Compe&ngordersinDiracfermions

Ins$tutfürTheore$schePhysikundAstrophysikUniversitätWürzburg

ToshihiroSato

TheWorkshop“TrendsinTheoryofCorrelatedMaterials(TTCM2018)”,October8-10,2018

FakherF.Assaad(UniversitätWürzburg)

Mar$nHohenadler(UniversitätWürzburg)

ZhenjiuWang(UniversitätWürzburg)

YuhaiLiu(BeijingNormalUniversity)

WenanGuo(BeijingNormalUniversity)

ChongWang(PerimeterIns$tuteforTheore$calPhysics)

Collaborators

2

Intertwinedorders

Landau-Ginzburg-Wilson(LGW)theoryoftwoorderparameters

Orderparameter

3

1storder

Coexistence

Con$nuous

Disorderedphase

g

!ϕ

!ϕ

!ϕ

!ϕ

!φ

!φ

!φ

g

g g

BreakdownofLGWtheory(nofinetuning)

!φ

PopularmasstermsofDiracfermions

4

Avarietyofquantumordersopensagapanddestroysthesemimetal

Néel(AFM)Semenoff(CDW)

Kekulé(VBS)Kane-Mele(QSH)

+m +m

+m

−m −m

−m

+δ +δ

+δ

−δ −δ

−δ

−δ

−δ

−δ

+δ

+δ

+δ

+i χ

n Theore$calinsightintermsofcombina$onsofDiracmasstermsn OuraimistostudywithexactQMCsimula$onscompe$ngorders　inDiracfermions.・Wedesignthemodelswherecompe$ngordersaredynamicallygenerated.n Ourgoalistounderstandtheresultsintermsofi)Symmetrygroupoftherespec$veorderedphases

ii)Algebraicproper$esofthecorrespondingDiracmassterms

IntertwinedordersinDiracfermions

5

S.Ryu,C.Mudry,C-Y.Hou,C.Chamon|PRB(2009)

Inthistalk…

6

I.Froman&ferromagne&smtoKekulévalencebondsolid:Non-Landautransi&onwithEmergentsymmetryTS,M.Hohenadler,F.F.Assaad,PRL119,197203(2017)II.Superconduc&vityfromCondensa&onofTopologicalDefectsinQuantumSpin-HallInsulatorY.Liu,Z.Wang,TS,M.Hohenadler,C.Wang,W.Guo,F.F.Assaad,manuscriptinprepara$on・AuxiliaryfieldquantumMonteCarlosimula$ons:　ALF@hkp://alf.physik.uni-wuerzburg.de|SciPostPhys.3,013(2017)

I.Froman&ferromagne&smtoKekulévalencebondsolid:Non-Landautransi&onwithEmergentsymmetry

7

TS,M.Hohenadler,F.F.Assaad,PRL119,197203(2017)

ChiralSO(3)Neelmassterm Z2KVBSmassterm

versus

WeintroducethemodelinDiracfermionwithdynamicallygenerated,an$-commu$ngchiralSO(3)NeelandZ2Kekulémassterms.

8

+m +m

+m

−m −m

−m −δ

−δ

−δ

+δ

+δ

+δ

!MAFM,MKVBS{ }= 0

MKVBS =1⊗ iγ0γ5!MAFM =

!σ ⊗γ0

U t

Jh

ξ

ξ

ξ

Fermion

Ising spin

0

0

0

-ξ

-ξ

-ξ

A1

A2

unitcell

sijz, sij

x

DesignerHamiltonian

9

Honeycomb-lamceHubbardmodelathalffilling

Transverse-fieldIsingspins(J = -1)

Fermion-Isingspincouplingξij = 0, ±ξ ξ = 0.5( )

ciσ†

H = −t ciσ† c jσ + h.c.( )

ij ,σ∑ +U ni↑ −

12

⎛

⎝⎜

⎞

⎠⎟

i∑ ni↓ −

12

⎛

⎝⎜

⎞

⎠⎟+ J sij

zsklz

ij,kl∑ − h sij

x

ij∑ + ξij sij

zciσ† c jσ

ij σ∑

・LamceswithL×Lunitcells:V=6L2

・SO(3)symmetry

・Z2symmetry:invarianceunderthecombinedopera$onofinversionand  sij

z →−sijz

DesignerHamiltonian

10

Relevantsymmetriesinfermionicsector:

SO(3)⊗ Z2

SO(3)spinrota$on

brokenbyNéel(AFM)orderatlargeenoughU

Z2inversion

brokenbykekulé(KVBS)orderatsmallenoughh

ChiralSO(3)Néelmassterm Z2KVBSmassterm

H = −t ciσ† c jσ + h.c.( )

ij ,σ∑ +U ni↑ −

12

⎛

⎝⎜

⎞

⎠⎟

i∑ ni↓ −

12

⎛

⎝⎜

⎞

⎠⎟+ J sij

zsklz

ij,kl∑ − h sij

x

ij∑ + ξij sij

zciσ† c jσ

ij σ∑

+m +m

+m

−m −m

−m −δ

−δ

−δ

+δ

+δ

+δ

QMCresult:1/h-Uphasediagram

0

2

4

6

0.1 0.2 0.31/h

UAFM

KVBS

Semimetal

T=0.05

PujariPRL2016

k0 : order wave vector   δk ~ 2π LRAFM KVBS =1−λAFM KVBS1 k0 +δk( )λAFM KVBS1 k0( )

 

λα

1 k( ) : largest eigenvalue of Cγδα k( ) =V −1 OiγOjδe

ik⋅ Ri−Rj( )

ij∑  

・L→∞: RO →1(RO →0)inorder(disordered)phase

・Scale-invariantforLatcri$calpoint

n Finite-sizescalingofcorrela$onra$os:

α =AFM : O = Si = ciσ† !σσσ 'c j ʹσσσ '∑

       KVBS : O = Bij = −t ciσ† c jσ + c jσ

† ciσ( )σ∑  

RAFM KVBS = FAFM KVBS L1 υ h−1 − hc,AFM KVBS

−1( )⎡⎣

⎤⎦ 11

00.20.40.60.81

L=3L=6L=9

0.26 0.28 0.3 0.32

R KVBS

1/h

00.20.40.60.81

L=3L=6L=9

R AFM

0

2

4

6

0.1 0.2 0.31/h

UAFM

KVBS

Semimetal

QMCresult:1/h-Uphasediagram

0.8

0.9

1

0.26 0.28 0.3 0.32

L=3L=6L=9

1/h

F/h

0

0.5

1

1.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

1/h=0.330.3130.303

0.2990.2940.25

1/L

sp

Single particle gap Δsp  at Dirac point

Largegap

Free- energy derivative 

Con$nuous

12

T=0.05

AFM

KVBS

U=6

n AFM-KVBSphasetransi$on

U=6

RAFM KVBS = FAFM KVBS L1 υ h−1 − hc,AFM KVBS

−1( )⎡⎣

⎤⎦

6L2 sites

6L2 sites

Field-theoryinterpreta$onandemergentsymmetry

13

n AFMandKVBSmassesan$-commute→superspin

→gap

n Thesymmetry-breakingtermsbecomeirrelevantatthecri$calpoint

→EmergentSO(4)symmetry

!φ =

!mAFM,mKVBS( )

γ0 =σ 0 ⊗σ z    γ1 =σ z ⊗σ y    γ2 =σ 0 ⊗σ x   γ5 =σ y ⊗σ y     γµ,γν{ }= 2δµν   σ : Pauli matrices

!mAFM

mKVBS

KVBSphaseAFMphase Cri$calpoint

φ !mAFM

mKVBS

!mAFM

mKVBS !φ ≠ 0

Δsp =!φ =

"mAFM2 +mKVBS

2

L = Ψσ

!x,τ( ) ∂µγµδσ ʹσ +

!mAFM!x,τ( )

mKVBS!x,τ( )

⎛

⎝⎜⎜

⎞

⎠⎟⎟⋅

!σσσ ' ⊗12iγ5δσ ʹσ

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥σ ʹσ

∑ Ψ ʹσ

!x,τ( )+Lφ

Fulllow-energytheory: !φ

EvidenceforAFM-KVBStransi$onwithemergentSO(4)symmetry

n Fluctua$onofAFM,KVBSorderparameters:

n Expectcircularhistogramforjointprobabilitydistribu$on N.Kawashima|PRL(2007)

14

σO = O2 − O 2    ,   O = S, s

IngeneralindependentbutlockedtogetherifSO(4)symmetryemerges.

ηAFM =ηKVBS→σ KVBS σ AFM = F h− hc( )L1 ν⎡⎣ ⎤⎦

1

10

0.26 0.28 0.3 0.32

L=3

L=6

L=9

1/h

KV

BS/

AF

M

0 max.

KVBS

AFM

AFM KVBS

U=6

A.Nahumetal.|PRL(2015)

6L2 sites

n RealizedinQMCsimula$onsofeasy-planeJQmodel

n ExponentsareconsistentwiththoseoftheeasyplanJQmodela

Cri$calityofAFM-KVBStransi$on

H = −J Pij −ΔSizSj

z( )ij∑ −Q Pij

ijklmn∑ PklPmn    ,   Pij =

14−!Si ⋅!Sj

Qinetal.|PRX2017

15

SO(3)⊗ Z2

0

1

2

0.26 0.28 0.3 0.32

L=3L=6L=9

AFML2

/

1/h-1 -0.5 0 0.5 1

L=3L=6L=9

L1/ (1/h-1/hc)

η = 0.10  2β ν =η + d − 2 =1.10  ν = 0.48  1 hc = 0.295

η = 0.13 3( )   ,   ν = 0.48 2( )

U=6

xy-AFMVBS

DQCPQ

→・Microscopicmodelofdeconfinedquantumcri$calpoint(DQCP)

・EmergentSO(4)symmetryunifyingxy-AFMandVBSorderparameters

U(1)⊗ Z4

6L2 sites

II.Superconduc&vityfromCondensa&onofTopologicalDefectsinQuantumSpin-HallInsulator

16

Y.Liu,Z.Wang,TS,M.Hohenadler,C.Wang,W.Guo,F.F.Assaad,manuscriptinprepara$on

SO(3)QSHmassterm s-waveSCmassterm

WeintroducethemodelinDiracfermionwithdynamicallygeneratedthequintupletofan$-commu$ngquantumspinHall,SO(3),

ands-waveSC,U(1),massterms.

17

!MQSH,

!MSSC{ }= 0

!MQSH =

τ z ⊗σ x ⊗ iγ0γ3γ5τ0 ⊗σ y ⊗ iγ0γ3γ5τ z ⊗σ z ⊗ iγ0γ3γ5

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

!MSSC =

τ y ⊗σ y ⊗ iγ0γ2γ3τ x ⊗ iσ y ⊗ iγ0γ2γ3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

H = −t ci†c j + h.c.( )

ij∑ −λ iν ijci

†σ c j + h.ci, j ∈      ∑

⎡

⎣⎢⎢

⎤

⎦⎥⎥

∑2

DesignerHamiltonian

18

Rij

ciσ†

Fermionathalffilling

t

SO(3)⊗U(1)i

ν ij =

!ez ⋅!Rik ×

!Rkj( )

!ez ⋅!Rik ×

!Rkj( )

= ±1 depending on its direction, sublattice, and spin 

・LamceswithL×Lunitcells:V=2L2・Con$nuousglobalsymmetries:

j

k

ci† = ci,↑

† , ci,↓†( )

n Finite-sizescalingofcorrela$onra$os:

QMCresult:groundstatephasediagram

0 0.01 0.02 0.03 0.04SemimetalQSHSSC

19

PujariPRL2016

RQSH SC =1−λQSH SC1 k0 +δk( )λQSH SC1 k0( )

  k0 : order wave vector   δk ~ 2π L

λα

1 k( ) : largest eigenvalue of Cγδα k( ) = dτV −1 Oiγ τ( )Ojδ 0( )eik⋅ Ri−Rj( )

ij∑0

β

∫  

・L→∞: RO →1(RO →0)inorder(disordered)phase

・Scale-invariantforLatcri$calpoint RQSH SC = FQSH SC L1 υ λ −λcQSH SC( )⎡

⎣⎤⎦

α =QSH : Oiδ = iciσ† !σσσ 'ci+δ ʹσ + h.c.σσ '∑    ,   SC : Oi =1 2 ci↑

† ci↓† + h.c.( ) 

QMCresult:groundstatephasediagram

20

n QuantumspinHallinsulator(QSH)byspontaneouslybrokenSO(3)symmetry

0 0.01 0.02 0.03 0.04SemimetalQSHSSC

0.5

0.6

0.7

0.8

0.017 0.018 0.019 0.020 0.021

RQSH

χ

λ

a

0.0180

0.0185

0.0190

0.0195

0.0 0.1 0.2

λQSH

c1

(L)

1/L

0.8

1.0

1.2

1.4

1.6

0.0 0.1 0.2

1/ν(L

)

1/L

b

0.5

0.6

0.7

0.8

0.9

0.0 0.1 0.2

1/η(L)

1/L

c

L = 6L = 9L = 12

L = 15L = 18L = 21

a+ b/Lc

a+ b/Lc a+ b/Lc

SemimetalQSH

(2L2 sites)

R QSH

QMCresult:groundstatephasediagram

21

n Quantumphasetransi$onbetweenQSHands-wavesuperconduc$vity(SSC)

0 0.01 0.02 0.03 0.04SemimetalQSHSSC

DQCP

21 22 23 24 25

0.031 0.032 0.033 0.034 0.035

∂F/∂λ

λ

L=6L=9

L=12L=15L=18L=21

0

0.6

1.2

0 0.1 0.2 0.3 0.4

ΔSP

1/L

λ=0.032λ=0.033λ=0.034Largegap

Con$nuous

(2L2 sites)

0.6

0.7

0.8

0.9

1.0

0.031 0.032 0.033 0.034 0.035

RQSH

χ

λ

a

0.6

0.7

0.8

0.9

1.0

0.031 0.032 0.033 0.034 0.035

RSC

χ

λ

b

0.030

0.032

0.034

0.036

0.0 0.1 0.2

λO c2(L

)

1/L

QSH

SC

L = 6L = 9L = 12

L = 15L = 18L = 21 (2L2 sites)

R QSH

R S

C

n AllowsforSkyrmiontopologicaldefects,Skyrmioncarriestheelectricchargetwo Lowenergyeffec$veac$oninthepresenceofachargegaugefield　:

Integrateoutthefermions,thelargemassexpansion:

Skyrmiondensity：

QSHinsulatorbyspontaneouslybrokenSO(3)symmetry

22

QSkyrmionc = 2eQ   ,   Q = dxdy 1

8πε 0νλ!N ⋅∂ν

!N ×∂λ

!N∫ : pontryagin index

S = d3x 1g∂µ!N( )

2+ 2iAµ

cJµT∫    ,   Jµ

T =18π

εµνλ!N ⋅∂ν

!N ×∂λ

!N

!N : order parameter in terms of threefold QSH mass

S = d3xψ γµ −i∂µ + Aµc( )+ imQSH

!σ ⋅!N !x,τ( )⎡

⎣⎤⎦∫ ψ

Aµc

T.GroverandT.Senthil|PRL(2008)

n Testthepictureonthelamce

Pontryaginindex:

H = −t ci†c j + h.c.( )

ij ,σ∑ +λ

!N    ⋅ iν ijci

†σ c j + h.c.i, j ∈      ∑

⎛

⎝⎜⎜

⎞

⎠⎟⎟∑

QSHinsulatorbyspontaneouslybrokenSO(3)symmetry

23

Q =18π!N !x

!x∑ ⋅

!N !x+!a1 −

!N !x( )×

!N !x −

!N !x+!a2 +

!N !x −

!N !x−!a1+!a2( )

Densityofstateforsingleskyrmionconfigura$onQ∼-1

L=36t=1λ=0.5periodicboundary

Densityofstateforuniformpolariza$onQ=0

!a1, !a2 : unit vector 

ci† = ci,↑

† , ci,↓†( )

n Testthepictureonthelamce

QSHinsulatorbyspontaneouslybrokenSO(3)symmetry

24

L=36t=1λ=0.5openboundary

・Theinser$onofaskyrmionpumpsapairofchargesfromthevalencetotheconduc$onbandthroughtheedge.

Densityofstateforsingleskyrmionconfigura$onQ∼-1

n AllowsforSkyrmiontopologicaldefects,Skyrmioncarriestheelectricchargetwo

　・Thecondensa$onofskyrmiondefectsinaninterac$on-generatedQSHinsulatorleadstogenerates-wavesuperconduc$vity.　・IncontrasttoaBCS-typesuperconduc$vity,itsvor$cesencloseaspin-1/2degreeoffreedomcorrespondingtothefrac$onalizedQSHorderparameter.・TheQSH-SCtransi$onisanexampleofadeconfinedquantumcri$calpoint.

QSHinsulatorbyspontaneouslybrokenSO(3)symmetry

25

I.Froman&ferromagne&smtoKekulévalencebondsolid:

II.Superconduc&vityfromtheCondensa&onofTopologicalDefectsinaQuantumSpin-HallInsulator

Summary

26

0

2

4

6

0.1 0.2 0.31/h

U

AFM

KVBS

Semimetal

・Con$nuousAFM-KVBStransi$onwithemergentSO(4)symmetry

0 0.01 0.02 0.03 0.04SemimetalQSHSSC

DQCP

TS,M.Hohenadler,F.F.Assaad,PRL119,197203(2017)

Y.Liu,Z.Wang,TS,M.Hohenadler,C.Wang,W.Guo,F.F.Assaad,manuscriptinprepara$on

・QSHinsulatorbyspontaneouslybrokenSO(3)spinsymmetry

QMCresult:groundstatephasediagram

27

n QuantumspinHallinsulator(QSH)byspontaneouslybrokenSO(3)symmetry

0 0.01 0.02 0.03 0.04SemimetalQSHSSC

η = 0.78 9( )   ,   1 ν =1.1 2( )

O(3) -GN universality class : η = 0.76 2( )   ,   1 ν =1.02 1( ) Otsukaetal.|PRX2016

O(3)-Gross-Neveu

QMCresult:groundstatephasediagram

28

n Quantumphasetransi$onbetweenQSHands-wavesuperconduc$vity(SSC)

0 0.01 0.02 0.03 0.04SemimetalQSHSSC

DQCP

ηQSH = 0.22 6( )   ,   ηSC = 0.23 4( )1 νQSH =1.8 4( )   ,   1 ν SC =1.79 6( )

AFM-VBS transition : ηAFM = 0.259 6( )   ,   ηVBS = 0.25 3( ) Nahumetal.|PRX2015

n NoncompactCP1model:　 Skyrmionnumber→fluxoftheCP1gaugefield:

QSH-SSCtransi$onintermsofDQCP

!N = z†

!σ z

z = z1, z2( ) : two component complex spinon field   Seff =

1g

d3x −i∂µ − aµ( ) z∫2   ,   aµ = −iz

†∂µz

Q =12π

dxdyε 0µν∂µaν∫

ConjecturedRGflowofpresentmodelSinglediverginglengthscale

ConjecturedRGflowofDQCPfortheAFM-VBStransi$on

T.Senthiletal|JPSJ(2005)

SMGN-O(3)QSHDQCPSSCλ

C4

29

T.GroverandT.Senthil|PRL(2008)