Spin gap in doped ladder systems [Elektronische Ressource] / vorgelegt von Lorenzo Campos Venuti
153 Pages
English
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Spin gap in doped ladder systems [Elektronische Ressource] / vorgelegt von Lorenzo Campos Venuti

Downloading requires you to have access to the YouScribe library
Learn all about the services we offer
153 Pages
English

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Spin gap in doped ladder systemsVon der Fakult¨at fur¨ Mathematik und Physikder Universit¨at Stuttgartzur Erlangung der Wurde¨ einesDoktors der Naturwissenschaften (Dr. rer. nat.)genehmigte AbhandlungVorgelegt vonLorenzo Campos Venutiaus Bologna (Italien)Hauptberichter: Prof. Dr. Alejandro MuramatsuMitberichter: Prof. Dr. Gun¨ ter MahlerTag der mundlic¨ hen Prufung:¨ 15. Dezember 2003Institut fur¨ Theoretische Physik der Universit¨at Stuttgart2004CONTENTS 1ContentsAbstract 5Introduction 71 Models for Copper-Oxide materials 151.1 Doped antiferromagnetic . . . . . . . . . . . . . . . 171.1.1 The two dimensional high-T superconductors . . . . . 17c1.1.2 One dimensional chains and ladders . . . . . . . . . . . 201.2 Microscopic models for doped antiferromagnets . . . . . . . . 211.2.1 The three-band Hubbard model . . . . . . . . . . . . . 211.2.2 Mapping to a Spin Fermion model. . . . . . . . . . . . 241.2.3 Mapping to a t−J model: the Zhang-Rice approach . 261.3 The undoped case: Non Linear Sigma Model approach . . . . 302 Effective theories for doped chains and ladders 372.1 The chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.1.1 Parametrization . . . . . . . . . . . . . . . . . . . . . . 412.1.2 Integration of the spin fluctuations . . . . . . . . . . . 442.1.3 Integration of the fermionic high energy modes . . . . . 452.1.3.1 Rotating reference frame . . . . . . . . . . . . 462 CONTENTS2.1.3.2 The free action . . . . . . .

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Spin gap in doped ladder systems
Von der Fakult¨at fur¨ Mathematik und Physik
der Universit¨at Stuttgart
zur Erlangung der Wurde¨ eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigte Abhandlung
Vorgelegt von
Lorenzo Campos Venuti
aus Bologna (Italien)
Hauptberichter: Prof. Dr. Alejandro Muramatsu
Mitberichter: Prof. Dr. Gun¨ ter Mahler
Tag der mundlic¨ hen Prufung:¨ 15. Dezember 2003
Institut fur¨ Theoretische Physik der Universit¨at Stuttgart
2004CONTENTS 1
Contents
Abstract 5
Introduction 7
1 Models for Copper-Oxide materials 15
1.1 Doped antiferromagnetic . . . . . . . . . . . . . . . 17
1.1.1 The two dimensional high-T superconductors . . . . . 17c
1.1.2 One dimensional chains and ladders . . . . . . . . . . . 20
1.2 Microscopic models for doped antiferromagnets . . . . . . . . 21
1.2.1 The three-band Hubbard model . . . . . . . . . . . . . 21
1.2.2 Mapping to a Spin Fermion model. . . . . . . . . . . . 24
1.2.3 Mapping to a t−J model: the Zhang-Rice approach . 26
1.3 The undoped case: Non Linear Sigma Model approach . . . . 30
2 Effective theories for doped chains and ladders 37
2.1 The chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1.1 Parametrization . . . . . . . . . . . . . . . . . . . . . . 41
2.1.2 Integration of the spin fluctuations . . . . . . . . . . . 44
2.1.3 Integration of the fermionic high energy modes . . . . . 45
2.1.3.1 Rotating reference frame . . . . . . . . . . . . 462 CONTENTS
2.1.3.2 The free action . . . . . . . . . . . . . . . . . 51
2.1.3.3 Action in the Reduced Brillouin Zone. . . . . 53
2.1.3.4 Continuum limit . . . . . . . . . . . . . . . . 54
2.1.3.5 Gaussian integration . . . . . . . . . . . . . . 55
2.2 Ladder case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.2.1 Two-leg ladder . . . . . . . . . . . . . . . . . . . . . . 61
2.2.2 Three leg ladder. . . . . . . . . . . . . . . . . . . . . . 63
3 Spin gap vs doping in a two-leg ladder 65
3.1 Integrate out the fermions . . . . . . . . . . . . . . . . . . . . 68
3.2 Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 Antiferromagnetic contribution . . . . . . . . . . . . . . . . . 74
3.4 Ferromagnetic and mixed contributions . . . . . . . . . . . . . 76
3.4.1 The effective bands . . . . . . . . . . . . . . . . . . . . 78
3.5 Result and comments . . . . . . . . . . . . . . . . . . . . . . . 79
3.6 Comparison with Experiments . . . . . . . . . . . . . . . . . . 83
4 Green’s function for the doped two-leg ladder 87
4.1 Schulz’s approach . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Green’s function for the two-leg ladder . . . . . . . . . . . . . 92
4.3 Integration of the ferromagnetic fluctuation. . . . . . . . . . . 95
4.4 Zero order propagator . . . . . . . . . . . . . . . . . . . . . . 97
4.5 First order self energy . . . . . . . . . . . . . . . . . . . . . . 98
4.6 Zero doping limit . . . . . . . . . . . . . . . . . . . . . . . . . 101
Conclusions 105CONTENTS 3
Zusammenfassung 107
A Effective action in the continuum limit 115
A.1 The trace reduction formula . . . . . . . . . . . . . . . . . . . 115
A.2 Calculation of the antiferromagnetic contributions . . . . . . . 117
A.2.1 Contribution I . . . . . . . . . . . . . . . . . . . . . . . 118
A.2.2 Contribution II . . . . . . . . . . . . . . . . . . . . . . 119
A.2.3 Contribution III . . . . . . . . . . . . . . . . . . . . . . 119
A.2.4 Contribution IV . . . . . . . . . . . . . . . . . . . . . . 122
A.2.5 All the AFM contributions . . . . . . . . . . . . . . . . 125
A.3 Calculation of ferromagnetic and mixed contribution . . . . . 126
A.4 Integration of the fluctuation. . . . . . . . . . . 130
B Self energy for the doped two-leg ladder 133
B.1 Matrix inversion up to second order . . . . . . . . . . . . . . . 133
B.2 Self energy contribution up to first order . . . . . . . . . . . . 135
B.2.1 Contribution I . . . . . . . . . . . . . . . . . . . . . . . 135
B.2.2 Contribution II . . . . . . . . . . . . . . . . . . . . . . 136
Bibliography 1394ABSTRACT 5
Abstract
A proper theoretical description of doped antiferromagnets is until now an
unresolved issue. In this work we study chain and ladder systems, and in
particularly we focus on the physically most interesting two leg ladder.
Our starting point is the Spin Fermion model in which an antiferromagnetic
background of localized spins interacts with mobile holes via a rotation in-
variant Kondo-like term. The interesting region of the phase diagram is that
close to the Mott-insulator transition where the doping δ is zero. At zero
dopinginfactthesystemisaninsulatorandthespinsorganizethemselvesin
a spin liquid, a rotational invariant state characterized by a finite correlation
length and an energy gap (a spin gap) above the ground state. Such a state
isalsothegroundstateofafieldtheory, thenonlinearsigmamodel(NLσM)
which is recognized as the low energy effective theory for antiferromagnetic
spin ladder.
Thefirstquestionweanswerishowsuchaspinstateevolvesasonemovesoff
from the Mott phase by increasing the doping. Integrating out the fermions
in our model we obtain an effective theory for the spins which we are able to
evaluate in the continuum limit. The effective theory is again a NLσM with
coupling constants which depend on the concentration of dopant holes. In
contrast to existing mean field calculation our theory predicts a lowering of
thespingapwithdopingandaconsequentincreaseinthecorrelationlength.
Indeed a lowering of the spin gap due to doping is also observed in numerical
simulation and on NMR experiments on Sr Ca Cu O with which we14−x x 24 41
obtain very good agreement.
Secondly we concentrated on the behavior of the fermions. The general6 ABSTRACT
paradigm of interacting fermions in one dimension is that of the Luttinger
Liquid characterized by bosonic excitations and spin charge separation. By
generalizing our approach we are able to access the one particle fermion
propagator as a quantity averaged over the NLσM which controls the spin
background. We see that in the limit of zero doping the quasiparticle weight
Z is non-zero in a neighborhood of the Fermi energy. This in turn implies
that the Luttinger liquid parameter K goes to one as the doping δ goes toρ
zero as was first argued by Schulz. Our stronger result allows us to assert
that in the very low doping regime the fermions constitute a Fermi liquid.INTRODUCTION 7
Introduction
Inthelate60’s, thankstothesuccessesofLandau’sFermiliquid(FL)theory
(dated 1956) [1, 2, 3] and BCS theory of superconductivity (dated 1957)
[4, 5], most researchers in the field of solid state physics thought that their
discipline was a very settled one and a very general framework was given
once and for all. The situation changed radically during the 80’s with the
discovery of high temperature superconductivity (1986) [6], heavy fermions
(80’s) [7] and quantum Hall effect (early 80’s) [8, 9]. The common feature
of all these systems is the constrained motion of the electrons as a result of
restricteddimensionalityand/orthestronginter-electroninteractions. These
constraints induce complicated correlations in the motion of the electrons,
and hence the term which is used to describe them collectively: ’Strongly
Correlated Electron Systems’.
Emblematic of the interplay between dimensionality and correlation is the
differentdegreeofsuccessexperiencedbyLandau’sFLtheorywithdecreasing
dimensionality. In the Fermi liquid theory the low energy spectrum of the
fully interacting many body system, is in one to one correspondence with
the (low energy part of) the spectrum of the free Fermi gas. Hence, in terms
of the quasiparticles –the quanta of collective excitations of the many body
system– the Fermi liquid is essentially a free theory. Although this theory
was originally formulated to provide a description of the energy levels of
3He, it was later extended to treat electrons in a metal and nuclear matter
andbasicallyitappliestomostmetalsinthreespatialdimension–andatlow
temperature–. Intwodimensionstheanomalouscharacteristicsofthenormal
(i.e. not superconducting) state of the copper-oxide superconductors (see8 INTRODUCTION
e.g. [10]), have motivated many researchers to look for theories alternative
to the FL one. Still the Fermi liquid state is quite robust in two dimensions
and to obtain a breakdown of it one must resort to particular situations
like the presence of (unscreened) long range forces, singularities in the free
propagator or very strong coupling [11]. In these cases one then speaks of
non-Fermi liquid (NFL) or marginal-Fermi liquid (MFL), a particular case
of NFL.
Instead the situation in one dimension is much more clear and it is now well
knownthatabreakdownoftheFLpictureinonedimensionistherulerather
thentheexception. Quantumfluctuationaresostrongin1Dthatawhatever
small interaction in a Fermi (-Dirac) gas, drives the system to a completely
new state, the Luttinger liquid (LL) (see e.g. the original paper [12] and
1the review [13]). This novel state of matter is characterized by the absence
of quasiparticle excitations manifested as a vanishing of the renormalization
constant Z, and a separation of the charge and spin degrees of freedom. The
Luttinger liquid only support bosonic excitations with either charge (charge-
density waves) or spin (spin-density waves).
Thusitbecomesclearthat1Dsystemsrepresenta“playground”forthestudy
of strongly correlated systems, where many concepts (like the breakdown of
the FL) can be developed and/or tested. In fact in 1D more theoretical
methods are available or more powerful than in higher dimensionality. To
cite only the most important, exact Bethe ansatz results are available for
some models and/or for particular couplings, bosonization techniques with
the help of which the concept of LL has been set on a clear grounds, and
numerical simulations which are superior in low dimension in that larger
lattices can be studied.
The study of low dimensional systems is not only interesting for theorists.
2Chemistshavelongbeenawarethatinmanyorganiccompoundsanisotropy
3can be as large as 10 so that these systems can be regarded as quasi one
1Tomonaga, already in 1950, wasthe first to identifybosonic excitations in 1D systems
[14].
2Anisotropy can be characterized by the ratio of the conductivities measured parallel
and perpendicular to the chain direction.