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

123 pages
Numerical approaches to complexquantum, semiclassical and classicalsystemsI n a u g u r a l d i s s e r t a t i o nzurErlangung des akademischen Gradesdoctor rerum naturalium (Dr. rer. nat.)an der Mathematisch-Naturwissenschaftlichen FakultatderErnst-Moritz-Arndt-Universitat Greifswaldvorgelegt vonGerald Schubertgeboren am 01.02.1976in RehauGreifswald, 03.11.2008Dekan: Prof. Dr. Klaus Fesser1. Gutachter: Prof. Dr. Holger Fehske2. Gutachter: Prof. Dr. Vladimir FilinovTag der Promotion: 15.12.2008IVContentsExpose 11 Quantum percolation in disordered structures 51.1 Local distribution approach . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.1 Conceptual background . . . . . . . . . . . . . . . . . . . . . . . . 71.1.2 Calculation of the local density of states . . . . . . . . . . . . . . . 91.1.3 Illustration of the method: Anderson localisation in 3D . . . . . . 111.2 Localisation e ects in quantum percolation . . . . . . . . . . . . . . . . . 131.2.1 3D site percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.2 2D site p . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3 Di usion and localisation in quantum random resistor networks . . . . . . 261.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Optical absorption and activated transport in polaronic systems 352.1 Holstein model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.
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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 percolation model to a quantum random
resistor model, we comment on the possible relevance of our results to the in uence of dis-
order on the conductivity in graphene sheets. Furthermore, we con rm the localisation
properties of the 2D percolation model by calculating the full quantum time evolution
of a given initial state. For the calculation of the LDOS as well as for the Chebyshev
expansion of the time evolution operator, the KPM is the key numerical technique.
In chapter 2 we examine how a single quantum particle is in uenced by retarded
bosonic elds that are inherent to the system. Within the Holstein model, these bosonic
degrees of freedom (phonons) give rise to an in nite dimensional Hilbert space, posing
a true many-particle problem. Constituting a minimal model for polaron formation,
the Holstein model allows us to study the optical absorption and activated transport
in polaronic systems. Using a two-dimensional variant of the KPM, we calculate for
the rst time quasi-exactly the optical absorption and dc-conductivity as a function
of temperature. Concerning the numerical technique, the close relation to the time
evolution in the other chapters get clear if we identify temperature with an imaginary
time.
In chapter 3 we come back to the time evolution of a quantum particle in an exter-
nal, static potential and investigate the capability of semiclassical approximations to
it. Considering various one-dimensional geometries, we address basic quantum e ects
as tunneling, interference and anharmonicity. The question is, to which extend and at
which numerical costs, several semiclassical methods can reproduce the exact result for
the quantum dynamics, calculated by Chebyshev expansion. To this end we consider
the linearised semiclassical propagator method, the Wigner-Moyal approach and the re-
cently proposed quantum tomography. A conceptually very interesting aspect of the
compared semiclassical methods is their relation to di eren t representations of quantum
mechanics (wave function/density matrix, Wigner function, quantum tomogram).
Finally, in chapter 4 we calculate the dynamics of a classical many-particle system
under the in uence of external elds. Considering a low-temperature rf-plasma, we
investigate the interplay of the plasma dynamics and the motion of dust particles, im-
mersed into the plasma for diagnostic reasons. In addition to the huge number of involved
particles, the numerical description of this systems faces the challenge of a large range of
involved time and length scales. Exploiting the mass di erences of plasma constituents
and dust particles allows for separating the PIC description of the from the MD
simulation of the dust particles in the e ectiv e surrounding plasma.
The challenge for the numerical methods presented in chapters 1-4 is to balance the
competing requirements for accuracy, treatable system sizes and computation time. Fu-
ture developments in this eld will be guided by the need for suitable numerical al-
gorithms, their e cien t coding and a successful porting to modern (super-) computer
architectures. Along this line, computational physics has emerged as an important,
independent branch of natural science, complementing traditional theoretical and ex-
perimental approaches.
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