Poster No. 10

Finite-Density Massless Two-Color QCDat Isospin Roberge-Weiss Pointand 't Hooft Anomaly
ABCDEF
古澤拓也 、谷崎佑弥 、伊藤悦子
ABCDEF
東工大理 、理研 、京大基研 、慶應自然セ , 高知大 , 阪大RCNPFurusawa, Tanizaki, Itou, PRResearch 2, 033253 (2020)
熱場の量子論, 2020年8月

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Outline1. Introduction
massless 2-color QCD at isospin RW point
2. ʼt Hooft anomaly at isospin RW point
3. Applications
Finite-T&-μ phase diagram, Similarity to AF
4. Summary & discussion

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Massless 2-color QCD
SU(2) gauge theory w/ 2 massless Dirac fermions(fund. rep.)
12μμ¯tr(F[a])+ūγ(∂+ia)u+dγ(∂+ia)dμνμμμμ22g

Nakamura (1984), Hands, Kogut, Lombardo, Morrison (1999),Kogut, Stephanov, Toublan (1999), Kogut, Stephanov, Toublan, Verbaarschot, Zhitnitsky (2000)

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Massless 2-color QCD
SU(2) gauge theory w/ 2 massless Dirac fermions(fund. rep.)
12μμ¯tr(F[a])+ūγ(∂+ia)u+dγ(∂+ia)dμνμμμμ22g
Good toy model for 3-color QCD at ﬁnite-T &-μShows asymptotic freedom, ChSB, …
No sign problem at ﬁnite density=> Lattice simulation is available.
Nakamura (1984), Hands, Kogut, Lombardo, Morrison (1999),Kogut, Stephanov, Toublan (1999), Kogut, Stephanov, Toublan, Verbaarschot, Zhitnitsky (2000)

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Massless 2-color QCD
SU(2) gauge theory w/ 2 massless Dirac fermions(fund. rep.)
12μμ¯tr(F[a])+ūγ(∂+ia)u+dγ(∂+ia)dμνμμμμ22g
Good toy model for 3-color QCD at ﬁnite-T &-μ
Lattice simulation needs much computational cost.
=> ʼt Hooft anomaly matching
Nakamura (1984), Hands, Kogut, Lombardo, Morrison (1999),Kogut, Stephanov, Toublan (1999), Kogut, Stephanov, Toublan, Verbaarschot, Zhitnitsky (2000)

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Anomalies in QCD-like theories
Center symmetry is key for anomaly at ﬁnite T.
e.g., Pure Yang-Mills gauge theory w/ theta term
Gaiotto, Kapustin, Komargodski, Seiberg (2017)

Anomaly btw. center and CP symmetries
Constraint on deconﬁnement & CPbreaking temperatures3/11

Anomalies in QCD-like theories
Center symmetry is key for anomaly at ﬁnite T.
e.g., Pure Yang-Mills gauge theory w/ theta term
Gaiotto, Kapustin, Komargodski, Seiberg (2017)

Fund. matters usually break the center.
=> Twisted b.c. of matter ﬁelds
e.g., Zn QCD, QCD w/ imaginary μ @ RW pt., …
Shimizu, Yonekura (2018), Tanizaki, Kikuchi, Misumi, Sakai (2018), Yonekura (2019), …

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Anomalies in QCD-like theories
Center symmetry is key for anomaly at ﬁnite T.
e.g., Pure Yang-Mills gauge theory w/ theta termGaiotto, Kapustin, Komargodski, Seiberg (2017)

Fund. matters usually break the center.
=> Twisted b.c. of matter ﬁeldse.g., Zn QCD, QCD w/ imaginary μ @ RW pt., …
Shimizu, Yonekura (2018), Tanizaki, Kikuchi, Misumi, Sakai (2018), Yonekura (2019), …

2cQCD w/ imaginary isospin chemical potential

4/11Imaginary isospin chemical potential
μI = θI /βμμ00¯¯ūγ (∂μ + iaμ)u + dγ (∂μ + iaμ)d + iμI(ūγ u − dγ d)Fermion action:

Boundary condition:u(τ + β) = − u(τ),

d(τ + β) = − d(τ),

Furusawa, Tanizaki, Itou, PRResearch 2, 033253 (2020)

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Imaginary isospin chemical potential
μI = θI /βμμ00¯¯ūγ (∂μ + iaμ)u + dγ (∂μ + iaμ)d + iμI(ūγ u − dγ d)Fermion action:

Boundary condition:iθIu(τ + β) = − e u(τ),

Absorbed into B.C.

d(τ + β) = − e−iθId(τ),

Field redeﬁnition:u(τ) → eiθI τ/βu(τ) d(τ) → e−iθI τ/βd(τ)

Furusawa, Tanizaki, Itou, PRResearch 2, 033253 (2020)

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Imaginary isospin chemical potential
μI = θI /βμμ00¯¯ūγ (∂μ + iaμ)u + dγ (∂μ + iaμ)d + iμI(ūγ u − dγ d)Fermion action:

Boundary condition:iθIu(τ + β) = − e u(τ),=> θI is periodic.

θI ∼ θI + πd(τ + β) = − e−iθId(τ),SU(2) gauge invarianceu(x) ∼ − u(x), d(x) ∼ − d(x),Furusawa, Tanizaki, Itou, PRResearch 2, 033253 (2020)5/11Isospin Roberge-Weiss PointθI = π/2 is special (isospin RW point).
The boundary condition is invariant under u(x) ↔ d(x)up to SU(2) gauge transformation.

θI = π/2 → − π/2 ∼ π/2θI ∼ θI + π

Emergent symmetry [(ℤ2)center] at the isospin RW pointu(x) ↔ d(x)

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Gauging ﬂavor symmetry
To ﬁnd anomaly, letʼs gauge ﬂavor subgroupat T, μ, & θI = π/2:

U(1)B × U(1)L,3 ⊂ Gμ,θI<= Gauged by AB

U(1)B: U(1) baryon symmetry

U(1)L,3: left-handed isospin U(1) symmetry <= Gauged by AL,3uL → eiλ3/2

uL, dL → e−iλ3/2dL,[See our paper for complete discussion.]
Furusawa, Tanizaki, Itou, PRResearch 2, 033253 (2020)

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ʼt Hooft anomaly at isospin RW point
AB & AL,3 violates (ℤ2)center symmetry:Sym.
ZQC2D[AB, AL,3] = ZQC D[AB, AL,3]
2

μI in action

μI in boundary conditionu(τ) → eiθI τ/βd(τ) → e−iθI τ/βu(τ)d(τ)

Furusawa, Tanizaki, Itou, PRResearch 2, 033253 (2020)

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ʼt Hooft anomaly at isospin RW point
AB & AL,3 violates (ℤ2)center symmetry:iSym.
ZQC2D[AB, AL,3] = ZQC D[AB, AL,3] × exp −AB ∧ dAL,32[ 4π ∫]
μI in action

μI in boundary conditionu(τ) → eiθI τ/βd(τ) → e−iθI τ/βu(τ)d(τ)

Redeﬁnition generates
additional phase!

Furusawa, Tanizaki, Itou, PRResearch 2, 033253 (2020)

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ʼt Hooft anomaly at isospin RW point
AB & AL,3 violates (ℤ2)center symmetry:iSym.
ZQC2D[AB, AL,3] = ZQC D[AB, AL,3] × exp −AB ∧ dAL,32[ 4π ∫](ℤ2)center w/ background gauge ﬁelds:i
ZQC2D[AB, AL,3] → ZQC2D[AB, AL,3] × exp
AB ∧ dAL,3[ 2π ∫]
Mixed ʼt Hooft anomaly btw. (ℤ2)center ⋉ U(1)B × U(1)L,3
Furusawa, Tanizaki, Itou, PRResearch 2, 033253 (2020)

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ʼt Hooft anomaly matchingʼt Hooft anomaly is invariant under RG ﬂow.
=> 2cQCD must be nontrivial at IR:
SSB, CFT, Topological order,
unique gappedʼt Hooft (1980), Frishman, Schwimmer, Banks, Yankielowicz (1981), Coleman, Grossman (1982)

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ʼt Hooft anomaly matchingʼt Hooft anomaly is invariant under RG ﬂow.
=> 2cQCD must be nontrivial at IR:
SSB, CFT, Topological order,
unique gappedʼt Hooft (1980), Frishman, Schwimmer, Banks, Yankielowicz (1981), Coleman, Grossman (1982)

One of (ℤ2)center ⋉ U(1)B × U(1)L,3 must be broken:(ℤ2)center: Quark gluon plasma
U(1)B : Baryon superﬂuidity
U(1)L,3 : Chiral symmetry breaking

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Possible phase diagram

TTTc

TBSF

QGPQGP⟨P⟩ ≠ 0, ⟨q̄q⟩ = 0, ⟨qq⟩ = 0

TQGP = TBSF

TcBCS

Coexisting phase(deconﬁned)⟨P⟩ ≠ 0, ⟨q̄q⟩ = 0, ⟨qq⟩ ≠ 0

Hadronic

Superﬂuid

Superﬂuid
BECBCSBEC

0is allowed.

BCS(deconﬁned)(conﬁned)

mπ /2(a)μ0⟨P⟩ = 0, ⟨q̄q⟩ = 0, ⟨qq⟩ ≠ 0

BCS(conﬁned)

μTQGP(b)TQGP ≤ TChSB or TBSF
Furusawa, Tanizaki, Itou, PRResearch 2, 033253 (2020)10/11

Similarity to antiferromagnet
2+1D CP1 model (QFT of antiferromagnets):
224† z2| (∂μ − ibμ)ϕ | + r | ϕ | + λ | ϕ | + λEP | ϕ σ ϕ |ϕ: 2-comp. complex scalar, bμ: U(1) dynamical gauge ﬁeld
† αSpin vector: Sα ∼ ϕ σ ϕEasy-plane potential breaks
SU(2)spin → O(2)spin.

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Similarity to antiferromagnet
2+1D CP1 model (QFT of antiferromagnets):
224† z2| (∂μ − ibμ)ϕ | + r | ϕ | + λ | ϕ | + λEP | ϕ σ ϕ |ϕ: 2-comp. complex scalar, bμ: U(1) dynamical gauge ﬁeld
† αSpin vector: Sα ∼ ϕ σ ϕEasy-plane potential breaks
SU(2)spin → O(2)spin.

“Hidden” global symmetry: U(1)magnetic
Current: Jμ = ϵμνρ∂νbρ /(2π)

Charged object: monopole

Metlitski,Thorngren (2018), Komargodski, Sulejmanpasic, Unsal (2018), Komargodski, Sharon, Thorngren, Zhou, (2019).

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Similarity to antiferromagnet
2+1D CP1 model (QFT of antiferromagnets):
224† z2| (∂μ − ibμ)ϕ | + r | ϕ | + λ | ϕ | + λEP | ϕ σ ϕ |O(2)spin × U(1)magnetic has the same anomaly as 2cQCD.
Identiﬁcation:O(2)spin ↔ (ℤ2)center ⋉ U(1)L,3U(1)magnetic ↔ U(1)B10/11

Similarity to antiferromagnet
2+1D CP1 model (QFT of antiferromagnets):
224† z2| (∂μ − ibμ)ϕ | + r | ϕ | + λ | ϕ | + λEP | ϕ σ ϕ |O(2)spin × U(1)magnetic has the same anomaly as 2cQCD.
Identiﬁcation:O(2)spin ↔ (ℤ2)center ⋉ U(1)L,3

U(1)magnetic ↔ U(1)BNovel quantum criticality: SO(5) at λEP = 0 & r = rc=> SO(5) symmetry in 2cQCD?

Wang, Nahum, Metlitski,Xu, Senthil (2017)

Summary & Discussion

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2 color QCD with imaginary isospin chemical potential.
・(ℤ2)center symmetry at isospin RW point
・(ℤ2)center ⋉ U(1)B × U(1)L,3 anomaly at ﬁnite T & μ
Nonperturbative constraint from anomaly matching
TQGP ≤ TChSB or TBSF
・Finite-T &-μ phase diagram
・The same anomaly as 2+1D antiferromagnets=> Unconventional quantum critical point at ﬁnite T?
[See our paper for details. PRResearch 2, 033253 (2020)]

Sign problem free => testable by lattice simulation!

Poster Preview

Poster No. 10Finite-Density Massless Two-Color QCDat Isospin Roberge-Weiss Pointand 't Hooft Anomaly
ABCDEF
古澤拓也 、谷崎佑弥 、伊藤悦子
ABCDEF
東工大理 、理研 、京大基研 、慶應自然セ , 高知大 , 阪大RCNPFurusawa, Tanizaki, Itou, PRResearch 2, 033253 (2020)
熱場の量子論, 2020年8月

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2-color QCD
QCD

2cQCD

Good toy model for 3-color QCD at ﬁnite-T &-μShows asymptotic freedom, ChSB, …
No sign problem at ﬁnite density=> Lattice simulation is available.

2/3Isospin RW point & ʼt Hooft Anomaly2c QCD w/ imaginary isospin chemical potential μI(sign-problem-free deformation)
Emergent ℤ2 symmetry for special μI (isospin RW point)2/3

Isospin RW point & ʼt Hooft Anomaly
2c QCD w/ imaginary isospin chemical potential μI(sign-problem-free deformation)
Emergent ℤ2 symmetry for special μI (isospin RW point)ʼt Hooft anomaly btw. ℤ2 & ﬂavor symmetries
Present even at ﬁnite temperature & density

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Applications
Finite-T and -μ phase diagram

Similarity to antiferromagnet

TQGP⟨P⟩ ≠ 0, ⟨q̄q⟩ = 0, ⟨qq⟩ = 0

Tc
BCS(deconﬁned)⟨P⟩ ≠ 0, ⟨q̄q⟩ = 0, ⟨qq⟩ ≠ 0
BCS(deconﬁned)

SuperﬂuiduidBCS(conﬁned)

BEC

μ0

⟨P⟩ = 0, ⟨q̄q⟩ = 0, ⟨qq⟩ ≠ 0

BCS(conﬁned)(b)

μTQGP ≤ TChSB or TBSF

Novel QCP at ﬁnite T?

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Applications
Finite-T and -μ phase diagram

Similarity to antiferromagnet

TQGP

Please come to my poster!
⟨P⟩ ≠ 0, ⟨q̄q⟩ = 0, ⟨qq⟩ = 0

Tc

BCS(deconﬁned)⟨P⟩ ≠ 0, ⟨q̄q⟩ = 0, ⟨qq⟩ ≠ 0
BCS(deconﬁned)

Superﬂuid

uid
BCS(conﬁned)

BEC

μ0⟨P⟩ = 0, ⟨q̄q⟩ = 0, ⟨qq⟩ ≠ 0

Poster No. 10

BCS(conﬁned)(b)

μTQGP ≤ TChSB or TBSF

Novel QCP at ﬁnite T?