new quantum phase transitions in a noisy mesoscopic...

theabdus salaminternational centre for theoretical physics

strada costiera, 11 - 34014 trieste italy - tel. +39 040 2240111 fax +39 040 224163 - [email protected] - www.ictp.trieste.it

united nationseducational, scientific

and culturalorganization

international atomicenergy agency

SMR.1572 - 5

Workshop onNovel States and Phase Transitions in Highly Correlated Matter

12 - 23 July 2004

------------------------------------------------------------------------------------------------------------------------

New quantum phase transitions in anoisy mesoscopic world

Karyn LE HURUniversite de SherbrookeDepartement de Physique

Cite UniversitaireQuebec

J1K 2R1 SherbrookeCANADA

------------------------------------------------------------------------------------------------------------------------These are preliminary lecture notes, intended only for distribution to participants

New quantum phase transitions in anoisy mesoscopic world

Karyn Le Hur

“Trieste 2004: Novel phase transitions in highly correlated matter”

New quantum phase transitions in a noisy mesoscopic world – p.1/23

In Brief

Target: Large quantum dot(single electron box)Charge fluctuations on a large dot coupled to a 2DEG:Mapping on Fermi-Kondo models

Dissipative environment: Bosonic bath?

Zero-point fluctuations?

Bose-Fermi Kondo models?

Quantum critical points? Z(ω)

Vg Vg

Z(ω)VgVg

Z(ω)Vg

Z(ω)

Cl

� ��

Cg

metallic grain

t

+ δ

=

L

Ct

Josephson junction arrayK. Le Hur, PRL 92, 196804, 2004


small/large dot?

Level spacing

Energy spectrum in dot

∆ ∼ L

Charging energy Ec ~ L Ec

λF

L

d−1∆ ~

Two−dimensional dot: <<Ec∆

−d

−1

Small dot (Nano): Spin-1/2Large dot (Micron): “Many-body”


Coulomb Blockade

Glazman and Matveev, JETP 71, 1031 (1990)

Cgd

GV

2DEG x

y

dot

aV

(large)

��

quantum dot

SET

pointcontact 1

pointcontact 2

gate500nm

leadlead

��

��

��

��

�

��

��

-0.5 0.0 0.5

-0.5

0.0

0.5

0.5

(c)

NSET=0

Vds=Ec/eCur

rent

(nA

)

SET Vds(mV)0 1 2

0.0

0.2

0.4

(d)

0.25

0.5

0.75

Ec/e

Vds=1.25Ec/e

NSET� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � ��

� � � � � �� Ashoori sample in GaAs

Single-mode QPC with 2 spin channels

In-plane magnetic field: Zeeman

� � ��

Strong-field: Spin filter!


Pepper’s groupZero field: extra plateau

��

?

Strong field: plateau

��

0.5 1 1.5 2

0.2

0.4

0.6

0.8

1

= 0.

2δ

δ= 0

.1

Tra

nsm

issi

on c

oeff

icie

nt (1

-R0

)

δ

δ

=0

ξ

T0 01- R=

= 0.

4

QPC; harmonic potentiallinear in �

K. Le Hur, PRB 161302R (2001)

Connor (1968)Glazman, Lesovik, Khmel’nitskii, Shekter (1988)

Cyclotron effects: Fertig and Halperin (1987), Büttiker (1990)New quantum phase transitions in a noisy mesoscopic world – p.5/23

� � � � � � � � � � � � � ��

� � ��


� � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� ��

� ��

� � � � � � � � � � � � � � � � � � ��

��

� ��

− −− −

K.A. Matveev

JETP 72, 892 (1991)PRB 51, 1734 (1995)

0


Capacitance versus field

Cgd

G

BV

2DEG

Va(B)

x

y

dotε 0

0.2 0.4 0.6 0.8 1

1

2

3

δ C/Cgd

N

Zero field

Strong field

n;Vg

∆

Magnetic field = anisotropic two-channel Kondo model!Strong magnetic field = one-channel Kondo model (Fermi liquid)

K. Le Hur and G. Seelig, PRB 65, 165338 (2002)

Coulomb peaks subsist until

� �


Observat

�

of 2-channel Kondo realm?

0

2

Gwc=0.55G=0.67

Γ=0.15UU*=0.43U

(a)

C/C op

en

1

2

U*=0.33U

Γ=0.32U

Gwc=1.08

Gsc=1.08

G=1.09(b)

0.0 0.5 1.0

1.0

1.5

U*=0.29U

Γ=0.44UGsc=1.41

G=1.50

N

(c)

C/C op

en

0.0 0.5 1.00.9

1.0

1.1

(e)

N

G=1.81 Gsc=1.9

Γ=1.0U

(d)

-0.46 -0.44 -0.42 -0.400

1

2 Gsc

Gwc

GG (e

2 /h)

Lead Voltage (V)

PRL 82, 161 1999MIT group - Ashoori

Datas from Berman et al.: No Magnetic field!Problem: To find a reasonable

�� !

dots: perturbative results of Kostya Matveev and Hermann Grabert


More experimentsWestervelt et al. (Harvard)

staircase still close to perfect transmission

T

Lehnert-Schoelkopf (SC qubit in field)(many modes)

Log-deviations from Curie law!!!


“Quantum noise”: A brief detour

Z(ω)Z(ω)VgVg

Z(ω)Vg

Z(ω)

Cl

Vg Vg+ δ

Z(ω ) = (L/Ct1/2 = R)

Ro

� ��

t=

L

Ct

dissipative environment

(t)

e.g. Josephson junction array

Identifications:After diagonalization,��

� � � � ��

sum of the momenta

� �

Important:

�� conjugate!

Quantum Hamiltonian for the transmission line:

�� " # $& '( ) * ), & ' ( � � (

Ohmic bath:

- �. � � ( � � � (/ � �. 0 . � � � �1 . � � � � 2.


Charge accumulation at the junction

� � � � � ��

Important:� �� ! � " �� # ! $

probability that an inelastic tunnel event occursor “photon” of energy E is emitted to the bath

% & ' () * + � , -/ �

0 �/ � 1� 2 3 4

/2 1 -/

% 5 & 6 2 2 1 � 1 -/and 7 8 9 %; &

<> � � -� ? � 3@ A Bquantum of resistance

Clear observation in Cleland et al. PRL (1990) using thin film resistorsDevoret, Esteve, Grabert et al. (1990,1998); Nazarov and Ingold (1992)


Zero-bias anomaly!


Here is a useful mapping

Cgd

GV

2DEG x

y

dot

aV

(large)

NOISY

STRONGB(t)

Ohmic dissipative bath:

�� and

��

� ��

� � �� ! ��

"� #$ %' ( ) #�* + , -. / )1 / 24 � 57 8 -: / ,< =< > � � -.

Map on anisotropic Bose-Fermi Kondo modelZarand

?

Demler, PRB 66, 024427 (2002)Zhu

?

Si, PRB 66, 024426 (2002)Kircan

?M. Vojta, cond-mat/0312150

K. Le Hur, cond-mat/0312292 and PRL 2004New quantum phase transitions in a noisy mesoscopic world – p.15/23

RG analysis

Pedestrian RG equations:� ��

� ��

��

� � � � � � � � ��

λ λz

γ γ

λ

exchange of fermions

exchange of bosons

� � ��

, �� ! , and

�# � $ % $� � � &

No transverse coupling between orbital spin and bosonsKosterlitz-Thouless flow in terms of

�� and

' �� ( � � � � )


Sketch of the story

XY BoseIsing Bose

Bose coupling

strongly dependenton bare conditions

fermion coupling

SU(2) Bose

SU(2) Fermi liquid

XY Bose Fermi

SU(2) Bose−Fermi

purely classical

quantum criticality

� � �� : SU(2) Fermi liquid ��

large and ��

� � �� : Ising Bose liquid ��

and ��

bare


Phase diagram Results

gC = = 1/ΤΚ(g)

Boson phase

exp[−1/( − νg/ 2)]

gc λ / ν= R/RΚ

No quantum critical behavior

χ

λKondo energy scale

Kondo regime

TΚ(g) = Do

= 2 = 4|t|Ndot / ν

Almost free spin =1/Tχ= gz

T

� � � � ��

R=0

(|t|ρ)=1

1

0.5

0.75

0.25

00.51 0

<Q>

N

R>R c[|t| ρ]

1<Q>

0.5

0.5 0N

0.25

0.75

01

R=0

(|t|ρ)<<1Already visible at accessible�

:Indeed suppression of the logcorrections in � � �

!


Another approach for mesoscopists

Absorb noise in the tunneling (Kondo) term��

�� will be renormalized by

� ��

� �� fermionic imaginary time propagator� �� " � � � #% � " & �' �' � �(* �+- .� . �

Scaling dimension

/� �0 &' �' �

Irrelevant for sure when

' �' 1 is 2 � � �

Restoration of a nice staircase by increasing


Universality in the results

Grain with dense spectrum

�

Small dot with 1 levelSemi- � lead

�

mesoscopic ring with 1 level

extZ

R Rt ,C

t ,CL L

IR

IL

Ud Ua1C C2

Φ

I1 -0.5

-0.25

0

0.25

0.5

-0.5 0 0.5ε/ωc

<σ

z>

P. Cedraschi and M. Buttiker, Annals of Physics 289, 1-23 (2001)P. Cedraschi, V. Ponomarenko, and M. Buttiker, PRL 84, 346 (2000)


� �� Analogy with a superconducting qubit

Time-evolution of the off-diagonal element� �� where� �� ! �" �$ %' �� ( �) � ( � * �� +, ��

decays like a power-law in time at zero temperatureat finite T, dephasing rate proportional to

-

Makhlin, G. Schon, and Shnirman, cond-mat/0309049Schoelkopf, Clerk, Girvin, Lehnert, and Devoret, cond-mat/0210247


Close to perfect transmission

Z(ω)Z(ω)VgVg

Z(ω)Vg

Vg

Vgδ = 0

ρ| t | = 1

� ��

� � �� dissipative environment

Z(ω ) = (L/C)t1/2

Cgperfect

transmission

Cg

−N

+ δVg

−N +N

+N

N00.51

0

0.25

0.75

0.5

<Q>

R=0

1

bath irrelevantdissipative

Friedel sum rule

��

� �� ! � " � " # � �Shift in energy# $� � � � �% � � � � � � �� # "& " ��

with

') ' *, -/ 01 '2 '3 *I. Aleiner and L. Glazman, PRB 57 9608 (1998)


In closing

Atomic system (dot) coupled to 2 baths:Photons (Bosons) and Matter (Electrons)

Zero-point fluctuations of “photons”affect the ground state of the dot

R=0(|t|ρ)=1

1

0.5

0.75

0.25

00.51 0

<Q>

N

R>R c[|t| ρ]

1<Q>

0.5

0.5 0N

0.25

0.75

01

R=0

(|t|ρ)<<1

New effects in small tunneling regime!

Effect visible already at finite

�

No log-corrections for capacitance(free spin at zero temperature)

Sub-Ohmic bath? Q. Si et al.Two-bath problem? Meirong Li and K. Le Hur cond-mat/0405039


new quantum phase transitions in a noisy mesoscopic...

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