Antiferromagnetism and d-wave superconductivity in the Hubbard Model [Elektronische Ressource] / by Hans Christian Krahl

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Publié par	ruprecht-karls-universitat_heidelberg
Publié le	01 janvier 2007
Nombre de lectures	11
Langue	Deutsch
Poids de l'ouvrage	1 Mo

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Dissertation
submitted to the Combined Faculties for the
Natural Sciences and Mathematics
of the Ruperto – Carola University of Heidelberg
for the degree of
Doctor of Natural Sciences
presented by
Diplom-Physiker Hans Christian Krahl
born in Konstanz, Germany
Oral Examination: July 25, 2007Antiferromagnetism and
d-wave Superconductivity in the
Hubbard Model
Referees: Prof. Dr. Christof Wetterich
PD. Dr. Jan Martin PawlowskiivAntiferromagnetismus und d-Wellen-Supraleitung
im Hubbard-Modell
Zusammenfassung
Das zweidimensionale Hubbard Modell gilt als vielversprechendes, eﬀektives Modell
zur Beschreibung der Freiheitsgrade der Elektronen in den Kupferoxidschichten von
Hochtemperatursupraleitern. Wir stellen einen Zugang zu diesem Modell mithilfe
der funktionalen Renormierungsgruppe mit Fokus auf Antiferromagnetismus und
d-Wellensupraleitung vor. Um die relevanten Freiheitsgrade auf allen L¨angenskalen
explizit zug¨anglich zu machen, fu¨hren wir zusammengesetzte bosonische Felder ein,
welchedieWechselwirkungzwischendenFermionenvermitteln. DasspontaneBrech-
en einer Symmetrie spiegelt sich in einem nichtverschwindenden Erwartungswert
eines bosonischen Feldes wieder. Wir zeigen wie durch Spinwellenﬂuktuationen eine
d-Wellenkopplung erzeugt wird. Des Weiteren berechnen wir die h¨ochste Temper-
atur, bei der die Wechselwirkungsst¨arke der Elektronen im Renormierungsgruppen-
ﬂuss divergiert sowohl fu¨r Antiferromagnetismus als auch fu¨r d-Wellensupraleitung
u¨ber einen weiten Dotierungsbereich. Diese “pseudokritische” Temperatur signal-
isiertdasEinsetzen vonlokaler Ordnung. AusserdemwirddieTemperaturabha¨ngig-
keit der d-Wellensupraleitung in einem vereinfachten Modell, welches sich durch
eine Kopplung lediglich im d-Wellenkanal auszeichnet, studiert. Wir ﬁnden einen
Phasenu¨bergang vom Kosterlitz-Thouless-Typ.
Antiferromagnetism and d-wave Superconductivity
in the Hubbard Model
Abstract
Thetwo-dimensionalHubbardmodelisapromisingeﬀectivemodelfortheelectronic
degrees of freedom in the copper-oxide planes of high temperature superconductors.
We present a functional renormalization group approach to this model with focus
on antiferromagnetism and d-wave superconductivity. In order to make the rele-
vant degrees of freedom more explicitly accessible on all length scales, we introduce
composite bosonic ﬁelds mediating the interaction between the fermions. Spon-
taneous symmetry breaking is reﬂected in a non-vanishing expectation value of a
bosonic ﬁeld. The emergence of a coupling in the d-wave pairing channel triggered
by spin wave ﬂuctuations is demonstrated. Furthermore, the highest temperature
at which the interaction strength for the electrons diverges in the renormalization
ﬂow is calculated for both antiferromagnetism and d-wave superconductivity over
a wide range of doping. This “pseudo-critical” temperature signals the onset of
local ordering. Moreover, the temperature dependence of d-wave superconducting
order is studied within a simpliﬁed model characterized by a single coupling in the
d-wave pairing channel. The phase transition within this model is found to be of
the Kosterlitz-Thouless type.viContents
1 Introduction 1
2 Interacting Electrons 5
2.1 High Temperature Superconductivity . . . . . . . . . . . . . . . 5
2.2 Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Functional Integral Representation of the Partition Function . . 9
2.4 Partial Bosonization . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.1 Hubbard-Stratonovich Transformation . . . . . . . . . . 14
2.4.2 d-wave Operators . . . . . . . . . . . . . . . . . . . . . . 16
3 Functional Renormalization Group Equations 21
3.1 Functional Integral Formulation of QFT . . . . . . . . . . . . . 21
3.2 Flow Equation for the Eﬀective Average Action . . . . . . . . . 24
3.3 Scale-Dependent Field Transformations . . . . . . . . . . . . . . 26
4 Competition of Antiferromagnetism andd-waveSuperconductivity 29
4.1 Truncation of the Eﬀective Action . . . . . . . . . . . . . . . . . 30
4.2 Regularization Scheme . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 1-PI Vertex Functions to One-loop Order . . . . . . . . . . . . . 37
4.3.1 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3.2 Yukawa Couplings . . . . . . . . . . . . . . . . . . . . . 46
4.3.3 Four Fermion Interactions . . . . . . . . . . . . . . . . . 48
4.4 Rebosonization of Fermionic Interactions . . . . . . . . . . . . . 50
4.5 Flow Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5.1 Eﬀective Potential . . . . . . . . . . . . . . . . . . . . . 55
4.5.2 Kinetic Terms of the Bosons . . . . . . . . . . . . . . . . 59
4.5.3 Yukawa Couplings . . . . . . . . . . . . . . . . . . . . . 62
4.6 GenerationofaCouplinginthed-waveChannel . . . . . . . . . . 63
4.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 Kosterlitz-Thouless Transition to d-wave Superconductivity 69
5.1 Eﬀective Model for d-wave Superconductivity . . . . . . . . . . 70
5.2 Essential Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Anomalous Dimension . . . . . . . . . . . . . . . . . . . . . . . 79
viiviii Contents
6 Summary and Outlook 81
A Conventions and Notation 87
B Matrices for Computation of 1-PI Vertex Functions 89
C Explicit Flow Equations 93
C.1 Eﬀective Potential . . . . . . . . . . . . . . . . . . . . . . . . . 93
C.2 Kinetic Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
C.3 Yukawa Couplings . . . . . . . . . . . . . . . . . . . . . . . . . 97
D Useful Formulae 99
D.1 Spinor Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
D.2 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
D.3 Matsubara Sums . . . . . . . . . . . . . . . . . . . . . . . . . . 101
D.4 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Bibliography 104Chapter 1
Introduction
Interacting many-body systems in condensed matter physics are usually dis-
cussed in terms of eﬀective models which incorporate only a comparatively
small number of degrees of freedom. Of course, the complexity of taking into
account all microscopic degrees of freedom such as the core atoms as well
as their electron shells and energy bands would be prohibitive. In order to
understand macroscopic properties, such as the phase diagram, it is further-
morequestionablewhetherallthesemicroscopicdegreesoffreedomareneeded.
Hence, one of the challenges is to identify the relevant microscopic degrees of
freedom and to formulate a microscopic theory which potentially comprises all
the information needed to explain the macroscopic behavior.
A famous example of such a microscopic model for interacting fermions
is the Hubbard model [1, 2, 3]. Besides many other applications the two-
dimensionalversionoftheHubbardmodelisregardedasapromisingcandidate
for a description of the electron degrees of freedom in the copper-oxide planes
of cuprate high temperature superconductors [4] since it resembles important
features of these materials. Particularly, the ground state of this model is
antiferromagneticallyorderedforahalf-ﬁlledlatticeandisexpectedtobecome
a superconductor withd-wave pairing symmetry slightly away from half-ﬁlling
[5, 6]. Recently, the Hubbard model was also proposed forultra-cold fermionic
atomsintwo-dimensional opticallattices [7,8]. These systems mayprovide an
ideal testing ground for this model since they are comparatively clean and as
a bonus the ratio of the parameters is tunable. Experiments in optical lattices
may also shed light on the question which degrees of freedom are responsible
for high temperature superconductivity.
A common feature of such an interacting many-body system is that its be-
havior depends qualitatively on the energy or momentum scale. For instance,
in high temperature superconductors ordering phenomena such as d-wave su-
perconductivity are separated by several orders of magnitude in energy from
thebareCoulombinteraction. Theintermediatescalesaredominatedbyshort-
range magnetic correlations. A calculation of macroscopic properties involves
an integration over ﬂuctuations on all scales. However, it is hard to treat the
12 Chapter 1 Introduction
ﬂuctuationsonallscalesatthesametimesincetheappropriatedegreesoffree-
dom change qualitatively with scale. In particular, perturbative approaches
are plagued by infrared divergences and therefore are often not applicable,
especially in low dimensions.
Among the most promising approaches to the Hubbard model are renor-
malization group studies [9, 10, 11, 12]. Fluctuations from diﬀerent scales are
treated successively starting from the highest present momentum scale. This
allows to very eﬃciently describe the transition from the microscopic scale
where the eﬀective model is deﬁned, to macroscopic scales where collective
phenomena may occur. In recent years an increasing number of renormaliza-
tiongroupstudies ofthetwo-dimensional Hubbardmodel have beenpublished
[13, 14, 15, 16, 17, 18, 19, 20]. The results are encouraging and indeed suggest
an antiferromagnetic instability at half-ﬁlling and the dominance of d-wave
superconductivity slightly away from half-ﬁlling. However, all of thes