Quantum criticality analysis by finite-size scaling and exponential basis sets

Abstract

We combine the finite-size scaling method with the mesh-free spectral method to calculate quantum critical parameters for a given Hamiltonian. The basic idea is to expand the exact wave function in a finite exponential basis set and extrapolate the information about system criticality from a finite basis to the infinite basis set limit. The used exponential basis set, though chosen intuitively, allows handling a very wide range of exponential decay rates and calculating multiple eigenvalues simultaneously. As a benchmark system to illustrate the combined approach, we choose the Hulthen potential. The results show that the method is very accurate and converges faster when compared with other basis functions. The approach is general and can be extended to examine near-threshold phenomena for atomic and molecular systems based on even-tempered exponential and Gaussian basis functions.

Kinetic energy density for orbital‐free density functional calculations by axiomatic approach

“…This class of potentials possesses the functional form: where a i , b i , and c i are generated randomly, and obey the following constraints: , and . An efficient spectral method is used to solve for the states and hence the densities with an accuracy greater than for the exact noninteracting kinetic energy ( T s ) …”

“…An efficient spectral method is used to solve for the states and hence the densities with an accuracy greater than 10 212 for the exact noninteracting kinetic energy (T s ). [34,35] 3.1 | T½q Calculated using the density numerically obtained through solving the Schr€ odinger equation…”

Scaling behaviour of the Hellmann potential with different strength parameters

“…In an atomic system, in comparison to thermal physics, one can define a rigors mapping [3,4] and examine the criticality of the bound state energy for nl-state,E nl , as a free energy, and the strength or screening parameters of a specific potential, D, as the temperature, T, in a thermal system. Consequently, we can write [5] E nl (D) = −γ nl D − D c,nl β (2) where γ nl is a constant and D c,nl is the critical parameter for nl-state. At D c,nl , the nl-bound state will simply disappear and re-appear as a resonance [6].…”

Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions

“…One of the complexities in applying spectral methods (SM) to the TF equation is the fact that it is defined on a semiinfinite domain. Significant research has been conducted on applying SM on infinite and semi-infinite domains [7][8][9][10][11][12][13][14][15]. This has been achieved by implementing a wide range of approaches varying from using suitable basis sets and truncating the numerical window to forcing size scaling.…”

“…In our approach to solve the TF equation, we apply the spectral method based on the exponential basis set. This basis set and its polynomial version have been recently used for solving several differential equations on semiinfinite domains [13][14][15]. The use of a similar basis set was initially presented in the 1970s by Raffenetti, Bardo, and Ruedenberg [25][26][27] for self-consistent field wave functions.…”