1/15

Boson-Fermion Duality
in Four Dimensions
Takuya Furusawa (TITech/RIKEN)
TF & Y. Nishida, PRD., 99, 101701(R)
YITP, September, 2019

2/15Outline
1. Introduction

Basic concepts, boson-fermion duality in 3D,…

2. Boson-fermion duality in 4D

Free Dirac <=> Abelian-Higgs model of Θ =3. Summaryπ

F & Y. Nishida, PRD., 99, 101701(R)3/15Duality

̶ equivalence between 2 diﬀerent QFTs.
QFT1

DualityQFT2

̶ a powerful tool to strongly-correlated systems.
Strong-weak duality
Mean ﬁeld approaches to topological orders…

We focus on boson-fermion dualities(= dualities btw. bosonic and fermonic theories)

Boson-fermion duality in 3D
Boson theory
224/15Fermion theory

4iμμνρ
ψ̄(γ (∂μ − iAμ) + m)ψ +ϵ Aμ∂ν Aρ
8π

Exchanging bosons:

The Chern-Simons term attaches
a ﬂux to the boson.

| Daϕ | − m | ϕ | + λ | ϕ |i μνρ+ ϵ (a + A)μ∂ν(a + A)ρ
4π∼= (+ 1)

EOM for a0 :Exchanging boson-ﬂux composites:∼= (+ 1) × e= (− 1) Fermion!!
iπDensity of ﬂux∇ × (a + A)
ρϕ += 02π
Density of boson

A. Karch, D. Tong, PRX (2016), N. Seiberg, T. Senthil, C. Wang, E. Witten, Ann. Phys. (2016)

5/15

Duality web
Boson-fermion duality
Particle-vortex duality
for bosons
M. E. Peskin, Ann. Phys. (1978)

Particle-vortex duality
for fermions

Many new dualities

D. T. Son, PRX (2015)

A. Karch, D. Tong, PRX (2016), N. Seiberg, T. Senthil, C. Wang, E. Witten, Ann. Phys. (2016)

Many applications to condensed matter systems:
D. T. Son, PRX (2015)
FQHS at ν = 1/2,
Surface states for strongly-correlated top. ins., C. Wang & T. Senthil, PRX (2015)
Deconﬁned criticality, C. Wang, A. Nahum, M. A. Metlitski, C. Xu & T. Senthil, PRX (2017)…

Duality web
Boson-fermion duality
Particle-vortex duality
for bosons

Particle-vortex duality
for fermions

Many new dualities
A. Karch, D. Tong, PRX (2016), N. Seiberg, T. Senthil, C. Wang, E. Witten, Ann. Phys. (2016)

The boson-fermion duality is derived based oncoupled-wire construction & lattice gauge theory.
D. F. Mross, J. Alicea & O. I. Motrunich, PRX (2017).

J.-Y. Chen, J. H. Son, C. Wang, & S. Raghu, PRL. (2018).

5/15

6/15Outline
1. Introduction

Basic concepts, boson-fermion duality,…

2. Boson-fermion duality in 4D

Free Dirac <=> Abelian-Higgs model of Θ =3. Summaryπ

TF & Y. Nishida, PRD., 99, 101701(R)Strategy

7/15

Parent Theory on LatticeIntegrating out
Bosonsfor Bosons & Fermions

Lattice Theoryfor FermionsContinuum Theory
for Fermions

EqualContinuum
LimitDual

Integrating outFermions

Lattice Theory
for Bosons

Continuum Theory
for Bosons

J.-Y. Chen, J. H. Son, C. Wang, & S. Raghu, PRL. (2018), TF & Y. Nishida, PRD., 99, 101701(R)Strategy

7/15① Parent Theory on LatticeIntegrating out
Bosonsfor Bosons & Fermions② Lattice Theory

for FermionsContinuum Theory
for Fermions

Equal
Continuum
Limit
Dual

Integrating out
Fermions③ Lattice Theory

for Bosons

Continuum Theory
for Bosons

J.-Y. Chen, J. H. Son, C. Wang, & S. Raghu, PRL. (2018), TF & Y. Nishida, PRD., 99, 101701(R)Parent theory̶ includes bosonic and fermionic sectors:••XY model:− SXY = βWilson fermion: − SWF =∑cos(Δμθx − ax,μ)x,μ+ iax,μ+ iAx,μ− − iax,μ− iAx,μhe+he+(M+4)χ̄
χx,μx,μxx]∑[∑x,μxμμ+γ−
1−γ−1+−χx+ μ hx,μ = χ̄x+ μχx
Hopping terms: hx,μ = χ̄x22

̶ is described by the following partition function:∫No action for ax,μZ[A; β, M] =[da][dθ][dχ̄dχ]e− SXY[θ,a;β]− SWF[ χ̄,χ,a+ A;M]

Inﬁnite gauge coupling.

8/15

Lattice action for fermionsIntegrations over the bosons are performed exactly.
Only shift the mass & generate local interactions.

Z[A; β, M] =∫[da][dθ][dχ̄dχ]e∝ [da][dψ̄dψ]e∫− SXY[θ,a;β]− SWF[ χ̄,χ,a+ A;M]

− SWF[ψ̄,ψ,A;M′]− SInt[ψ̄,ψ,A;1/β]

Wilson fermion w\ self interactions
Weakly coupled when β ≫ 1 :SInt[ψ̄, ψ, A; 1/β] = 𝒪(1/β)(β ≫ 1)

9/15

Continuum theory for fermion

Phase diagram for − SWF[ψ̄, ψ, A; M′] − SInt[ψ̄, ψ, A; 1/β] ??
In the non-interacting case (1/β = 0, M ∼ 0),

TopologicalTrivial

Θ= π

Θ= 00Single DiraciΘ μνρσℒeﬀ =ϵF[A]F[A]+⋯μν
ρσ32π 2
X.-L. Qi, T. L. Hughes, & S.-C. Zhang, PRB (2008)

M

10/15

Continuum theory for fermion

Phase diagram for − SWF[ψ̄, ψ, A; M′] − SInt[ψ̄, ψ, A; 1/β] ??
In the non-interacting case (1/β ≠0, M ∼ 0),
M = − 1.94/β + ⋯1/β

Critical line

TopologicalTrivial

Θ= π

Θ= 00

Single DiraciΘ μνρσ
ℒeﬀ =ϵF[A]F[A]+⋯μνρσ32π 2
X.-L. Qi, T. L. Hughes, & S.-C. Zhang, PRB (2008)

0 = M′Re = M′ + Σ(k = 0)
M = − 1.94/β + ⋯M

10/15

Continuum theory for fermion

Phase diagram for − SWF[ψ̄, ψ, A; M′] − SInt[ψ̄, ψ, A; 1/β] ??
In the non-interacting case (1/β ≠0, M ∼ 0),
M = − 1.94/β + ⋯1/β

Critical line

TopologicalTrivial

Θ= π

Θ= 00

IrrelevantSingle DiraciΘ μνρσℒeﬀ =ϵF[A]F[A]+⋯μν
ρσ32π 2X.-L. Qi, T. L. Hughes, & S.-C. Zhang, PRB (2008)

0 = M′Re = M′ + Σ(k = 0)
M = − 1.94/β + ⋯MContinuum theory:a free Dirac fermionμψ̄[γ (∂μ − iAμ) + m]ψ
m > 0 : trivial
m < 0 : topological

10/15

Lattice action for boson

11/15

Integrations over the fermions:
Z[A; β, M] =∫[da][dθ][dχ̄dχ]e∝ [da][dθ]e∫− SXY[θ,a;β]− SWF[ χ̄,χ,a+ A;M]

− SXY[θ,a;β]− SGauge[a+ A;M]

Fermion bilinear∂ SWF− SGauge[a + A; M] = − Tr log( ∂χ̄x∂χy )Abelian-Higgs model with induced gauge action

Its partition function is equal to that of the fermion theory.
1/β
The same phase diagram!
M = − 1.94/β + ⋯

Θ= π0

Θ= 0

M

2

Phase diagram for boson

12/15

Interpretation of the phases in terms of the bosons ??
Abelian-Higgs model with induced gauge action− SXY[θ, a; β] − SGauge[a + A; M]
Natural possibilities:

Coulomb (gapless), Higgs, Conﬁnement
M = − 1.94/β + ⋯Θ= πE. Fradkin & S. H. Shenker, PRD (1979)

1/β0

Θ= 0M

Phase diagram for boson

12/15

Interpretation of the phases in terms of the bosons ??
Abelian-Higgs model with induced gauge action− SXY[θ, a; β] − SGauge[a + A; M]
Natural possibilities:

Coulomb (gapless), Higgs, Conﬁnement
M = − 1.94/β + ⋯Located on the lower
temperature side

E. Fradkin & S. H. Shenker, PRD (1979)

1/βHiggs

Conﬁnement

Θ= π

Θ= 00

Continuously connectedto the limit M → ∞ .

MContinuum theory for boson
The continuum theory for the phase transition?
The continuum limit is nontrivial ….

13/15

1/β

Higgs

Conﬁnement

Θ= π

Θ= 00M

Continuum theory for boson
The continuum theory for the phase transition?
The continuum limit is nontrivial ….
Take the limit by parts:
2

1/β

Higgs

Conﬁnement

Θ= π

Θ= 0

λ44
SXY[θ, a; β] → d x | Daϕ | + m | ϕ | + | ϕ |[]∫2
1i42μνρσSG[a; M]→ d x
F[a]+ϵF[a]F[a]μνμνρσ2[]∫4g32πM< 0213/150M

Continuum theory for boson
The continuum theory for the phase transition?
The continuum limit is nontrivial ….
Naive

Take the limit by parts:
21/βHiggs

Conﬁnement

Θ= π

Θ= 0λ44SXY[θ, a; β] → d x | Daϕ | + m | ϕ | + | ϕ |[]∫21i
42μνρσSG[a; M]→ d x
F[a]+ϵF[a]F[a]μνμνρσ2[]∫4g32πM< 0213/15

Abelian-Higgs model w\ theta angle π :λ1 2i μνρσ224| Daϕ | + m | ϕ | + | ϕ | +F
[a+A]+ϵF[a+A]F[a+A]μνμνρσ24g 232π0M

Continuum theory (contʼd)Abelian-Higgs model w\ theta angle π :λ1 2i μνρσ4| Daϕ | + m | ϕ | + | ϕ | +F
[a+A]+ϵF[a+A]F[a+A]μνμνρσ224g32π2•2Low-energy EFT is consistently reproduced:iΘ μνρσℒeﬀ =ϵF[A]F[A]+⋯μνρσ232π•Θ = 0 (m > 0){Θ =π (m < 0)Conﬁnement
Higgs

The gauge ﬁeld must be “compact” (include monopoles) to provide
the conﬁnement phase.
In the ordinary Abelian-Higgs model,
conﬁnement phase = monopole condensate

14/16

Fermions from bosons
Abelian-Higgs model w\ theta angle π :λ1 2i μνρσ4| Daϕ | + m | ϕ | + | ϕ | +F
[a+A]+ϵF[a+A]F[a+A]μνμνρσ224g32π2<latexit sha1_base64="JFCKnyx80Esw6hgKNGQasEJPeUs=">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</latexit>

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15/16

Fermions from bosons

15/16

Abelian-Higgs model w\ theta angle π :λ1 2i μνρσ4| Daϕ | + m | ϕ | + | ϕ | +F
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E. Witten, PLB (1976)

Fermions from bosons

15/16

Abelian-Higgs model w\ theta angle π :λ1 2i μνρσ4| Daϕ | + m | ϕ | + | ϕ | +F
[a+A]
+ϵF
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Monopoles get electric charges 1/2& become dyons due to the Witten eﬀect.
Time-reversal partner

E. Witten, PLB (1976)

Fermions from bosons

15/16

Abelian-Higgs model w\ theta angle π :λ1 2i μνρσ4| Daϕ | + m | ϕ | + | ϕ | +F
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DDTStat.
F
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F

Monopoles get electric charges 1/2& become dyons due to the Witten eﬀect.
E. Witten, PLB (1976)

Time-reversal partner

Spin-statistics theorem| L EM | = 1/2Fermion!
A. S. Goldhaber, PRL (1976)

Fermions from bosons

15/16

Abelian-Higgs model w\ theta angle π :λ1 2i μνρσ4| Daϕ | + m | ϕ | + | ϕ | +F
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DT
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10
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1/2
11

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ϕ*DDTStat.
FB
B
B
FFMonopoles get electric charges 1/2& become dyons due to the Witten eﬀect.
E. Witten, PLB (1976)

Time-reversal partner

Spin-statistics theorem| L EM | = 1/2Fermion!

ψA. S. Goldhaber, PRL (1976)

Summary

16/16

Boson-fermion duality in 3+1D is constructed:
Free Dirac fermion:μψ̄[γ (∂μ − iAμ) + m]ψ

m > 0 : Trivial insulator <=> Conﬁnement phasem < 0 : Topological insulator <=> Higgs phase
Dirac fermion <=> Scalar + Dyon pair

For details, see

TF & Y. Nishida,PRD., 99, 101701(R) (2019)

Abelian-Higgs model w\ theta angle: πλ1 2i μνρσ4| Daϕ | + m | ϕ | + | ϕ | +F
[a+A]+ϵF[a+A]F[a+A]μνμνρσ24g 232π2•2Partially speculative… Comments are welcome!