K理論とトポロジカル結晶物質

YITP,KyotoUniversityMasatoshiSato

• KenShiozaki(Riken)• KiyonoriGomi(ShinshuUniversity)

Outline

1.  Topological(crystalline)insulators/superconductors

2.  K-groupclassificaIon

3.  Nonsymmorphicsymmetryandnewtopologicalphases

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Mobiustwistedsurfacestates

Shiozaki-MS,Phys.Rev.B90,166114(2014).[41pages]Shiozaki-MS-Gomi,Phys.Rev.B91,155120(2015).Shiozaki-MS-Gomi,Phys.Rev.B93,195413(2016).Shiozaki-MS-Gomi,Phys.Rev.B95,235425(2017).[Editor’ssuggesIon,54pages]

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TI/TSC𝐸

Ordinaryinsulator/SC𝐸

Theideaoftopologicalinsulator/superconductor(TI/TSC)hasbeensuccessfullyestablishedwithmanyexperimentalsupportsforsurfacestates

Introduc.on

TI/TSC =Non-trivialtopological#ofoccupiedstate

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gaplesssurfacestate

Thenon-trivialtopologicalstructurepredictstheexistenceofgaplesssurfacestates

gapofexcitaIon

vacuum

TIs/TSCshavegaplessboundarystatesensuredbybulktopologicalnumbers

bulk-boundarycorrespondence

TI/TSC

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Topologicalinsulator(TI)

insulatorsuppor.ngsurfaceDiracfermions

Bi1-xSbx Angleresolvedphotoemissionspectroscopy(ARPES)(photoelectroneffect)

Bi2Se3

Energy(eV)

wavenumber(Å-1)

valenceband

conducIonband

Fermilevel

surfaceDiracfermions

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TopologicalSCs

3He-B

Sr2RuO4

CuxBi2Se3

CuxBi2Se3[Fu-Berg(10)]

[Sasaki-Kriener-Segawa-Yada-Tanaka-MS–Ando(11)]

[Yamakage-Yada-MS-Tanaka(12)]

[MS(10)]

CuxBi2Se3

Energy　E

kT-invarianttopologicalSC

T-breakingtopologicalSCchiral

helical

[Kashiwayaetal(11)]

[Murakawa,Nomuraetal(09)]

[Sasakietal(09)]

Experiment

Theory

spin-triplet(odd-parity)SC [MS(09,10),Fu-Berg(10)]

•  Diracfermion+s-wavecondensate MS(03)

Hsiehetal

Topologicalinsulator

S-waveSCDiracfermion+s-waveSC

S-waveSCscanhosttopologicalsuperconducIvityifaspinlesssystemisrealizedeffecIvely

Fu-Kane(08)

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•  S-wavesuperconducIngstatewithRashbaSO+Zeemanfield

Zeemanfield

FermiLevel

Nonspin-degeneratesingleFermisurface

[MS-Takahashi-Fujimoto(09),J.Sauetal(10)]

TopolgiocalSC

Mouriketal.,Science(2012)

InSb/NbTiN

1DNanowireZeemanfield

MFnanowire Lutchynetal(10),Oregetal(10)

MajoranaFermion

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KeytorealizeTI/TSC=SymmetryTime-reversalsymmetry(TRS)

Par.cle-holesymmetry(PHS)

Kramerspair

•  Nobackscajering•  topologicallystable

Majoranafermion

A

AIII

AIBDIDDIIIAIICIICCI

TRS PHS CS

0 0 0

d=1 d=2 d=3

0 Z 0

0 0 1 Z 0 Z

110-1-1-101

01110-1-1-1

01010101

0ZZ2Z202Z00

00ZZ2Z202Z0

000ZZ2Z202Z

IQHS

p+ipchiralpSr2RuO4,3He-A

3He-B

3DTICuxBi2Se3

Majoranananowire

QSH

[Schnyder-Ryu-Furusaki-Ludwig(12)][Avron-Seiler-Simon(83)]

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TopologicalCrystallineInsulator

Recently,ithasbeenrecognizedthatpointgroupsymmetryalsoprovidesnoveltopologicalsurfacestates

SnTe

[L.Fu(11),Hsiehetal(12)]

MirrorreflecIon

BZ

surfaceBZ

(110)

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Idea

Usingtheeigenvalueofmirroroperator,ky=0planecanbeseparatedintotwoQHstates.

[Y.Tanakaetal(12)]twoDiracfermions

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mirrorreflecIonsymmetrySr2RuO4

BZ

TopologicalCrystallinesuperconductor[Ueno-MSetal(13),Zhang-Kane-Mele(13))

Sr2RuO4

[Ueno-MSetal(13)]

ApairofMajoranafermions

[Ueno-MSetal(13)]

Applica.ontoSr2RuO4

Odd

WecanexpectMajoranaFermions!!

InthepresenceofmagneIcfieldsalongthez-direcIon,NMRdatesuggests

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Ques.ons

•  IsitpossibletoclassifysuchtopologicalphasessystemaIcally?

•  HowmanynewtopologicalphasescanweobtaininthepresenceofaddiIonalsymmetry?

Toanswertheseques.ons,weemploytheK-theory.

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Shiozaki-MS,Phys.Rev.B90,166114(2014).Shiozaki-MS-Gomi,Phys.Rev.B91,155120(2015).Shiozaki-MS-Gomi,arXiv:1511.01463.

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Ourse\ng

Insteadofoccupiedstates,weclassifyflajenedHamiltonians

emptyband occupiedband

wavefn. qfilledbands

pemptybands

+1

-1

onlydis.nc.onbetweenemptyandoccupiedbandisimportant

fla^enedHamiltonian

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Momentum space

Matrix (classifying space)

Map

TheflajenedHamiltoniandefinesamapfrommomentumspacetoHilbertspace

Ifthemapdefinesanon-trivialhomotopy,wemayhaveanon-trivialtopologicalphase

Theproblemishowsymmetryaffectsclassifyingspace

Addingtopologicallytrivialbandsmakestheclassifica.onsimpler

WhyweuseK-theory?

stableequivalence

[Kitaev(09)]

InaddiIontosimpledeformaIonofHamiltonians,theK-theoryapproachallowsustoaddtopologicallytrivialbandsduringthedeformaIonofHamiltonians

deformablebyaddingextratrivialbands

Thestable-equivalenceclassesdefinestopologicalphases

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Importantly,usingstableequivalence,wecanavoidannoyinginterferencebetweentopologicalcharges

trivialbandpair-annihila.onofmonopoles

NointerferenceThevortexandmonopolebelongtodifferentsectorsofHamiltonian

interferenceduetoABeffect

Butifweusestableequivalence…

monopole

vortex

TheK-theorysimplifiestheclassifica.onoftopologicalphases

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Ingeneral,topologicalphasescanbeunderstoodastheexistenceoftopologicalobjectsinthemomentumspace

Classifica.onofTCIsandTCSCs:K-theoryapproach

1dimtop.phase 2dimtop.phase

“vortex” “monopole”

1dimBZ 2dimBZ

=1dimcircle =2dimtorus

windingnumber Chernnumber

chiralsymmetry

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IntheframeworkofK-theory,onecanincreasethedimensionofthesystemsystemaIcally

chiralcase

non-chiralcase

Teo-Kane(10)

(d+1)-dimtop.phase

d-dimtop

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Thismapkeepsthetopologicalnumberbutitshitsthesymmetryofthesystem

classDIII(TRISCs)

classAII(TRIInsulators)

AIBDIDDIIIAIICIICCI

TRS PHS CS d=1 d=2110-1-1-101

01110-1-1-1

01010101

0ZZ2Z202Z00

00ZZ2Z202Z0

000ZZ2Z202Z

d=3

ThistermbreaksPHS

hierarchyoftop#

WegeneralizethisideatosystemswithaddiIonalsymmetry

classDIII(TRISCs)+mirrorreflec.on

classAII(TRIInsulators)+mirrorreflec.on

However,thisisnottheonlypossibility….

mirrorsym.

mirrorsym.

Thistermkeepsmirrorsym.

Shiozaki-MS(14)

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ThemappedHamiltonianalsohasadifferentaddiIonalsymmetry

mirrorreflec.on

2-foldrota.on

classDIII classAII

Differentaddi.onalsymmetrybutthesametopologicalstructure

Thesametop#

Thissymmetryflipskd+1=2-foldrota.on

d-dim (d+1)-dim

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Inthismanner,wecanchangethenumberofflippedcoordinatesd||underthesymmetry,withkeepingthetopologicalstructure

reflecIon two-foldrotaIon

d||=1 d||=2 d||=3

inversionZ2global

d||=0

・・・・

d||=4…

UsingtheserelaIons,wecompletetheclassificaIonofTCIsandTCSCsprotectedbyorder-twospace(andmagneIcspace)groups

glideZ2global+halftrans

two-foldscrew

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SnTe

Sr2RuO4

ExtendedtopologicalTable [Shiozaki-MS(14),Shiozaki-MS-Gomi(15)]

Asingleperiodictablewith10differenttopologicalclass

6periodictableswith27classes+

6periodictableswith10classes

222class

[Schnyderetal(08)]

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Ques.on

•  IsitpossibletorealizeessenIallynewstructureofsurfacestates?

Mobiustwistinsurfacestates

GlidesymmetrygivesrisetodisIncttopologicalstates

Shiozaki-MS-Gomi(15)

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glide=mirrorreflecIon+non-primiIvelavcetranslaIon

MirrorreflecIon

HalftranslaIon

NonsymmorphicSymmetry

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AAIII

AIBDIDDIIIAIICIICCI

d=1 d=2 d=3

0 Z 0

Z 0 Z

0ZZ2Z202Z00

00ZZ2Z202Z0

000ZZ2Z202Z

AAIII

AIBDIDDIIIAIICIICCI

d=1 d=2 d=3

Z2 0 Z20 Z2 0

Z2Z2Z4Z2Z2000

0Z2Z2Z4Z2Z200

00Z2Z2Z4Z2Z20

withglidesymmetry

[Schnyder-Ryu-Furusaki-Ludwig(08)]

[Shiozaki-MS-Gomi(15)]

GlidesymmetrygivesessenIallynewstructuresintopologicaltable

Z2#withoutan.-unitarysymmetry

Withan.-unitarysymmetry,wehaveZ4#

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GlidesymmetricHamiltonian

Liketopologicalcrystallineinsulators,kz=0planecanbeseparatedintotwoglidesubsectors

But,thereisanimportantdifference

[Shiozaki-MS-Gomi(15),Fang-Fu(15)]

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KeypointEigenvaluesofglideoperatordonothave2𝜋periodicityinkx

Asaresult,thesetwosubsectorsareconnectedwithatwist

Mobiusstripstructure

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Correspondingly,asurfacestatealsoshouldhaveaMobiustwiststructurewhenkz=0,

AlongthekxdirecIon,chiraledgestatein𝐺(𝑘↓𝑥 )= 𝑒↑𝑖𝑘↓𝑥 /2 sectorissmoothlyconnectedtoanI-chiralonein𝐺(𝑘↓𝑥 )=− 𝑒↑𝑖𝑘↓𝑥 /2 sector.

Chiralstate

An.-chiralstate

Mobiustwist

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InaddiIon,fromtheMobiustwiststructure,thetwoglidesubsectorscanbeexchangedadiabaIcally

Chiralstate

An.-chiralstate

Chiralstate

An.-chiralstate

Chiralmode(N=1)=An.-chiralmode(N=-1)Z2topologicalphase

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InteresIngly,thesurfacestatewithMobiustwistcanbeopeninthekx-direcIon

TopologicalInsulator

Bi2Se3

[Hsiehetal(09)]

Surfacestateisalwaysa^achedtothebulkstate

Surfacestateisdetachedfromthebulkstateatkz=0

Thepresentcase

ThisfeaturecanbeusedtoidenIfythisphasebyARPESexperiments

atkz=0

InthepresenceofTRS,theKramersdegeneracyatthezoneboundaryrequirestwohelicalmodestohavetheopenstructure.

Kramersdegeneracyatboundary

Z4phase

twohelicalmode

(TRS+glide) Shiozaki-Gomi-MS,arXiv:1511.01463(15)

Twochiralstate

Twoan.-chiraledgestate

Atkz=0

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Twochiralmode(N=2)=Twoan.-chiralmode(N=-2)

Z4topologicalphase

Usingasimilarargumentasbefore, Shiozaki-Gomi-MS,arXiv:1511.01463(15)

Twochiralstate

Twoan.-chiralstate

Twoan.-chiralstate

Twochiralstate

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Wangetal,Nature(16)KHgSb Po-YaoChangetal,arXiv:1603.03435CeNiSn

•  Hourglassfermion •  MobiusKondoInsulator

Materialrealiza.on

MaterialrealizaIonswerequicklyproposedaterourfinding

Wecanexpectothercandidates..•  HeavyfermionSCs

Summary

•  SpacegroupaswellasIme-reversalandparIcle-hole(=charge

conjugaIon)symmetriesgivenoveltopologicalphases.

•  UsingtheHamiltonianmappingshitingsymmetry,asystemaIcclassificaIonofTCIsandTCSCscanbedone

•  ThenonsymmorphicsymmetryprovidesaninteresIngMobius

twiststructureofthesurfacestate.

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