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arXiv:1304.7433 · Physical Review E

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

Fahhad H. Alharbi, Sabre Kais

Published Apr 30, 2013• 10 citations
arXivHTMLPDF

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.

DOI
10.1103/physreve.87.043308

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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.
Finite size scaling for the Schrödinger equation is a systematic approach to calculate the quantum critical parameters for a given Hamiltonian. This approach has been shown to give very accurate results for critical parameters by using a systematic expansion with global basis-type functions. Recently, the finite-element method was shown to be a powerful numerical method for ab initio electronic-structure calculations with a variable real-space resolution. In this work, we demonstrate how to obtain quantum critical parameters by combining the finite-element method (FEM) with finite size scaling (FSS) using different ab initio approximations and exact formulations. The critical parameters could be atomic nuclear charges, internuclear distances, electron density, disorder, lattice structure, and external fields for stability of atomic, molecular systems and quantum phase transitions of extended systems. To illustrate the effectiveness of this approach we provide detailed calculations of applying FEM to approximate solutions for the two-electron atom with varying nuclear charge; these include Hartree-Fock, local density approximation, and an "exact" formulation using FEM. We then use the FSS approach to determine its critical nuclear charge for stability; here, the size of the system is related to the number of elements used in the calculations. Results prove to be in good agreement with previous Slater-basis set calculations and demonstrate that it is possible to combine finite size scaling with the finite-element method by using ab initio calculations to obtain quantum critical parameters. The combined approach provides a promising first-principles approach to describe quantum phase transitions for materials and extended systems.
We compare two methods of obtaining critical parameters for a quantum Hamiltonian using a finite size scaling approach. A finite element and basis set method were used in conjunction with the finite size scaling to obtain the critical parameters for the Hulthen potential. The critical parameters obtained analytically were the coupling constant λc=12, the critical exponents for the energy α=2 and for the “correlation length” ν=1. The extrapolated results for finite size scaling with the basis set method are λc=0.49999, α=1.9960, and ν=0.99910. The results for the finite element solutions are λc=0.50184, α=1.99993, and ν=1.00079 for the linear interpolation and λc=0.50000, α=2.00011, and ν=1.00032 for the Hermite interpolation. The results for each method compare very well with the analytical results obtained for the Hulthen potential. However, the finite element method is easier to implement and may be combined with ab initio and density functional theory to obtain quantum critical parameters for more complex systems.
We apply the theory of Finite-size scaling (FSS) to an atomic and a non-equilibrium system in order to extract critical parameters. In atomic systems, we look at the energy dependence on the binding charge near threshold between bound and free states, where we seek the critical nuclear charge for stability. We use different ab initio methods, such as Hartree-Fock, Density Functional Theory, and exact formulations implemented numerically with the finite-element method (FEM). Using Finite-size scaling formalism, where in this case the size of the system is related to the number of elements used in the basis expansion of the wavefunction, we predict critical parameters in the large basis limit. Results prove to be in good agreement with previous Slater-basis set calculations and demonstrate that this combined approach provides a promising first-principles approach to describe quantum phase transitions for materials and extended systems. In the second part we look at non-equilibrium one-dimensional model known as the raise and peel model describing a growing surface which grows locally and has non-local desorption. For a specific values of adsorption ( ua) and desorption (ud) the model shows interesting features. At ua = ud, the model is described by a conformal field theory (with conformal charge c = 0) and its stationary probability can be mapped to the ground state of a quantum chain and can also be related a two dimensional statistical model. For ua ≥ ud, the model shows a scale invariant phase in the avalanche distribution. In this work we study the surface dynamics by looking at avalanche distributions using FSS formalism and explore the effect of changing the boundary conditions of the model. The model shows the same universality for the cases with and with our the wall for an odd number of tiles removed, but we find a new exponent in the presence of a wall for an even number of avalanches released. We provide new conjecture for the probability distribution of avalanches with a wall obtained by using exact diagonalization of small lattices and Monte-Carlo simulations.
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.
In this review we discuss quantum phase transitions and the mapping between symmetry breaking of electronic structure configurations at the large-dimension limit and mean-field theory of phase transitions. We show that the finite size scaling method can be used for the calculations of the critical parameters of the few-body Schrodinger equation. In this approach, the finite size corresponds to the number of elements in a complete basis set used to expand the exact eigenfunction of a given Hamiltonian. The critical parameters such as the critical nuclear charges will be used to explain and predict the stability of atomic and molecular negative ions. For N-electron atoms with 2 N 86, results show that, at most, only one electron can be added to a free atom in the gas phase. However, doubly charged atomic negative ions might exist in a strong magnetic field.