a holographic model of the kondo effect...the kondo effect 34 p hysics w orld j anuary 2001 below tk...

Andy O’Bannon

University of OxfordOctober 29, 2013

A Holographic Modelof the

Kondo Effect

71 High Street, Oxford

OUTOF

BUSINESS

Credits

Based on 1310.3271

Jackson Wu

Johanna Erdmenger

National Center for Theoretical Sciences, Taiwan

Max Planck Institute for Physics, Munich

Carlos HoyosTel Aviv University

Outline:

• The Kondo Effect• The CFT Approach•A Top-Down Holographic Model•A Bottom-Up Holographic Model• Summary and Outlook

July 10, 1908

Heike Kamerlingh Onnes liquifies helium

Leiden, the Netherlands

(1 atm)T � 4.2 K

Shortly Thereafter

Leiden, the Netherlands

Begins studying low-temperature properties of metals

T � 1 to 10 K

April 8, 1911

Heike Kamerlingh Onnes discovers superconductivity

R

“for his investigations on the properties of matter at low temperatures which led, inter alia,

to the production of liquid helium”

1913

Onnes receives the Nobel Prize in Physics

Smith and Fickett, J. Res. NIST, 100, 119 (1995)

Volume 100, Number 2, March–April 1995Journal of Research of the National Institute of Standards and Technology

158

Ag

�D � 200 K


158

Ag

Resistivity measures electron scattering cross section

�D � 200 K

Debye Temperature

Quantized vibrational modes of a solid = Phonons

It is reasonable to assume that the minimum wavelength of a phonon is twice the atom separation, asshown in the lower figure. There are atoms in a solid. Our solid is a cube, which means there are

atoms per edge. Atom separation is then given by , and the minimum wavelength is

making the maximum mode number (infinite for photons)

This is the upper limit of the triple energy sum

For slowly-varying, well-behaved functions, a sum can be replaced with an integral (also known asThomas-Fermi approximation)

So far, there has been no mention of , the number of phonons with energy Phonons obeyBose-Einstein statistics. Their distribution is given by the famous Bose-Einstein formula

Debye model - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Debye_model

3 of 10 10/13/12 11:20 PM

Minimum wavelength:2 x (lattice spacing)

Maximal Frequency

lowest temperature at which maximal-energy

phonon excited

�D


158

T � �D

electron-phonon scattering

�D � 200 K

Ag

�(T ) � T

�(T ) = �0 + aT 2 + b T 5electron-phonon scattering


158

T � �D

�D � 200 K

Ag

�(T ) = �0 + aT 2 + b T 5


158

T � �D

electron-electron scattering

�D � 200 K

Ag

�(T ) = �0 + aT 2 + b T 5


158

T � �D

electron-impurity scattering

�D � 200 K

Ag


158

T � �D

increasing concentration of impurities

�D � 200 K

Ag

The Kondo Effect

P H Y S I C S W O R L D J A N U A R Y 2 0 0 134

below TK emerged in the late 1960s from Phil Anderson’s ideaof “scaling” in the Kondo problem. Scaling assumes that thelow-temperature properties of a real system are adequatelyrepresented by a coarse-grained model. As the temperature is lowered, the model becomes coarser and the number ofdegrees of freedom it contains is reduced. This approach canbe used to predict the properties of a real system close toabsolute zero.

Later, in 1974, Kenneth Wilson, who was then at CornellUniversity in the US, devised a method known as “numericalrenormalization” that overcame the shortcomings of conven-tional perturbation theory, and confirmed the scaling hypo-thesis. His work proved that at temperatures well below TK,the magnetic moment of the impurity ion is screened entirelyby the spins of the electrons in the metal. Roughly speaking,this spin-screening is analogous to the screening of an electriccharge inside a metal, although the microscopic processes arevery different.

The role of spinThe Kondo effect only arises when the defects are magnetic –in other words, when the total spin of all the electrons in theimpurity atom is non-zero. These electrons coexist with themobile electrons in the host metal, which behave like a seathat fills the entire sample. In such a Fermi sea, all the stateswith energies below the so-called Fermi level are occupied,while the higher-energy states are empty.

The simplest model of a magnetic impurity, which wasintroduced by Anderson in 1961, has only one electron levelwith energy εo. In this case, the electron can quantum-mechanically tunnel from the impurity and escape providedits energy lies above the Fermi level, otherwise it remainstrapped. In this picture, the defect has a spin of 1/2 and its z-component is fixed as either “spin up” or “spin down”.

However, so-called exchange processes can take place thateffectively flip the spin of the impurity from spin up to spindown, or vice versa, while simultaneously creating a spin ex-citation in the Fermi sea. Figure 2 illustrates what happenswhen an electron is taken from the localized impurity stateand put into an unoccupied energy state at the surface of the

Fermi sea. The energy needed for such a process is large,between about 1 and 10 electronvolts for magnetic impur-ities. Classically, it is forbidden to take an electron from thedefect without putting energy into the system. In quantummechanics, however, the Heisenberg uncertainty principleallows such a configuration to exist for a very short time –around h/|εo|, where h is the Planck constant. Within thistimescale, another electron must tunnel from the Fermi seaback towards the impurity. However, the spin of this electronmay point in the opposite direction. In other words, the initialand final states of the impurity can have different spins.

This spin exchange qualitatively changes the energy spec-trum of the system (figure 2c). When many such processes aretaken together, one finds that a new state – known as theKondo resonance – is generated with exactly the same energyas the Fermi level.

The low-temperature increase in resistance was the firsthint of the existence of the new state. Such a resonance isvery effective at scattering electrons with energies close to theFermi level. Since the same electrons are responsible for thelow-temperature conductivity of a metal, the strong scatter-ing contributes greatly to the resistance.

The Kondo resonance is unusual. Energy eigenstates usu-ally correspond to waves for which an integer number of halfwavelengths fits precisely inside a quantum box, or aroundthe orbital of an atom. In contrast, the Kondo state is gener-ated by exchange processes between a localized electron andfree-electron states. Since many electrons need to be involved,the Kondo effect is a many-body phenomenon.

It is important to note that that the Kondo state is always “onresonance” since it is fixed to the Fermi energy. Even thoughthe system may start with an energy, εo, that is very far awayfrom the Fermi energy, the Kondo effect alters the energy ofthe system so that it is always on resonance. The only require-ment for the effect to occur is that the metal is cooled to suffi-ciently low temperatures below the Kondo temperature TK.

Back in 1978 Duncan Haldane, now at Princeton Universityin the US, showed that TK was related to the parameters ofthe Anderson model by TK = 1/2(ΓU )1/2exp[πεo(εo +U )/ΓU ],where Γ is the width of the impurity’s energy level, which

1 The Kondo effect in metals and in quantum dots

resi

stan

ce

temperature

~10 K

cond

ucta

nce

temperature

~0.5 K

2e2/h

(a) As the temperature of a metal is lowered, its resistance decreases until it saturates at some residual value (blue). Some metals become superconducting at acritical temperature (green). However, in metals that contain a small fraction of magnetic impurities, such as cobalt-in-copper systems, the resistance increasesat low temperatures due to the Kondo effect (red). (b) A system that has a localized spin embedded between metal leads can be created artificially in asemiconductor quantum-dot device containing a controllable number of electrons. If the number of electrons confined in the dot is odd, then the conductancemeasured between the two leads increases due to the Kondo effect at low temperature (red). In contrast, the Kondo effect does not occur when the dot containsan even number of electrons and the total spin adds up to zero. In this case, the conductance continuously decreases with temperature (blue).

a b

476 A . M . T s v e l i c k and P. B. W i e g m a n n

Fig. 1.13

E o

Q_

9

8 -

7 -

6 '

5 -

4 -

5 -

2 -

1

*o (L._gq, Ce) B 6

% . * ° Oat %Ce ~****** ~ - 0.61 a t % C e

. . . . 1.20 at % Ce • • • • v v v 1 . 8 0 at % C e °

k . . - 2 .90 at % Ca : ~ " . +.. .

..,. -,q.,,,,. • J j ~ , , p • . . . . . . . . . . . . . . . . . . . . . . . ~ Z . ~

I 1 ! I | I I I I I 1 I 0 .05 0.1 0.2 0.5 1 2 5 I0 20 5 0 1 0 0 2 0 0

T~ K

Electrical resistivity of LaB s and four (La, Ce)B 6 samples as a function of temperature (after Samwer and Winzer 1976).

Fig. 1.14

E o

q_

0 + ° ° ° Jl°,,o°,.,%

( L__q, Ce] B 6 0.5÷ ........ "--.[5.. 0 .8 T . . . . "%";, 1.2 at % Ce

° " " " ° ' ° ' , , , ~ . , ° ° , ' ~ ° °

1 T . . . . . . . ~ . . . . ~, "x."-: ' .

.5 T ................................. ~..'.'.!.~::,.% 2 T . . . . . . . . . . . . . " " - " "~ I:~ ."

. . . . . . . . . : : : : ' : ! ! t ~ . . . . .. 4 T . . . . . . . . . . . . . . . . . . . . . . " ..... ~ . . . - "~ ' " " " .......

e t o

6 T

' ' o ' i ' ' ' ' 'o 'o 0 .02 0 .05 O.I 2 0 5 1 2 5 I0 2 5 I 00 T/K

Electrical resistivity of an (La, Ce)B 6 sample with 1"2 a t .~ Ce versus temperature for various magnetic fields (after Samwer and Winzer 1976).

Dow

nloa

ded

by [U

nive

rsity

of C

ambr

idge

] at 0

9:40

04

Oct

ober

201

2

Samwer and Winzer, Z. Phys B, 25, 269, 1976

MAGNETIC Impurities

Curie:Exact results in the theory of magnetic alloys

Fig. 1.1

469

80

"T 60 (.~

o E 2 40 13)

t

20 / , / #

/ / /

/ /

/ /

/

,r " / I

- 2 0

/

__, t ~' (La Ce) B 6 ~xod x° at % Ce xoo

o o

* 0.072 °×,8 o0.1.3 xoX o

exo

×o~ x° t21 x o o x

x o × ×~ 1 ~oo ~ .~o ~'° 420~

°xx° ~°x°

The Kondo Hamiltonian

Conduction electronsck�c†k�

� =�, �

Dispersion relation

,

Spin SU(2)

HK =�

k,�

�(k) c†k�ck� + gK �S ·�

k�k��

c†k�12��ck��

�(k) =k2

2m� �F

ck� � ei�ck� Charge U(1)

gK > 0 Anti-Ferromagnetic

gK Kondo couplinggK < 0 Ferromagnetic

Spin of magnetic impurity�S

�� Pauli matrices

The Kondo Hamiltonian

HK =�

k,�

�(k) c†k�ck� + gK �S ·�

k�k��

c†k�12��ck��

concentration of impurities

DECREASES�(T )decreasesT�

UV cutoff=

c, c̃ �

as

�(T ) = �0 + aT 2 + b T 5 + c g2K � c̃ g3K

ln (T/�F )

�F

gK < 0Ferromagnetic

DECREASES�(T )decreasesT� asgK < 0

Ferromagnetic

�(T ) = �0 + aT 2 + b T 5 + c g2K � c̃ g3K

ln (T/�F )


UV cutoff=

c, c̃ �

�F

�(T ) = �0 + aT 2 + b T 5 + c g2K � c̃ g3K

ln (T/�F )


INCREASES�(T )decreasesT�

UV cutoff=

c, c̃ �

as

�F

Anti-Ferromagnetic

gK > 0

�(T ) = �0 + aT 2 + b T 5 + c g2K � c̃ g3K

ln (T/�F )

“Kondo temperature”

O(g3K) O(g2K)term is same order as term when

TK � �F e�cc̃

1gK

�(T ) = �0 + aT 2 + b T 5 + c g2K � c̃ g3K

ln (T/�F )

Breakdown of Perturbation Theory

Cross section for electron scattering off aMAGNETIC impurity

INCREASES as energy DECREASES

�(T ) = �0 + aT 2 + b T 5 + c g2K � c̃ g3K

ln (T/�F )

�gK� �g2

K+O(g3

K)

Asymptotic freedom!

TK � �QCD

The Kondo Problem

What is the ground state?

We know the answer!

The coupling diverges at low energy!

Solutions of the Kondo Problem

Numerical RG (Wilson 1975)

Fermi liquid description (Nozières 1975)

Bethe Ansatz/Integrability(Andrei, Wiegmann, Tsvelick, Destri, ... 1980s)

Conformal Field Theory (CFT)(Affleck and Ludwig 1990s)

Large-N expansion(Anderson, Read, Newns, Doniach, Coleman, ...1970-80s)

Quantum Monte Carlo(Hirsch, Fye, Gubernatis, Scalapino,... 1980s)

UV

IR

The electrons SCREEN the impurity’s spin

Fermi liquid+

decoupled spin

“Kondo resonance”

A MANY-BODY effect

Produces a MANY-BODY RESONANCE

UV

IR

A SINGLE electron binds with the impurity

Anti-symmetric singlet of SU(2)

1�2

(|�i �e� � |�i �e�)

Fermi liquid+

decoupled spin

“Kondo singlet”

Intuitive SINGLE-BODY Description

Fermi liquid+

decoupled spin

UV

IR

Fermi liquid

+ electrons EXCLUDED from impurity location

+ NO spin

Fermi liquid+

NON-MAGNETIC impurity

Fermi liquid+

decoupled spin

UV

IR

476 A . M . T s v e l i c k and P. B. W i e g m a n n

Fig. 1.13

E o

Q_

9

8 -

7 -

6 '

5 -

4 -

5 -

2 -

1

*o (L._gq, Ce) B 6

% . * ° Oat %Ce ~****** ~ - 0.61 a t % C e

. . . . 1.20 at % Ce • • • • v v v 1 . 8 0 at % C e °

k . . - 2 .90 at % Ca : ~ " . +.. .

..,. -,q.,,,,. • J j ~ , , p • . . . . . . . . . . . . . . . . . . . . . . . ~ Z . ~

I 1 ! I | I I I I I 1 I 0 .05 0.1 0.2 0.5 1 2 5 I0 20 5 0 1 0 0 2 0 0

T~ K

Electrical resistivity of LaB s and four (La, Ce)B 6 samples as a function of temperature (after Samwer and Winzer 1976).

Fig. 1.14

E o

q_

0 + ° ° ° Jl°,,o°,.,%

( L__q, Ce] B 6 0.5÷ ........ "--.[5.. 0 .8 T . . . . "%";, 1.2 at % Ce

° " " " ° ' ° ' , , , ~ . , ° ° , ' ~ ° °

1 T . . . . . . . ~ . . . . ~, "x."-: ' .

.5 T ................................. ~..'.'.!.~::,.% 2 T . . . . . . . . . . . . . " " - " "~ I:~ ."

. . . . . . . . . : : : : ' : ! ! t ~ . . . . .. 4 T . . . . . . . . . . . . . . . . . . . . . . " ..... ~ . . . - "~ ' " " " .......

e t o

6 T

' ' o ' i ' ' ' ' 'o 'o 0 .02 0 .05 O.I 2 0 5 1 2 5 I0 2 5 I 00 T/K

Electrical resistivity of an (La, Ce)B 6 sample with 1"2 a t .~ Ce versus temperature for various magnetic fields (after Samwer and Winzer 1976).

Dow

nloa

ded

by [U

nive

rsity

of C

ambr

idge

] at 0

9:40

04

Oct

ober

201

2

Samwer and Winzer, Z. Phys B, 25, 269, 1976

Kondo Effect in Many Systems

Quantum dots

SERC School on Magnetism and Superconductivity ’06

VBS The Kondo Effect – 9

Not Impressed? How about Quantum Dots?

Regions that can hold a few hundred electrons!

Can drive a current through these!

This is Nano!

Nature © Macmillan Publishers Ltd 1998

8

letters to nature

156 NATURE | VOL 391 | 8 JANUARY 1998

3. Butler, R. P. & Marcy, G. W. A planet orbiting 47 Ursae Majoris. Astrophys. J. 464, L153–L156 (1996).4. Butler, R. P. et al. Three new ‘‘51 Peg-type’’ planets. Astrophys. J. 474, L115–L118 (1997).5. Cochran, W. D., Hatzes, A. P., Butler, R. P. & Marcy, G. W. The discovery of a planetary companion to

16 Cygni B. Astrophys. J. 483, 5457–463 (1997).6. Noyes, R. W. et al. A planet orbiting the star r Coronae Borealis. Astrophys. J. 483, L111–L114 (1997).7. Hatzes, A. P., Cochran, W. D. & Johns-Krull, C. M. Testing the planet hypothesis: a search for

variability in the spectral line shapes of 51 Peg. Astrophys. J. 478, 374–380 (1997).8. Henry, G. W., Baliunas, S. L., Donahue, R. A., Soon, W. H. & Saar, S. H. Astrophys. J. 474, 503–510 (1997).9. Gray, D. F. Absence of a planetary signature in the spectra of the star 51 Pegasi. Nature 385, 795–796

(1997).10. Gray, D. F. & Hatzes, A. P. Nonradial oscillation in the solar-temperature star 51 Pegasi. Astrophys. J.

490, 412–424 (1997).11. Tull, R. G., MacQueen, P. J., Sneden, C. & Lambert, D. L. The high-resolution cross-dispersed echelle

white pupil spectrometer of the McDonald Observatory 2.7-m telescope. Publ. Astron Soc. Pacif. 107,251–264 (1995).

12. Marcy, G. W. et al. The planet around 51 Pegasi. Astrophys. J. 481, 926–935 (1997).

Acknowledgements. This work was supported by NASA’s Origins of the Solar System Program.

Correspondence should be addressed to A.P.H. (e-mail: [email protected]).

Kondoeffect ina

single-electron transistor

D. Goldhaber-Gordon*†, Hadas Shtrikman†, D. Mahalu†,David Abusch-Magder*, U. Meirav† & M. A. Kastner** Department of Physics, Massachusetts Institute of Technology, Cambridge,Massachussetts 02139, USA

†Braun Center for Submicron Research, Department of Condensed Matter

Physics, Weizmann Institute of Science, Rehovot 76100, Israel

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

How localized electrons interact with delocalized electrons is acentral question to many problems in sold-state physics1–3. Thesimplest manifestation of this situation is the Kondo effect, whichoccurs when an impurity atom with an unpaired electron is placedin a metal2. At low temperatures a spin singlet state is formedbetween the unpaired localized electron and delocalized electronsat the Fermi energy. Theories predict4–7 that a Kondo singletshould form in a single-electron transistor (SET), which containsa confined ‘droplet’ of electrons coupled by quantum-mechanicaltunnelling to the delocalized electrons in the transistor’s leads. Ifthis is so, a SET could provide a means of investigating aspects ofthe Kondo effect under controlled circumstances that are notaccessible in conventional systems: the number of electrons can bechanged from odd to even, the difference in energy between thelocalized state and the Fermi level can be tuned, the coupling tothe leads can be adjusted, voltage differences can be applied toreveal non-equilibrium Kondo phenomena7, and a single localizedstate can be studied rather than a statistical distribution. But forSETs fabricated previously, the binding energy of the spin singlethas been too small to observe Kondo phenomena. Ralph andBuhrman8 have observed the Kondo singlet at a single accidentalimpurity in a metal point contact, but with only two electrodesand without control over the structure they were not able toobserve all of the features predicted. Here we report measure-ments on SETs smaller than those made previously, which exhibitall of the predicted aspects of the Kondo effect in such a system.

When the channel of a transistor is made very small and isisolated from its leads by tunnel barriers it behaves in an unusualway. A transistor can be thought of as an electronic switch that is onwhen it conducts current and off when it does not. Whereas aconventional field-effect transistor, such as one in a computermemory, turns on only once when electrons are added to it, theSET turns on and off again every time a single electron is added toit9,10. This increased functionality may eventually make SETs tech-nologically important.

The unusual behaviour of SETs is a manifestation of the quanti-zation of charge and energy caused by the confinement of thedroplet of electrons in the small channel. As similar quantizationoccurs when electrons are confined in an atom, the small droplet ofelectrons is often called an artificial atom11,12.

We have fabricated SETs using multiple metallic gates (electrodes)deposited on a GaAs/AlGaAs heterostructure (Fig. 1a) containing atwo-dimensional electron gas, or 2DEG. First, the electrons aretrapped in a plane by differences in the electronic properties of theheterostructure’s layers. Second, they are excluded from regions ofthe plane beneath the gates when negative voltages are applied to

Figure 1 a, Scanning electron microscope image showing top view of sample.

Three gate electrodes, the one on the right and the upper and lower ones on the

left, control the tunnel barriers between reservoirs of two-dimensional electron

gas (at top and bottom) and the droplet of electrons. The middle electrode on the

left is used as a gate to change the energy of the droplet relative to the two-

dimensional electron gas. Source and drain contacts at the top and bottom are

not shown. Although the lithographic dimensions of the confined region are

150 nm square, we estimate lateral depletion reduces the electron droplet to

dimensions of 100nm square. The gate pattern shown was deposited on top of a

shallow heterostructure with the following layer sequence grown on top of a thick

undoped GaAs buffer: 5 nm Al0.3Ga0.7As, 5 3 1012 cm2 1 Si d-doping, 5 nm

Al0.3Ga0.7As, d-doping, 5 nm Al0.3Ga0.7As, 5 nm GaAs cap (H.S., D.G.-G. and U.M.,

manuscript in preparation). Immediately before depositing the metal, we etched

off the GaAs cap in the areas where the gates would be deposited, to reduce

leakage between the gates and the electron gas. b, Schematic energy diagram of

the artificial atom and its leads. The situation shown corresponds to Vds , kT=e,

for which theFermi energies in sourceanddrainarenearlyequal, and to avalueof

Vg near a conductance minimum between a pair of peaks corresponding to the

same spatial state. For this case there is an energy cost ,U to add or remove anelectron. To place an extra electron in the lowest excited state costs ,U þ De.

Cu, Ag, Au, Mg, Zn, ... doped with Cr, Fe, Mo, Mn, Re, Os, ...

200nm

Alloys

Goldhaber-Gordon, et al., Nature 391 (1998), 156-159.

Cronenwett, et al., Science 281 (1998), no. 5376, 540-544.

Enhance the spin group

Generalizations

SU(2)� SU(N)

Observation of the SU(4) Kondo state in a

double quantum dot

A. J. Keller1, S. Amasha1,†, I. Weymann2, C. P. Moca3,4, I. G. Rau1,‡, J. A. Katine5,

Hadas Shtrikman6, G. Zaránd3, and D. Goldhaber-Gordon1,*

1Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA2Faculty of Physics, Adam Mickiewicz University, Poznań, Poland

3BME-MTA Exotic Quantum Phases “Lendület” Group, Institute of Physics, Budapest University

of Technology and Economics, H-1521 Budapest, Hungary4Department of Physics, University of Oradea, 410087, Romania

5HGST, San Jose, CA 95135, USA6Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 96100, Israel

†Present address: MIT Lincoln Laboratory, Lexington, MA 02420, USA‡Present address: IBM Research – Almaden, San Jose, CA 95120, USA

*Corresponding author; [email protected]

Central to condensed matter physics are quantum impurity models,

which describe how a local degree of freedom interacts with a continuum.

Surprisingly, these models are often universal in that they can quantitatively

describe many outwardly unrelated physical systems. Here we develop a

double quantum dot-based experimental realization of the SU(4) Kondo

model, which describes the maximally symmetric screening of a local four-

fold degeneracy. As demonstrated through transport measurements and

detailed numerical renormalization group calculations, our device a↵ords

exquisite control over orbital and spin physics. Because the two quan-

tum dots are coupled only capacitively, we can achieve orbital state- or

“pseudospin”-resolved bias spectroscopy, providing intimate access to the

interplay of spin and orbital Kondo e↵ects. This cannot be achieved in the

few other systems realizing the SU(4) Kondo state.

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gro

up

calcu

latio

ns,

ourdev

icea↵ord

s

exquisite

contro

lov

erorb

italand

spin

physics.

Beca

use

the

two

quan-

tum

dots

are

coupled

only

capacitiv

ely,we

can

ach

ieve

orb

italsta

te-or

“pseu

dosp

in”-reso

lved

biassp

ectrosco

py,

prov

iding

intim

ate

access

toth

e

interp

layofsp

inandorb

italKondoe↵

ects.This

cannotbeach

ieved

inth

e

fewoth

ersy

stemsrea

lizingth

eSU(4)Kondosta

te.

1

arXiv:1306.6326v1 [cond-mat.mes-hall] 26 Jun 2013


Generalizations

SU(2)� SU(N)

arXiv:1310.6563v1 [cond-mat.str-el] 24 Oct 2013

SU(1

2)KondoEffect

inCarb

on

Nanotu

beQuantu

mDot

IgorKuzm

enko

1an

dYshai

Avish

ai 1,2

1Depa

rtmen

tofPhysics,

Ben

-Gurio

nUniversity

oftheNegev

Beer-S

heva

,Isra

el2Depa

rtmen

tofPhysics,

HongKongUniversity

ofScien

ceandTech

nology

,Kowloo

n,HongKong

(Dated

:Octob

er25,

2013)

Westu

dytheKon

doeff

ectin

aCNT(left

lead)-C

NT(Q

D)-C

NT(righ

tlead

)stru

cture.

Here

CNTis

asin

gle-wall

metallic

carbon

nan

otube,

forwhich

1)thevalen

cean

dcon

duction

ban

dsof

electrons

with

zeroorb

italan

gular

mom

entum

(m=

0)coalesc

atthetw

ovalley

poin

tsK

and

K′of

the

first

Brillou

inzon

ean

d2)

theen

ergysp

ectrum

ofelectron

swith

m!=

0has

agap

whose

sizeis

prop

ortional

to|m

|.Follow

ingad

sorption

ofhydrogen

atomsan

dap

plication

ofan

approp

riately

design

edgate

poten

tial,electron

energy

levelsin

theCNT(Q

D)are

tunab

leto

have:

1)tw

o-foldspin

degen

eracy;2)

two-fold

isospin

(valley)degen

eracy;3)

three-fold

orbital

degen

eracym

=0,±

1.As

aresu

lt,an

SU(12)

Kon

doeff

ectis

realizedwith

remarkab

lyhigh

Kon

dotem

peratu

re.Apart

from

this

prop

erty,thepertin

enttunnel

conductan

cehas

asim

ilartem

peratu

redep

enden

ceas

forthe

usual

SU(2)

Kon

doeff

ect.Ontheoth

erhan

d,apecu

liarresu

ltrelated

totheSU(12)

symmetry

is

that

themagn

eticsuscep

tibilities

forparallel

andperp

endicu

larmagn

eticfield

sdisp

layan

isotropy

with

auniversal

ratioχ‖ /χ

⊥=

ηthat

dep

endson

lyon

theelectron

’sorb

italan

dspin

gfactors.

PACSnumbers:

73.21.Hb,73.21.L

a,73.22.D

j,73.23.H

k,73.40.R

w,73.63.F

g,73.63.K

v

I.IN

TRODUCTIO

N

Back

gro

und:

Kon

dotunnellin

gthrou

ghcarb

onnan

otubequan

-

tum

dots

CNT(Q

D)has

recently

becom

easubject

ofinten

setheoretical 1

–5an

dexperim

ental 6

–11stu

d-

ies.Oneof

themotivation

sfor

pursu

ingthisresearch

direction

isthequest

forach

ievingan

exotic

Kon

do

effect

with

SU(N

)dynam

icalsymmetry, 1

2–16based

on

thepecu

liarprop

ertiesof

electronspectru

min

CNT. 1

Achiev

ingSU(4)

symmetry

isnatu

ralbecau

setheen-

ergyspectru

mof

metallic

CNT

consists

oftw

oinde-

pendentvalley

sthat

touch

attheK

andK

′poin

tsof

theBrillou

inzon

e.Theenergy

levelspossess

degen

-

eracyin

both

spin

(↑,↓)an

disosp

in(or

valleyK,K

′)

quan

tum

numbers.

Thus,dueto

both

spin

andisosp

in

degen

eracy,an

SU(4)

Kon

doeff

ecttakes

place. 2

–4,9–11

Motiv

atio

n:

Achiev

ing

evenhigh

erdegen

eracy

SU(N

>4)

oftheQD

ishigh

lydesirab

le.Firstly,

the

Kon

do

temperatu

redram

aticallyincreases

with

N.

Secon

dly,

there

isahop

eto

expose

novel

physical

ob-

servables

that

arepecu

liarto

these

high

ersymmetries.

Inthepresen

tdevice,

high

erdegen

eracymay

beob

-

tained

byem

ploy

ingtheorb

italsymmetry

ofelectron

statesin

CNT,an

option

which

sofar

has

not

been

effectively

employed

inthisquest.

Inord

erto

man

ip-

ulate

these

orbital

features,

weuse

thefact

that

ad-

sorption

ofox

ygen

,hydrogen

orfluorin

eatom

sgives

riseto

gapop

eningin

thespectru

mof

themetallic

CNT. 17,18Realization

ofSU(N

>4)

Kon

doeff

ectthen

becom

esfeasib

le,sin

cethere

isnow

spin,isosp

in(val-

ley)an

dorb

italdegen

eracy.

Themain

objectiv

es:

Themain

goalsof

thepresen

t

work

are:1)

Toshow

that

SU(12)

Kon

doeff

ectin

the

CNT(left

lead)-C

NT(Q

D)-C

NT(righ

tlead

)stru

cture

isindeed

achievab

lean

d2)

Toelu

cidate

thephysi-

calcon

tentof

this

structu

reat

theKon

doregim

eas

encoded

bytunnelin

gcon

ductan

cean

dthemagn

etic

suscep

tibility.

Thefirst

goalob

tainsbydesign

ingthe

electronspectru

min

theCNT(Q

D)to

have

a12-fold

degen

eracyfollow

ingad

sorption

ofhydrogen

atoms

combined

with

anap

plication

ofanon

-uniform

gate

poten

tial.Nam

ely,theenergy

levelsof

thecen

tral

elementCNT(Q

D)are

tunab

leinto

athree-fold

or-

bital

degen

eracyfor

m=

0,±1.

Thesecon

dgoal

is

achieved

throu

ghquan

titativean

alysis,

based

onper-

turbation

theory

athigh

temperatu

resan

dmean

field

slaveboson

formalism

atlow

temperatu

re.

The

main

resu

lts:Theenergy

spectru

mof

the

CNT(Q

D)gated

byaspacially

modulated

poten

tialis

elucid

ated,an

dthepossib

ilityto

getaCNT(Q

D)with

twelve-fold

degen

eratequan

tum

statesis

substan

ti-

arX

iv:1

310.

6563

v1 [

cond

-mat

.str-e

l] 2

4 O

ct 2

013

SU(12) Kondo Effect in Carbon Nanotube Quantum Dot

Igor Kuzmenko1 and Yshai Avishai1,2

1 Department of Physics, Ben-Gurion University of the Negev Beer-Sheva, Israel2 Department of Physics, Hong Kong University of Science and Technology, Kowloon, Hong Kong

(Dated: October 25, 2013)

We study the Kondo effect in a CNT(left lead)-CNT(QD)-CNT(right lead) structure. Here CNT is

a single-wall metallic carbon nanotube, for which 1) the valence and conduction bands of electrons

with zero orbital angular momentum (m = 0) coalesc at the two valley points K and K′ of the

first Brillouin zone and 2) the energy spectrum of electrons with m != 0 has a gap whose size is

proportional to |m|. Following adsorption of hydrogen atoms and application of an appropriately

designed gate potential, electron energy levels in the CNT(QD) are tunable to have: 1) two-fold spin

degeneracy; 2) two-fold isospin (valley) degeneracy; 3) three-fold orbital degeneracy m = 0,±1. As

a result, an SU(12) Kondo effect is realized with remarkably high Kondo temperature. Apart from

this property, the pertinent tunnel conductance has a similar temperature dependence as for the

usual SU(2) Kondo effect. On the other hand, a peculiar result related to the SU(12) symmetry is

that the magnetic susceptibilities for parallel and perpendicular magnetic fields display anisotropy

with a universal ratio χ‖/χ⊥ = η that depends only on the electron’s orbital and spin g factors.

PACS numbers: 73.21.Hb, 73.21.La, 73.22.Dj, 73.23.Hk, 73.40.Rw, 73.63.Fg, 73.63.Kv

I. INTRODUCTION

Background:

Kondo tunnelling through carbon nanotube quan-

tum dots CNT(QD) has recently become a subject

of intense theoretical1–5 and experimental6–11 stud-

ies. One of the motivations for pursuing this research

direction is the quest for achieving an exotic Kondo

effect with SU(N) dynamical symmetry,12–16 based on

the peculiar properties of electron spectrum in CNT.1

Achieving SU(4) symmetry is natural because the en-

ergy spectrum of metallic CNT consists of two inde-

pendent valleys that touch at the K and K′ points of

the Brillouin zone. The energy levels possess degen-

eracy in both spin (↑, ↓) and isospin (or valley K,K′)quantum numbers. Thus, due to both spin and isospin

degeneracy, an SU(4) Kondo effect takes place.2–4,9–11

Motivation: Achieving even higher degeneracy

SU(N>4) of the QD is highly desirable. Firstly, the

Kondo temperature dramatically increases with N .

Secondly, there is a hope to expose novel physical ob-

servables that are peculiar to these higher symmetries.

In the present device, higher degeneracy may be ob-

tained by employing the orbital symmetry of electron

states in CNT, an option which so far has not been

effectively employed in this quest. In order to manip-

ulate these orbital features, we use the fact that ad-

sorption of oxygen, hydrogen or fluorine atoms gives

rise to gap opening in the spectrum of the metallic

CNT.17,18 Realization of SU(N> 4) Kondo effect then

becomes feasible, since there is now spin, isospin (val-

ley) and orbital degeneracy.

The main objectives: The main goals of the present

work are: 1) To show that SU(12) Kondo effect in the

CNT(left lead)-CNT(QD)-CNT(right lead) structure

is indeed achievable and 2) To elucidate the physi-

cal content of this structure at the Kondo regime as

encoded by tunneling conductance and the magnetic

susceptibility. The first goal obtains by designing the

electron spectrum in the CNT(QD) to have a 12-fold

degeneracy following adsorption of hydrogen atoms

combined with an application of a non-uniform gate

potential. Namely, the energy levels of the central

element CNT(QD) are tunable into a three-fold or-

bital degeneracy for m = 0,±1. The second goal isachieved through quantitative analysis, based on per-

turbation theory at high temperatures and mean field

slave boson formalism at low temperature.

The main results: The energy spectrum of the

CNT(QD) gated by a spacially modulated potential is

elucidated, and the possibility to get a CNT(QD) with

twelve-fold degenerate quantum states is substanti-

Multiple “channels” or “flavors”


Representation of impurity spin

Generalizations

SU(2)� SU(N)

c� c� � = 1, . . . , k

simp = 1/2 � Rimp

U(1)� SU(k)

IR fixed point:

“Non-Fermi liquids”

NOT alwaysa fermi liquid

Generalizations

Kondo model specified by

Apply the techniques mentioned above...

N, Rimp, k

Open Problems

Entanglement Entropy

Quantum Quenches

Multiple Impurities

Kondo:

Form singlets with each other

Competition between these can produce a

QUANTUM PHASE TRANSITION

Form singlets with electrons

�Si · �Sj

Open Problems

UBe13 UPt3

CeCu6YbAl3

CePd2Si2YbRh2Si2

Multiple Impurities

Heavy fermion compounds

NpPd5Al2 CeCoIn5

Open Problems

Example

J. Custers et al., Nature 424, 524 (2003)

Kondo lattice

YbRh2Si2

0 2 4 6 80

5

10

15

20

25

0 10.0

0.1

0.2

0.3

LFLAF

NFL

YbRh2Si

2

H || c

2

T (

K)

H (T)

c

YbRh2(Si

0.95Ge

0.05)

2

!(µ"

cm

)

T (K) 0 1 2 30

5

10T

N

SCAF

d

CePd2Si

2

T (

K)

P (GPa)

0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

2.5

TN

AF

aCeCu

6-xAu

x

T (

K)

x

FIG. 1: Quantum critical points in heavy fermion metals. a: AF ordering temperature TN vs. Au

concentration x for CeCu6−xAux (Ref.7), showing a doping induced QCP. b: Suppression of the

magnetic ordering in YbRh2Si2 by a magnetic field. Also shown is the evolution of the exponent α in

∆ρ ≡ [ρ(T )−ρ0] ∝ Tα, within the temperature-field phase diagram of YbRh2Si2 (Ref.9). Blue and

orange regions mark α = 2 and 1, respectively. c: Linear temperature dependence of the electrical

resistivity for Ge-doped YbRh2Si2 over three decades of temperature (Ref.9), demonstrating the

robustness of the non-Fermi liquid behavior in the quantum critical regime. d: Temperature

vs. pressure phase diagram for CePd2Si2, illustrating the emergence of a superconducting phase

centered around the QCP. The Néel- (TN ) and superconducting ordering temperatures (Tc) are

indicated by closed and open symbols, respectively.10

sitions. At the melting point, ice abruptly turns into water, absorbing latent heat. In other

words, the transition is of first order. A piece of magnet, on the other hand, typically “melts”

into a paramagnet through a continuous transition: The magnetization vanishes smoothly,

and no latent heat is involved. In the case of zero temperature, the point of such a second-

2

� � T 2

0 2 4 6 80

5

10

15

20

25

0 10.0

0.1

0.2

0.3

LFLAF

NFL

YbRh2Si

2

H || c

2

T (

K)

H (T)

c

YbRh2(Si

0.95Ge

0.05)

2

!(µ"

cm

)

T (K) 0 1 2 30

5

10T

N

SCAF

d

CePd2Si

2

T (

K)

P (GPa)

0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

2.5

TN

AF

aCeCu

6-xAu

x

T (

K)

x

FIG. 1: Quantum critical points in heavy fermion metals. a: AF ordering temperature TN vs. Au

concentration x for CeCu6−xAux (Ref.7), showing a doping induced QCP. b: Suppression of the

magnetic ordering in YbRh2Si2 by a magnetic field. Also shown is the evolution of the exponent α in

∆ρ ≡ [ρ(T )−ρ0] ∝ Tα, within the temperature-field phase diagram of YbRh2Si2 (Ref.9). Blue and

orange regions mark α = 2 and 1, respectively. c: Linear temperature dependence of the electrical

resistivity for Ge-doped YbRh2Si2 over three decades of temperature (Ref.9), demonstrating the

robustness of the non-Fermi liquid behavior in the quantum critical regime. d: Temperature

vs. pressure phase diagram for CePd2Si2, illustrating the emergence of a superconducting phase

centered around the QCP. The Néel- (TN ) and superconducting ordering temperatures (Tc) are

indicated by closed and open symbols, respectively.10

sitions. At the melting point, ice abruptly turns into water, absorbing latent heat. In other

words, the transition is of first order. A piece of magnet, on the other hand, typically “melts”

into a paramagnet through a continuous transition: The magnetization vanishes smoothly,

and no latent heat is involved. In the case of zero temperature, the point of such a second-

2

� � T

Multiple Impurities

Heavy fermion compounds

The Kondo Lattice

The Kondo Lattice...

“... remains one of thebiggest unsolved problems

in condensed matter physics.”Alexei Tsvelik

QFT in Condensed Matter Physics(Cambridge Univ. Press, 2003)

“... remains one of thebiggest unsolved problems

in condensed matter physics.”Alexei Tsvelik

QFT in Condensed Matter Physics(Cambridge Univ. Press, 2003)

Let’s try AdS/CFT!

The Kondo Lattice...

GOAL

Find a holographic descriptionof the

Kondo Effect

Outline:


Kondo interaction preserves spherical symmetry

gK �3(�x) �S · c†(�x) 1

2�� c(�x)

Reduction to one dimension

restrict to momenta near

restrict to s-wave

kF

CFT Approach to the Kondo EffectAffleck and Ludwig 1990s

c(�x) � 1r

�e�ikF r�L (r)� e+ikF r�R (r)

�

LL

L

R

FIG. 4. Reflecting the left-movers to the negative axis.

D. Fermi Liquid Approach at Low T

What is the T → 0 behavior of the antiferromagetic Kondo model? The simplest assumption isλeff → ∞. But what does that really mean? Consider the strong coupling limit of a lattice model,

2

for convenience, in spatial dimension D = 1. (D doesn’t really matter since we can always reducethe model to D = 1.)

H = t∑

i

(ψ†iψi+1 + ψ†i+1ψi) + λ#S · ψ

†0

#σ2ψ0 (1.20)

Consider the limit λ >> |t|. The groundstate of the interaction term will be the following con-figuration: one electron at the site 0 forms a singlet with the impurity: | ⇑↓〉 − | ⇓↑〉. (We as-sume SIMP = 1/2). Now we do perturbation theory in t. We have the following low energy states:an arbitary electron configuration occurs on all other sites-but other electrons or holes are forbiddento enter the site-0, since that would destroy the singlet state, costing an energy, ∆E ∼ λ >> t.Thus we simply form free electron Bloch states with the boundary condition φ(0) = 0, where φ(i) isthe single-electron wave-function. Note that at zero Kondo coupling, the parity even single particlewave-functions are of the form φ(i) = cos ki and the parity odd ones are of the form φ(i) = sin ki.On the other hand, at λ → ∞ the parity even wave-functions become φ(i) = | sin ki|, while theparity odd ones are unaffected.

The behaviour of the parity even channel corresponds to a π/2 phase shift in the s-wave channel.

φj ∼ e−ik|j| + e+2iδeik|j|, δ = π/2. (1.21)

In terms of left and right movers on r > 0 we have changed the boundary condition,

ψL(0) = ψR(0), λ = 0,

ψL(0) = −ψR(0), λ = ∞. (1.22)

The strong coupling fixed point is the same as the weak coupling fixed point except for a changein boundary conditions (and the removal of the impurity). In terms of the left-moving descriptionof the P -even sector, the phase of the left-mover is shifted by π as it passes the origin. Imposinganother boundary condition a distance l away quantizes k:

ψ(l) = ψL(l) + ψR(l) = ψL(l) + ψL(−l) = 0,

λ = 0 : k =πl(n + 1/2)

λ = ∞ : k = πnl

(1.23)

6

r

LL

L

R




2


H = t∑

i

(ψ†iψi+1 + ψ†i+1ψi) + λ#S · ψ

†0

#σ2ψ0 (1.20)





ψL(0) = ψR(0), λ = 0,

ψL(0) = −ψR(0), λ = ∞. (1.22)


ψ(l) = ψL(l) + ψR(l) = ψL(l) + ψL(−l) = 0,

λ = 0 : k =πl(n + 1/2)

λ = ∞ : k = πnl

(1.23)

6

r

�L(�r) � �R(+r)

r = 0

r = 0

RELATIVISTIC chiral fermions

“speed of light”vF

CFT Approach to the Kondo Effect

g̃K �k2F

2�2vF� gK

CFT!

=

HK =vF2�

� +�

��dr

��†Li�r�L + �(r) g̃K �S · �

†L�� L

�

k � 1Spin SU(N)

U(1)

SU(k)

SU(N)

J = �†L�L

JA = �†L tA �L

�J = �†L �� L

Kac-Moody Current Algebra

z � � + ir

JA(z) =�

n�Zz�n�1JAn

[JAn , JBm] = if

ABCJCn+m + Nn

2�AB�n,�m

SU(k)N

N counts net number of chiral fermions


Full symmetry:

(1 + 1)d conformal symmetry

SU(N)k � SU(k)N � U(1)kN

HK =vF2�

� +�

��dr

��†Li�r�L + �(r) g̃K �S · �

†L�� L

�


J = �†L�L

JA = �†L tA �L

Kondo coupling: �S · �J

U(1)

SU(k)

SU(N)

HK =vF2�

� +�

��dr

��†Li�r�L + �(r) g̃K �S · �

†L�� L

�

�J = �†L �� L

UV

IR SU(N)k � SU(k)N � U(1)Nk

SU(N)k � SU(k)N � U(1)Nk

Eigenstates are representations of the Kac-Moody algebra

UV



|c, s, f�

|c, s�, f�

Fusion Rules

s� simp = s�

UV



Fusion Rules

s� simp = s�Example:

|s� simp| � s� � min{s + simp, k � (s + simp)}

(for k � (s + simp) > 0)

SU(2)k

UV

IR

decoupled spin at r = 0

LL

L

R




2


H = t∑

i

(ψ†iψi+1 + ψ†i+1ψi) + λ#S · ψ

†0

#σ2ψ0 (1.20)





ψL(0) = ψR(0), λ = 0,

ψL(0) = −ψR(0), λ = ∞. (1.22)


ψ(l) = ψL(l) + ψR(l) = ψL(l) + ψL(−l) = 0,

λ = 0 : k =πl(n + 1/2)

λ = ∞ : k = πnl

(1.23)

6

r

�L(0�) = �L(0+)

�L(0�) = ��L(0+)

�/2 phase shift

UV

IR

LL

L

R




2


H = t∑

i

(ψ†iψi+1 + ψ†i+1ψi) + λ#S · ψ

†0

#σ2ψ0 (1.20)





ψL(0) = ψR(0), λ = 0,

ψL(0) = −ψR(0), λ = ∞. (1.22)


ψ(l) = ψL(l) + ψR(l) = ψL(l) + ψL(−l) = 0,

λ = 0 : k =πl(n + 1/2)

λ = ∞ : k = πnl

(1.23)

6

r

� (r) = A cos kr + B sin kr

� (r) = A�| sin kr| + B� sin kr

decoupled spin at r = 0

�/2 phase shift


Take-Away Messages

Central role of theKac-Moody Algebra

PHASE SHIFT

Kondo coupling: �S · �J

Outline:


GOAL

Find a holographic descriptionof the

Kondo Effect

What classical action do we writeon the gravity side of the correspondence?

How do we describe holographically...

1

2

3

The chiral fermions?

The impurity?

The Kondo coupling?

Top-down:

AdS solution to a string or supergravity theory

Bottom-up:

AdS solution of some ad hoc Lagrangian

Holography

Open strings

0 1 2 3 4 5 6 7 8 9Nc D3 X X X XN7 D7 X X X X X X X XN5 D5 X X X X X X

3-3 5-5 7-73-7 7-33-57-5 5-7

and and

and

and

and

5-3

Top-Down Model


3-3 5-5 7-73-7 7-33-57-5 5-7

and and

and

and

andCFT with holographic dual

5-3

Top-Down Model


3-3 5-5 7-73-7 7-33-57-5 5-7

and and

and

and

and

Decouple5-3

Top-Down Model


3-3 5-5 7-73-7 7-3

5-33-57-5 5-7

and and

and

and

and

(1+1)-dimensionalchiral fermions

Top-Down Model


3-3 5-5 7-73-7 7-33-57-5 5-7

and and

and

and

and

the impurity

5-3

Top-Down Model


3-3 5-5 7-73-7 7-33-57-5 5-7

and and

and

and

and

Kondo interaction

5-3

Top-Down Model

Previous work

Mück 1012.1973

Kachru, Karch, Yaida 0909.2639, 1009.3268

Faraggi and Pando-Zayas 1101.5145

Jensen, Kachru, Karch, Polchinski, Silverstein 1105.1772

Karaiskos, Sfetsos, Tsatis 1106.1200

Harrison, Kachru, Torroba 1110.5325

Benincasa and Ramallo 1112.4669, 1204.6290

Faraggi, Mück, Pando-Zayas 1112.5028

Itsios, Sfetsos, Zoakos 1209.6617


3-3 5-5 7-73-7 7-33-57-5 5-7

and and

and

and

and

Absent in previous constructions5-3

Top-Down Model

The D3-branes

N = 4 SU(Nc) YMSUSY

0 1 2 3 4 5 6 7 8 9Nc D3 X X X X

� � g2Y MNc

3-3 strings

�� = 0

(3 + 1)- dimensional

CFT!

The D3-branes

N = 4 SU(Nc) YMSUSY

0 1 2 3 4 5 6 7 8 9Nc D3 X X X X

� � g2Y MNc

3-3 strings(3 + 1)- dimensional

g2Y M � 0Nc �� fixed

The D3-branes

N = 4 SU(Nc) YMSUSY

0 1 2 3 4 5 6 7 8 9Nc D3 X X X X

� � g2Y MNc

3-3 strings(3 + 1)- dimensional

g2Y M � 0Nc ��

0 1 2 3 4 5 6 7 8 9Nc D3 X X X X

N = 4 SYMType IIB Supergravity

Nc �⇥ =

The D3-branes

�� AdS5 � S5

g2Y MNc � L4AdS/��2g2Y M � gs

LAdS � 1

0 1 2 3 4 5 6 7 8 9Nc D3 X X X X

The D3-branes

�

S5F5 � Nc F5 = dC4

N = 4 SYMType IIB Supergravity

Nc �⇥ =�� AdS5 � S

5

Anti-de Sitter Space

r =�

r = 0

boundary

Figure 1: The two slicings of AdS5. The horizontal axis is the direction x transverse to thebrane and the vertical axis is the radial direction of AdS interpolating from the boundary(solid line) to the horizon (dashed line). The figure on the left shows lines of constant ρwhile the figure on the right shows lines of constant r.

Every AdSd+1 bulk field φd+1(#y, w, r) of mass M , transforming in some representationof SO(d, 2), decomposes into a tower of AdSd modes φd,n(#y, w) inhabiting representationsof the preserved isometry group SO(d − 1, 2). Each mode is multiplied by an appropriatewavefunction of the r-direction:

φd+1(#y, w, r) =∑

n

ψn(r) φd,n(#y, w) . (15)

Among the data of the SO(d − 1, 2) representation is an AdSd-mass mn for each φd,n,

∂2dφd,n = m2nφd,n , (16)

where ∂2d is the AdSd-Laplacian. The mass mn and the wavefunction ψn(r) may be deter-mined by solving the wave equation for φd+1(#y, x, r). In general the backreaction of thebrane may produce a more general warp factor A(r), ds2 = dr2 + e2A(r)ds2AdSd, although (13)will continue to hold at large |r|; this more general metric still preserves AdSd isometriesassociated with dual dCFT. To linear order the wave equation then reduces to an ordinarydifferential equation for the wavefunction ψn(r),

∂2rψn(r) + dA′(r)∂rψn(r) + e

−2A(r)m2nψn(r) − M2ψn(r) = 0 . (17)

This will receive corrections from various interactions in the brane worldvolume theory,7 allof which affect the calculation of the masses mn.

The field φd+1 of mass M is dual to an ambient operator Od(#y, x) of dimension ∆d (with∆d(∆d − d) = M2) in the dCFT. Analogously, since the φd,n inhabit an effective AdSd the-ory (they are representations of SO(d − 1, 2)), they are related to dual “defect operators”

7The brane interactions will generally cause a mixing between the modes corresponding to different bulkfields φd+1, though we neglect this here. However, precisely the same phenomenon occurs also in the BOPE,and it is easy to generalize our discussion to incorporate it.

11

x

ds2 =dr2

r2+ r2

��dt2 + dx2 + dy2 + dz2

�

Poincaré horizon

Anti-de Sitter Space

Figure 1: The two slicings of AdS5. The horizontal axis is the direction x transverse to thebrane and the vertical axis is the radial direction of AdS interpolating from the boundary(solid line) to the horizon (dashed line). The figure on the left shows lines of constant ρwhile the figure on the right shows lines of constant r.

Every AdSd+1 bulk field φd+1(#y, w, r) of mass M , transforming in some representationof SO(d, 2), decomposes into a tower of AdSd modes φd,n(#y, w) inhabiting representationsof the preserved isometry group SO(d − 1, 2). Each mode is multiplied by an appropriatewavefunction of the r-direction:

φd+1(#y, w, r) =∑

n

ψn(r) φd,n(#y, w) . (15)

Among the data of the SO(d − 1, 2) representation is an AdSd-mass mn for each φd,n,

∂2dφd,n = m2nφd,n , (16)

where ∂2d is the AdSd-Laplacian. The mass mn and the wavefunction ψn(r) may be deter-mined by solving the wave equation for φd+1(#y, x, r). In general the backreaction of thebrane may produce a more general warp factor A(r), ds2 = dr2 + e2A(r)ds2AdSd, although (13)will continue to hold at large |r|; this more general metric still preserves AdSd isometriesassociated with dual dCFT. To linear order the wave equation then reduces to an ordinarydifferential equation for the wavefunction ψn(r),

∂2rψn(r) + dA′(r)∂rψn(r) + e

−2A(r)m2nψn(r) − M2ψn(r) = 0 . (17)

This will receive corrections from various interactions in the brane worldvolume theory,7 allof which affect the calculation of the masses mn.

The field φd+1 of mass M is dual to an ambient operator Od(#y, x) of dimension ∆d (with∆d(∆d − d) = M2) in the dCFT. Analogously, since the φd,n inhabit an effective AdSd the-ory (they are representations of SO(d − 1, 2)), they are related to dual “defect operators”

7The brane interactions will generally cause a mixing between the modes corresponding to different bulkfields φd+1, though we neglect this here. However, precisely the same phenomenon occurs also in the BOPE,and it is easy to generalize our discussion to incorporate it.

11

x

ds2 =dr2

r2+ r2

��dt2 + dx2 + dy2 + dz2

�

UV

IR


3-3 5-5 7-73-7 7-33-57-5 5-7

and and

and

and

and

Decouple5-3

Top-Down Model


5-57-7SYMU(N5)(5 + 1)-dim.SYM(7 + 1)-dim. U(N7)

g2Dp � gs ��p�32

g2Y M � gs

g2Y MNc � 1/��2


g2D5N5 � gYMN5�Nc

g2D7N7 �N7Nc

5-57-7SYMU(N5)(5 + 1)-dim.SYM(7 + 1)-dim. U(N7)

g2Dp � gs ��p�32

Probe Limit

Nc �⇥ g2Y M � 0

g2D5N5 � gY MN5�Nc� 0

g2D7N7 �N7Nc� 0

N7 ,N5 fixed

N7/Nc � 0 and N5/Nc � 0

Probe Limit

becomes a global symmetryU(N7)� U(N5)

Total symmetry:

(plus R-symmetry)

SU(Nc)� �� U(N7)� U(N5)� �� gauged global

SYM theories on D7- and D5-branes decouple

3-3 5-5 7-73-7 7-3

5-33-57-5 5-7

and and

and

and

and

(1+1)-dimensionalchiral fermions


Top-Down Model

0 1 2 3 4 5 6 7 8 9Nc D3 X X X XN7 D7 X X X X X X X X

8 Neumann-Dirichlet (ND) intersection

The D7-branes

Harvey and Royston 0709.1482, 0804.2854Buchbinder, Gomis, Passerini 0710.5170

Neumann

Dirichlet


The D7-branes


1/4 SUSY

(1+1)-dimensional chiral fermionsN7 �L

SUSY

8 Neumann-Dirichlet (ND) intersection

N = (0, 8)


The D7-branes

(1+1)-dimensional chiral fermionsN7

L

�L


The D7-branes

S3-7 =�

dx+dx��†L (i�� A�) �L


SU(Nc)� U(N7)� U(N5)Nc N7 singlet


The D7-branes

SU(Nc)N7 � SU(N7)Nc � U(1)NcN7Kac-Moody algebra


S3-7 =�

dx+dx��†L (i�� A�) �L


The D7-branes

Do not come from reduction from (3+1) dimensions

Genuinely relativistic

Differences from Kondo



The D7-branes

SU(Nc) is gauged!

�J = �†L�� L

Differences from Kondo


Gauge Anomaly!

The D7-branes0 1 2 3 4 5 6 7 8 9

Nc D3 X X X XN7 D7 X X X X X X X X

SU(Nc) is gauged!


Gauge Anomaly!

The D7-branes0 1 2 3 4 5 6 7 8 9

Nc D3 X X X XN7 D7 X X X X X X X X

SU(Nc) is gauged!

Probe Limit!

SU(Nc) SU(Nc)

gY M gY M

In the probe limit, the gauge anomaly is suppressed...

N7

� g2Y MN7

Nc

gD7gD7� g2D7Nc

... but the global anomalies are not.

g2D7 � 1/Nc

U(N7)U(N7)

In the probe limit, the gauge anomaly is suppressed...

... but the global anomalies are not.

SU(Nc)N7 � SU(Nc)

SU(N7)Nc � U(1)NcN7 � SU(N7)Nc � U(1)NcN7

AdS3 � S5Probe D7-branes

N = 4 SYMNc �⇥ =��

Probe �L =

Type IIB Supergravity

AdS5 � S5


N = 4 SYMNc �⇥ =��

Probe �L =


AdS5 � S5

ds2 =dr2

r2+ r2

��dt2 + dx2 + dy2 + dz2

�+ ds2S5

J = ACurrent Gauge fieldU(N7) U(N7)


N = 4 SYMNc �⇥ =��

Probe �L =


AdS5 � S5

Kac-Moody Algebra Chern-Simons Gauge Field

Gukov, Martinec, Moore, Stromingerhep-th/0403225

Kraus and Larsenhep-th/0607138

=rank and level

ofalgebra

= rank and levelofgauge field

J = ACurrent Gauge field

J = ACurrent Gauge field

Gauge field on D7-brane

Decouples on field theory side...

...but not on the gravity side!

U(N7)Nc

AdS3 � S5Probe D7-branes along

= �12TD7(2��)2

�P [F5] � tr

�A � dA + 2

3A �A �A

�+ . . .

SD7 = +12TD7(2��)2

�P [C4] � trF � F + . . .

= �Nc4�

�

AdS3

tr�

A � dA + 23A �A �A

�+ . . .

U(N7)Nc Chern-Simons gauge field

Answer #1

Chern-Simons Gauge Field in AdS3

The chiral fermions:


3-3 5-5 7-73-7 7-33-57-5 5-7

and and

and

and

and

the impurity

5-3

Top-Down Model

0 1 2 3 4 5 6 7 8 9Nc D3 X X X XN5 D5 X X X X X X

The D5-branes

Gomis and Passerini hep-th/0604007

(0+1)-dimensional fermionsN5 �

1/4 SUSY

8 ND intersection


The D5-branes

Gomis and Passerini hep-th/0604007


8 ND intersection


The D5-branes

S3-5 =�

dt �†(i�t �At � �9)�

SU(Nc)� U(N7)� U(N5)Nc singlet N5



The D5-branes

SU(Nc) is “spin”

“Abrikosov pseudo-fermions”

Abrikosov, Physics 2, p.5 (1965)

“slave fermions”

�S = �†��

Integrate out

N5 = 1Gomis and Passerini hep-th/0604007

�

...R = }Det (�D) = TrRP exp

�i

�dt (At + �9)

�

Q = �†�

chargeU(N5) = U(1)

Probe D5-branes

N = 4 SYMNc �⇥ =��

Probe =


AdS5 � S5

AdS2 � S4�

ds2 =dr2

r2+ r2

��dt2 + dx2 + dy2 + dz2

�+ ds2S5

Probe D5-branes

N = 4 SYMNc �⇥ =��

Probe =


AdS5 � S5

AdS2 � S4�

Probe D5-branes

N = 4 SYMNc �⇥ =��

Probe =


AdS5 � S5

AdS2 � S4�

Q = Electric fluxJ =Current Gauge field aU(N5) U(N5)

Probe D5-brane along AdS2 � S4

Camino, Paredes, Ramallo hep-th/0104082

electric field AdS2

��gf tr

��AdS2

= Q = �†�

QDissolve strings into the D5-brane

frt = �rat � �tar

Answer #2

Yang-Mills Gauge Field in AdS2

electric flux=Rimp

The impurity:

3-3 5-5 7-73-7 7-33-57-5 5-7

and and

and

and

and

Kondo interaction

5-3


Top-Down Model

0 1 2 3 4 5 6 7 8 9N5 D5 X X X X X XN7 D7 X X X X X X X X

The Kondo Interaction

2 ND intersection

Complex scalar!

SU(Nc)� U(N7)� U(N5)singlet

O � �†L�

N5N7

TACHYON

The Kondo Interaction0 1 2 3 4 5 6 7 8 9

N5 D5 X X X X X XN7 D7 X X X X X X X X

m2tachyon = �1

4��

D5 becomes magnetic flux in the D7

SUSY completely broken

The Kondo InteractionSU(Nc) is “spin”

�S · �J = |�†L�|2 + O(1/Nc)

“double trace”

�J = �†L�� L �S = �†��

�S · �J = �†�� · �†L�� L

��ij · ��kl = �il�jk �1

Nc�ij�kl


N = 4 SYMNc �⇥ =��

Probe �L =


AdS5 � S5

Probe D5-branes Probe = AdS2 � S4�

= Bi-fundamental scalarAdS2 � S4O � �†L�

Answer #3

Bi-fundamental scalar in

The Kondo interaction:

AdS2

r =�

r = 0

x

tr f2�

AdS2

�

AdS2

|D�|2+V (�†�)

Nc

�

AdS3

A � F

D� = �� + iA�� ia�

Top-Down Model

r =�

r = 0

x

tr f2�

AdS2

�

AdS2

|D�|2+V (�†�)

Nc

�

AdS3

A � F

Top-Down Model

N, k, Rimp

r =�

r = 0

x

tr f2�

AdS2

�

AdS2

|D�|2+V (�†�)

Nc

�

AdS3

A � F

Top-Down Model

U(k)N

N, k, Rimp

r =�

r = 0

x

tr f2�

AdS2

�

AdS2

|D�|2+V (�†�)

Nc

�

AdS3

A � F

What is V (�†�) ?

Top-Down Model

We don’t know.

r =�

r = 0

x

tr f2�

AdS2

�

AdS2

|D�|2+V (�†�)

Nc

�

AdS3

A � F

Top-Down Model


Calculation in

Gava, Narain, Samadi hep-th/9704006

Aganagic, Gopakumar, Minwalla, Strominger hep-th/0009142

Difficult to calculate in AdS5 � S5

R9,1

Top-Down Model


Calculation in

Gava, Narain, Samadi hep-th/9704006

Aganagic, Gopakumar, Minwalla, Strominger hep-th/0009142

Switch to bottom-up model!

R9,1

Top-Down Model

Outline:


r =�

r = 0

x

tr f2�

AdS2

�

AdS2

|D�|2+V (�†�)

Bottom-Up Model

Nc

�

AdS3

A � F

D� = �� + iA�� ia�

r =�

r = 0

x

tr f2�

AdS2

�

AdS2

|D�|2+V (�†�)

Nc

�

AdS3

A � F

We pick V (�†�)

Bottom-Up Model

r =�

r = 0

x

tr f2�

AdS2

�

AdS2

|D�|2+V (�†�)

Nc

�

AdS3

A � F

V (�†�) = m2�†�

Bottom-Up Model

S = SCS + SAdS2

SCS = �N

4�

�tr

�A � dA + 2

3A �A �A

�

SAdS2 = ��

d3x �(x)��g

�14trf2 + |D�|2 + V (�†�)

�

D� = �� + iA�� ia�V (�†�) = m2�†�

Bottom-Up Model

S = SCS + SAdS2

SCS = �N

4�

�tr

�A � dA + 2

3A �A �A

�

SAdS2 = ��

d3x �(x)��g

�14trf2 + |D�|2 + V (�†�)

�

Bottom-Up Model


N, Rimp, k

SCS = �N

4�

�tr

�A � dA + 2

3A �A �A

�

SAdS2 = ��

d3x �(x)��g

�14trf2 + |D�|2 + V (�†�)

�

Bottom-Up Model


N, Rimp, k

U(k)N

F = dA f = da

Single channel ...Rimp =

U(1) gauge fields

Probe limit

Chern-Simons AdS2

U(1)Nc

Equations of Motion

µ, � = r, t, x m,n = r, t� = ei��

�mµ�Fµ� = �4�N

�(x)Jm

�n��g gnqgmpfqp

�= �Jm

�mJm = 0

�m��g gmn�n�

�=��g gmn(Am � am + �m�)(An � an + �n�)� +

12��g �V

��

Jm � 2��g gmn (An � an + �n�) �2

Ansatz:

Equations of Motion

at(r)�(r) Ax(r)

Static solution

After gauge fixing, only non-zero fields:

frt = a�t(r) Frx = A�x(r)

J t(r) = �2��g gttat�2

Equations of Motion

�r��g grr �r�

��g gtt a2t ��

��g m2 � = 0

�trxFrx = �4�N

�(x)J t(r)

�r��g grrgttfrt

�= �J t(r)


Boundary Conditions

c = g̃K c̃Witten hep-th/0112258

� (r) = c r�1/2 log r + c̃ r�1/2 + . . .

We choose Breitenlohner-Freedman boundm2 =

��gfrt

��AdS2

= Q

Our double-trace (Kondo) coupling:

� (r) = 0T > Tc

��†L�� = 0

��†L�� = 0

A holographic superconductor in

Hawking temperature T=

AdS2

T < Tc

AdS-Schwarzschild black hole

��gf tr

��AdS2

�= 0

��gf tr

��AdS2

�= 0 �(r) �= 0

Hawking temperature T=AdS-Schwarzschild black hole

Superconductivity???

� (r) = 0T > Tc

��†L�� = 0

��†L�� = 0

T < Tc

��gf tr

��AdS2

�= 0

��gf tr

��AdS2

�= 0 �(r) �= 0


The large-N Kondo effect!

� (r) = 0T > Tc

��†L�� = 0

��†L�� = 0

T < Tc

��gf tr

��AdS2

�= 0

��gf tr

��AdS2

�= 0 �(r) �= 0

Large-N Approach to the Kondo Effect

Spin SU(N) Rimp = anti-symm. k = 1

N �� fixedwith NgK

SU(N)� �� U(1)� U(1)� �� singlet bi-fundamental

�S = �†��

O(�) � c†(0, �)�(�)


�O� �= 0

�O� = 0Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, 216403 (2003)



T > Tc

Tc � TK

T < Tc

�O� �= 0


Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, 216403 (2003)



T > Tc

Represents the binding of an electron to the impurity

T < Tc

�O� = 0


“(0+1)-DIMENSIONAL SUPERCONDUCTIVITY”




T > Tc

U(1)� U(1)� U(1)

T < Tc �O� �= 0

�O� = 0





T > Tc

The phase transition is an ARTIFACT of the large-N limit!The actual Kondo effect is a crossover

T < Tc �O� �= 0

�O� = 0


The large-N Kondo effect!

� (r) = 0T > Tc

��†L�� = 0

��†L�� = 0

T < Tc

��gf tr

��AdS2

�= 0

��gf tr

��AdS2

�= 0 �(r) �= 0

�r��g grr �r�

��g gtt a2t ��

��g m2 � = 0


�(x)J t(r)


�= �J t(r)


The Phase Shift


�(x)J t(r)


�= �J t(r)


The Phase Shift

T > Tc �(r) = 0 J t(r) = 0

��gfrt = Q

��gfrt = Q

T > Tc �(r) = 0 J t(r) = 0

The Phase Shift

UV

IR


�(x)J t(r)


�= �J t(r)


The Phase Shift

T > TcT < Tc �(r) �= 0 J t(r) �= 0

Screening of the Impurity

(a.) (b.)

Figure 6: Plots of our numerical results for the electric flux at the horizon,√−gf tz|z=1 =

z2a′t(z)∣

∣

z=1= a′t(z = 1), as a function of T/Tc, for Q = −1/2. (a.) Between T/Tc = 1 and

T/Tc = 0.012, a′t(z = 1) decreases by only about 40%, from −Q = 1/2 to about 0.30. (b.) The sameas (a.) but a log-linear plot, revealing that a′t(z = 1) decreases only logarithmically for T/Tc ! 0.20:the solid red line is the curve 0.522 + 0.048 ln (T/Tc), obtained from a fit to the data.

from critical or over-screening. Roughly speaking, the impurity’s representation in the UV is

dual to at(z)’s electric flux at the boundary, limz→0√−gf tz = −Q, as discussed in subsec-

tion 4.1. When T > Tc and φ(z) = 0, the electric flux is constant from the boundary to the

horizon. When T ≤ Tc, the non-trivial φ(z) draws electric charge away from at(z), reducingthe electric flux at the horizon. At T/Tc = 0, if φ(z) does not draw all the charge away from

at(z), then we may interpret the remaining non-zero flux of at(z) at z → ∞ as an impurityin the IR in a representation smaller than in the UV. This is under-screening. If φ(z) does

draw all the charge away from at(z), so that at(z) has zero electric flux at z → ∞, then noimpurity survives in the IR, as occurs in critical and over-screening.

Fig. 6 shows our numerical results for the electric flux at the horizon, which after the

re-scaling in eq. (4.19) is simply√−gf tz

∣

∣

z=1= z2a′t(z)

∣

∣

z=1= a′t(z = 1), as a function of T/Tc

for Q = −1/2. Fig. 6 (a.) shows that a′t(z = 1) indeed decreases as T/Tc decreases, althoughbetween T/Tc = 1 and T/Tc = 0.012 the decrease is only ≈ 40%, from −Q = 1/2 to about0.30. Fig. 6 (b.) shows that the decrease in a′t(z = 1) is only logarithmic for T/Tc ! 0.20.

Our numerical results for a′t(z = 1) are insufficient to extrapolate reliably to T/Tc = 0, so we

will leave the fate of a′t(z = 1) at low T as an open question for the future.

How does a phase shift at the IR fixed point appear

a holographic model of the kondo effect...the kondo effect 34 p hysics w orld j anuary 2001 below tk...

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