Finite-size scaling: new results

Derek Larson, Helmut G. Katzgraber, M A Moore +1 more

We study the existence of a line of transitions of an Ising spin glass in a magnetic field---known as the de Almeida-Thouless line---using one-dimensional power-law diluted Ising spin-glass models. We choose the power-law exponent to have values that approximately correspond to three- and four-dimensional nearest-neighbor systems and perform a detailed finite-size scaling analysis of the data for large linear system sizes, using both a new approach proposed recently [Phys. Rev. Lett. 103, 267201 (2009)], as well as traditional approaches. Our results for the model corresponding to a three-dimensional system are consistent with there being no de Almeida-Thouless line, although the new finite-size scaling approach does not rule one out. For the model corresponding to four space dimensions, the new and traditional finite-size scaling analyses give conflicting results, indicating the need for a better understanding of finite-size scaling of spin glasses in a magnetic field.

Werner Fettes, I. Morgenstern

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 dx2-y2-wave symmetry).

E. Brézin

Etienne Hofstetter, Michael Schreiber

Based on finite-size scaling ideas a new approach to analyse the singularities associated with second-order phase transitions is presented. Under certain requirements, the method allows to derive an upper limit for the critical exponent from finite-size data that describe the system far away from the transition. By extrapolation the critical exponent ν is determined. As an example, the approach is applied to the metal-insulator transition in disordered systems. The result ν = 1.35 ± 0.15 is independent of the employed random distribution. With field theoretical arguments we discuss and generalize this result. Our results are consistent with the assumption that one-parameter scaling and universality are valid in the 3D Anderson model.

A. P. Young