QCD Kondo effect - Perturbative to Non-perturbative - Sho Ozaki - - PowerPoint PPT Presentation

QCD Kondo effect

Sho Ozaki (Keio Univ.)

Strangeness and charm in hadron and dense mater, YITP ,15-26 May, 2017

• Perturbative to Non-perturbative -

Contents

Introduction QCD Kondo effect from perturbative RG Summary

• S. O., K. Itakura and
• Y. Kuramoto, PRD94 (2016) 074013
• K. Hattori, K. Itakura, S. O. and S.

Yasui, PRD92 (2015) 065003

I) II) III) QCD Kondo effect from CFT IV)

• T. Kimura and S. O., in preparation
• T. Kimura and S.O., arXiv: 1611.07284

Kondo effect

近藤効果: 磁性不純物の入った金の電気抵抗の低温で の振る舞い

近藤効果

出典: フリー百科事典『ウィキペディア（Wikipedia）』 近藤効果（こんどうこうか、Kondo effect）と は、磁性を持った極微量な不純物（普通磁性のある 鉄原子など）がある金属では、温度を下げていくと ある温度以下で電気抵抗が上昇に転じる現象であ る。これは通常の金属の、温度を下げていくとその 電気抵抗も減少していくという一般的な性質とは異 なっている。現象そのものは電気抵抗極小現象とよ ばれ、1930年頃から知られていたが、その物理的 機構は1964年に日本の近藤淳が初めて理論的に解 明した[1]。近藤はこの仕事により1973年に日本学 士院恩賜賞を受章した。

目次

1 現象 2 理論 3 理論の拡張と応用 4 脚注 5 参考文献 6 関連項目 7 外部リンク

現象

金属は電圧を加えると、金属内の伝導電子が加速され電流が流れる。これを電気伝導という。 一方で、この伝導電子には電気抵抗がはたらく。金属の電気抵抗の主な要因は、金属内に含まれる不純 物などによる格子欠陥と、原子の熱振動の2つである。不純物による抵抗は温度に依存せず一定であ る。熱振動による抵抗は、温度を下げると小さくなり、低温では抵抗は温度Tの5乗に比例する。そのた め、金属の電気抵抗は通常、温度を下げると減少し、絶対零度で、一定値(=不純物による抵抗値）に落 ち着く。 しかし、金属によっては、ある温度までは温度が下がると電気抵抗も減少するが、さらに温度を下げる と電気抵抗は逆に増大するという、通常では起こりえないふるまいを見せる。この現象は、1933年、 ド・ハース、ド・ブール、ファン・デン・バーグが、金の電気抵抗を測定したときに初めて観測され た[2]。 その後の研究により、この現象は金(Au)、銀(Ag)、銅(Cu)などに鉄(Fe)、マンガン(Mn)、クロム(Cr)な どの磁性不純物を微量に加えた金属で起こることが明らかになった。

T 2 (Classical) logT

(Quantum) By “infrared divergence”

Kondo effect is firstly observed in experiment as an enhancement of electrical resistivity of impure metals.

Jun Kondo (1930-)

• J. Kondo has explained the phenomenon based on the

second order perturbation of interaction between conduction electron and impurity.

Asymptotic freedom in Kondo effect and QCD

Λ

Fermi Surface

ΛK

G(Λ)

Λ Kondo effect Running coupling of QCD

q q’ P P’ q’-q

q

¯ q

g g

Asymptotic freedom in Kondo effect and QCD

Λ

Fermi Surface

ΛK

G(Λ)

Λ

Kondo singlet Color singlet (Hadron)

q q’ P P’ q’-q

q

¯ q

g g

Asymptotic freedom in Kondo effect and QCD

Λ

Fermi Surface

ΛK

G(Λ)

Λ

Kondo singlet Color singlet (Hadron) QCD Kondo

ΛK

G(Λ)

Conditions for the appearance of Kondo effect 0) Heavy impurity i) Fermi surface ii) Quantum fluctuation (loop effect) iii) Non-Abelian property of interaction (spin-flip int.)

Conditions for the appearance of QCD Kondo effect 0) Heavy quark impurity i) Fermi surface of light quarks ii) Quantum fluctuation (loop effect) iii) Color exchange interaction in QCD

QCD Kondo effect

• K. Hattori, K. Itakura, S. O. and S.

Yasui, PRD92 (2015) 065003

Heavy quark impurity

(light) quark matter with

charm or bottom quark

µ ΛQCD

Q

(light) quark matter with µ ΛQCD

Q

q q

q q’ P P’ q’-q

q k q’ P P+q-k P’

(a)

q-k q’-k q k q’ P P-q’+k P’ q-k q’-k

(b)

q k q’ P P+q-k P’

(a)

q-k q’-k q k q’ P P-q’+k P’ q-k q’-k

(b)

+ +

MQ → large

Heavy quark:

−iM = q

Q

Quark propagator at finite density

iS(q|µ) = i/ q 2✏q ✓ ✓(|~ q| − kF ) 1 q0 − ✏+

q + i" + ✓(kF − |~

q|) 1 q0 − ✏+

q − i"

◆

particle hole anti-particle

(massless quark)

iS(q|µ) = i/ q 2✏q ✓ − 1 q0 − ✏−

q − i"

◆

✏q = |~ q| ✏±

q = ±|~

q| − µ

iS(q; µ)

Gluon propagator at finite density

Screening effect

= GP µν

T

+ FP µν

L

Screening masses

G(q) = i⇡q0 2|~ q|m2

D

F(q) = m2

D = g2µ2

2π2

iDij = i(ij − ˆ qiˆ qj) (q0)2 − |~ q|2 − i π

2 m2 D|q0|/|~

q|

Gluon propagator

P 00

T = P 0i T = 0

P ij

T = ij − qiqj/|~

q|2

P µν

L

= qµqν/q2 − gµν − P µν

T

Vacuum polarization

, iD00 = i q2 − m2

D

Heavy quark propagator and vertex

P µ = MQvµ + kµ

• n-shell
• ff-shell

vµ = ( p 1 + |~ v|2,~ v)

MQ → large

Heavy quark:

Heavy quark propagator

i / P + MQ P 2 − M 2

Q

→ i 1 v · k 1 + / v 2 γµ → 1 + / v 2 γµ 1 + / v 2 = vµ 1 + / v 2 → vµ

Spin-indep.

Q

q q’ P P’ q’-q

Vertex

Q Q

Gluon exchange interactions Color electric interaction Color magnetic interaction

Suppressed by 1/M

q q’ P P’ q’-q q q’ P P’ q’-q

gγ0 gv0 iD00 gvj iDij gγi

Dominant contribution

Q

Tree amplitude

q q’ P P’ q’-q

iD00(q0 − q)

gγ0(T A)a0a

g(T B)b0,b

δAB −iMBorn = −ig2D00(q0 − q)(T A)a0a(T A)b0bγ0 ⊗ 1 + γ0 2

S-wave projection (partial wave decomposition)

−iMS−wave

Born

= 1 2 Z 1

−1

d(cosθ)Pl=0(cosθ) (−iMBorn)

G = 1 2 Z 1

1

d(cosθ)Pl=0(cosθ)g2iD00(q0 − q)

= 1 2 Z 1

1

d(cosθ)Pl=0(cosθ) −g2 (q0 − q)2 − m2

D

= g2 4µ2 log 4µ2 m2

D

S-wave projected gluon exchange int.

≡ −iMS−wave

Born

= −iG(T A)a0a(T A)b0b

1-loop amplitudes (S-wave projected)

q k q’ P P+q-k P’

(a)

q-k q’-k q k q’ P P-q’+k P’ q-k q’-k

(b)

q k q’ P P+q-k P’

(a)

q-k q’-k q k q’ P P-q’+k P’ q-k q’-k

(b)

particle hole

G2ρF T (a)

a0a;b0b

Z ΛUV

Λ

1 E dE G2ρF T (b)

a0a;b0b

Z ΛUV

Λ

1 E dE

(a) (b)

−i i

ρF = k2

F

(2π)2 : density of state

• n Fermi surface

1-loop amplitudes (S-wave projected)

q k q’ P P+q-k P’

(a)

q-k q’-k q k q’ P P-q’+k P’ q-k q’-k

(b)

q k q’ P P+q-k P’

(a)

q-k q’-k q k q’ P P-q’+k P’ q-k q’-k

(b)

particle hole

G2ρF T (a)

a0a;b0b

Z ΛUV

Λ

1 E dE G2ρF T (b)

a0a;b0b

Z ΛUV

Λ

1 E dE

(a) (b)

T (a)

a0a;b0b = (T A)a0a00(T B)a00a(T A)b0b00(T B)b00b = N 2 c − 1

4N 2

c

δa0aδb0b − 1 Nc (T A)a0a(T A)b0b

T (b)

a0a;b0b = (T A)a0a00(T B)a00a(T B)b0b00(T A)b00b = N 2 c − 1

4N 2

c

δa0aδb0b − ✓ 1 Nc − Nc 2 ◆ (T A)a0a(T A)b0b

−i i

ρF = k2

F

(2π)2 : density of state

• n Fermi surface

Color factors (Non-abelian property of the QCD interaction)

1-loop amplitudes (S-wave projected)

q k q’ P P+q-k P’

(a)

q-k q’-k q k q’ P P-q’+k P’ q-k q’-k

(b)

q k q’ P P+q-k P’

(a)

q-k q’-k q k q’ P P-q’+k P’ q-k q’-k

(b)

particle hole

G2ρF T (a)

a0a;b0b

Z ΛUV

Λ

1 E dE G2ρF T (b)

a0a;b0b

Z ΛUV

Λ

1 E dE

(a) (b)

T (a)

a0a;b0b = (T A)a0a00(T B)a00a(T A)b0b00(T B)b00b = N 2 c − 1

4N 2

c

δa0aδb0b − 1 Nc (T A)a0a(T A)b0b

T (b)

a0a;b0b = (T A)a0a00(T B)a00a(T B)b0b00(T A)b00b = N 2 c − 1

4N 2

c

δa0aδb0b − ✓ 1 Nc − Nc 2 ◆ (T A)a0a(T A)b0b

−i i ,

ρF = k2

F

(2π)2 : density of state

• n Fermi surface

−iNc 2 G2ρF logΛUV Λ (T A)a0a(T A)b0b

Color factors (Non-abelian property of the QCD interaction)

Renormalization group equation of scattering amplitude

q q’ P P’ q q’ P P’

q q’ k P P+q-k P’ (a)

q q’ k P P-q’ +k P’ (b)

Q

Q Q Q Q Q

G(Λ − dΛ)

G(Λ)

= + +

G(Λ) G(Λ)

Λ ∼ Λ − dΛ Λ ∼ Λ − dΛ

G(Λ) G(Λ)

Λ Λ − dΛ Λ0

· · · · · ·

Λ0 = ΛUV ' kF

Initial scale

~poor man’s scaling~

Renormalization group equation of scattering amplitude

ΛdG(Λ) dΛ = −Nc 2 ρF G2(Λ)

Solution

G(Λ) = G(Λ0) 1 + Nc

2 ρF G(Λ0)log(Λ/Λ0)

Kondo scale (from the Landau pole)

ΛK ' kF exp ✓

• 8π

Ncαslog(π/αs) ◆

Λ0 = ΛUV ' kF

Initial scale

Λ

Fermi Surface

Λ0

ΛK

q q Q

QCD Kondo effect

The strength of the q-Q interaction increases as the energy scale decreases, and the system becomes non-perturbative one below the Kondo scale. This indicates a change of mobility of light quarks.

Several transport coefficients will be largely affected by QCD Konde effect.

' kF exp ✓

• 8π

Ncαslog(π/αs) ◆

G(Λ)

Heavy ion collisions

Q

@ J-PARC, GSI-FAIR

¯ Q

Q

¯ Q

D

QCD Kondo effect would affect properties of QGP at high density, and also production rate of D mesons.

Magnetically induced QCD Kondo effect

• S. O., K. Itakura and
• Y. Kuramoto, PRD94 (2016) 074013

Non-central heavy ion collisions

B

|eB| > ∼ Λ2

QCD

@ RHIC, LHC

Extremely strong magnetic fields are generated in high energy heavy ion collisions. Charm quarks are also produced.

ρF

ρLLL

3+1 D 3+1 D 1+1 D 1+1 D

S-wave projection (Partial wave decomposition) LLL projection

Degenerate fermions on the fermi surface Degenerate fermions in the Landau levels

pz |~ p|

kF

eB

|~ p|

Super conductivity Kondo effect Magnetic catalysis

B B

(Dimensional reduction)

Conditions for the appearance of QCD Kondo effect 0) Heavy quark impurity i) Fermi surface of light quarks ii) Quantum fluctuation (loop effect) iii) Color exchange interaction in QCD

0) Heavy quark impurity i) Strong magnetic field ii) Quantum fluctuation (loop effect) iii) Color exchange interaction in QCD

The magnetic field does not affect color degrees of freedom.

“Magnetically induced QCD Kondo effect” Conditions for the appearance of

q q’ P P’ q’-q

q k q’ P P+q-k P’

(a)

q-k q’-k q k q’ P P-q’+k P’ q-k q’-k

(b)

q k q’ P P+q-k P’

(a)

q-k q’-k q k q’ P P-q’+k P’ q-k q’-k

(b)

+ + −iM = q

Q

B

Quark propagator of the Lowest Landau Level (LLL)

iSLLL(k; µ|eqB) = e−

k2 ⊥ eqB i

✏k ⇢ ✓(k3 − kF ) k0 − ✏+

k + i" + ✓(kF − k3)✓(k3)

k0 − ✏+

k − i"

• ⇢

− ✓(−k3) k0 − ✏−

k − i"

k00 − k33 P0

P0 = 1 + iγ1γ2 2

with spin projection operator

Vertex

γµ → P0γµP0 = P0γ ¯

µP0

¯ µ = 0, 3

,

q q’ P P’ q’-q

qLLL qLLL

Gluon propagator in strong magnetic fields

Magnetic screening effect Gluon propagator Vacuum polarization

iDAB

µν (p) = −i

gk

µν

p2 + p2

kΠ(p2 ?, p2 k) + g? µν

p2 − p?

µ p? ν + p? µ pk ν + pk µp? ν

p4 ! δAB

= (p2

kgk µν − pk µpk ν)Π(p2 ?, p2 k)

Π(p2

?, p2 k) = −exp

✓ − p2

?

2eqB ◆ m2

g

p2

k

m2

g = αs

π eqB

with Gluon mass

• V. P

. Gusynin,

• V. A. Miransky and I. A. Shovkovy, NPB 563 (1999)

G(T A)a0a(T A)b0b (1 + sgn(q0

z))

Leading order amplitude

The 1+1 dimensional gluon exchange int.

After integrating the transverse momentum, we get

−i

G ' g2 eqB log ✓eqB m2

g

◆

G ≡ 1 eqBπ Z d2(q0

? − q?)e(q0

?q?)2/eqB ⇥

(ig)2iD(q0 − q; eqB) ⇤ ' g2 eqBπ Z d2(q0

? q?) e(q0

?q?)2/eqB

(q0

? q?)2 + m2 g

q k q’ P P+q-k P’

(a)

q-k q’-k q k q’ P P-q’+k P’ q-k q’-k

(b)

q k q’ P P+q-k P’

(a)

q-k q’-k q k q’ P P-q’+k P’ q-k q’-k

(b)

particle anti-particle

NLO (LLL projected 1-loop amplitudes)

independent of the chemical potential G2ρLLLT (a)

a0a;b0b

Z ΛUV

Λ

1 E dE

−G2ρLLLT (b)

a0a;b0b

Z ΛUV

Λ

1 E dE

−G2ρLLL Nc 2 logΛUV Λ (1 + sgn(q0

z))(T A)a0a(T A)b0b

ρLLL = eqB (2π)2

: density of state of the LLL

−i −i −i

Renormalization group equation Kondo scale (from the Landau pole)

ΛdG(Λ) dΛ = −Nc 2 ρLLLG2(Λ)

solution

ΛK ' p eqBα1/2

s

exp (

• 2π

Ncαslog(π/αs) + log ✓ π αs ◆1/6)

' p eqBα1/3

s

exp ⇢

• 2π

Ncαslog(π/αs)

• G(Λ) =

G(Λ0) 1 + Nc

2 ρLLLG(Λ0)log(Λ/Λ0)

Non-central heavy ion collisions

B

|eB| > ∼ Λ2

QCD

@ RHIC, LHC

Lattice QCD simulation (Numerical experiment of QCD)

D

B

D

q ¯ Q

QCD Kondo effect from CFT

• T. Kimura and S. O, in preparation
• T. Kimura and S. O, arXiv:1611.07284

Non-perturbative region

Λ

G(Λ)

ΛK

ΛUV

Perturbative region Fermi surface IR fixed point

μ

≈ ≈

ΛQCD

QCD Kondo effect

In order to investigate QCD Kondo effect in IR region below Kondo scale, we have to rely on non-perturbative method.

Non-perturbative approach for Kondo effect

Numerical renormalization group lattice QCD Bethe ansatz 1+1 dim. conformal field theory (CFT) approach

[Wilson] [Andrei] [Wiegmann] …. [Affleck-Ludwig]

k(multi)-channel SU(2) Kondo effect

Non-perturbative approach for Kondo effect

Numerical renormalization group lattice QCD Bethe ansatz 1+1 dim. conformal field theory (CFT) approach

[Wilson] [Andrei] [Wiegmann] …. [Affleck-Ludwig]

k(multi)-channel SU(2) Kondo effect k-channel SU(N) Kondo effect

• T. Kimura and S. O, arXiv:1611.07284

Non-perturbative approach for Kondo effect

Numerical renormalization group lattice QCD Bethe ansatz 1+1 dim. conformal field theory (CFT) approach

[Wilson] [Andrei] [Wiegmann] …. [Affleck-Ludwig]

k(multi)-channel SU(2) Kondo effect k-channel SU(N) Kondo effect

• T. Kimura and S. O, arXiv:1611.07284

High density QCD in the presence of the heavy quark 1+1 dim. (Dimensional reduction)

G = αs log4µ2 m2

g

= αs log4π αs ⌧ 1

[E. Shuster & D. T. Son, and T. Kojo et al.]

S1+1

eff =

Z d2x ¯ Ψ [iΓµ∂µ] Ψ − GΨ†taΨQ†taQ

Effective 1+1 dim. theory at high density

G = Z d2q (2π)2 (ig)2 q2 − m2

D

= ρ2D

F

Z dΩq 4π (ig)2 q2 − m2

D

ρ2D

F

= k2

F

π = µ2 π

kF

Dimensionless coupling G is obtained from gluon exchange

D

density of state

• n the Fermi surface

s-wave

High density QCD in the presence of the heavy quark 1+1 dim. s-wave (Dimensional reduction) with G = αs log4µ2

m2

g

= αs log4π αs ⌧ 1

[E. Shuster & D. T. Son, and T. Kojo et al.]

S1+1

eff =

Z d2x ¯ Ψ [iΓµ∂µ] Ψ − GΨ†taΨQ†taQ

Effective 1+1 dim. theory at high density

High density QCD in the presence of the heavy quark 1+1 dim. s-wave

Ψ is light quark fields with 2Nf components of flavor

and Nc colors. The 2 comes from spin d.o.f. in 4 dim. (Dimensional reduction) with G = αs log4µ2

m2

g

= αs log4π αs ⌧ 1

[E. Shuster & D. T. Son, and T. Kojo et al.]

S1+1

eff =

Z d2x ¯ Ψ [iΓµ∂µ] Ψ − GΨ†taΨQ†taQ

Effective 1+1 dim. theory at high density

High density QCD in the presence of the heavy quark 1+1 dim. This is nothing but k-channel SU(N) Kondo model in 1+1 dim., where s-wave k = 2Nf, N = Nc

Ψ is light quark fields with 2Nf components of flavor

. and Nc colors. The 2 comes from spin d.o.f. in 4 dim. (Dimensional reduction) with G = αs log4µ2

m2

g

= αs log4π αs ⌧ 1

[E. Shuster & D. T. Son, and T. Kojo et al.]

S1+1

eff =

Z d2x ¯ Ψ [iΓµ∂µ] Ψ − GΨ†taΨQ†taQ

Effective 1+1 dim. theory at high density

Boundary CFT

H = iΨ† ∂Ψ ∂x + GΨ†taΨCaδ(x)

Hamiltonian density of QCD Kondo effect

where Q†taQ

Caδ(x)

MQ → large

Boundary CFT

H = iΨ† ∂Ψ ∂x + GΨ†taΨCaδ(x)

Hamiltonian density of QCD Kondo effect

where Q†taQ

Caδ(x)

MQ → large

Currents

color flavor charge

Ja = : Ψ†taΨ :

JA = : Ψ†T AΨ : J = : Ψ†Ψ :

: OO(x) := lim

✏→0 {OO(x + ✏) hOO(x + ✏)i}

Boundary CFT

H = iΨ† ∂Ψ ∂x + GΨ†taΨCaδ(x)

Hamiltonian density of QCD Kondo effect

where Q†taQ

Caδ(x)

MQ → large

Currents

color flavor charge

H = 1 Nc + 2Nf JaJa + 1 2Nf + Nc JAJA + 1 4NcNf J2 + GJaCaδ(x)

The Sugawara form of the Hamiltonian density

Ja = : Ψ†taΨ :

JA = : Ψ†T AΨ : J = : Ψ†Ψ :

: OO(x) := lim

✏→0 {OO(x + ✏) hOO(x + ✏)i}

x

J a = Ja + G 2 (Nc + 2Nf)Caδ(x) H = 1 Nc + 2Nf JaJa + 1 2Nf + Nc JAJA + 1 4NcNf J2 + GJaCaδ(x)

H = 1 Nc + 2Nf J aJ a + 1 2Nf + Nc JAJA + 1 4NcNf J2

Impurity effect Boundary of the theory

x = 0

τ

Boundary CFT

g-factor and Impurity entropy

Partition function

gRimp × e

πcL 6β

Z(L, β)

bulk boundary

Free energy Entropy Impurity entropy at IR fixed point (T=0)

gRimp = SRimp0 S00

: central charge : g-factor

Smn : modular S-matrix

,

( )

c = 2NcNf

S(T) = −∂F ∂T

L → ∞

F = − 1 β log Z Simp = S(T) − Sbulk(T)|T =0 = log (gRimp)

universal quantities

(impurity)

g-factor and Impurity entropy

g-factor provides the impurity entropy:

Simp = log(gRimp)

Rimp : (anti-)fundamental representation

ex) SU(2), k=1 (standard Kondo effect)

Simp

log2

IR UV

T

Simp = log(2s + 1) s = 1/2 Simp → log2

In UV, In IR, s → 0 (Kondo singlet)

Simp → 0 ,

Overscreening Kondo effect in multi-channel SU(2) Kondo model

k = 1

Fermi liquid at IR fixed point : integer

g

(single channel)

k = 2

non-Fermi liquid at IR fixed point : non-integer

g

(two channel)

Standard Kondo effect Overscreening Kondo effect

g-factor in QCD Kondo effect @ IR fixed point

In general Nc and Nf, the g-factor is non-integer, and thus QCD Kondo effect has non-Fermi liquid IR fixed point.

In large Nc limit:

Nc = 3 g → 1 g = 1 + √ 5 2 g = 2.24598... g = 2.53209... (Nf = 1) (Nf = 2) (Nf = 3) Nc → ∞

Fermi liquid at IR fixed point

Nf ,

: fixed

k = 2Nf

(zero temperature)

1+1 dim. (boundary) CFT approach

Correlation functions are exactly determined by

Conformal symmetry in 1+1 dim. Kac-Moody algebra

From the correlation functions, one can evaluate T

• dep. of

several observables of QCD Kondo effect in IR regions.

hO1(x)O2(y)i = C1,2 |x y|∆

C1,2

∆

determined by conformal symmetry determined by KM algebra [J a(x), J b(y)] = if abcJ c(x)δ(x − y) + Nfδab ∂ ∂xδ(x − y)

Specific heat, susceptibility and the Wilson ratio in QCD Kondo effect

Bulk contributions to C

χ

&

Cbulk = π 3 Nk T

χbulk = k 2π

These are well known properties of free Nk (bulk) fermions in 1+1 dim. with k = 2Nf N = Nc ,

Impurity contributions to C

χ

& i) k=1 & arbitrary N [Affleck 1990]

Leading irrelevant operator From the perturbation w.r.t.

with k = 1

δH1 = λ1J aJ a(x)δ(x) λ1 ∼ 1/TK δH1

Cimp = −λ1 k(N 2 − 1) 3 π2T

χimp = −λ1 k(N + k) 2

Typical Fermi liquid behaviors

Leading irrelevant operator :adjoint operator, appearing when k = 2Nf >1 scaling dimension is :Fourier mode of . From the perturbation with respect to the leading irrelevant operator, we can evaluate . Cimp χimp , φa ∆ = Nc/(Nc + 2Nf)

δH = λJ a

−1φa(x)δ(x)

J a

n

J a(x) λ ∼ 1/T ∆

K

ii) k > 1, Overscreening case(relevant for QCD Kondo effect)

∆ < 1

Z = e−βF (T,λ,h)

= Z DΨD ¯ Ψ e−

R d2x Hexp

( + Z β/2

−β/2

dτ " λJ a

−1φa(τ, 0) + h

2π Z L

−L

dx J 3(τ, x) #)

= Z0 * exp ( + Z β/2

−β/2

dτ " λJ a

−1φa(τ, 0) + h

2π Z L

−L

dx J 3(τ, x) #)+

F = Lfbulk + fimp C = −T ∂2F ∂T 2 χ = −∂2F ∂h2 Free energy can be divided in to bulk and impurity parts which is expressed in terms of the correlation functions

• f the leading irrelevant operators.

,

Observables

h = 0

L → ∞

Specific heat of QCD Kondo effect

Cimp = 8 > > > > > > > > < > > > > > > > > : λ2 2 π1+2∆(2∆)2(N 2

c − 1)(Nc + Nf)

✓1 − 2∆ 2 ◆ Γ(1/2 − ∆)Γ(1/2) Γ(1 − ∆) T 2∆ (2Nf > Nc) λ2π1+2∆(N 2 − 1)(Nc + Nf)(2∆)2 T log ✓TK T ◆ (2Nf = Nc) −λ1 2 3(N 2

c − 1)π2T + 2λ2π2(N 2 c − 1)(Nc + Nf)

2∆ 1 + 2∆ β−2∆+1

K

2∆ − 1 ! T (Nc > 2Nf)

δH = λJ a

−1φa(x)δ(x)

δH1 = λ1J aJ a(x)δ(x)

Specific heat of QCD Kondo effect

Cimp = 8 > > > > > > > > < > > > > > > > > : λ2 2 π1+2∆(2∆)2(N 2

c − 1)(Nc + Nf)

✓1 − 2∆ 2 ◆ Γ(1/2 − ∆)Γ(1/2) Γ(1 − ∆) T 2∆ (2Nf > Nc) λ2π1+2∆(N 2 − 1)(Nc + Nf)(2∆)2 T log ✓TK T ◆ (2Nf = Nc) −λ1 2 3(N 2

c − 1)π2T + 2λ2π2(N 2 c − 1)(Nc + Nf)

2∆ 1 + 2∆ β−2∆+1

K

2∆ − 1 ! T (Nc > 2Nf)

Cimp ∝      T 2∆ (2Nf > Nc) T log(TK/T) (2Nf = Nc) T (Nc > 2Nf)

Non-Fermi Non-Fermi Fermi

For Nc > 2Nf, although the g-factor (at IR fixed point) exhibits non-Fermi liquid signature, T

• dep. of Cimp shows Fermi liquid

behavior.

Fermi/non-Fermi mixing [T. Kimura and S. O, arXiv:1611.07284]

Low T scaling

Susceptibility of QCD Kondo effect

χimp = 8 > > > > > > > > < > > > > > > > > : λ2 2 π2∆−1(Nc + Nf)2(1 − 2∆)Γ(1/2 − ∆)Γ(1/2) Γ(1 − ∆) T 2∆−1 (2Nf > Nc) 2λ2(Nc + Nf)2 log ✓TK T ◆ (2Nf = Nc) −λ1Nf(Nc + 2Nf) + 2λ2(Nc + Nf)2 β−2∆+1

K

2∆ − 1 ! (Nc > 2Nf)

χimp =      T 2∆−1 (2Nf > Nc) log(TK/T) (2Nf = Nc) const. (Nc > 2Nf)

Non-Fermi Non-Fermi Fermi

Low T scaling

The Wilson ratio of QCD Kondo effect

For Nc >= 2Nf, the Wilson ratio is no longer universal, which depends on the detail of the system, such as

RW = ✓ χimp Cimp ◆ , ✓ χbulk Cbulk ◆

= (Nc + NF )(Nc + 2Nf)2 3Nc(N 2

c − 1)

(2Nf ≥ Nc)

RW = (Nc + Nf)(Nc + 2Nf/3) N 2

c − 1

γ − 2Nf(Nc + 2Nf) (Nc + Nf)2 γ − 2Nf(Nc + 2Nf/3) Nc(Nc + Nf)

Unknown parameters are canceled, and thus the Wilson ratio is universal.

(Nc > 2Nf)

γ = 4λ2 λ1 T 2∆−1

K

with

λ, TK

2Nf >= Nc Nc > 2Nf g-factor (IR fixed point) non-Fermi non-Fermi Low T scaling non-Fermi Fermi Wilson ratio universal non-universal

• T. Kimura and S. O, arXiv:1611.07284

IR behaviors of QCD Kondo effect (k-channel SU(N) Kondo effect) (k >= N) (N > k >1)

Fermi/non-Fermi mixing

Summary

We apply CFT approach to QCD Kondo effect and investigate its IR behaviors below the Kondo scale. In the vicinity of IR fixed point, QCD Kondo effect shows Fermi/ non-Fermi mixing for Nc > 2Nf, while it shoes non-Fermi liquid behaviors for 2Nf >= Nc. We propose QCD Kondo effect appearing at high density quark matter and in strong magnetic fields.