Numerical approaches to complex quantum, semiclassical and classical systems [Elektronische Ressource] / vorgelegt von Gerald Schubert

Sujets

• Physik

Informations

• Publié par: ernst-moritz-arndt-universitat_greifswald
• Publié le: 01 janvier 2008
• Nombre de lectures: 11
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

Numerical approaches to complex
quantum, semiclassical and classical
systems
I n a u g u r a l d i s s e r t a t i o n
zur
Erlangung des akademischen Grades
doctor rerum naturalium (Dr. rer. nat.)
an der Mathematisch-Naturwissenschaftlichen Fakultat
der
Ernst-Moritz-Arndt-Universitat Greifswald
vorgelegt von
Gerald Schubert
geboren am 01.02.1976
in Rehau
Greifswald, 03.11.2008Dekan: Prof. Dr. Klaus Fesser
1. Gutachter: Prof. Dr. Holger Fehske
2. Gutachter: Prof. Dr. Vladimir Filinov
Tag der Promotion: 15.12.2008IVContents
Expose 1
1 Quantum percolation in disordered structures 5
1.1 Local distribution approach . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 Conceptual background . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 Calculation of the local density of states . . . . . . . . . . . . . . . 9
1.1.3 Illustration of the method: Anderson localisation in 3D . . . . . . 11
1.2 Localisation e ects in quantum percolation . . . . . . . . . . . . . . . . . 13
1.2.1 3D site percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2 2D site p . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Di usion and localisation in quantum random resistor networks . . . . . . 26
1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Optical absorption and activated transport in polaronic systems 35
2.1 Holstein model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Calculation of the optical conductivity via kernel polynomial expansion . 36
2.3 Numerical results and discussion . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Comparative study of semiclassical approaches to quantum dynamics 47
3.1 Computational schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.1 Chebyshev expansion of the time evolution operator . . . . . . . . 48
3.1.2 Linearised semiclassical propagator method . . . . . . . . . . . . . 50
3.1.3 Wigner-Moyal approach . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.4 Tomographich . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.1 Discussion of the time evolution . . . . . . . . . . . . . . . . . . . 62
3.2.2 Details of implementation . . . . . . . . . . . . . . . . . . . . . . . 68
3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4 Dynamics of complex classical many-body systems 75
4.1 Dust particles as micro-probes for plasma sheaths . . . . . . . . . . . . . . 77
4.1.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1.2 Structure formation above the adaptive electrode . . . . . . . . . . 79
4.1.3 Determination of particle charges using resonance measurements . 82
4.1.4 Particle dynamics upon bias switching . . . . . . . . . . . . . . . . 83
4.2 Complementary calculation of the sheath structure . . . . . . . . . . . . . 87
V4.2.1 Characteristics of the PIC code . . . . . . . . . . . . . . . . . . . . 87
4.2.2 Lateral sheath structure . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.3 Vertical . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Bibliography 97
VIExpose
Physics is the only science that provides theoretical models for the description of matter
over more than ft y orders of magnitude for energy, length and time scales. While this
range { from the Planck scale to cosmic extensions { seems exotic to most of us, also
in daily life we encounter situations where physical quantities vary over sixty orders
of magnitude. Here, a prominent example is the resistivity which can be changed by
external elds or system parameters from (super-) conducting to insulating behaviour.
Furthermore, physical theories apply to the description of isolated particles as well as to
19 23dense systems such as plasmas or solids, built up of 10 10 particles.
For each system certain physical e ects are dominant while others are of minor impor-
tance. Consequently, the success of a physical model/theory depends on capturing the
essential features of the system. For instance, in celestial mechanics there is no need for
a full quantum description; the motion of the earth around the sun is well approximated
by Kepler’s laws. Corrections to this trajectory, caused by the presence of the other
planets, have been calculated analytically some hundred years ago using sophisticated
perturbation series. However, perturbation theories crucially depend on the existence of
a small expansion parameter, and fail without it. Thus, analytically describing the full
dynamics of many objects with comparable masses, e.g., a galaxy or a globular cluster,
is impossible.
In absence of suitable analytical approaches, we have to resort to numerical methods
for the description of such kind of systems. During the last decades, a steady progress
was achieved in both the development and implementation of di eren t computational
algorithms. Nowadays, molecular dynamics (MD) and particle-in-cell (PIC) based codes
allow for the thorough simulation of a wide class of classical many-particle systems.
Ranging from astrophysical applications over plasma physics to classical statistical me-
chanics, they also apply to the simulation of biochemical processes as protein folding,
situated at the borderline to quantum systems.
In the realm of quantum mechanics, a full many-particle description is not always
necessary and various physical e ects may be explained on the basis of an e ectiv e
single-particle description. Introducing the concept of quasiparticles, we may calculate
the electronic band structure of solids or the properties of elementary excitations in
electron and spin systems. If the residual interactions between the quasiparticles are
weak, further corrections may be calculated perturbatively. However, such an approach
encounters its limitations when correlations and interaction e ects become dominant. As
a result, in strongly correlated electron systems, subtle many-particle e ects appear such
as high temperature superconductivity or colossal magnetoresistance. Here the interplay
of di eren t physical aspects is crucial, and their comparable importance prohibits the
use of a perturbative treatment.
1interaction effects
quantum systems
exact diagonalisation (DMRG)
kernel polynomial expansion
propagator techniques
semiclassical Wigner−Moyal approach
methods
quantum tomography
accessible particle−in−cell method
system size &
molecular dynamicsnumber of particles
classical systems
particle density
Figure 0.1: Numerical methods used in this work for di eren t physical systems.
To tackle the problem of strongly correlated many-particle systems, the quasi-exact
numerical investigation of microscopic models is especially appealing. Starting from a
microscopic model Hamiltonian, we can calculate the ground state properties of nite
systems in principle by full diagonalisation. Spectral or dynamical quantities can be
treated by the kernel polynomial method (KPM) quasi-exactly. To correctly extrapolate
from the obtained nite-system data to the thermodynamic limit is a challenging task.
Here, we face the drawback of a full microscopic modelling: the exponential growth of
the Hilbert space dimension with the number of considered particles. In order to access
larger systems, we have to turn to approximate methods, such as quantum-chemical
approaches, density-functional based theories or quantum Monte Carlo simulations.
For one-dimensional systems, the diagonalisation-based density matrix renormalisation
group (DMRG) is especially appealing, where the Hilbert space is restricted to its most
important subspace. Alternatively, semiclassical approximations have been conceived for
quantum systems in which not all correlations have to be taken into account. Aiming at
a coarse grained modelling with a more favourable scaling of the computation time with
the number of considered particles they close the gap between a quantum and classical
description of a system.
In this work we will analyse the capabilities of several of the above mentioned numerical
techniques (see Fig. 0.1). Thereby, the considered systems range from quantum over
semiclassical to classical and from few- to many-particle systems. For each case we
address an interesting, partly unsolved question. Despite the di eren t topics we address
in the individual chapters, the problems under study are somehow related because we
focus on the time evolution of the system.
In chapter 1 we investigate the behaviour of a single quantum particle in the pres-
ence of an external disordered background (static potentials). Starting from the quan-
tum percolation problem, we address the fundamental question of a disorder induced
2
external fields
temperature
dimension(Anderson-) transition from extended to localised single-particle eigenstates. Distin-
guishing isolating from conducting states by applying a local distribution approach for
the local density of states (LDOS), we detect the quantum percolation threshold in two-
and three-dimensions. Extending the quantum