Theoretical Study on Correlation Effects in Topological Matter

143 Pages·2017·5.037 MB·English

by  Hiroki Isobe (auth.)

Springer Theses
Recognizing Outstanding Ph.D. Research
Hiroki Isobe

Theoretical Study
on Correlation Effects
in Topological Matter
Springer Theses
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Hiroki Isobe
Theoretical Study
on Correlation Effects
in Topological Matter
Doctoral Thesis accepted by
the University of Tokyo, Tokyo, Japan
123
Author Supervisor
Dr. Hiroki Isobe Prof. NaotoNagaosa
Massachusetts Institute of Technology TheUniversity of Tokyo
Cambridge, MA Tokyo
USA Japan
ISSN 2190-5053 ISSN 2190-5061 (electronic)
SpringerTheses
ISBN978-981-10-3742-9 ISBN978-981-10-3743-6 (eBook)
DOI 10.1007/978-981-10-3743-6
LibraryofCongressControlNumber:2017934863
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’
Supervisor s Foreword
Diracinventedaningeniouswaytomergetherelativityandquantummechanicsto
express the relation E ¼mc2 in terms of the four-component wavefunction. The
relativistic wave equation called Dirac equation constitutes the most fundamental
basis of high-energy physics. When the mass of the particle is zero, the
four-component Dirac equation is split into two two-component equations called
Weyl equation. On the other hand, relativistic fermions appearing in the electronic
band structures of solids are attracting intensive recent interests. Dirac fermion in
graphene is a representative two-dimensional example, where many novel phe-
nomena have been predicted theoretically and also observed experimentally.
Massive Dirac fermion offers a good description of the bulk states of the
three-dimensionaltopologicalinsulator,whichresultsinthemasslessWeylfermion
on its surface. Weyl semimetals with broken spatial inversion or time-reversal
symmetries are also the new emerging field of research.
Topology is closely related to these relativistic fermions. For example, Weyl
fermion with two-component spinor is characterized as the Dirac monopole or
anti-monopole of the Berry curvature in momentum space depending on its chi-
rality. One can define the topological index defined by the integral of this Berry
curvature over the surface enclosing the Weyl point. Chiral anomaly is another
topologicalaspectwherethechiral charge isnotconservedinthepresenceofboth
magnetic and electric fields. These topological features can be observed in solids
through the measurements of the magnetic, transport, thermal, and optical
properties.
However, studies of relativistic fermions in solids are not limited to the simple
application of ideas developed in high-energy physics. They have much richer
physicsuniquetosolidssuchasthedifferentvelocityofelectronsfromthatoflight
c,anisotropyofthevelocities,generalizeddispersionsdifferentfromthelinearone
such as the quadratic dispersion along some direction. Moreover, dimensionality
can be controlled, i.e., one-, two-, and three-dimensional systems can be prepared.
The small velocity of electrons compared with that of light enhances the
effective fine structure constant a. In the quantum electrodynamics (QED), this
v
vi Supervisor’sForeword
dimensionlesscouplingconstantaisgivenbytheuniversalconstantsasa¼ e2 with
hc
h being the Planck constant. The smallness of a(cid:2)1=137 justifies the perturbative
renormalization group analysis of QED. In solids, on the other hand, c is replaced
by the electron velocity v, and hence, a can be of the order of unity and the
interaction effect becomes very important.
Dr.Isobehasaddressedexactlythisissue,i.e.,theinteractioneffectinrelativistic
fermions in solids, in terms of the renormalization group (RG) method. First, he
formulated the QED with the different velocities between the electrons and light,
and found that these two approaches due to the renormalization effect in the low
energy limit. Namely, the Lorentz invariance is recovered asymptotically in the
fixed point. Correspondingly, there are two energy scales in the problem, which
correspond to the crossovers of the dielectric constant and magnetic susceptibility,
respectively.
Next, Dr. Isobe extended the analysis to the tilted Weyl fermion, i.e., the case
withthedifferentvelocitiesdependingonthedirection.Thissituationisrealizedin
anorganicsemimetala-BEDTsalt.HefoundtheRGequationsforthevelocitiesas
thefunctionoftheenergy,whichleadstothenontrivialenergydispersionduetothe
interaction effect. This leads to the prediction of the nontrivial temperature
dependence of the magnetic susceptibility measured in NMR, and the good
agreement between theory and experiment is obtained. Furthermore, as a potential
possibility,hepredictstheCherenkovradiationandevenmoreexoticdispersionin
the tilted Weyl systems.
Another possible generalization of Weyl fermions is the k2-dispersion in some
direction.Thisisarealisticsituationinsolids,e.g.,atthequantumphasetransition
point where the two Weyl fermions collide and pair-annihilate. Dr. Isobe studied
the interaction effect in this generalized Weyl fermion and found that it strongly
dependsonthedimensionalityofthemodel.Inthreedimensions,theinteractionis
irrelevantandthesystemisaFermiliquidwithoutthelogarithmiccorrectiondueto
marginallyirrelevantoperators.Intwodimensions,ontheotherhand,hefoundthe
system becomes a non-Fermi liquid in the lowest energy/temperature. This is a
remarkable example where the solid RG analysis predicts the nontrivial ground
state of the many-body system.
Studies of relativistic fermions will continue to produce interesting physics and
beanactivefield,andIbelieveDr.Isobe’sthesiswillbeaniceintroductiontothis
exciting topic.
Tokyo, Japan Prof. Naoto Nagaosa
December 2016
List of Publications
Parts of this thesis have been published in the following journal articles:
(cid:129) HirokiIsobeandNaotoNagaosa,Theoryofaquantumcriticalphenomenonina
topological insulator: (3+1)-dimensional quantum electrodynamics in solids,
Physical Review B 86, 165127 (2012).
(cid:129) Hiroki Isobe and Naoto Nagaosa, Renormalization group study of electro-
magnetic interaction in multi-Dirac-node systems, Physical Review B 87,
205138 (2013).
(cid:129) Hiroki Isobe and Naoto Nagaosa, Renormalization Effects on Quasi-Two-
Dimensional Organic Conductor a-(BEDT-TTF) I , Journal of the Physical
2 3
Society of Japan 81, 113704 (2012).
(cid:129) HirokiIsobeandNaotoNagaosa,CoulombInteractionEffectinWeylFermions
withTiltedEnergyDispersioninTwoDimensions,PhysicalReviewLetters116,
116803 (2016).
(cid:129) Hiroki Isobe and Naoto Nagaosa, Generalized Hund’s rule for two-atom sys-
tems, Physical Review B 90, 115122 (2014).
(cid:129) Hiroki Isobe and Naoto Nagaosa, Enhancement of spin-orbit interaction by
competitionbetweenHund’scouplingandelectronhopping,JournalofPhysics:
Conference Series 592, 012058 (2015).
(cid:129) Hiroki Isobe and Liang Fu, Theory of interacting topological crystalline insu-
lators, Physical Review B 92, 081304(R) (2015).
Someworksabouttopologicalphasetransitionandquantumcriticalphenomena,
which not presented here, are published in the following articles:
(cid:129) Bohm-Jung Yang, Mohammad Saeed Bahramy, Ryotaro Arita, Hiroki Isobe,
Eun-Gook Moon, and Naoto Nagaosa, Theory of Topological Quantum Phase
Transitions in 3D Noncentrosymmetric Systems, Physical Review Letters 110,
086402 (2013).
(cid:129) Bohm-Jung Yang, Eun-Gook Moon, Hiroki Isobe, and Naoto Nagaosa,
Quantum criticality of topological phase transitions in three-dimensional
interacting electronic systems, Nature Physics 10, 774–778 (2014).
(cid:129) HirokiIsobe,Bohm-JungYang,AndreyChubukov,JörgSchmalian,andNaoto
Nagaosa, Emergent Non-Fermi-Liquid at the Quantum Critical Point of a
TopologicalPhaseTransitioninTwoDimensions,PhysicalReviewLetters116,
076803 (2016).
vii
Acknowledgements
This thesis is based on the works I have done during my graduate course and
writingthethesiswouldhavebeenimpossiblewithoutthehelpandencouragement
of many people.
First of all, I would like to express my gratitude to my supervisor Prof. Naoto
Nagaosaforhiskindguidanceandsupportformyresearch.Goingbackfiveyears,
itwasobviouslyoneofmotivatorsformetostarttheoreticalresearchofcondensed
matter physics that his invitation to this exciting field. Through discussions and
conversations with him, I have learned a lot about physics. What he taught me is,
more importantly, the way to tackle a problem from a broader and intuitive
viewpoint and to survive this severe “battle field.”
I appreciate stimulating discussions and conversations with the present and
former members of Nagaosa group: Prof. Masahito Mochizuki, Prof. Yuki
Kawaguchi, Dr. Motohiko Ezawa, Dr. Naoyuki Sugimoto, Dr. Akihito Takeuchi,
Dr. Atsuo Shitade, Dr. Ryuji Takahashi, Dr. Sho Nakosai, Mr. Daichi Asahi,
Mr. Yoshihiro Shimada, Mr. Junichi Iwasaki, Mr. Ko Kikutake, Mr. Ryohei
Wakatsuki,Mr.XiaoxiaoZhang,Mr.KeitaHamamoto,Mr.ToshiyukiOshima,and
Mr.FanYiteng.Throughchatsandmorningmeetings,Ihavebeeninspiredtoforge
ahead with my research. Besides, I extend thanks to Ms. Tae Tokuyoshi and
Ms. Rie Tsukui for their unstinting generosity and administrative support. I have
enjoyedafulfillinglifeoncampuswiththeassistanceofthemembersofcondensed
matter theory groups, including Dr. Takahiro Misawa, Dr. Yohei Yamaji, Dr. Joji
Nasu, Dr. Satoru Hayami, Dr. Yusuke Nomura, Mr. Junki Yoshitake, Mr. Ryo
Ozawa, Mr. Amane Uehara, Mr. Yusuke Sugita, and Mr. Petr Mishchenko.
Discussions and conversations with members of other group have expanded the
breadth of knowledge, especially on strongly correlated systems.
During my graduate course, I have enrolled in and have been supported by the
Advanced Leading Graduate Course for Photon Science (ALPS) course.
IacknowledgefruitfuldiscussionswithProf.TetsuoHatsudaassecondaryadvisor.
His comments from the viewpoint of high-energy physics gave a new perspective
ix
x Acknowledgements
oncondensedmatterphysics;theboundarybetweencondensedmatterphysicsand
high-energy physics is blurring recently.
Supported by the ALPS course, I had the privilege to stay at Massachusetts
InstituteofTechnologyfor threemonths.On referral fromProf. Nagaosa,Ivisited
Prof. Liang Fu and had a lot of exciting discussions with him. The work on
topological crystalline insulators has been conducted in collaboration with him.
IappreciateusefulcommentsfromProf.SenthilTodadriandDr.ChongWang,and
I warmly thank Dr. Timothy Hsieh, Dr. Maksym Serbyn, Dr. Justin Song,
Dr. Junwei Liu, and Ms. Margaret O’Meara for their hospitality.
I would like to thank Prof. Kazushi Kanoda and Dr. Michihiro Hirata for
introducingmeintriguingexperimentalresults,whichinspiredtheworkonelectron
correlationeffectinorganicconductors.IamgratefultoProf.HidetoshiFukuyama
fordiscussionsandhisdeepinsightintothediagrammaticanalysis.Iwouldliketo
expressmygratitudetoDr.Bohm-JungYang fordiscussionsontopologicalphase
transition and renormalization group calculations. I am thankful for collaborations
on the work about topological quantum phase transitions of Dr. Eun-Gook Moon,
Dr. Mohammad Saeed Bahramy, and Prof. Ryotaro Arita. About numerical cal-
culations for microscopic models, I appreciate useful comments from Dr. Satoshi
Okamoto.
In completing this thesis, I am grateful to important comments from the mem-
bers of the dissertation committee: Prof. Naoto Nagaosa, Prof. Kazushi Kanoda,
Prof. Yukitoshi Motome, Prof. Hosho Katsura, and Prof. Takashi Oka.
IacknowledgethesupportbyGrant-in-AidforJSPSFellowsduringmydoctoral
course.
Finally, I appreciate my family for their encouragement and understanding.
Throughout my life in Tokyo, I have enjoyed their unflagging support. I cannot
expresshowgratefulIamformyfamily,especiallyformyparents.Iwouldliketo
devote this thesis to them.