Matrix product state clculations for one-dimensional quantum chains and quantum impurity models [Elektronische Ressource] / Wolfgang Münder. Betreuer: Jan von Delft

Wolfgang Münder

Sujets

• Physik

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2011
• Nombre de lectures: 204
• Poids de l'ouvrage: 4 Mo

Extrait

Matrix product state calculations for
one-dimensional quantum chains
and quantum impurity models
Wolfgang Münder
Dissertation
an der Fakultät für Physik
der Ludwig–Maximilians–Universität
München
vorgelegt von
Wolfgang Münder
aus München
München, den 30.08.2011Erstgutachter: Prof. Dr. Jan von Delft
Zweitgutachter: Prof. Dr. Ulrich Schollwöck
Tag der mündlichen Prüfung: 28.09.2011Contents
Abstract v
I General Introduction 1
1 Introduction 3
2 Strongly correlated electron systems 7
2.1 Fermi liquid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Luttinger liquid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Renormalization group approaches . . . . . . . . . . . . . . . . . . . . . . 9
3 Quantum impurity problems 13
3.1 Kondo effect and Kondo temperature . . . . . . . . . . . . . . . . . . . . . 13
3.2 Quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Single-impurity Anderson model . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Fermi sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.5 Anderson orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5.1 Static quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5.2 Dynamical quantities after a quantum quench . . . . . . . . . . . . 19
3.6 Population switching in quantum dots . . . . . . . . . . . . . . . . . . . . 21
4 Numerical renormalization group (NRG) 25
4.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 From the electronic band to the Wilson chain . . . . . . . . . . . . . . . . 27
4.2.1 Logarithmic discretization . . . . . . . . . . . . . . . . . . . . . . . 27
4.2.2 Mapping to the Wilson chain . . . . . . . . . . . . . . . . . . . . . 29
4.2.3 Wilson chain in the original form . . . . . . . . . . . . . . . . . . . 30
4.2.4 Improved discretization scheme . . . . . . . . . . . . . . . . . . . . 30
4.3 Key points of the Wilson chain mapping . . . . . . . . . . . . . . . . . . . 32
4.4 NRG solution of the Wilson chain . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.1 Description of the iteration procedure . . . . . . . . . . . . . . . . . 36
4.4.2 Renormalization group flow . . . . . . . . . . . . . . . . . . . . . . 37iv Contents
4.4.3 Matrix product state structure of the basis states . . . . . . . . . . 38
4.5 Anders-Schiller basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5.2 Local operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.6 Density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.7 Spectral functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.8 Expectation values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.9 Overlap calculations in the context of Anderson Orthogonality . . . . . . . 54
4.10 Scattering phase shifts from the energy flow diagram . . . . . . . . . . . . 55
II Results 59
5 The correlation density matrix 61
6 Anderson orthogonality 113
6.1 Anderson Orthogonality and the Numerical Renormalization Group . . . . 113
6.2y in the Dynamics After a Local Quantum Quench . 123
III Miscellaneous 145
Bibliography 147
List of Publications 153
Deutsche Zusammenfassung 155
Acknowledgements 157Abstract
This thesis contributes to the field of strongly correlated electron systems with studies
in two distinct fields thereof: the specific nature of correlations between electrons in one
dimension and quantum quenches in quantum impurity problems. In general, strongly
correlated systems are characterized in that their physical behaviour needs to be described
in terms of a many-body description, i.e. interactions correlate all particles in a complex
way. The challenge is that the Hilbert space in a many-body theory is exponentially
large in the number of particles. Thus, when no analytic solution is available - which is
typically the case - it is necessary to find a way to somehow circumvent the problem of
such huge Hilbert spaces. Therefore, the connection between the two studies comes from
our numerical treatment: they are tackled by the density matrix renormalization group
(DMRG) [1] and the numerical renormalization group (NRG) [2], respectively, both based
on matrix product states.
The first project presented in this thesis addresses the problem of numerically finding
the dominant correlations in quantum lattice models in an unbiased way, i.e. without using
prior knowledge of the model at hand. A useful concept for this task is the correlation
density matrix (CDM) [3] which contains all correlations between two clusters of lattice
sites. We show how to extract from the CDM, a survey of the relative strengths of the
system’s correlations in different symmetry sectors as well as detailed information on the
operators carrying long-range correlations and the spatial dependence of their correlation
functions. We demonstrate this by a DMRG study of a one-dimensional spinless extended
Hubbard model [4], while emphasizing that the proposed analysis of the CDM is not
restricted to one dimension.
The second project presented in this thesis is motivated by two phenomena under
ongoing experimental and theoretical investigation in the context of quantum impurity
models: optical absorption involving a Kondo exciton [5, 6, 7] and population switching
in quantum dots [8, 9, 10, 11, 12, 13, 14, 15]. It turns out that both phenomena rely
on the various manifestations of Anderson orthogonality (AO) [16], which describes the
fact that the response of the Fermi sea to a quantum quench (i.e. an abrupt change of
some property of the impurity or quantum dot) is a change of the scattering phase shifts
of all the single-particle wave functions, therefore drastically changing the system. In
this context, we demonstrate that NRG, a highly accurate method for quantum impurity
models, allows for the calculation of all static and dynamic quantities related to AO and
present an extensive NRG study for population switching in quantum dots.vi AbstractPart I
General IntroductionChapter 1
Introduction
The field of strongly correlated electronic systems has emerged from the awareness that
for the correct description of many phenomena interactions play the key role, thereby
correlating all particles in the system in a complex way. Prominent examples are Mott
insulators [17], which cannot be explained within the conventional band theories, and
the Kondo effect [18]. The latter was observed already in the 1930’s by de Haas et al.
[19] as a resistivity anomaly at low temperatures: instead of a decrease of the resistivity
with decreasing temperature it increases again when the temperature is lowered below
what is nowadays called the Kondo temperature T . The reason for this behaviour isK
that the local moment of magnetic impurities in some metals (the first observation was
about iron impurities in gold and silver) is screened by the conduction electrons, forming
a strongly correlated state. This explanation has been found by Jun Kondo in the 1960’s
[18]. Nevertheless, the question which specific realization of his model yields a realistic
description of the observations of de Haas et al. could be answered only recently [20].
The necessity to include (electron-electron) interactions into models describing solid
state systems caused an enormous challenge: the description of these systems in terms
of single particles is not appropriate any longer and instead many-body theories had to
be established. Evidently, the complexity of these systems increases with the number
of particles, manifesting in the scaling of the dimension of the Hilbert space: it grows
exponentially with the number of particles. The problem is now that for most systems no
analytic solution is available and exact diagonalization requires huge computational effort
even for small systems. Thus, several numerical methods have been established, which
somehow circumvent the treatment of the full, high-dimensional Hilbert space.
The two studies presented in this work are tackled by two of these methods, namely the
density matrix renormalization group (DMRG) [1, 21] and the numerical renormalization
group (NRG) [2, 22, 23]. Both methods take advantage, albeit each in somewhat different
form, oftheconceptoftherenormalizationgroup(RG)[2]whichearnedKennethG.Wilson
the Nobel prize in 1982. In general, RG proposes a way to iteratively consider all degrees
of freedom of a problem by systematically thinning out irrelevant degrees of freedom at
each step.
The first project presented in this thesis generally addresses correlations in quantum4 1. Introduction
latticesystems. Wheninvestigatingcorrelationstheproblemarisesthataprioritheformof
the dominant correlations is unknown. One needs to guess in advance - based on experience
or prior knowledge - which operators could carry long range correlations and check whether
this is really the case. The concept of the correlation density matrix (CDM) [3] is intended
as an unbiased way to numerically find the operators carrying the dominant correlations.
For two disjoint, separated clusters the CDM is defined b