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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
Volume 100, Number 2, March–April 1995Journal of Research of the National Institute of Standards and Technology
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
Volume 100, Number 2, March–April 1995Journal of Research of the National Institute of Standards and Technology
158
T � �D
electron-phonon scattering
�D � 200 K
Ag
�(T ) � T
�(T ) = �0 + aT 2 + b T 5electron-phonon scattering
Volume 100, Number 2, March–April 1995Journal of Research of the National Institute of Standards and Technology
158
T � �D
�D � 200 K
Ag
�(T ) = �0 + aT 2 + b T 5
Volume 100, Number 2, March–April 1995Journal of Research of the National Institute of Standards and Technology
158
T � �D
electron-electron scattering
�D � 200 K
Ag
�(T ) = �0 + aT 2 + b T 5
Volume 100, Number 2, March–April 1995Journal of Research of the National Institute of Standards and Technology
158
T � �D
electron-impurity scattering
�D � 200 K
Ag
Volume 100, Number 2, March–April 1995Journal of Research of the National Institute of Standards and Technology
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 )
concentration of impurities
UV cutoff=
c, c̃ �
�F
�(T ) = �0 + aT 2 + b T 5 + c g2K � c̃ g3K
ln (T/�F )
concentration of impurities
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.
1
arX
iv:1
306.
6326
v1 [
cond
-mat
.mes
-hal
l] 2
6 Ju
n 20
13
Observa
tion
ofth
eSU(4
)Kondosta
tein
a
double
quantu
mdot
A.J.
Keller
1,S.Amash
a1,†,
I.Weym
ann2,C.P.Moca
3,4,I.G.Rau
1,‡,J.
A.Katin
e5,
Had
asShtrikm
an6,G.Zarán
d3,an
dD.Gold
hab
er-Gord
on1,*
1Geb
alleLab
oratoryfor
Advan
cedMaterials,
Stan
fordUniversity,
Stan
ford,CA
94305,USA
2Facu
ltyof
Physics,
Adam
Mickiew
iczUniversity,
Pozn
ań,Polan
d3B
ME-M
TA
Exotic
Quantu
mPhases
“Len
dület”
Grou
p,Institu
teof
Physics,
Budap
estUniversity
ofTech
nology
andEcon
omics,
H-1521
Budap
est,Hungary
4Dep
artment
ofPhysics,
University
ofOrad
ea,410087,
Rom
ania
5HGST,San
Jose,CA
95135,USA
6Dep
artment
ofCon
den
sedMatter
Physics,
Weizm
annInstitu
teof
Scien
ce,Reh
ovot96100,
Israel†P
resentad
dress:
MIT
Lincoln
Lab
oratory,Lexin
gton,MA
02420,USA
‡Present
address:
IBM
Research
–Alm
aden
,San
Jose,CA
95120,USA
*Corresp
ondingau
thor;
goldhab
er-gordon
@stan
ford.ed
u
Cen
tralto
conden
sedmatter
physics
are
quantu
mim
purity
models,
which
describ
ehow
aloca
ldeg
reeoffreed
om
intera
ctswith
aco
ntin
uum.
Surp
risingly,
these
models
are
often
universa
lin
thatth
eyca
nquantita
tively
describ
emany
outw
ard
lyunrela
tedphysica
lsy
stems.
Here
wedev
elop
a
double
quantu
mdot-b
ased
experim
entalrea
lizatio
nofth
eSU(4)Kondo
model,
which
describ
esth
emaxim
ally
symmetric
screeningofaloca
lfour-
fold
deg
enera
cy.Asdem
onstra
tedth
rough
transp
ort
mea
surem
ents
and
deta
ilednumerica
lren
orm
aliza
tion
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
Enhance the spin group
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”
Enhance the spin group
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
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
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)
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
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)
Outline:
• The Kondo Effect• The CFT Approach•A Top-Down Holographic Model•A Bottom-Up Holographic Model• Summary and Outlook
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
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
�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
CFT Approach to the Kondo Effect
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
�
CFT Approach to the Kondo Effect
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
IR SU(N)k � SU(k)N � U(1)Nk
SU(N)k � SU(k)N � U(1)Nk
|c, s, f�
|c, s�, f�
Fusion Rules
s� simp = s�
UV
IR SU(N)k � SU(k)N � U(1)Nk
SU(N)k � SU(k)N � U(1)Nk
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
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
�L(0�) = �L(0+)
�L(0�) = ��L(0+)
�/2 phase shift
UV
IR
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
� (r) = A cos kr + B sin kr
� (r) = A�| sin kr| + B� sin kr
decoupled spin at r = 0
�/2 phase shift
CFT Approach to the Kondo Effect
Take-Away Messages
Central role of theKac-Moody Algebra
PHASE SHIFT
Kondo coupling: �S · �J
Outline:
• The Kondo Effect• The CFT Approach•A Top-Down Holographic Model•A Bottom-Up Holographic Model• Summary and Outlook
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
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
andCFT with holographic dual
5-3
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 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
Decouple5-3
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 XN5 D5 X X X X X X
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 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
the impurity
5-3
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 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
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
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
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
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
Decouple5-3
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 XN5 D5 X X X X X X
5-57-7SYMU(N5)(5 + 1)-dim.SYM(7 + 1)-dim. U(N7)
g2Dp � gs ��p�32
g2Y M � gs
g2Y MNc � 1/��2
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
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
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
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
0 1 2 3 4 5 6 7 8 9Nc D3 X X X XN7 D7 X X X X X X X X
The D7-branes
Harvey and Royston 0709.1482, 0804.2854Buchbinder, Gomis, Passerini 0710.5170
1/4 SUSY
(1+1)-dimensional chiral fermionsN7 �L
SUSY
8 Neumann-Dirichlet (ND) intersection
N = (0, 8)
0 1 2 3 4 5 6 7 8 9Nc D3 X X X XN7 D7 X X X X X X X X
The D7-branes
(1+1)-dimensional chiral fermionsN7
L
�L
0 1 2 3 4 5 6 7 8 9Nc D3 X X X XN7 D7 X X X X X X X X
The D7-branes
S3-7 =�
dx+dx��†L (i�� �A�) �L
(1+1)-dimensional chiral fermionsN7 �L
SU(Nc)� U(N7)� U(N5)Nc N7 singlet
0 1 2 3 4 5 6 7 8 9Nc D3 X X X XN7 D7 X X X X X X X X
The D7-branes
SU(Nc)N7 � SU(N7)Nc � U(1)NcN7Kac-Moody algebra
(1+1)-dimensional chiral fermionsN7 �L
S3-7 =�
dx+dx��†L (i�� �A�) �L
0 1 2 3 4 5 6 7 8 9Nc D3 X X X XN7 D7 X X X X X X X X
The D7-branes
Do not come from reduction from (3+1) dimensions
Genuinely relativistic
Differences from Kondo
(1+1)-dimensional chiral fermionsN7 �L
0 1 2 3 4 5 6 7 8 9Nc D3 X X X XN7 D7 X X X X X X X X
The D7-branes
SU(Nc) is gauged!
�J = �†L�� �L
Differences from Kondo
(1+1)-dimensional chiral fermionsN7 �L
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!
Harvey and Royston 0709.1482, 0804.2854Buchbinder, Gomis, Passerini 0710.5170
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
AdS3 � S5Probe D7-branes
N = 4 SYMNc �⇥ =���
Probe �L =
Type IIB Supergravity
AdS5 � S5
ds2 =dr2
r2+ r2
��dt2 + dx2 + dy2 + dz2
�+ ds2S5
J = ACurrent Gauge fieldU(N7) U(N7)
AdS3 � S5Probe D7-branes
N = 4 SYMNc �⇥ =���
Probe �L =
Type IIB Supergravity
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:
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
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
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 �
8 ND intersection
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
S3-5 =�
dt �†(i�t �At � �9)�
SU(Nc)� U(N7)� U(N5)Nc singlet N5
(0+1)-dimensional fermionsN5 �
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
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 =
Type IIB Supergravity
AdS5 � S5
AdS2 � S4�
ds2 =dr2
r2+ r2
��dt2 + dx2 + dy2 + dz2
�+ ds2S5
Probe D5-branes
N = 4 SYMNc �⇥ =���
Probe =
Type IIB Supergravity
AdS5 � S5
AdS2 � S4�
Probe D5-branes
N = 4 SYMNc �⇥ =���
Probe =
Type IIB Supergravity
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
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
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
AdS3 � S5Probe D7-branes
N = 4 SYMNc �⇥ =���
Probe �L =
Type IIB Supergravity
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
What is V (�†�) ?
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
What is V (�†�) ?
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:
• The Kondo Effect• The CFT Approach•A Top-Down Holographic Model•A Bottom-Up Holographic Model• Summary and Outlook
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
Kondo model specified by
N, Rimp, k
SCS = �N
4�
�tr
�A � dA + 2
3A �A �A
�
SAdS2 = ��
d3x �(x)��g
�14trf2 + |D�|2 + V (�†�)
�
Bottom-Up Model
Kondo model specified by
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)
J t(r) = �2��g gttat�2
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
Hawking temperature T=AdS-Schwarzschild black hole
The large-N Kondo effect!
� (r) = 0T > Tc
��†L�� = 0
��†L�� �= 0
T < Tc
��gf tr
���AdS2
�= 0
��gf tr
���AdS2
�= 0 �(r) �= 0
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)
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, �)�(�)
Large-N Approach to the Kondo Effect
�O� �= 0
�O� = 0Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, 216403 (2003)
Spin SU(N) Rimp = anti-symm. k = 1
N �� fixedwith NgK
T > Tc
Tc � TK
T < Tc
�O� �= 0
Large-N Approach to the Kondo Effect
Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, 216403 (2003)
Spin SU(N) Rimp = anti-symm. k = 1
N �� fixedwith NgK
T > Tc
Represents the binding of an electron to the impurity
T < Tc
�O� = 0
Large-N Approach to the Kondo Effect
“(0+1)-DIMENSIONAL SUPERCONDUCTIVITY”
Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, 216403 (2003)
Spin SU(N) Rimp = anti-symm. k = 1
N �� fixedwith NgK
T > Tc
U(1)� U(1)� U(1)
T < Tc �O� �= 0
�O� = 0
Large-N Approach to the Kondo Effect
Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, 216403 (2003)
Spin SU(N) Rimp = anti-symm. k = 1
N �� fixedwith NgK
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
Hawking temperature T=AdS-Schwarzschild black hole
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
�trxFrx = �4�N
�(x)J t(r)
�r���g grrgttfrt
�= �J t(r)
J t(r) = �2��g gttat�2
The Phase Shift
�trxFrx = �4�N
�(x)J t(r)
�r���g grrgttfrt
�= �J t(r)
J t(r) = �2��g gttat�2
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
�trxFrx = �4�N
�(x)J t(r)
�r���g grrgttfrt
�= �J t(r)
J t(r) = �2��g gttat�2
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