Kondo effects in multilevel quantum dots (a renormalization group - - PowerPoint PPT Presentation

David Logan Oxford University (a renormalization group study) Martin Galpin, Chris Wright and

NATO ARW, Yalta, 20.9.07.

Kondo effects in multilevel quantum dots

• 1. Introduction.

Recent years have seen strong renewal of interest in Kondo physics, due in large part to the advent of quantum dot systems – from semiconducting QDs based on GaAs/n-doped GaAs heterostructures, through e.g. carbon nanotube dots, to molecular electronic devices. Semiconducting QDs in particular provide a tunable, controlled realisation of ‘mesoscopic atoms’ – cf the spin-1/2 Kondo effect in an odd-electron QD, manifest e.g. in the unitarity limit for the zero-bias conductance ......

Vt l Vbl

Vr

Vgl

van der Wiel et al, Science (2000).

Odd electron valleys:– – spin-1/2 Kondo effect; – unitarity limit in zero-bias conductance for T ¿ TK : – Kondo progressively killed by increasing T. – associated in effect with a single dot level, singly occupied. Spin-1/2 Kondo effect ......

source lead drain

van der Wiel et al, Science (2000).

Odd electron valleys: – spin-1/2 Kondo effect; Even electron valleys: – no Kondo; suppressed conductance

van der Wiel et al, Science (2000).

This odd/even (or ‘Kondo/non-Kondo’) alternation is common in QDs, but not ubiquitous: – if the dot level spacing is sufficiently small, might expect instead to see S=1 Kondo physics in an even valley, i.e. S=1 (ferromag Hund’s coupling).

• dd

even Such behaviour is indeed

• bserved ....... e.g......

Kogan et al, PRB 67, 113309 (2003)

Triplet Kondo:

Odd valley: spin -½ Kondo, strongly enhanced zero-bias conductance. Even valley: also strongly enhanced zero-bias conductance, indicative of (underscreened) spin-1 Kondo.

¢ Vg

Vt l Vbl

Talk about aspects of the S=1 and S=1/2 Kondo regimes, and the transition between them ……

Theory: Experiment:

Kogan, Granger, Kastner, Goldhaber-Gordon, Shtrikman Singlet-triplet transition in a single-electron transistor at zero magnetic field, PRB 67, 113309 (2003) Mehta, Andrei, Coleman, Borda, Zarand, Regular and singular Fermi-liquid fixed points in quantum impurity models, PRB 72, 014430 (2005) Koller, Hewson, Meyer, Singular dynamics of underscreened magnetic impurity models, PRB 72, 045117 (2005) Conductance of a spin-1 quantum dot: the two-stage Kondo effect, PRB 75, 245329 (2007) Pustilnik, Borda, Phase transition, spin-charge separation and spin filtering in a quantum dot, PRB 73, 201301 (R) (2006) Hofstetter, Schoeller, Quantum phase transition in a mulitlevel dot, PRL 88, 016803 (2002) Pustilnik, Glazman, Kondo effect in real quantum dots, PRL 87, 216601 (2001) Posazhennikova, Coleman, Anomalous conductance of a spin-1 quantum dot, PRL 94, 036802 (2005); Vojta, Bulla, Hofstetter, Quantum phase transitions in models of coupled magnetic impurities, PRB 65, 140405 (R) (2002)

Some background refs:

Outline:-

• 2. Model

– 2-level dot coupled to two conducting leads. – some issues/questions.

• 3. Results

– static/thermodynamic properties. – evolution of Kondo scale. – phase diagram(s). – single-particle dynamics, and transport. – experiment. Results shown here obtained mainly from Wilson’s numerical renormalization group (NRG)* as method of choice – non-perturbative RG, providing essentially exact results on low-temperature/energy scales central to the physics. [* use both ‘standard’ NRG and full density matrix/complete Fock space NRG (Peters, Pruschke, Anders; Weichselbaum, von Delft …)]

V1R R L 1 2 V2R V2L V1L

H0 = X

¸ = L ;R

X

k;¾

²k cy

k¸ ¾ck¸ ¾

HD = ²1^ n1 + ²2^ n2 + U (^ n1" ^ n1# + ^ n2" ^ n2#) + U0^ n1^ n2 ¡ JH ^ s1 ¢^ s2

(JH > 0 – ferromagnetic [Hund’s rule] coupling)

MODEL:

2-level dot, coupled to two conducting leads.

Model both ‘general’ and rich. Look first at the dot states arising in the ‘atomic limit’, where dot decouples from leads .......

HV = X

¸ = L ;R

X

i = 1;2

X

k;¾

Vi ¸ (cy

k¸ ¾di ¾ + dy i ¾ck¸ ¾)

– two non-interacting, metallic leads – dot/leads tunnel coupling

Atomic limit: dot states

HD = ²1^ n1 + ²2^ n2 + U (^ n1" ^ n1# + ^ n2" ^ n2#) + U0^ n1^ n2 ¡ JH ^ s1 ¢^ s2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

²2 ²1

effectively a one-level problem; ‘filling level 1’.

Atomic limit: dot states

HD = ²1^ n1 + ²2^ n2 + U (^ n1" ^ n1# + ^ n2" ^ n2#) + U0^ n1^ n2 ¡ JH ^ s1 ¢^ s2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

²2 ²1

effectively a one-level problem; ‘filling level 1’. effectively a one-level problem; ‘filling level 2’.

Atomic limit: dot states

HD = ²1^ n1 + ²2^ n2 + U (^ n1" ^ n1# + ^ n2" ^ n2#) + U0^ n1^ n2 ¡ JH ^ s1 ¢^ s2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

²2 ²1

States shown are spin S=1/2 or 0. Finally, there’s the S=1 state.

Atomic limit: dot states

HD = ²1^ n1 + ²2^ n2 + U (^ n1" ^ n1# + ^ n2" ^ n2#) + U0^ n1^ n2 ¡ JH ^ s1 ¢^ s2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

²2 ²1

Obvious question: what happens on coupling to the leads? .... – ‘deep’ in the S=1/2 regimes, low-energy model is usual spin-1/2 Kondo. So ultimate stable fixed point (FP) is usual strong coupling FP: dot spin quenched on scale TK ; strongly enhanced T=0 zero-bias conductance, System a ‘regular’ Fermi liquid. Gc(0) » 2e2=h:

Atomic limit: dot states

HD = ²1^ n1 + ²2^ n2 + U (^ n1" ^ n1# + ^ n2" ^ n2#) + U0^ n1^ n2 ¡ JH ^ s1 ¢^ s2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

²2 ²1

Obvious question: what happens on coupling to the leads? ... – ‘deep’ in the S=1 regime, low-energy model will be spin-1 Kondo. Is it 1-channel or 2-channel spin-1 Kondo? Depends on coupling to leads….. If 2-channel, S=1 wholly quenched on coupling to leads (strong coupling FP); (Pustilnik/Glazman). If 1-channel, S=1 only partially quenched

• n coupling to leads (‘underscreened’ S=1

FP, Nozieres/Blandin);in this case, strongly enhanced System a ‘singular’ Fermi liquid (Mehta et al). and strong suppressed dc, Gc(0) » 0 Gc(0) » 2e2=h:

Questions contd: – for S=1 regime, two-channel behaviour is generic. So apparent puzzle (Coleman et al): why do exps see strongly enhanced Answer: low-energy model is generically 2-channel spin-1 Kondo with channel anisotropy:

L0 R0 J+ J¡

2-stage quenching of spin-1: TK + » D exp(¡ 1=½ J+ ) – on ‘high’ scale , quench spin S=1 S=1/2 by coupling to ! R0

0:

Here flow towards underscreened S=1 fixed point. – then on ‘low’ scale TK ¡ » D exp(¡ 1=½ J¡ ); quench spin S=1/2 S=0 by coupling to ! L0

0:

(flowing then to fully screened, strong coupling FP) Gc(0) » 2e2=h? But the two scales may be vastly different in magnitude. TK ¡ And if ‘irrelevantly small’, in – experiment will then ‘see’ the underscreened spin-1 FP. practice might as well be 0: We’ll take this for granted here; in practice, consider effective 1-channel set-up from beginning.

: J+ > J¡

Atomic limit: dot states

HD = ²1^ n1 + ²2^ n2 + U (^ n1" ^ n1# + ^ n2" ^ n2#) + U0^ n1^ n2 ¡ JH ^ s1 ¢^ s2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

²2 ²1

Questions contd: – where does exp fit in to figure? – so (Pustilnik/Borda) does there exist a quantum phase transition (QPT) between ‘regular’ spin-1/2 Kondo and spin-1 Kondo? If so, what is the nature of the QPT? – Can exp be explained?

H0 = X

¸ = L ;R

X

k;¾

²k cy

k¸ ¾ck¸ ¾

HD = ²1^ n1 + ²2^ n2 + U (^ n1" ^ n1# + ^ n2" ^ n2#) + U0^ n1^ n2 ¡ JH ^ s1 ¢^ s2 HV = X

¸ = L ;R

X

i = 1;2

X

k;¾

Vi ¸ (cy

k¸ ¾di ¾ + dy i ¾ck¸ ¾)

So consider explicitly:- – two non-interacting, metallic leads – dot/leads tunnel coupling

Vcosµ Vsinµ

2 R L 1

Vsinµ Vcosµ

[two leads of course, but of 1-channel form: ]

1 2

V V

R

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

²2 ²1

Recall discussion of ‘atomic limit’:- What of the phase boundaries on coupling to leads? – all phases that were S=1/2 or 0 in the atomic limit are continuously connected (as we know..); here the stable fixed point is the usual strong coupling (or frozen impurity) FP. – but the underscreened triplet state has a distinct FP. So we expect a quantum phase transition; and hence the qualitative behaviour shown. Indeed.

1 2 1 2

²2 ²1

²1, and progressively decrease ²2

‘high up’ in the spin-1/2 Kondo regime, down into the spin-1 Kondo regime, to ²2 = ²1: We’ll chose specifically ²1 = ¡ U=2¡ U0 (‘midpoint’); the point ²2 = ²1 is then particle-hole symmetric. Will consider sequentially: – static/thermodynamic properties. – evolution of the Kondo scale. – phase diagram(s). – single-particle dynamics. – transport/differential conductance. – and experiment. For purposes of illustration here, will now fix from

Entropy, Simp(T) vs T=D:

U = 20; JH = 5; U0 = 0

Fixed ²1 = ¡ U=2 , with (energies i.t.o. ‘hybridization strength’ ¡ = ¼

V 2½ );

– and progressively lowering ²2:

1 2

TK

Screened spin-1/2 Kondo phase. ln(2) Simp(T = 0) = 0: – high ²2 À 0; so essentially T . U: single-level S=1/2 Kondo for – intermediate ln(2) plateau, characteristic of spin-1/2 local moment FP. – characteristic Kondo scale TK ; below which dot spin quenched: Usual spin-1/2 strong coupling FP.

²1 = ¡ 10)

(and

²2 = + 9 ²2 ²1

screened USC

Entropy, Simp(T) vs T=D:

1 2

Now decrease ²2 : ¡

TK

Screened spin-1/2 Kondo phase. ln(2)

screened USC

TK TK

²2 = 5; 4; 3:

– Kondo scale progressively decreases.

²2 ²1

Entropy, Simp(T) vs T=D:

1 2

TK vanishes at a critical ²2;c ²2 = 2:8

(here ²2;c = 2:775) – symptomatic of a QPT to the underscreened triplet phase. Screened spin-1/2 Kondo phase. ln(2) For ²2 < ²2;c have the underscreened spin-1 phase ......

screened USC

²2 ²1

Entropy, Simp(T) vs T=D:

1 2

Undercreened spin-1 Kondo phase.

²2 = 0

Throughout this phase,

Simp(T = 0) = ln 2

ln(2) – characteristic of the USC spin-1 fixed point. But there is no low-energy scale in this phase that vanishes as transition approached.

screened USC

²2 ²1

Entropy, Simp(T) vs T=D:

1 2

Undercreened spin-1 Kondo phase. ln(2) ln(3)

²2 = ¡ 4 TK +

There is of course a characteristic low-energy scale in the USC phase – TK + :

screened USC

²2 ²1

Entropy, Simp(T) vs T=D:

1 2

Undercreened spin-1 Kondo phase.

TK + ²2 = ¡ 10 (= ²1)

There is of course a characteristic low-energy scale in the USC phase – TK + : But it barely varies throughout the phase.

screened USC

²2 ²1

Spin susceptibility TÂimp(T); vs T=D:

1 4

TÂimp(T) ! TÂimp(T) !

1 4

For screened Kondo phase, as T ! 0: For underseened spin-1 phase, as T ! 0: Âimp(T) »

S(S+ 1) 3T

(i.e. with S=1/2).

screened Kondo USC Kondo

²2 nimp ²2;c

“Impurity” charge, nimp ('

h^ n1 + ^ n2i ):

– very boring, evolves smoothly through the transition. But key re transport (see on)!

screened USC

²2 ²1

Vanishing of Kondo scale as ²2 ! ²2;c+

from screened Kondo phase: -

²2;c

²2

ln( TK D )

screened USC

²2 ²1

ln( TK D )

(²2 ¡ ²2;c)¡

1 2

TK / exp µ ¡ a p ²2 ¡ ²2;c ¶

²2 ! ²2;c:

as – QPT of Kosterlitz-Thouless type (consistent with line of FPs in USC phase; and with no evidence for a separate critical FP). [behaviour quite generic.........]

Vanishing of Kondo scale as ²2 ! ²2;c+

from screened Kondo phase: -

screened USC

²2 ²1

and JH = 5:

Phase diagram:-

²2 ²1

JH=4 ¡ U ¡ JH =4

For U = 20; U0 = 0 Phase boundary: solid line

screened USC

Phase diagram:-

EXPT

x = ²2 + 1

2U + U0

= ²1 + 1

2U + U0

y y = x y = ¡ x

nimp = 2

Phase diagram:-

x = ²2 + 1

2U + U0

= ²1 + 1

2U + U0

y y = x Phase transition boundaries in general continuous KT-transitions.

screened USC

Exception: on the line ²1 = ²2 (y = x) – where a level crossing QPT arises (as permitted by invariance of ^ H dy

1¾ $ dy 2¾

under ‘1-2’ transformation )

Single-particle dynamics and transport.

Di j (! ) = ¡

1 ¼ ImGi j (! )

Can consider the single-particle spectra (with i,j referring to dot levels (1 or 2), and Gi j (t) = ¡ iµ(t)h n di ¾(t); dy

j ¾

• i

Gi j (! ) $ ); Equivalently, take even/odd combinations of dot levels, ; and work with Vsinµ

1 2 R L

Vsinµ Vcosµ Vcosµ Gee(! ) =

1 2 [ G11(! ) + G22(! ) § 2G12(! )]

• dy

e¾ = 1 p 2

³ dy

1¾ § dy 2¾

´

• ¾

(¡ = ¼ V 2½ ) Dee(! ) = ¡

1 ¼ImGee(! )

Focus here on e-e spectrum – which determines the zero-bias conductance: ( and Geo(! ) =

1 2[G11(! ) ¡ G22(! )] ):

and ´ 4¡ L

¡ R

(1 +

¡ L ¡ R )2

[where: ¡ R = ¡ cos2(µ) with ¡ L = ¡ sin2(µ); ; G0 = 2e2 h sin22µ

sin22µ

¡ L = ¡ R for equivalent coupling to leads, .] s.t. G0 = 2e

2=h

Gc(T) = G0 Z 1

¡ 1

d! ¡ @ f (! ) @ ! 2¼ ¡ D ee(! ; T)

Single-particle dynamics and transport.

Equivalently, take even/odd combinations of dot levels, ; and work with

1 2 R L

Vsinµ Vsinµ Vcosµ Vcosµ Gee(! ) =

1 2 [ G11(! ) + G22(! ) § 2G12(! )]

• dy

e¾ = 1 p 2

³ dy

1¾ § dy 2¾

´

• ¾

(¡ = ¼ V 2½ ) Dee(! ) = ¡

1 ¼ImGee(! )

Focus here on e-e spectrum – which determines the zero-bias conductance: s.t. – probes spectrum at Fermi level, ! = 0: ( and Geo(! ) =

1 2[G11(! ) ¡ G22(! )] ):

Gc(T) = G0 Z 1

¡ 1

d! ¡ @ f (! ) @ ! 2¼ ¡ D ee(! ; T)

Gc(T = 0) G0 = 2¼ ¡ Dee(! = 0)

Single-particle dynamics.

U = 20; JH = 5; U0 = 0

Fix ²1 = ¡ U=2, with – and progressively lower ²2: – ‘all scales’ overview. 2¼ ¡ Dee(! ) vs ! =¡ (¡ = ¼ V 2½ ) For screened Kondo phase. ²1 = ¡ 10 and ²2 = (²2;c = 2:775) 4.5, 3.7, 3.5.

²2 ²1

screened Kondo

USC

(T = 0)

Single-particle dynamics.

For screened Kondo phase. (²2;c = 2:775) ²1 = ¡ 10 and ²2 = 4.5, 3.7, 3.5. Key low-energy feature is of course the Kondo resonance;

Kondo resonance

with characteristic scale TK : Kondo resonance progressively narrows as transition approached, and ........

²2 ²1

screened Kondo

USC

Single-particle dynamics.

..... vanishes as the transition to the underscreened spin-1 Kondo phase is crossed. For underscreened Kondo phase. (²2;c = 2:775) ²1 = ¡ 10 and ²2 = 2.7

[Again, there is no low-energy scale in the USC phase that vanishes as transition approached from that side.]

²2 ²1

screened Kondo

USC

Single-particle dynamics.

(²2;c = 2:775) ²1 = ¡ 10 and ²2 = 4.5, 3.7, 3.5, 2.7 How does the Kondo resonance ‘collapse’ as transition approached from screened Kondo phase? ......

²2 ²1

screened Kondo

USC

Single-particle dynamics.

(²2;c = 2:775) ²1 = ¡ 10 and ²2 = 4.5, 3.7, 3.5, 2.7 – resonance narrows progressively, and vanishes ‘on the spot’. – and as it does so exhibits scaling in terms of the Kondo scale TK : How does the Kondo resonance ‘collapse’ as transition approached from screened Kondo phase? ......

Single-particle dynamics.

(²2;c = 2:775) ²1 = ¡ 10 and ²2 = 4.5, 3.7, 3.5. Scaling spectrum in screened Kondo phase.

Single-particle dynamics.

(²2;c = 2:775) ²1 = ¡ 10 and ²2 = 4.5, 3.7, 3.5, 2.7 How does the Kondo resonance ‘collapse’ as transition approached from screened Kondo phase? ...... – resonance narrows progressively, and vanishes ‘on the spot’; leaving an incoherent background in underscreened phase (dashed line).

Dee(! = 0), and hence the (T=0)

zero bias conductance / D ee(0); jumps discontinuously across the transition ........

²1 = ¡ U=2 = ¡ 10)

(for

2¼ ¡ Dee(! = 0) vs ²2 (U = 20; U0 = 0; JH = 5)

– discontinuity in zero-bias conductance as cross transition

²2 ²1

screened Kondo

USC

2¼ ¡ D ee(0)

²2

²2;c

underscreened phase screened phase

– discontinuity here a ‘drop’ from screened underscreened; as expect from collapsing Kondo resonance. Gc(T = 0) G0 = 2¼ ¡ Dee(! = 0)

Single-particle dynamics.

For underscreened Kondo phase.

U = 20; JH = 5; U0 = 0

Fix ²1 = ¡ U=2, with – and progressively lowering ²2 = ¡ 2; ¡ 4; ¡ 6; ¡ 8; ¡ 10:

²2 = ¡ 2 ²2 = ¡ 10

²2 ²1

screened USC

²1 = ¡ U=2 = ¡ 10)

(for

2¼ ¡ Dee(! = 0) vs ²2 (U = 20; U0 = 0; JH = 5)

– discontinuity in zero-bias conductance as cross transition

²2 ²1

screened Kondo

USC

2¼ ¡ D ee(0)

²2

²2;c

underscreened phase screened phase

– discontinuity here a ‘drop’ from screened underscreened; as expect from collapsing Kondo resonance. But a ‘drop’ isn’t ubiquitous…. Gc(T = 0) G0 = 2¼ ¡ Dee(! = 0)

²1 ²2 ²2

²2;c

2¼ ¡ Dee(0) 2¼ ¡ Dee(! = 0) vs ²2

– along trajectory shown on r.h.s, with ²2 = ²1 + 8

(U = 10; JH = 1)

How do we understand this in general terms? It’s quite subtle ……

underscreened screened phase

I L = Im Z 0

¡ 1

d! Tr ½@ § (! ) @ ! G(! ) ¾ 2¼ ¡ Dee(! = 0) = sin2 ¡ ¼

2 nimp

¢

: screened Kondo Recall with the (Fermi level) scattering phase shift. ± = sin2± Can show quite generally – independently of phase (screened/usc) – that [and § (! ) the 2x2 self-energy matrix, likewise for ]

G(! ); ± =

¼ 2 nimp + I L

with the Luttinger integral:

I L

(1) (2) For a normal Fermi liquid – the screened phase – Luttinger (integral) theorem holds So (2) gives – i.e. the Friedel sum rule – and hence from (1): ± =

¼ 2 nimp

I L = 0

throughout the phase:

“Luttinger sum rules”

Well known (+ can check directly from NRG that I L = 0): Gc(T = 0)=G0 = 2¼ ¡ Dee(! = 0)

I L = Im Z 0

¡ 1

d! Tr ½@ § (! ) @ ! G(! ) ¾

Recall with the (Fermi level) scattering phase shift. ± Can show quite generally – independently of phase (screened/usc) – that [and § (! ) the 2x2 self-energy matrix, likewise for ]

G(! ); ± =

¼ 2 nimp + I L

with the Luttinger integral:

I L

(1) (2)

“Luttinger sum rules”

But the underscreened phase is a ‘singular Fermi liquid’. There is no reason to suppose Luttinger’s theorem is satisfied in it. (I L = 0) Throughout the USC phase (i.e. regardless of bare parameters U, J etc), we find: jI L j = ¼ 2 So (1),(2) give directly that:

2¼ ¡ D ee(! = 0) = sin2 ¡ ¼

2 [nimp ¡ 1]

¢

: underscreened phase And it isn’t. So overall we have: = sin2± Gc(T = 0)=G0 = 2¼ ¡ Dee(! = 0)

2¼ ¡ D ee(! = 0) = sin2 ¡ ¼

2 nimp

¢

: screened phase

sin2 ¡ ¼

2 [nimp ¡ 1]

¢

: underscreened phase

“Luttinger sum rules”

with: – and since n im p in general evolves smoothly across the transition, the zero-bias conductance jumps discontinuously. Can test independently, since can determine and Dee(! ) nimp separately via NRG …… Gc(T = 0)=G0 = 2¼ ¡ Dee(! = 0)

dotted: results from direct NRG crosses: from NRG calcn of calcn of nimp:

2¼ ¡ D ee(! = 0) = sin2 ¡ ¼

2 nimp

¢

: screened phase

sin2 ¡ ¼

2 [nimp ¡ 1]

¢

: underscreened phase

2¼ ¡ Dee(0)

²2

²2;c

D ee(! = 0):

Kogan et al, PRB 67, 113309 (2003)

EXPERIMENT:

Odd valley: spin -½ Kondo, strongly enhanced diff con. Even valley: also strongly enhanced differential conductance, indicative of spin-1 Kondo effect.

¢ Vg (mV) Vds (mV)

EXPERIMENT:

Kogan et al.

Kogan et al, PRB 67, 113309 (2003)

EXPERIMENT:

take cut along Vds = 0 For zero-bias conductance

EXPERIMENT:

Experimental T . 40 mK:

Experiment

EXPERIMENT / THEORY

Comparison to theory (T=0):

sin2 ¡ ¼

2 nimp

¢

: screened phase

sin2 ¡ ¼

2 [nimp ¡ 1]

¢

: underscreened phase Gc(0)=G0 =

screened screened USC

Comparison to theory (T>0):

EXPERIMENT / THEORY

screened screened USC

T=¡ = 3 £ 10¡ 3 (» 30mK)

Comparison to theory (T>0):

EXPERIMENT / THEORY

screened screened USC screened screened USC

T=¡ = 7:5 £ 10¡ 3 (» 75mK)

Comparison in (VG; Vds) -plane: Theory Experiment

Theory Experiment ...... and what you’ll see if you just ‘miss’ the transition:

Concluding remarks.

H D = ²1^ n1 + ²2^ n2 + U (^ n1" ^ n1# + ^ n2" ^ n2#) + U0^ n1^ n2 ¡ JH ^ s1 ¢^ s2 Considered a ‘simple’ two-level quantum dot: – QPT usually of KT type, with a vanishing Kondo scale as approach transition from screened side.

screened USC

²1 ²2 – zero-bias conductance explicable i.t.o. “Luttinger sum rules” for two phases; including anomalous transport as transition crossed. – both phases seen experimentally; and appear explicable by theory. the usual screened phases: Continuous line of QPTs in (²1; ²2) -plane separating an underscreened spin-1 phase from – rich range of behaviour in static & dynamic properties.

Vcosµ Vsinµ

2 R L 1

Vsinµ Vcosµ

Acknowledgements

Andrew Mitchell, Michael Pustilnik. David Goldhaber-Gordon, Andrei Kogan. Funding: EPSRC Programme Grant, Condensed Matter Theory Group, Oxford.