CFT approach to multi-channel SU(N) Kondo effectyasutake/matter/ozaki.pdfCFT approach to multi-channel SU(N) Kondo effect Sho Ozaki (Keio Univ.) Seminar @ Chiba Institute of Technology,

CFT approach to multi-channel SU(N) Kondo effect

Sho Ozaki (Keio Univ.)

Seminar @ Chiba Institute of Technology, 2017 July 8

In collaboration with

Taro Kimura (Keio Univ.)

Contents

Introduction

CFT approach to multi-channel SU(N) Kondo effect

Summary

I)

II)

III) Application to high energy physics: QCD Kondo effect

IV)

T. Kimura and S. O., in preparation

T. Kimura and S.O., arXiv: 1611.07284 to be published in JPSJ

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 betweenconduction electron and impurity.

Conditions for the appearance of Kondo effect

0) Localized (Heavy) impurity

i) Fermi surface

ii) Quantum fluctuation (loop effect)

iii) Non-Abelian property of interaction

(spin-flip int.)

s-d model (Kondo model)

Born term

q q’

P P’

J

|k "i = c†k"|FSi

Scattering amplitude T (k "! k0 ")

k " k0 "

M M

Hsd =X

k,�

✏kc†k,�ck,� � J

X

k,k0

~sk0k · ~S (J < 0)

T (1) = hk0 " |� J~sk0k · ~S|k "i = �JSz

Second order perturbation theory

q q’k

P P+q-k P’

(a)

q q’

kP

P-q’ +k

P’

(b)

- only spin flip processes -

T (2)b = hk0 " |

X

k00

⇣J2S

+c†k00#ck"⌘⇣

J2S

�c†k0"ck00#

⌘

✏k00 � ✏� i⌘|k "iT (2)

a = hk0 " |X

k00

⇣J2S

�c†k0"ck00#

⌘⇣J2S

+c†k00#ck"⌘

✏� ✏k00 + i⌘|k "i

✏ = ✏k0 = ✏k

S+ S� S� S+

⇢F : density of state on the Fermi surface, D : Bandwidth

k " k "k0 " k0 "

M M + 1 M � 1M MM

k00 #

k00 #

particle hole

T (2)=

J2

4

⇥S+, S�⇤ ⇢F log

✓D

T

◆

T = T (1) + T (2) + · · ·

At low temperature (IR) regions: T ⌧ D

The perturbative expansion breaks down.

Total amplitude

We need a some non-perturbative method to analyze the Kondo effect at IR regions.

T ' TK@

' T (1)

✓1 +

J

2

⇢F log

✓T

D

◆◆

1 ' J

2

⇢F log

✓T

D

◆(J < 0)

Non-perturbative approach for Kondo effect

Numerical renormalization group

Bethe ansatz

1+1 dim. conformal field theory (CFT) approach

[Wilson]

[Andrei] [Wiegmann] ….

[Affleck-Ludwig]

k(multi)-channel SU(2) Kondo




[Wilson]

[Affleck-Ludwig]


k-channel SU(N) Kondo with k >= N

Bethe ansatz [Andrei] [Wiegmann] ….




[Wilson]

[Affleck-Ludwig]


k-channel SU(N) Kondo with k >= N

k-channel SU(N) Kondo including N > k >1T. Kimura and S. O, arXiv:1611.07284

Bethe ansatz [Andrei] [Wiegmann] ….

Boundary CFT

H = i

†(x)@ (x)

@x

+ �K †(x)ta (x)Sa

�(x)

Assuming that the impurity is sufficiently dilute, and the interaction is a contact-type one, the s-wave approx. is valid, which leads to the following one-dim. effective theory:

Boundary CFT

Currents

color

flavor

charge

H = i

†(x)@ (x)

@x


�(x)

J

a(x) =: †(x)ta (x) :

J

A(x) =: †(x)TA (x) :

J(x) =: †(x) (x) :

: OO(x) := lim✏!0

{O(x)O(x+ ✏)� hO(x)O(x+ ✏)i}

Normal order product


Boundary CFT

Currents

color

flavor

charge

The Sugawara form of the Hamiltonian density

H = i

†(x)@ (x)

@x


�(x)

J

a(x) =: †(x)ta (x) :

J

A(x) =: †(x)TA (x) :

J(x) =: †(x) (x) :

: OO(x) := lim✏!0

{O(x)O(x+ ✏)� hO(x)O(x+ ✏)i}

Normal order product

H =1

N + k

J

aJ

a +1

k +N

J

AJ

A +1

2kNJJ + �KJ

aS

a�(x)


x

Impurity effect

Boundary of the theory

x = 0

⌧

Boundary CFT

H =1

N + k

J

aJ

a +1

k +N

J

AJ

A +1

2kNJJ + �KJ

aS

a�(x)

J a = J

a +�K

2(N + k)Sa

�(x)

H =1

N + kJ aJ a +

1

k +NJAJA +

1

2kNJJ

g-factor and Impurity entropy

Partition function

gRimp ⇥ e⇡cL6�Z(L,�)

bulkboundary

Free energy

Entropy

Impurity entropy at IR fixed point (T=0)

gRimp =SRimp0

S00

: central charge

: g-factor

Smn : modular S-matrix

,

( )

S(T ) = �@F

@T

L ! 1

F = � 1

�logZ

Simp = S(T )� Sbulk(T )|T=0 = log (gRimp)

universal quantities

(impurity)

c = Nk

g-factor and Impurity entropy

g-factor provides the impurity entropy:

Simp = log(gRimp)

Rimp : fundamental representation

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

Simp

log2

0IR 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

: integerg

(single channel)

k = 2

non-Fermi liquid at IR fixed point

: non-integerg

(two channel)

Standard Kondo effect

Overscreening Kondo effect

N k = 1 k = 2 k = 3 k = 4 k = ∞

2 1 1.4142... 1.6180... 1.7320... 2

3 1 1.6180... 2 2.2469... 3

4 1 1.7320... 2.2469... 2.6131... 4

∞ 1 2 3 4 ∞

TABLE II. Numerical values of the g-factor for the (anti)fundamental representation. It approaches

to an integer in the large (N, k) limit.

where ρ is the Weyl vector (A4), and the q-number [x]q is defined by

[x]q =qx/2 − q−x/2

q1/2 − q−1/2. (25)

This q-number is reduced to the ordinary number [x]q → x in the limit q → 1, so that

dimq R → dimR. Therefore, in this limit, the quantum dimension becomes

dimq Nq→1−→ N. (26)

We remark the anti-fundamental representation N gives the same quantum dimension.

Let us consider the large N behavior of the g-factor. Expanding the expression (24) with

respect to the large N at a fixed k, we obtain

g = k −k(k2 − 1)

N2

π2

6+O(N−3) . (27)

In the large N limit, the g-factor is approximated to g = k, and the correction starts with

O(N−2). This implies that the SU(N)k Kondo effect is described as the Fermi liquid in

the large N limit, and thus the low-temperature scaling of the specific heat and so on is

expected to exhibit the Fermi liquid behavior.

Table II shows the numerical values of the g-factor for the (anti)fundamental representa-

tion. Although there is an accidental case giving an integer value for three-channel SU(3)

system SU(3)k=3, we obtain irrational values for k > 1 in general cases. This is a signature

of the non-Fermi liquid behavior at the IR fixed point of the multi-channel Kondo system,

corresponding to zero temperature (ground state). We remark that the coincidence of g-

factor for SU(N)k and SU(k)N reflects the level-rank duality of the Kac–Moody algebra.

The SU(3)3 system is self-dual in this sense (N, k) = (3, 3).

9

g-factor in SU(N) Kondo effect @ IR fixed point(zero temperature)

For k=1, g-factor is always unity, and thus the system becomes the Fermi liquid.

In general with k > 1, the g-factor becomes non-integer, which indicates that the system is described by non-Fermi liquid.

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 k-channel SU(N) 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)�(x� y) +i

2⇡

✓k

2

◆�

ab @

@x

�(x� y)

Specific heat, susceptibility and the Wilson ratio

Bulk contributions to C �&

Cbulk =⇡

3Nk T

�bulk =k

2⇡

These are well known properties of free Nk (bulk) fermions in 1+1 dim.

Impurity contributions to C �&

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

Leading irrelevant operator

From the perturbation w.r.t.

with k = 1

�H1 = �1J aJ a(x)�(x)

�1 ⇠ 1/TK

�H1

Cimp = ��1k(N2 � 1)

3⇡2T

�imp = ��1k(N + k)

2

Typical Fermi liquid behaviors

Leading irrelevant operator

:adjoint operator, appearing when k >1 scaling dimension is

:Fourier mode of

.

From the perturbation with respect to the leading irrelevant operator, we can evaluate .Cimp �imp,

�a

�H = �J a�1�

a(x)�(x)

J an J a(x)

� ⇠ 1/T�K

ii) k > 1, Overscreening case

� < 1

� =N

N + k

Z = e��F (T,�,h)

=

ZD D ¯

e

� Rd

2xH

exp

(+

Z�/2

��/2d⌧

"�J a

�1�a

(⌧, 0) +

h

2⇡

ZL

�L

dx J 3(⌧, x)

#)

= Z0

*exp

(+

Z �/2

��/2d⌧

"�J a

�1�a(⌧, 0) +

h

2⇡

Z L

�Ldx J 3

(⌧, x)

#)+

F = Lfbulk + fimp

C = �T@2F

@T 2� = �@2F

@h2

Free energy can be divided in to bulk and impurity parts

which is expressed in terms of the correlation functions of the leading irrelevant operators.

,

Observables

h = 0

L ! 1

D D †

�H = �J a�1�

a(x)�(x)�H1 = �1J aJ a(x)�(x)

Cimp =

8>>>>>>>><

>>>>>>>>:

�2

2

⇡1+2�(2�)

2(N2 � 1)(N + k/2)

✓1� 2�

2

◆�(1/2��)�(1/2)

�(1��)

T 2�(k > N)

�2⇡1+2�(N2 � 1)(N + k/2)(2�)

2 T log

✓TK

T

◆(k = N)

��1k

3

(N2 � 1)⇡2T + 2�2⇡2(N2 � 1)(N + k/2)

2�

1 + 2�

��2�+1K

2�� 1

!T (N > k > 1)

Specific heat of k-channel SU(N) Kondo effect

Specific heat of k-channel SU(N) Kondo effect

Cimp =

8>>>>>>>><

>>>>>>>>:

�2

2

⇡1+2�(2�)

2(N2 � 1)(N + k/2)

✓1� 2�

2

◆�(1/2��)�(1/2)

�(1��)

T 2�(k > N)

�2⇡1+2�(N2 � 1)(N + k/2)(2�)

2 T log

✓TK

T

◆(k = N)

��1k

3

(N2 � 1)⇡2T + 2�2⇡2(N2 � 1)(N + k/2)

2�

1 + 2�

��2�+1K

2�� 1

!T (N > k > 1)

Low T scaling

Cimp /

8><

>:

T 2�(k > N)

T log(TK/T ) (k = N)

T (N > k > 1)

For N > k >1, 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]

Non-Fermi

Non-FermiFermi

�imp =

8>>>>>>>><

>>>>>>>>:

�2

2

⇡2��1(N + k/2)2(1� 2�)

�(1/2��)�(1/2)

�(1��)

T 2��1(k > N)

2�2(N + k/2)2 log

✓TK

T

◆(k = N)

��1k(N + k)

2

+ 2�2(N + k/2)2

��2�+1K

2�� 1

!(N > k > 1)

Low T scaling

Susceptibility of k-channel SU(N) Kondo effect

�imp =

8><

>:

T 2��1(2k > N)

log(TK/T ) (2k = N)

const. (N > k > 1)

Non-Fermi

Non-FermiFermi

The Wilson ratio of QCD Kondo effect

For N >= k, the Wilson ratio is no longer universal, which depends on the detail of the system, such as

RW =

✓�imp

Cimp

◆,✓�bulk

Cbulk

◆

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

� = 4�2

�1T 2��1Kwith

�, TK

=(N + k/2)(N + k)2

3N(N2 � 1)(k � N)

RW =(N + k/2)(N + k/3)

N2 � 1

� � k(N + k)

(N + k/2)2

� � k(N + k/3)

N(N + k/2)

(N > k > 1)

k >= N N > k > 1

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 k-channel SU(N) Kondo effect

Fermi/non-Fermi mixing

Application to high energy physics: QCD Kondo effect

Strong interaction

QCD Lagrangian

LQCD = �1

4F aµ⌫F

aµ⌫ + q (i�µDµ �Mq) q

Dµ = @µ � igAaµt

a

F aµ⌫ = @µA

a⌫ � @⌫A

aµ + gfabcAb

µAc⌫

Quarks

ColorRed Green Blue

Flav

or up

down

strage

charm

bottom

top

u

d d d

s

c

u u

s s

c c

b b b

t t t

2.3MeV

4.8MeV

95MeV

⇤QCD ⇠ 200MeV

1200MeV

4200MeV

173⇥ 103MeV

Electron

e�

⇠ 0.5MeV

ColorRed Green Blue

Flav

or up

down

strage

charm

bottom

top

u

d d d

s

c

u u

s s

c c

b b b

t t t

2.3MeV

4.8MeV

95MeV

⇤QCD ⇠ 200MeV

1200MeV

4200MeV

173⇥ 103MeV

Impurity effectby heavy flavors

Kondo effect induced by color d.o.f.

Electron

e�

⇠ 0.5MeV

Quarks

Asymptotic freedom in Kondo effect and QCD

⇤Fermi

Surface

0 ⇤K

G(⇤)

0 ⇤

Kondo effect Running coupling of QCD

q q’

P P’

q’-q

FIG. 2: The tree diagram. Solid and double solid lines are massless and heavy quarks, respectively.

Here ⇧(p2k/m2) has a rather complicated form (see Refs. [11–15] for the explicit expression).

But in our analysis with the massless QCD, it is su�cient to know that ⇧(p2k/m2) ! 1 in

the massless limit m ! 0. It should be emphasized that there is a strong screening e↵ect

in the first term of the gluon propagator (4) owing to the gluon mass. The propagator (4)

is an analog of that employed in analyses of the Schwinger model, namely, 1+1 dimensional

QED [16].

A. Tree amplitude

Now we compute the amplitude for scattering between a light (massless) quark near the

Fermi surface and a heavy quark impurity. Under the strong magnetic field, the light quark

moves only in the direction parallel to the magnetic field. In the LLL with e

q

> 0, the spin

of the light quark is fixed to the magnetic field direction. We set the momentum of the

initial quark as positive direction of the z-axis: qz

> 0. Then, the leading order amplitude

as shown in Fig. 2 is given by

�iM0

= (ig)2⇥u

LLL

(q0)�µ(TA)a

0a

u

LLL

(q)⇤DAB

µ⌫

(q0 � q|eq

B)⇥U(P 0)�⌫(TB)

b

0b

U(P )⇤, (7)

where the color indices of quarks can take a, a0, b, b0 = 1, 2, · · ·Nc

. The spinors are defined by

u

LLL

(q) = Nq

⇣�",

�zqz

|qz | �"

⌘t

with �

z

�" = +�", and U(P ) = NQ

(⇠�

, 0)t with �

z

⇠± = ±⇠±.

Nq

and NQ

are normalization constants. By using these spinors, we find u

LLL

�

µ

u

LLL

=

u

LLL

�

µ

u

LLL

where µ = 0, 3, and U�

⌫

U = U�

0

U . Then, in the gluon propagator (4), only

the first term proportional to g

k00

contributes to the amplitude. Furthermore, as we will

see soon, only forward scattering is allowed in the massless limit of the light quark. Then,

7

q

q

g

g


⇤Fermi

Surface

0 ⇤K

G(⇤)

0 ⇤

Kondosinglet

Colorsinglet

(Hadron)

q q’

P P’

q’-q







QED [16].

A. Tree amplitude




q

> 0, the spin





�iM0

= (ig)2⇥u

LLL

(q0)�µ(TA)a

0a

u

LLL

(q)⇤DAB

µ⌫

(q0 � q|eq

B)⇥U(P 0)�⌫(TB)

b

0b

U(P )⇤, (7)



u

LLL

(q) = Nq

⇣�",

�zqz

|qz | �"

⌘t

with �

z

�" = +�", and U(P ) = NQ

(⇠�

, 0)t with �

z

⇠± = ±⇠±.

Nq

and NQ


LLL

�

µ

u

LLL

=

u

LLL

�

µ

u

LLL


⌫

U = U�

0



k00



7

q

q

g

g


⇤Fermi

Surface

0 ⇤K

G(⇤)

0 ⇤

Kondosinglet

Colorsinglet

(Hadron)

QCD Kondo

⇤K

G(⇤)

Conditions for the appearance of Kondo effect

0) Heavy impurity

i) Fermi surface


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


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 � ⇤QCD

Q

(light) quark matter with µ � ⇤QCD

Q

q q

q q’

P P’

q’-q







QED [16].

A. Tree amplitude




q

> 0, the spin





�iM0

= (ig)2⇥u

LLL

(q0)�µ(TA)a

0a

u

LLL

(q)⇤DAB

µ⌫

(q0 � q|eq

B)⇥U(P 0)�⌫(TB)

b

0b

U(P )⇤, (7)



u

LLL

(q) = Nq

⇣�",

�zqz

|qz | �"

⌘t

with �

z

�" = +�", and U(P ) = NQ

(⇠�

, 0)t with �

z

⇠± = ±⇠±.

Nq

and NQ


LLL

�

µ

u

LLL

=

u

LLL

�

µ

u

LLL


⌫

U = U�

0



k00



7

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)

FIG. 3: One-loop diagrams.

the momentum transfer q

0 � q carried by a gluon equals to zero. Accordingly, the gluon

propagator contributing to the leading order amplitude simplifies to

D

AB

00

=ig

00

m

2

g

�

AB

, (8)

and the other componets are vanishing. Since in this study we concentrate on the quarks

near the Fermi surface, we set q3 = q

03 = k

F

. Then, the four momentum vectors of the initial

and final state quarks are given by q

0µ = (q00, 0, 0, q03) = q

µ = (q0, 0, 0, q3) = (✏F

, 0, 0, kF

) with

✏

F

= k

F

. [need to mention about transverse momentum] The leading scattering amplitude,

thus, reads

�iM0

= �iG

⇥u

LLL

(q0)�0(TA)a

0a

u

LLL

(q)⇤ ⇥

U(P 0)�0(TA)b

0b

U(P )⇤

= �iG (TA)a

0a

(TA)b

0b

N 2

q

⇣1 + sgn(q0

z

)⌘�

†"�" N 2

Q

⇠

†�

0⇠�

, (9)

where we have introduced a dimensionful coupling G as

G =g

2

m

2

g

. (10)

In the large mass limit of the heavy quark impurity: M ! 1, the heavy-quark spin is frozen

as ⇠†�

0⇠�

= �

�

0�

(thus does not play a role in the QCD Kondo e↵ect). The tree amplitude (9)

is proportional to the factor (1 + sgn(q0z

)), and thus only the forward scattering is allowed.

This is due to the helicity conservation of the massless quarks.

8

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)

FIG. 3: One-loop diagrams.

the momentum transfer q

0 � q carried by a gluon equals to zero. Accordingly, the gluon

propagator contributing to the leading order amplitude simplifies to

D

AB

00

=ig

00

m

2

g

�

AB

, (8)

and the other componets are vanishing. Since in this study we concentrate on the quarks

near the Fermi surface, we set q3 = q

03 = k

F

. Then, the four momentum vectors of the initial

and final state quarks are given by q

0µ = (q00, 0, 0, q03) = q

µ = (q0, 0, 0, q3) = (✏F

, 0, 0, kF

) with

✏

F

= k

F

. [need to mention about transverse momentum] The leading scattering amplitude,

thus, reads

�iM0

= �iG

⇥u

LLL

(q0)�0(TA)a

0a

u

LLL

(q)⇤ ⇥

U(P 0)�0(TA)b

0b

U(P )⇤

= �iG (TA)a

0a

(TA)b

0b

N 2

q

⇣1 + sgn(q0

z

)⌘�

†"�" N 2

Q

⇠

†�

0⇠�

, (9)

where we have introduced a dimensionful coupling G as

G =g

2

m

2

g

. (10)

In the large mass limit of the heavy quark impurity: M ! 1, the heavy-quark spin is frozen

as ⇠†�

0⇠�

= �

�

0�

(thus does not play a role in the QCD Kondo e↵ect). The tree amplitude (9)

is proportional to the factor (1 + sgn(q0z

)), and thus only the forward scattering is allowed.

This is due to the helicity conservation of the massless quarks.

8

+ +

MQ ! largeHeavy quark:

�iM =

q

Q

ta

ta

tb

ta

ta

tb ta

tatb

tb

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’

kP

P-q’ +k

P’

(b)

Q Q Q Q

Q

Q

G(⇤� d⇤) G(⇤)

=

+ +

G(⇤) G(⇤)

⇤ ⇠ ⇤� d⇤

⇤ ⇠ ⇤� d⇤

G(⇤) G(⇤)

0 ⇤⇤� d⇤ ⇤0

· · · · · ·

⇤0 = ⇤UV ' kF

Initial scale

~poor man’s scaling~

Renormalization group equation of scattering amplitude

⇤dG(⇤)

d⇤= �Nc

2⇢FG

2(⇤)

SolutionG(⇤) =

G(⇤0)

1 +

Nc2 ⇢FG(⇤0)log(⇤/⇤0)

Kondo scale (from the Landau pole)

⇤K ' kF exp

✓� 8⇡

Nc↵slog(⇡/↵s)

◆

⇤0 = ⇤UV ' kF

Initial scale

⇤

FermiSurface

⇤00 ⇤K

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

✓� 8⇡

Nc↵slog(⇡/↵s)

◆

G(⇤)

Magnetically induced QCD Kondo effect

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

⇢F

⇢LLL

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

BB(Dimensional reduction)

Conditions for the appearance of QCD Kondo effect


i) Fermi surface of light quarks




i) Strong magnetic field



The magnetic field does not affect color degrees of freedom.

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

Renormalization group equation

Kondo scale (from the Landau pole)

⇤dG(⇤)

d⇤= �Nc

2⇢LLLG

2(⇤)

solution

⇤K 'peqB↵1/2

s exp

(� 2⇡

Nc↵slog(⇡/↵s)+ log

✓⇡

↵s

◆1/6)

'peqB↵1/3

s exp

⇢� 2⇡

Nc↵slog(⇡/↵s)

�

G(⇤) =

G(⇤0)

1 +

Nc2 ⇢LLLG(⇤0)log(⇤/⇤0)

QCD Kondo effect from CFT

T. Kimura and S. O, in preparation

Non-perturbativeregion

Λ

G(Λ)

ΛK ΛUV

Perturbative region

Fermi surface

IR fixedpoint

μ

Figure 1: Schematic picture of the flow of the e↵ective interaction G(⇤). The black solid

line is a perturbative flow of G(⇤), while the blue dashed line is a non-perturbative flow. ⇤K

stands for the Kondo scale. fig:scale_dependence

related to the resistivity

⇢(T ) ⇠ ⇢u

1� ReS(1)

2+ C�T�

!(2.25) eq:rho_NFL

where ⇢u

is the resistivity at the unitary limit, and C is a dimensionless constant which can

be specified by explicit computation of the first order perturbation theory. This factor is

|S(1)| < 1 for the non-Fermi liquid case, |S(1)| = 1 for the Fermi liquid case, and S(1) ! 1

in the large k limit. Although we cannot determine the sign of the temperature dependence

in this formalism, which is given by the sign of the coupling constant �, it is expected that

the correction would be positive for strong coupling and negative for weak coupling region.

We also remark that the resistivity for the Fermi liquid case is given by

⇢(T ) ⇠ ⇢u

⇥1� C 0�2T 2

⇤(2.26) eq:rho_FL

where C 0 is again a dimensionless constant. In this case the finite temperature correction is

negative, and second order in the coupling constant �, while it is first order for the non-Fermi

liquid case.

3 Application to QCD Kondo e↵ect

sec:QCD-Kondo

The QCD Kondo e↵ect is the Kondo e↵ect induced by the color exchange interaction between

light quarks near the Fermi surface and a heavy quark impurity. One of the present authors

8

⇡⇡

⇤QCD

QCD Kondo effect

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

High density QCD in the presence of the heavy quark

1+1 dim.

This is nothing but k-channel SU(N) Kondo modelin 1+1 dim., where

s-wave

k = 2Nf , N = Nc

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 = ↵s log4µ2

m2g

= ↵s log4⇡

↵s⌧ 1

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

S

1+1eff =

Zd

2x [i�µ@µ] �G †

t

a Q†t

aQ

Effective 1+1 dim. theory at high density

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 +

p5

2

g = 2.24598...

g = 2.53209...

(Nf = 1)

(Nf = 2)

(Nf = 3)

Nc ! 1Fermi liquid at IR fixed point

Nf, : fixed

k = 2Nf

(zero temperature)

g ! k = 2Nf

Specific heat of QCD Kondo effect

Cimp =

8>>>>>>>><

>>>>>>>>:

�2

2

⇡1+2�(2�)

2(N2

c � 1)(Nc +Nf )

✓1� 2�

2

◆�(1/2��)�(1/2)

�(1��)

T 2�(2Nf > Nc)

�2⇡1+2�(N2 � 1)(Nc +Nf )(2�)

2 T log

✓TK

T

◆(2Nf = Nc)

��12

3

(N2c � 1)⇡2T + 2�2⇡2

(N2c � 1)(Nc +Nf )

2�

1 + 2�

��2�+1K

2�� 1

!T (Nc > 2Nf )

Cimp /

8><

>:

T 2�(2Nf > Nc)

T log(TK/T ) (2Nf = Nc)

T (Nc > 2Nf )

Non-FermiNon-FermiFermi

For Nc > 2Nf, QCD Kondo effect shows Fermi/non-Fermi mixing.

Low T scaling

Susceptibility of QCD Kondo effect

�imp =

8>>>>>>>><

>>>>>>>>:

�2

2

⇡2��1(Nc +Nf )

2(1� 2�)

�(1/2��)�(1/2)

�(1��)

T 2��1(2Nf > Nc)

2�2(Nc +Nf )

2log

✓TK

T

◆(2Nf = Nc)

��1Nf (Nc + 2Nf ) + 2�2(Nc +Nf )

2

��2�+1K

2�� 1

!(Nc > 2Nf )

�imp =

8><

>:

T 2��1(2Nf > Nc)

log(TK/T ) (2Nf = Nc)

const. (Nc > 2Nf )

Non-Fermi

Non-FermiFermi

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 =

✓�imp

Cimp

◆,✓�bulk

Cbulk

◆

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

3Nc(N2c � 1)

(2Nf � Nc)

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

N2c � 1

� � 2Nf (Nc + 2Nf )

(Nc +Nf )2

� � 2Nf (Nc + 2Nf/3)

Nc(Nc +Nf )

Unknown parameters are canceled, and thus the Wilson ratio of QCD Kondo effect is universal for 2Nf >= Nc.

(Nc > 2Nf )

� = 4�2

�1T 2��1Kwith

�, 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

IR behaviors of QCD Kondo effect

(k >= N) (N > k >1)

Fermi/non-Fermi mixing

Summary

We apply CFT approach to QCD Kondo effect and determineits IR behaviors below the Kondo scale.

In the vicinity of IR fixed point, the Kondo system shows Fermi/non-Fermi mixing for N > k > 1, while it shoes non-Fermi liquid behaviors for k >= N.

We develop the CFT approach to general k-channel SU(N)Kondo effect and investigate its IR behaviors.

Our CFT analysis for k-channel SU(N) Kondo effect can be alsoapplied to SU(3) Kondo effect in cold atom and SU(4) Kondo effect in Quantum dot systems with multi-channels.

CFT approach to multi-channel SU(N) Kondo effectyasutake/matter/ozaki.pdfCFT approach to multi-channel SU(N) Kondo effect Sho Ozaki (Keio Univ.) Seminar @ Chiba Institute of Technology,

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