Finite Size Scaling in Quantum Mechanics

Winton Moy, Marcelo A. Carignano, Sabre Kais

We combined the finite-size scaling method with the finite element method to provide a systematic procedure for obtaining quantum critical parameters for a quantum system. We present results for the Yukawa potential solved with the finite element approach. The finite-size scaling approach was then used to find the critical parameters of the system. The critical values lambda c, alpha, and nu were found to be 0.83990345, 2.0002, and 1.002, respectively, for l = 0. These results compare well with the theoretically exact values for alpha and nu and with the best numerical estimations for lambda c. The finite element method is general and can be extended to larger systems.

Dhurjati Prasad Datta

Aadi Shenoy

Tatsuhiko N. Ikeda, Yu Watanabe, Masahito Ueda

By calculating correlation functions for the Lieb-Liniger model based on the algebraic Bethe ansatz method, we conduct a finite-size scaling analysis of the eigenstate thermalization hypothesis (ETH), which is considered to be a possible mechanism of thermalization in isolated quantum systems. We find that the ETH in a weak sense holds in the thermodynamic limit even for an integrable system, although it does not hold in the strong sense. Based on the result of the finite-size scaling analysis, we compare the contribution of the weak ETH to thermalization with that of yet another thermalization mechanism, the typicality, and show that the former gives only a logarithmic correction to the latter.

Winton Moy

The study of phase transitions and critical phenomena is an area of great interest in science. Liquid to gas, ferromagnetic to paramagnetic, fluid to superfluid, insulator to conductor are a few examples of physical systems exhibiting phase transition and critical phenomena. Classical phase transitions have thermal fluctuations as the main driving force for the transition. In statistical mechanics phase transitions are associated with singularities in the free energy. These singularites only occur in the thermodynamic limit where the volume (V) and particles (N) go to infinity with the density held constant (N/V). By examining the partition function for a finite system, the function is a sum of a set number of terms. The partition function then would be analytical. It is only when an infinite number of terms are added is there a singularity in the partition function. The subject of Finite Size Scaling theory is the relation of the phenomena in a finite systems to the true phase transition of an infinite system. Finite Size Scaling theory provides a numerical method to obtain accurate results for infinite systems simply by studying corresponding small systems. The focus of this thesis is on transitions of a different nature. Quantum phase transitions (transitions that occur at absolute zero) have Heisenberg’s uncertainty principle as the driving force for the transition. In quantum mechanics, the finite size problem occurs when looking at the critical behaviour of the Hamiltonian as a function of a set of parameters. The interest is where the bound states energies become non-analytical. The size of the system is related to the number of elements in a complete basis set used to expand the exact wave function. In this thesis we present the use of the Finite Element Method (a numerical technique commonly used in engineering problems to solve partial differential equations or integral equations). Previous finite size scaling studies in quantum mechanics employed a basis set expansion of the wave function and the problem is solved variationally. Slater type basis sets were used for this purpose. In the effort to apply finite size scaling calculations for larger systems Gaussian type functions were employed however the a larger number of Gaussian functions are needed to obtain accurate results. The use of standard electronic structure packages was also explored, however, the lack of consistently increasing size for basis sets limits the application for finite size scaling purposes. We first apply the finite element element method (FEM) to solve for the short range Yukawa potential and obtained the critical parameters for the model potential. We develop the framework necessary to use FEM for finite size scaling. As mentioned before the 'size' of the system is the number of basis functions needed in the expansion of the wave function. However, using FEM the size of the system is the number of elements used. We then studied the Hulthen potential in comparison to the basis set results and also compare the results to the analytical solutions for the both methods. Finally we explore the use of FEM in combination with Hartree-Fock theory to solve 2 to 4 electron systems. The use of Hartree-Fock theory allows for the implementation of finite size scaling to larger systems. We used the FEM Hartree-Fock to calculate for the critical parameters of 2 to 4 electron systems. Although the results are not exact (as to be expected due to the use of Hartree-Fock theory) we get an estimate to the critical points of these systems. FEM can be combined with other electronic structure theories, which would allow for the study of larger systems exhibiting critical phenomena.

M. A. Contínentino