrections to scaling are tremendously large compared to classical systems [33,26,34]. However in view of the new stripe phase picture of the crystalline High-T c material, the simulated cluster can be reinterpreted as a single finite length stripe as seen in STM data [13], with typical dimensions of roughly 12×4 unit cells, i.e.…”

“…However in view of the new stripe phase picture of the crystalline High-T c material, the simulated cluster can be reinterpreted as a single finite length stripe as seen in STM data [13], with typical dimensions of roughly 12×4 unit cells, i.e. even smaller than our largest system sizes (16×16 in [34]). Scaling to an infinite system is inappropriate as the PQMC cluster calculation is now descriptive of the actual quantum systems present in the crystal (although they are treated as totally isolated).…”

“…The qualitative behavior of our results is not influenced by the choice of r c as long as we suppress the short range correlations (i. e. r c ≥ 1.9). Due to finite size effects (see [35,34] or in more detail [26]) the shape of the curves in respect to the filling in figure 5 is influenced by n and L.…”