FieldtheoryandTopologicalPhases

XiWu,D3High-energytheorygroup

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Quantumfieldtheoryisaveryusefultool

Quantumfieldtheory(QFT)• describesasystemwithinfinited.o.fbysmallnumberofd.o.f

• haveparticlepicture:1. creationandannihilation2. masseigenstatesasparticles

• isappliedtohigh-energyphysics,condensedmattersystem,Atomic,molecular,opticalphysics

TopologicalphasescanbestudiedbyQFT

Anewtypeofphasesofmatter(quantumphases)

• hasdistinctivegroundstatesbytopologicalinvariantinmomentumspace

• appearsinlowtemperature—>quasiparticleexcitations

• hasveryverylargenumberofd.o.f QFT

Boundary condition analysis

Thesummaryofmyresearch

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2. New exotic states—edge-of-edge states

3. Equivalence between wave function formalism(TKNN) and field theoretical formalism

1. Boundary conditions in topological phases

Thebenefitofboundaryconditionsanalysis

• Helpsusunderstandedgestatesintopologicalphases

• Helpsusunderstandtherelationbetweentheoriesindifferentdimensions

• It’safirstprinciplecalculation

1BoundaryConditionsinTopologicalPhases

XiWu(OsakaUniv.)

IncollaborationwithKojiHashimoto(OsakaU.)andTaroKimura(KeioU.)

Prog.Theor.Exp.Phys2017,053I01

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Doboundaryconditions affectedgestates?

• Doboundaryconditionsaffecttopologicalphasesingeneral

x

Topologicalmaterial

x

Wavefunctionofedgestates

0

Vacuum

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• Topologicalphasesofmatterarecharacterizedbytopologicalinvariantinmomentumspace

• Edgestatesareprotectedbythetopologicalinvariant

Doboundaryconditions affectedgestates?

Ourapproach:

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Continuumanalysis+latticesimulation

•Leastactionprinciple

•HermiticityofHamiltonian

•Hamiltonianeigenequation

(mypart)

Boundarycondition

Edgestatessolution

Boundaryconditionsaffectedgestates

Therotationdirectionisdictatedbythetopologicalnumber

1

2

3

Ourfindings:BoundaryconditionsofWeylsemimetalsaredictatedbyonesingleparameter

Thisparameterrotatesedgestatesdispersion

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Boundaryconditionsaffectedgestates

• Curveswithdifferentcolorscorrespondtodifferent

• Boundaryconditionsdon’tchangetheexistenceofedgestates.

Constantenergyslice

-1 +1

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anewexplanationofbulk-edgecorrespondence

2Edge-of-edgeStatesPhys.RevB95,165443(2017)

XiWu(OsakaUniv.)

IncollaborationwithKojiHashimoto(OsakaU.)andTaroKimura(KeioU.)

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Doboundaryconditions affectedgestates?

• Doboundaryconditionsaffecttopologicalphasesingeneral

x

Topologicalmaterial

x

Wavefunctionofedgestates

0

Vacuum

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• Topologicalphasesofmatterarecharacterizedbytopologicalinvariantinmomentumspace

• Edgestatesareprotectedbythetopologicalinvariant

Whatisanedge-of-edgestate?

Ifsomeedgestatehasanontrivialtopologyinmomentumspace

x�

Topologicalmaterial�

0�

Vacuum

edge�

edge�

edge-of-edge�

n 6= 0

Bulk �! edge �! e-o-e?

Energydispersiondiffersfrombulkandedgestates1

2

Whatweexpectforthee-o-e:

/ e�↵x��y

y Vacuum

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3 TKNN number and Ishikawa-Matsuyama(IM)

RelationsAn equivalence between wave function formalism

and field theory formalism

In collaboration with Prof. T Onogi(Osaka U.)

Xi Wu

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Two formalisms of Hall conductivity

• Wave function formalism: Hall conductivity is related with wave functions

�xy =X

EI<✏F

Z

BZ

d2p

(2⇡)2(@xa

Iy � @ya

Ix);

where aIi = �ihEI |@

@pi|EIi

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• Field theory formalism: Hall conductivity is related with the free fermion propagator

�xy / ✏↵µ⌫Z

d3p

(2⇡)3tr(S@↵S

�1S@µS�1S@⌫S

�1)

S(p) : fermion propagator Dirac sea

Energy bands

Our contribution

Dirac seaEnergy bands

1Wavefunctionformalismandfieldtheoryformalismareprovedtobeequivalentrigorously

2FieldtheoryformulaisshowntobevalidforarbitrarybandHamiltonian,notlimitedtoDiracfermions

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SummaryandFutureplan

• Interacting theory?

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Boundaryconditions

E-o-estates

TKNN formalism

Free fermion propagatorformalism

• Free theory—no electron-electron coupling

equivalent

Green’sfunction

Topologicalinvariant

Appendix:Boundaryconditionrotates edgestatesdispersion

• Obtainingedgestatesbysolving

• Edgestatesdispersionisaplanetangenttothebulkdispersion

• B.C.parameterappearsinthedispersionasarotationangle

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