Aoki phases in the lattice Gross-Neveu modelwithstaggered Wilson fermion
Taro Kimura
Department of Basic Science, Univ of Tokyo
and
Mathematical Physics Laboratory, RIKEN

Feb 9, 2012
Workshop on New-type of Fermions on the Lattice

Collaboration with M. Creutz (BNL) and T. Misumi (YITP)
Phys. Rev. D83 (2011) 094506 [arXiv:1101.4239]
Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Lattice fermions
Naive

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Lattice fermions
NaiveWilson−→Wilson[Wilson]Taro Kimura (U Tokyo/RIKEN)

Feb 2012

2 / 26Lattice fermions
Naive−→

WilsonWilson [Wilson]
Domain-wall & Overlap

Taro Kimura (U Tokyo/RIKEN)−→

Overlap (DW)

[Kaplan] [Furman-Shamir] [Neuberger]

Feb 2012

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Lattice fermions
Naive↓
Staggered−→

WilsonWilson [Wilson]
Domain-wall & Overlap
Staggered [Susskind]

Taro Kimura (U Tokyo/RIKEN)−→

Overlap (DW)

[Kaplan] [Furman-Shamir] [Neuberger]

Feb 2012

2 / 26

Lattice fermions
Naive↓
Staggered−→

Wilson−→St Wilson

Wilson [Wilson]
Domain-wall & Overlap
Staggered [Susskind]
Staggered Wilson

Taro Kimura (U Tokyo/RIKEN)−→

Overlap (DW)

[Kaplan] [Furman-Shamir] [Neuberger][Adams] [Hoelbling]Feb 2012

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Lattice fermions
Naive↓
Staggered−→

Wilson−→Overlap (DW)−→

St Wilson−→

St Overlap

Wilson [Wilson]
Domain-wall & Overlap
Staggered [Susskind][Kaplan] [Furman-Shamir] [Neuberger]

Staggered Wilson & Overlap

Taro Kimura (U Tokyo/RIKEN)[Adams] [Hoelbling]Feb 2012

2 / 26

Lattice fermions
Naive↓
Staggered−→

Wilson−→Overlap (DW)−→

St Wilson−→

St Overlap

Wilson [Wilson]
Domain-wall & Overlap
Staggered [Susskind][Kaplan] [Furman-Shamir] [Neuberger]

Staggered Wilson & Overlap

Taro Kimura (U Tokyo/RIKEN)[Adams] [Hoelbling]Feb 2012

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Wilson fermion# of doublers reduced 16 → 1
Chiral symmetry broken due to “mass term”

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Wilson fermion# of doublers reduced 16 → 1
Chiral symmetry broken due to “mass term”
Phase structure
g2

g2 =A

Aψ̄γ5 ψ

B= 0 (A phase)
6= 0 (B phase)g 2c1

Parity broken phase(Aoki phase)

g 2c2

A-8A-6Taro Kimura (U Tokyo/RIKEN)

A-4

A-20m0Feb 2012

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Wilson fermion# of doublers reduced 16 → 1
Chiral symmetry broken due to “mass term”
Phase structure
g2

g2 =A

Aψ̄γ5 ψB= 0 (A phase)
6= 0 (B phase)g 2c1

Parity broken phase(Aoki phase)

g 2c2

A-8

A-6

A-4

A-2

0m0

Chiral limit at 2nd order phase boundary

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Staggered Wilson fermion
Staggered fermion with Wilson term (doubler sensitive mass)# of doublers reduced
4→24→1

Adams’ type
Hoelbling’s type

Chiral symmetry broken

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

4 / 26

Staggered Wilson fermion
Staggered fermion with Wilson term (doubler sensitive mass)# of doublers reduced
4→24→1

Adams’ type
Hoelbling’s type

Chiral symmetry broken

Chiral limit?

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

4 / 26

Staggered Wilson fermion
Staggered fermion with Wilson term (doubler sensitive mass)# of doublers reduced
4→24→1

Adams’ type
Hoelbling’s type

Chiral symmetry broken

Chiral limit?
↓Phase structure

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Phase structure with staggered Wilson

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Phase structure with staggered Wilson
Gross-Neveu model
Asymptotic free
Spontaneous chiral symmetry breaking
Exactly solvable in 2d at large Nf

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

5 / 26

Phase structure with staggered Wilson
Gross-Neveu model
Asymptotic free
Spontaneous chiral symmetry breaking
Exactly solvable in 2d at large Nf

Toy model for 4d QCD

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

5 / 26

Phase structure with staggered Wilson
Gross-Neveu model
Asymptotic free
Spontaneous chiral symmetry breaking
Exactly solvable in 2d at large Nf

Toy model for 4d QCD

Phase structure of 4d QCD with staggered Wilson[Nakano et al.]Taro Kimura (U Tokyo/RIKEN)

Feb 2012

5 / 26

Phase structure with staggered Wilson
Gross-Neveu model
Asymptotic free
Spontaneous chiral symmetry breaking
Exactly solvable in 2d at large Nf

Toy model for 4d QCD

Phase structure of 4d QCD with staggered Wilson[Nakano et al.]GN model with Wilson fermion
Chiral limit

Taro Kimura (U Tokyo/RIKEN)[Aoki-Higashijima][Izubuchi-Noaki-Ukawa] cf. [Izubuchi-Nagai]

Feb 2012

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Contents

1Introduction

2

Gross-Neveu model with staggered Wilson fermion

3

Phase structure and Aoki phase

4

Chiral limit

5

Summary

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

6 / 26

Contents

1Introduction

2

Gross-Neveu model with staggered Wilson fermion

3

Phase structure and Aoki phase

4

Chiral limit

5

Summary

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

6 / 26

Gross-Neveu model
L = ψ̄a (i∂/ − m)ψa −g2(ψ̄a ψa )22N

N -flavor system −→ Global U(N ) symmetry
2d version of NJL model

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Gross-Neveu model
L = ψ̄a (i∂/ − m)ψa −g2(ψ̄a ψa )22N

N -flavor system −→ Global U(N ) symmetry
2d version of NJL model
Asymptotic freeππChiral Z4 (Z2 ) symmetry: ψ → ei 2 γ5 ψ, ψ̄ → ψ̄ei 2 γ5
Exactly solvable in 2d at large N

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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“Chiral” Gross-Neveu model
L = ψ̄a (i∂/ − m)ψa −
g2 (ψ̄a ψa )2 + (ψ̄a iγ5 ψa )22N

Chiral U(1) symmetry at m = 0:
ψ −→ eiθγ5 ψ,

Taro Kimura (U Tokyo/RIKEN)ψ̄ −→ ψ̄eiθγ5Feb 20128 / 26

“Chiral” Gross-Neveu model
L = ψ̄a (i∂/ − m)ψa −
g2 (ψ̄a ψa )2 + (ψ̄a iγ5 ψa )22N

Chiral U(1) symmetry at m = 0:
ψ −→ eiθγ5 ψ,ψ̄ −→ ψ̄eiθγ5

Spontaneous chiral U(1) breaking −→ NG mode
Remark: IR behavior

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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GN model on the lattice?

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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GN model on the lattice?

[Aoki-Higashijima]

Gross-Neveu model with Wilson fermion
SWGN =X
1X
ψ̄n γµ (ψn+µ̂ − ψn−µ̂ ) +ψ̄n M ψn + SW
2 n,µn2
2 i
g2 X h−ψ̄n ψn + ψ̄n iγ5 ψn
2N n

Wilson term
SW = −Taro Kimura (U Tokyo/RIKEN)rX
ψ̄n (ψn+µ̂ − 2ψn + ψn−µ̂ )2 n,µFeb 20129 / 26

Staggered fermion

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Staggered fermion
spin-diagonalization: ψ −→ χchirality: γ5 −→ n = (−1)n ' γ5 ⊗ ξ5

ψ̄n iγ5 ψnTaro Kimura (U Tokyo/RIKEN)

2−→(χ̄n in χn )2 = − (χ̄n χn )2 ?

Feb 2012

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Staggered fermion
spin-diagonalization: ψ −→ χchirality: γ5 −→ n = (−1)n ' γ5 ⊗ ξ5

ψ̄n iγ5 ψn

2−→(χ̄n in χn )2 = − (χ̄n χn )2 ?

Summation over spinor components −→ sublattice

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

10 / 26Staggered fermion
spin-diagonalization: ψ −→ χchirality: γ5 −→ n = (−1)n ' γ5 ⊗ ξ5ψ̄n iγ5 ψn2−→(χ̄n in χn )2 = − (χ̄n χn )2 ?

Summation over spinor components −→ sublattice!2ψ̄n iγ5 ψn2−→XiA χ̄A χA

ASublattice structure: n = 2N + A

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Wilson term for staggered fermion[Adams] [Hoelbling]cf. [Goltermann-Smit] [Goltermann]

SstW ' −in η1 η2 cos p1 cos p2 −→ 1 ⊗ τ3taste-splitting mass

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Gross-Neveu model with staggered Wilson fermion
SstWGN =X
1Xηµ χ̄n (χn+µ̂ − χn−µ̂ ) +χ̄n M χn + SstW
2 n,µn"!2X
g2 X−χ̄2N +A χ2N +A
2NN
A!2 #X+iA χ̄2N +A χ2N +AA

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Gross-Neveu model with staggered Wilson fermion
SstWGN =X
1Xηµ χ̄n (χn+µ̂ − χn−µ̂ ) +χ̄n M χn + SstW
2 n,µn"!2Xg2 X−χ̄2N +A χ2N +A
2NN
A!2 #X+iA χ̄2N +A χ2N +A
A

Analysis with auxiliary fields to eliminate the 4pt interactions

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Contents

1Introduction

2

Gross-Neveu model with staggered Wilson fermion

3

Phase structure and Aoki phase

4

Chiral limit

5

Summary

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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How to study?

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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How to study?
1

Introduce auxiliary fields (mesons) to eliminate the 4ptinteractions

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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How to study?
1

Introduce auxiliary fields (mesons) to eliminate the 4pt
interactions

2

Integrate out the fermionic fields, and then obtain theeffective action for mesons

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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How to study?
1

Introduce auxiliary fields (mesons) to eliminate the 4pt
interactions

2

Integrate out the fermionic fields, and then obtain theeffective action for mesons

3

Solve the saddle point equation, giving the gap equation

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

14 / 26BosonizationZ
Z=−SstWGN

Dχ̄Dχ eAuxiliary “mesonic” fields:

Z=DσDπ e−Seff (σ,π)χ̄n χn → σ,n χ̄n χn → π

Effective action for mesons1 X 22Seff = 2
σ N + πN
− Tr log D2gN

1Dn,m = (σN + in πN )δn,m + ηµ (δn+µ̂,m − δn−µ̂,m ) + (SstW )n,m2

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Saddle point equation:δSeff
δSeff==0δσδπWe now find a spatially constant solution σ0 , π0

Impose the critical condition: π = 0Zd2 k2c21 c22 σ0(2π)2 ((σ0 + c1 c2 )2 + π02 + s2 )((σ0 − c1 c2 )2 + π02 + s2 )Zd2 k
σ02 + s2 + c21 c22(2π)2 ((σ0 + c1 c2 )2 + π02 + s2 )((σ0 − c1 c2 )2 + π02 + s2 )

Mc
=4g2
1
=4g2

kcµ = cos 2µ , s2 = sin2 k21 + sin2 k22
Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Phase diagramπ= 0 (A phase)
6= 0 (B phase)

Parity broken phase(Aoki phase)

Pion mass 2δ Seff2mπ ∝δπn δπmTaro Kimura (U Tokyo/RIKEN)=0−→

Critical at phase boundary

M =Mc
Feb 2012

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Contents

1Introduction

2

Gross-Neveu model with staggered Wilson fermion

3Phase structure and Aoki phase

4

Chiral limit

5

Summary

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Critical phase boundary suggests the chiral limit for thestaggered Wilson fermion

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Critical phase boundary suggests the chiral limit for thestaggered Wilson fermion
but, even through renormalization?
chiral symmetry broken due to the (staggered) Wilson term

Renormalization effectTaro Kimura (U Tokyo/RIKEN)

Feb 2012

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“Generic” Gross-Neveu model with staggered Wilson fermion
SstWGN =X
1Xηµ χ̄n (χn+µ̂ − χn−µ̂ ) +χ̄n M χn + SstW
2 n,µn!2gσ2 X X
χ̄2N +A χ2N +A−2NNA!2gπ2 X X
iA χ̄2N +A χ2N +A−2NNA

Two kinds of coupling constants: (gσ , gπ )Chiral symmetry is preserved only when gσ = gπ at the classicallevel, but broken due to the Wilson term.

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Effective action1
1M + 1/a 2
2+ C1 σ0 +− C̃0 + log 4a π02
S̃eff = −gσ2a2gπ2π
11+− C̃0 + 2C2 + log 4a2 σ0222gσπ2
1σ + π02+ (σ02 + π02 ) log 0+ O(a)πeTaro Kimura (U Tokyo/RIKEN)

Feb 2012

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Effective action1
1M + 1/a 2
2+ C1 σ0 +− C̃0 + log 4a π02S̃eff = −gσ2a2gπ2π
11+− C̃0 + 2C2 + log 4a2 σ0222gσπ21σ + π02+ (σ02 + π02 ) log 0+ O(a)πeσ2 + π21 2Chiral limit(σ0 + π02 ) log 0 2 0→πeΛTaro Kimura (U Tokyo/RIKEN)

Feb 2012

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Effective action




1
1M + 1/a 22+ C1 σ0 +− C̃0 + log 4a π02S̃eff = −gσ2a2gπ2π
11+− C̃0 + 2C2 + log 4a2 σ0222gσπ21σ + π02+ (σ02 + π02 ) log 0+ O(a)πeσ2 + π21 2Chiral limit(σ0 + π02 ) log 0 2 0→πeΛtuning couplings & Λ-parameter11
1= C̃0 −2C2 + log,2gσ2π4Λ2 a2M =−Taro Kimura (U Tokyo/RIKEN)

2gσ2
C1 − 1,a1
1
1= C̃0 + log
2gπ2π4Λ2 a2π
π2aΛ = exp
C̃0 − πC2 − 224gσ
Feb 2012

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Gap equation: π0 = 0 with gσ2 = 0.4“2nd order” phase boundary −→ other possibilities?

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Potential analysis for σ0

1st order σ-transition

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Potential analysis for σ0

1st order σ-transition

Order of π-transition

1st & 2nd order transition

Taro Kimura (U Tokyo/RIKEN)

Feb 2012

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Phase structure & Chiral limit

We can safely take chiral & continuum limit

cf. Chial perturbation approach
Taro Kimura (U Tokyo/RIKEN)[Sharpe-Singleton]
Feb 2012

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Contents

1Introduction

2

Gross-Neveu model with staggered Wilson fermion

3

Phase structure and Aoki phase

4

Chiral limit

5Summary

Taro Kimura (U Tokyo/RIKEN)Feb 201225 / 26

SummaryStaggered Wilson Gross-Neveu model as a toy model for QCD
Phase structure of staggered Wilson GN model−→ Parity broken phase (Aoki phase)
Chiral limit of the model−→ Renormalization effect, safety of chiral limit

Phase structure in QCD

Taro Kimura (U Tokyo/RIKEN)[Nakano et al.]Feb 2012

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