FieldtheoryandTopologicalPhases
XiWu,D3High-energytheorygroup
1
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
4
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
6
Doboundaryconditions affectedgestates?
• Doboundaryconditionsaffecttopologicalphasesingeneral
x
Topologicalmaterial
x
Wavefunctionofedgestates
0
Vacuum
7
• Topologicalphasesofmatterarecharacterizedbytopologicalinvariantinmomentumspace
• Edgestatesareprotectedbythetopologicalinvariant
Doboundaryconditions affectedgestates?
Ourapproach:
8
Continuumanalysis+latticesimulation
•Leastactionprinciple
•HermiticityofHamiltonian
•Hamiltonianeigenequation
(mypart)
Boundarycondition
Edgestatessolution
Boundaryconditionsaffectedgestates
Therotationdirectionisdictatedbythetopologicalnumber
1
2
3
Ourfindings:BoundaryconditionsofWeylsemimetalsaredictatedbyonesingleparameter
Thisparameterrotatesedgestatesdispersion
9
Boundaryconditionsaffectedgestates
• Curveswithdifferentcolorscorrespondtodifferent
• Boundaryconditionsdon’tchangetheexistenceofedgestates.
Constantenergyslice
-1 +1
10
anewexplanationofbulk-edgecorrespondence
2Edge-of-edgeStatesPhys.RevB95,165443(2017)
XiWu(OsakaUniv.)
IncollaborationwithKojiHashimoto(OsakaU.)andTaroKimura(KeioU.)
11
Doboundaryconditions affectedgestates?
• Doboundaryconditionsaffecttopologicalphasesingeneral
x
Topologicalmaterial
x
Wavefunctionofedgestates
0
Vacuum
12
• 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
13
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
14
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
15
• 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
16
SummaryandFutureplan
• Interacting theory?
17
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
18