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

CFT approach to multi-channel SU(N) Kondo effect Sho Ozaki (Keio Univ.) In collaboration with Taro Kimura (Keio Univ.) Seminar @ Chiba Institute of Technology, 2017 July 8

Contents I) Introduction II) CFT approach to multi-channel SU(N) Kondo effect T. Kimura and S.O., arxiv: 1611.07284 to be published in JPSJ III) Application to high energy physics: QCD Kondo effect T. Kimura and S. O., in preparation IV) Summary

Kondo effect logt (Quantum) T 2 (Classical) 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.

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) H sd = X k, k c k, c k, J X k,k 0 ~s k0 k ~S (J <0) Scattering amplitude T (k "! k 0 ") Born term k " k 0 " J T (1) = hk 0 " J~s k0 k ~S k "i = JS z M M k "i = c k" FSi

Second order perturbation theory - only spin flip processes - particle k " k 00 # k " hole k 0 " k 0 " M M +1 M 1 M S + S S S + M k 00 # M T (2) a = hk 0 " X k 00 J 2 S c k 0 " c J k 00 # 2 S+ c k 00 # c k" k "i k 00 + i T (2) b = hk 0 " X k 00 J 2 S+ c k 00 # c J k" 2 S c k 0 " c k 00 # k "i k 00 i T (2) = J 2 4 = k 0 S +,S F log D T = k F : density of state on the Fermi surface, D : Bandwidth

Total amplitude T = T (1) + T (2) + ' T (1) 1+ J 2 F log T D At low temperature (IR) regions: T D 1 ' J 2 F log T D T ' T K (J <0) The perturbative expansion breaks down. We need a some non-perturbative method to analyze the Kondo effect at IR regions.

Non-perturbative approach for Kondo effect Numerical renormalization group [Wilson] Bethe ansatz [Andrei] [Wiegmann]. 1+1 dim. conformal field theory (CFT) approach [Affleck-Ludwig] k(multi)-channel SU(2) Kondo k-channel SU(N) Kondo with k >= N

Boundary CFT 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: Currents H = i (x) @ (x) @x + K (x)t a (x)s a (x) J a (x) =: (x)t a (x) : J A (x) =: (x)t A (x) : J(x) =: (x) (x) : color flavor charge Normal order product : OO(x) :=lim!0 {O(x)O(x + ) ho(x)o(x + )i}

Boundary CFT H = 1 N + k J a J a + 1 k + N J A J A + 1 2kN JJ + KJ a S a (x) J a = J a + K 2 (N + k)sa (x) H = 1 N + k J a J a + 1 k + N J A J A + 1 2kN JJ Impurity effect Boundary of the theory x =0 x

g-factor and Impurity entropy Partition function Z(L, ) Free energy Entropy L!1 F = S(T )= g Rimp e cl 6 boundary (impurity) 1 log Z @F @T bulk Impurity entropy at IR fixed point (T=0), universal quantities c = Nk g Rimp = S R imp 0 S 00 S imp = S(T ) S bulk (T ) T =0 =log(g Rimp ) : central charge : g-factor ( S mn : modular S-matrix )

g-factor and Impurity entropy g-factor provides the impurity entropy: S imp = log(g Rimp ) R imp : fundamental representation ex) SU(2), k=1 (standard Kondo effect) S imp = log(2s + 1), In UV, s =1/2 S imp! log2 log2 S imp In IR, s! 0 (Kondo singlet) S imp! 0 0 IR UV T

Overscreening Kondo effect in multi-channel SU(2) Kondo model Standard Kondo effect k =1 (single channel) Fermi liquid at IR fixed point g : integer Overscreening Kondo effect k =2 (two channel) non-fermi liquid at IR fixed point g : non-integer

g-factor in SU(N) Kondo effect @ IR fixed point (zero temperature) 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 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 J a (x), J b (y) = if abc J c (x) (x y)+ i 2 k 2 ab @ @x (x y) ho 1 (x)o 2 (y)i = C 1,2 x y C 1,2 determined by conformal symmetry determined by KM algebra From the correlation functions, one can evaluate T-dep. of several observables of k-channel SU(N) Kondo effect in IR regions.

Specific heat, susceptibility and the Wilson ratio Bulk contributions to C & C bulk = 3 NkT 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 H 1 = 1 J a J a (x) (x) 1 1/T K From the perturbation w.r.t. C imp = k(n 2 1) 1 2 T 3 imp = k(n + k) 1 2 H 1 with k =1 Typical Fermi liquid behaviors

ii) k > 1, Overscreening case Leading irrelevant operator H = J a 1 a (x) (x) a J a n :adjoint operator, appearing when k >1 scaling dimension is. :Fourier mode of J a (x) = N N + k 1/T K < 1 From the perturbation with respect to the leading irrelevant operator, we can evaluate, imp. C imp

Observables Z =e = Z F (T,,h) D D e = Z 0 *exp ( + R d 2 x H exp Z /2 /2 d " ( + Z /2 J a 1 /2 d " J a 1 a (, 0) + h 2 a (, 0) + h 2 Z L L Z L L dx J 3 (,x) L!1 dx J 3 (,x) #)+ #) Free energy can be divided in to bulk and impurity parts F = Lf bulk + f imp which is expressed in terms of the correlation functions of the leading irrelevant operators. C = T @2 F = @2 F @T 2, @h 2 h =0

Specific heat of k-channel SU(N) Kondo effect C imp = 8 >< >: 2 1 2 2 1+2 (2 ) 2 (N 2 1)(N + k/2) 2 2 1+2 (N 2 1)(N + k/2)(2 ) 2 T log TK T 1 k 3 (N 2 1) 2 T +2 2 2 (N 2 2 1)(N + k/2) 1+2 (1/2 ) (1/2) (1 ) 2 +1 K 2 1 T 2! T (k>n) (k = N) (N >k>1) Low T scaling C imp / 8 >< T 2 (k>n) T log(t K /T ) (k = N) >: T (N >k>1) Non-Fermi Non-Fermi Fermi 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]

Susceptibility of k-channel SU(N) Kondo effect imp = 8 2 2 1 (N + k/2) 2 (1 2 ) >< 2 2 (N + k/2) 2 TK log T >: 2 1 k(n + k) 2 (1/2 ) (1/2) (1 ) +2 2 (N + k/2) 2 2 +1 K 2 1! T 2 1 (k>n) (k = N) (N >k>1) Low T scaling imp = 8 >< T 2 1 (2k >N) log(t K /T ) (2k = N) >: const. (N >k>1) Non-Fermi Non-Fermi Fermi

The Wilson ratio of QCD Kondo effect R W = imp C imp, bulk C bulk Unknown parameters are canceled, and thus the Wilson ratio is universal. R W = = (N + k/2)(n + k)2 3N(N 2 1) (N + k/2)(n + k/3) N 2 1 (k N) k(n + k) (N + k/2) 2 k(n + k/3) N(N + k/2) with =4 For N >= k, the Wilson ratio is no longer universal, which depends on the detail of the system, such as,t K (N >k>1) 2 1 T 2 1 K

IR behaviors of k-channel SU(N) Kondo effect T. Kimura and S. O, arxiv:1611.07284 Fermi/non-Fermi mixing

Application to high energy physics: QCD Kondo effect

Strong interaction QCD Lagrangian L QCD = 1 4 F a µ F aµ + q (i µ D µ M q ) q D µ = @ µ F a µ = @ µ A a iga a µt a @ A a µ + gf abc A b µa c

Quarks Color Electron Red Green Blue e 0.5MeV Flavor up down u u u d d d 2.3MeV 4.8MeV strage s s s 95MeV QCD 200MeV charm c c c 1200MeV bottom top b b b t t t 4200MeV 173 10 3 MeV

Quarks Color Electron Red Green Blue e 0.5MeV Flavor up down u u u d d d 2.3MeV 4.8MeV strage s s s 95MeV QCD 200MeV Impurity effect by heavy flavors charm bottom c c c b b b 1200MeV 4200MeV top t t t 173 10 3 MeV Kondo effect induced by color d.o.f.

Asymptotic freedom in Kondo effect and QCD Kondo effect Running coupling of QCD G( ) q g q g 0 K Fermi Surface 0

Asymptotic freedom in Kondo effect and QCD Kondo singlet Color singlet (Hadron) G( ) q g q g 0 K Fermi Surface 0

Asymptotic freedom in Kondo effect and QCD Kondo singlet Color singlet (Hadron) G( ) QCD Kondo G( ) 0 K Fermi Surface K 0

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 Q charm or bottom quark (light) quark matter with µ QCD

q q Q (light) quark matter with µ QCD

t a q im = t a t a t b Q t a t b + + t a t b t b t a Heavy quark: M Q! large

Renormalization group equation of scattering amplitude ~poor man s scaling~ = G( d ) G( ) + + Q Q Q Q G( ) d G( ) 0 d 0 G( ) Initial scale 0 = UV ' k F Q d Q G( )

Renormalization group equation of scattering amplitude dg( ) d = N c 2 F G 2 ( ) Solution G( ) = G( 0 ) 1+ N c 2 F G( 0 )log( / 0 ) Kondo scale (from the Landau pole) Initial scale 0 = UV ' k F K ' k F exp 8 N c s log( / s )

QCD Kondo effect G( ) q q Q 0 Fermi Surface K ' k F exp 8 N c s log( / s ) 0 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.

Magnetically induced QCD Kondo effect S. O., K. Itakura and Y. Kuramoto, PRD94 (2016) 074013

3+1 D ~p k F S-wave projection (Partial wave decomposition) 1+1 D ~p F Degenerate fermions on the fermi surface Super conductivity Kondo effect 3+1 D 1+1 D B LLL projection (Dimensional reduction) B p z Degenerate fermions in the Landau levels eb LLL Magnetic catalysis

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

Conditions for the appearance of Magnetically induced QCD Kondo effect 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.

Renormalization group equation dg( ) d = N c 2 LLLG 2 ( ) solution G( ) = G( 0 ) 1+ N c 2 LLLG( 0 )log( / 0 ) Kondo scale (from the Landau pole) ( ' p e q B 1/3 exp K ' p e q B 1/2 exp s s 2 N c s log( / s ) 2 N c s log( / s ) + log s 1/6 )

QCD Kondo effect from CFT T. Kimura and S. O, in preparation

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

Effective 1+1 dim. theory at high density High density QCD in the presence of the heavy quark s-wave S 1+1 eff = Z 1+1 dim. d 2 x [i µ @ µ ] (Dimensional reduction) [E. Shuster & D. T. Son, and T. Kojo et al.] with G = s log 4µ2 m 2 g is light quark fields with 2Nf components of flavor and Nc colors. The 2 comes from spin d.o.f. in 4 dim. This is nothing but k-channel SU(N) Kondo model in 1+1 dim., where k =2N f, N = N c. G t a Q t a Q = s log 4 s 1

g-factor in QCD Kondo effect @ IR fixed point (zero temperature) N c =3 k =2N f g = 1+p 5 2 g =2.24598... g =2.53209... (N f = 1) (N f = 2) (N f = 3) 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: g! k =2N f N c!1, N f : fixed Fermi liquid at IR fixed point

Specific heat of QCD Kondo effect C imp = 8 >< >: 2 1 2 2 1+2 (2 ) 2 (Nc 2 1)(N c + N f ) 2 2 1+2 (N 2 1)(N c + N f )(2 ) 2 T log TK T 1 2 3 (N c 2 1) 2 T +2 2 2 (Nc 2 2 1)(N c + N f ) 1+2 (1/2 ) (1/2) (1 ) 2 +1 K 2 1 T 2 (2N f >N c )! (2N f = N c ) T (N c > 2N f ) Low T scaling C imp / 8 >< T 2 (2N f >N c ) T log(t K /T ) (2N f = N c ) >: T (N c > 2N f ) Non-Fermi Non-Fermi Fermi For Nc > 2Nf, QCD Kondo effect shows Fermi/non-Fermi mixing.

Susceptibility of QCD Kondo effect imp = 8 2 2 1 (N c + N f ) 2 (1 2 ) >< 2 2 (N c + N f ) 2 TK log T >: 2 (1/2 ) (1/2) (1 ) 1N f (N c +2N f )+2 2 (N c + N f ) 2 2 +1 K 2 1! T 2 1 (2N f >N c ) (2N f = N c ) (N c > 2N f ) Low T scaling imp = 8 >< T 2 1 (2N f >N c ) log(t K /T ) (2N f = N c ) >: const. (N c > 2N f ) Non-Fermi Non-Fermi Fermi

The Wilson ratio of QCD Kondo effect R W = imp C imp, bulk C bulk = (N c + N F )(N c +2N f ) 2 3N c (N 2 c 1) R W = (N c + N f )(N c +2N f /3) N 2 c 1 (2N f N c ) Unknown parameters are canceled, and thus the Wilson ratio of QCD Kondo effect is universal for 2Nf >= Nc. 2N f (N c +2N f ) (N c + N f ) 2 2N f (N c +2N f /3) N c (N c + N f ) with (N c > 2N f ) =4 For Nc >= 2Nf, the Wilson ratio is no longer universal, which depends on the detail of the system, such as,t K 2 1 T 2 1 K

IR behaviors of QCD Kondo effect (k >= N) (N > k >1) Fermi/non-Fermi mixing

Summary We develop the CFT approach to general k-channel SU(N) Kondo effect and investigate its IR behaviors. 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 apply CFT approach to QCD Kondo effect and determine its IR behaviors below the Kondo scale. Our CFT analysis for k-channel SU(N) Kondo effect can be also applied to SU(3) Kondo effect in cold atom and SU(4) Kondo effect in Quantum dot systems with multi-channels.