Interplay of Broken Symmetries and Quantum Criticality in Correlated Electronic Systems

292 Pages·2016·16.98 MB·English

Interplay of Broken Symmetries and Quantum
Criticality in Correlated Electronic Systems
A dissertation presented
by
Debanjan Chowdhury
to
The Department of Physics
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the subject of
Physics
Harvard University
Cambridge, Massachusetts
April 2016
c 2016 - Debanjan Chowdhury
⃝
All rights reserved.
Thesis advisor Author
Subir Sachdev Debanjan Chowdhury
Interplay of Broken Symmetries and Quantum Criticality in
Correlated Electronic Systems
Abstract
This thesis delves into a study of phases of strongly correlated quantum matter con-
fined to two spatial dimensions. The thesis can broadly be divided into three parts.
In the first part, comprising of chapters 2 and 3, we investigate some interesting as-
pects of symmetry breaking and quantum criticality in the superconducting phase
of the iron-based superconductors. In particular, motivated by tunneling microscopy
measurements on FeSe, in chapter 2 we study the effect of spontaneously broken ro-
tational symmetry on the structure of the superconducting vortex. In chapter 3, we
study the critical singularities associated with the superfluid-density at a wide class
of symmetry-breaking and topological phase transitions in a clean superconductor.
Inspired by experiments on BaFe (As P ) , we also analyze the effect of quenched
2 1 x x 2
−
disorder on the superfluid-density in the vicinity of magnetic quantum critical points.
The second part of this thesis, consisting of chapters 4 and 5, is devoted to a study
of the pseudogap phase in the underdoped cuprates. In chapter 4 we study the effect
of thermal fluctuations of various competing order parameters, including preformed
superconductivity and short-ranged charge-density wave, on the electronic excita-
tions. In chapter 5 we analyze the feedback of pairing fluctuations on the landscape
iii
Abstract
of various competing charge-density wave order parameters within the framework of
fermi-liquid theory.
In the final part of the thesis, consisting of chapters 6 and 7, we propose an
alternative picture for describing the pseudogap metal. In chapter 6, we study a
quantum-disordered phase of matter—the fractionalized fermi-liquid (FL*)—where
the electrons are coupled to the fractionalized excitations of a strongly fluctuating
antiferromagnet and propose it to be a candidate state for the pseudogap. We inves-
tigate instabilities of the FL* to density-wave order and compare with experiments.
Inchapter7, wedescribeaframeworkfordescribinganovelquantumphasetransition
without any broken-symmetries—a Higgs transition—that describes a transition from
a conventional fermi-liquid to a parent phase of the FL* state via an intermediate
non-fermi liquid. We discuss its possible connection to the optimal doping critical
point in the cuprates.
iv
Contents
Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Citations to Previously Published Work . . . . . . . . . . . . . . . . . . . viii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
1 Introduction 1
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Quantum Criticality . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Fermi-liquids and Luttinger’s theorem . . . . . . . . . . . . . 6
1.2 Beyond conventional condensed-matter physics . . . . . . . . . . . . . 13
1.2.1 High-temperature superconductivity . . . . . . . . . . . . . . 14
1.2.2 Quantum criticality in the pnictide superconductors . . . . . . 19
1.2.3 The enigma of the cuprate superconductors . . . . . . . . . . 24
1.3 Correlated antiferromagnetic metals . . . . . . . . . . . . . . . . . . . 38
1.3.1 Fractionalized Fermi liquid . . . . . . . . . . . . . . . . . . . . 39
1.3.2 Instabilities of a Fermi liquid with antiferromagnetic correlations 49
1.4 Organization of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2 Nematic order in the vicinity of a superconducting vortex 58
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.3.1 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.3.2 Vortex profile in the different regimes . . . . . . . . . . . . . . 66
2.3.3 Analytical Treatment . . . . . . . . . . . . . . . . . . . . . . . 70
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3 Quantum Phase Transitions beneath the superconducting dome 79
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
v
Contents
3.2 Singularity of the penetration depth at quantum critical points . . . . 81
3.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.3 Effect of quenched disorder in the vicinity of putative quantum critical
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.3.1 Insights from optical sum-rules . . . . . . . . . . . . . . . . . 99
3.3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4 Precursor thermal fluctuations in underdoped cuprates 110
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2.1 Onset of charge-order and superconductivity . . . . . . . . . . 115
4.2.2 Order parameter fluctuations coupled to Fermi-surface . . . . 118
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5 Feedback of pairing fluctuations on charge-order 129
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2.1 Instabilities of metal with antiferromagnetic exchange interaction133
5.2.2 Metal with SC and CDW fluctuations . . . . . . . . . . . . . . 139
5.2.3 Low-energy theory . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.3.1 Linearized hot-spot theory . . . . . . . . . . . . . . . . . . . . 146
5.3.2 Effect of fermi-surface curvature . . . . . . . . . . . . . . . . . 156
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6 Fractionalized Fermi-liquids 164
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.2.1 Fractionalized Fermi liquid (FL*) . . . . . . . . . . . . . . . . 173
6.2.2 Charge-order instabilities via T-matrix . . . . . . . . . . . . . 175
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7 Quantum phase transitions in metals without broken symmetries 187
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.2.1 Field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
vi
Contents
7.2.2 DC transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.2.3 SU(2) gauge theory of antiferromagnetic metals . . . . . . . . 200
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.3.1 Mean field phase diagram . . . . . . . . . . . . . . . . . . . . 203
7.3.2 Low-energy field theory . . . . . . . . . . . . . . . . . . . . . . 205
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
A Appendices for Chapter 2 219
A.1 Instabilities of the free energy . . . . . . . . . . . . . . . . . . . . . . 219
A.2 Effect of boundary terms . . . . . . . . . . . . . . . . . . . . . . . . . 220
A.3 Asymptotics of φ in the critical case . . . . . . . . . . . . . . . . . . . 222
B Appendices for Chapter 3 224
B.1 Superfluid-density for one-band problem . . . . . . . . . . . . . . . . 224
B.2 Numerical computation of λ2(0) . . . . . . . . . . . . . . . . . . . . . 224
L
B.3 Estimate of λ2(0) from Homes’ law . . . . . . . . . . . . . . . . . . . 225
L
C Appendices for Chapter 4 227
C.1 Saddle point equations of the O(6) Model . . . . . . . . . . . . . . . 227
C.2 Bosonic self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
C.3 Real and imaginary part of Fermion self-energy . . . . . . . . . . . . 230
D Appendices for Chapter 5 231
D.1 Feynman diagrams for linearized hot-spot theory . . . . . . . . . . . . 231
D.2 Feynman diagrams for hot-spot theory with a finite curvature . . . . 238
E Appendices for Chapter 6 245
E.1 CDW instabilities of FL . . . . . . . . . . . . . . . . . . . . . . . . . 245
E.2 Instabilities in the presence of strong Coulomb repulsion . . . . . . . 246
F Appendices for Chapter 7 249
F.1 Spiral order and Z gauge theory . . . . . . . . . . . . . . . . . . . . 249
2
F.2 Feynman diagram computations . . . . . . . . . . . . . . . . . . . . . 251
F.2.1 Self-energy: Gauge-field . . . . . . . . . . . . . . . . . . . . . 251
F.2.2 Self-energy: Higgs’ field . . . . . . . . . . . . . . . . . . . . . 251
F.2.3 Fermion self-energy at the hot-spot . . . . . . . . . . . . . . . 252
F.2.4 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Bibliography 254
vii
Citations to Previously Published Work
Most of the chapters of this thesis have appeared in print elsewhere. By chapter
number, they are:
Chapter 1:
•
Partly based on “The Enigma of the Pseudogap Phase of the Cuprate Super-
conductors,” in Quantum criticality in condensed matter, 50th Karpacz Winter
SchoolofTheoreticalPhysics, editedbyJ.Jedrzejewski(WorldScientific, 2015),
ISBN: 978-981-4704-08-3; arXiv:1501.00002.
Chapter 2:
•
“Nematic order in the vicinity of a vortex in superconducting FeSe,” Debanjan
Chowdhury, E. Berg and Subir Sachdev, Phys. Rev. B 84, 205113 (2011);
arXiv:1109.2600
Chapter 3:
•
“Singularity of the London penetration depth at quantum critical points in su-
perconductors,” Debanjan Chowdhury, B. Swingle, E. Berg and Subir Sachdev,
Phys. Rev. Lett. 111, 157004 (2013); arXiv:1305.2918.
“Phase transition beneath the superconducting dome in BaFe (As P ) ,” De-
2 1 x x 2
−
banjan Chowdhury, J. Orenstein, S. Sachdev and T. Senthil, Phys. Rev. B
92, 081113 (R) (2015); arXiv:1502.04122.
Chapter 4:
•
Partly based on “Connecting high-field quantum oscillations to zero-field elec-
tron spectral functions in the underdoped cuprates,” A.Allais, DebanjanChowd-
huryandSubirSachdev,NatureCommunications5, 5771(2014); arXiv:1406.0503.
Chapter 5:
•
“Feedback of superconducting fluctuations on charge order in the underdoped
cuprates,” Debanjan Chowdhury and Subir Sachdev, Phys. Rev. B 90,
134516 (2014); arXiv:1404.6532.
Chapter 6:
•
“Density wave instabilities of fractionalized Fermi liquids,” Debanjan Chowd-
huryandSubirSachdev,Phys. Rev. B90, 245136(2014); arXiv:1409.5430.
Chapter 7:
•
“Higgs criticality in a two-dimensional metal,” Debanjan Chowdhury and Subir
viii
Contents
Sachdev, Phys. Rev. B 91, 115123 (2015); arXiv:1412.1086.
Some additional work, which is not presented in this thesis, has appeared in:
“Breakdown of Fermi liquid behavior at the (π,π) = 2k spin-density wave
F
•
quantum critical point: the case of electron-doped cuprates,” D. Bergeron, De-
banjan Chowdhury, M. Punk, S. Sachdev and A.-M.S. Tremblay, Phys. Rev.
B 86, 155123 (2012); arXiv:1207.1106.
“Multipoint correlators of conformal field theories: implications for quantum
•
critical transport,” Debanjan Chowdhury, S. Raju, S. Sachdev, A. Singh and P.
Strack, Phys. Rev. B 87, 085138 (2013); arXiv:1210.5247.
“Topological excitations and the dynamic structure factor of spin liquids on
•
the kagome lattice,” M. Punk, Debanjan Chowdhury and S. Sachdev, Nature
Physics 10, 289-293 (2014); arXiv:1308.2222.
“Fluctuating charge order in the cuprates: spatial anisotropy and feedback from
•
superconductivity,”Y.Wang, DebanjanChowdhuryandA.V.Chubukov, Phys.
Rev. B 92, 161103(R) (2015); arXiv:1508.03636.
“Confinement transition to density wave order in metallic doped spin liquids”,
•
A.A. Patel, Debanjan Chowdhury, A. Allais and S. Sachdev, Phys. Rev. B
93, 165139 (2016); arXiv:1602.05954.
Electronic preprints (shown in typewriter font) are available on the Internet at the
following URL:
http://arXiv.org
ix
Acknowledgments
IthasbeenalmostsixyearssinceIcametoHarvardanditonlyfeelslikeyesterday!
My journey in graduate school has been an enriching experience, with its fair share
of bittersweet moments. We were told at the outset that the pursuit of a PhD is not
a sprint but a marathon. A number of people have helped me reach the finish line
and the least I can do now is express my gratitude towards them. Omissions, if any,
are entirely unintentional!
Theworkreportedinthisthesiswouldnothavebeenpossiblewithouttheguidance
of my advisor Professor Subir Sachdev. I consider myself to be extremely fortunate
to have been his student. Subir’s intuition for any given problem, combined with
his extraordinary technical skills are awe-inspiring. I must thank him for igniting
my interest in superconductivity and spin-liquids from early on, since it has been
great fun thinking about these subjects! Subir has supported me in far too many
endeavors to list here and I will remain eternally grateful to him for them. I hope
that our association will continue in the future and lead to many more successful
collaborations. I would also like to thank Subir and Usha for having been wonderful
hosts of several get-togethers that kept us in good spirits during this arduous journey!
I thank the other members of my thesis committee: Professor Eugene Demler,
Professor Amir Yacoby and Professor Philip Kim. They have always provided useful
feedback during the course of my PhD. In particular, I would like to thank Eugene for
teaching us three advanced quantum many-body courses during my first two years at
Harvard; my training in the subject would have remained incomplete had it not been
for these courses. I also thank Professor Bertrand Halperin for giving me multiple
opportunities to be a teaching fellow in his courses. It has been an honor to learn
x