International Symposium in Honor of Professor Nambu
for the 10th Anniversary of his Nobel Prize in Physics
Osaka City University, December 12-13, 2018

Non-equilibrium
Fluctuations in StronglyCorrelated Quantum Liquids
Kensuke Kobayashi
Graduate School of Science, Osaka University
Center for Spintronics Research Network, Osaka University
Quantum Information and Quantum Biology Division, Institute for Openand Transdisciplinary Research Initiatives, Osaka University

1Outline

1. Mesoscopic systems
2. Fluctuation = Noise
3. Kondo effect & FluctuationQuantum LiquidSymmetry crossover

On-chip collision experiment toprobe non-equilibrium quantum liquids

2

Mesoscopic Systems

3

Mesoscopic System

Solid-state device where quantum
mechanics manifests itself.

1980ʼs- Stage forfundamental physics

Quantum computing

ExampleElectron interferenceSingle electron control
etc.
~ 0.8μm

ChargeSpin
Phase
Coherence
Interaction…

Exotic materials

nanotube, graphene, topological…

MEMS

micro-electro mechanical systems

Spintronics
etc.

Webb et al. PRL 54, 2696 (1985)

 Degree of freedom in the design
 Controllability

Strongly-correlated
physics / Quantum liquid
this talk4

“Conductance is Transmission.”
electronlead

Mesoscopicsystemlead

Rolf Landauer(1927–1999)

transmit

Landauer formula∼.

Conductance gives you information on the electronic properties of single sitequantum systems (interference, single-level transport, Kondo physics…).

5

Quantum Point Contact (QPC)

n-AlGaAs
i-AlGaAs
i-GaAselectron gas

Constriction width ~ Fermi wave
length of electrons (~50 nm)→ Conductance quantized due toformation of perfect quantumchannel.
Width increases. plateauplateau

2 nd channel
0 → 100 %1st channel
0 → 100 %Landauer Formula2van Wees et al., PRL 60, 848 (1988).

6

Quantum Dot (Artificial Atom)Electrons in a small box↓LeadQD

Lead

Discrete energy levels in QD.

ElectronGateQD# of electrons in QD is fixed.
Lead

Electron can pass QD only when thelevel coincides with those of the leads.
Current

Lead

Charging effect & Confinement↓Gate Voltage

Gate voltage (V)7Mesoscopic Systems (from our work)
Quantum Dot(artificial atom)

Atom in Interferometer
Wave-particle duality

Particle nature

Phys. Rev. Lett. 106, 176601 (2011);
J. Phys. Soc. Jpn 73, L3235 (2004)

ElectronInterferometerWave nature
Phys. Rev. Lett 104, 080602 (2010); Phys. Rev. B 79,161306 (R) (2009); 83, 155431 (2011); J. Phys. Soc.
Jpn 71, L2094 (2002); Physica E 42, 1091 (2010).

Phys. Rev. Lett. 88, 256806 (2002); 92, 176802(2004); 95, 066801 (2005); Phys. Rev. B 68,235304 (2003); 70, 035319 (2004); 73, 195329(2006).

8

Fluctuation = Noise9Some electrons cannot transmit
leadelectron

Mesoscopic
systemleadtransmit

Reflected

You can’t avoid this.
Noise = Current fluctuation
10

Noise = Variance of # of electrons
Review: Blanter-Büttiker, Phys. Rep. 336, 1 (2000).

IVI

Resistor

R = 1/G
FFTt

Current noise power spectral[ / ]density
Different Unit = New Info.

≡Current:
Noise:
11

Shot Noise = Noneq. Current Fluctuation
injecttransmit

barrierA

reflected

Noise

Not so simple!

Schottky (1918)

Poisson process: indep. event
Ave. = Var. →〈〉
〈 〉
〈 〉

∝〈 〉

: time2〈〉

2〈〉

Current 〈 〉

2

〈 〉

2 〈 〉

Electrons arecorrelated.
Fano factor
Effective charge〈 〉2 ∗〈 〉
12

Shot noise in QPC
transmitted

reflected→ shot noiseExp.

Nakamura, et al. PRB 79,201308(R) (2009).

Current
Noise

Fano factor∑

1∑Blanter and Büttiker, Phys. Rep. 336, 1 (2000).

13

Cooper pair in SN junction∗Fractional quantum Hall
effectcurrent （mA）

Shot noise

Shot noise∗

current （pA）

NormalSuper.

Jehl et al., Nature 405, 50 (2000).

Saminadayar et al., PRL 79, 2526 (1997);
de-Picciotto et al., Nature 389, 162 (1997).
Nobel Prize in Physics (1998)

14

“The noise is the signal.”

Nature 392, 658 (1998)

Fluctuation Theorem

Spin polarization

Mesoscopic
nonequilibrium
statistical physics. System

Stern–Gerlach on chip

Rolf Landauer(1927–1999)

Reservoir

Nakamura et al., PRL 104, 080602 (2010);
PRB 83, 155431 (2011)

Kohda et al. Nature Comm. 3, 1082 (2012).
Nishihara et al. APL 100, 203111 (2012).

Spin shot noise

Edge dynamics
Edge mixing ingraphene pn junction
in QH regime

Arakawa et al., PRL 114, 016601 (2015);
Iwakiri, Niimi, Kobayashi, APEX 10, 053001 (2017).

Matsuo et al. Sci. Rep. 5, 11723 (2015).
Matsuo et al. et al. Nature Comm. 6, 8066 (2015).

15

Kondo effect & Fluctuation
Ferrier et al., Nature Physics 12, 230 (2016).
Teratani et al., J. Phys. Soc. Jpn. 85, 094718 (2016).
Ferrier et al., Phys. Rev. Lett. 118, 196803 (2017).
Hata et al., Phys. Rev. Lett. (in press): Noise of Andreev-Kondo effect16Collaborators
Experiment:

T. Arakawa, T. Hata, S. –H. Lee, R. Fujiwara(Osaka University)
M. Ferrier, R. Delagrange, R. Weil, R. Deblock,H. Bouchiat (CNRS, Université Paris sud)

Theory:R. Sakano (University of Tokyo)
Y. Teratani, A. Oguri (Osaka City University)
Financial support:
Grant-in-Aid for Scientific Research (S) (26220711)
JSPS KAKENHI (26400319, 16K17723, 15K17680, 25103003,15H05854)
Yazaki Memorial Foundation for Science and Technology

17Quantum liquid
Single particle

electron, neutron, atom,molecule…

Many particles

Quantum liquid
Particle flow~de Broglie lengthinteraction

 Quantum liquid: Essentially different from a single
particle ex. liquid He, BEC, supercondutivity….
 Non-equilibrium quantum liquid: strongly-correlated
systems, cold-atom physics, spintronics …

18

Kondo effect

Magnetic impurity in metals
High T1964

Quantum dot

J. Kondo 1930-

High TKondo state = Quantum liquid
A localized spin screened by many electrons

Low T

Low TRGTT

Kondo effect in QD: Gordhaber-Gordon et al. Nature 391, 156 (1998); Cronenwett et al., Science 281,540 (1998); Schmid et al. Physica B 256-258, 182 (1998). van der Wiel et al., Science 289, 2105 (2000).

19

Why Kondo effect in QD?
Impact of Kondo effect (1964)
 A monument in solid state physics(1) Single
Kondo state(2) External control

 Typical quantum many body effect Extensively studied & understood(but limited within equilibrium tolinear response regime)(3) Eq. ↔ Non-eq. tunable

Realization of Kondo effect in a QD (1998-)→Ideal test bed to explore nonequibilirium
quantum many-body systems

Gordhaber-Gordon et al. Nature 391, 156 (1998); Cronenwett et al., Science 281, 540 (1998);
Schmid et al. Physica B 256-258, 182 (1998). van der Wiel et al., Science 289, 2105 (2000).

20

Kondo shot noise
Kondo state

Theory predicts enhanced shot noise due to
Kondo correlation.
Theory：Sela, Oreg, von Oppen, and Koch, PRL 97, 086601 (2006); Golub, PRB73, 233310 (2006); Gogolin and Komnik, PRL 97, 016602 (2006); Mora,Leyronas, and Regnault PRL 100, 036604 (2008); Vitushinsky, Clerk, and Le Hur,PRL 100, 036603 (2008). Fujii, JPSJ 79, 044714 (2010); Sela and Malecki, PRB
80, 233103 (2010); Sakano, Fujii, and Oguri, PRB 83, 075440 (2011)…
Only a few experiments : Zarchin et al., PRB 77, 241303 (2008); Delattre et al.,Nat. Phys. 5, 208 (2009); Yamauchi, KK et al., PRL 106, 176601 (2011)21

Device & Measurements
Dilution fridge: 15 ~ 800 mK

Pd/Al

Carbon nanotube
quantum dotSource

Drain

Magnetic field

Gate electrode

1 MΩ
Mixer

MeasureVV

Measure G22

Ideal Kondo state realized
Kondo resonance

∼4N

4N+1

4N+2

4N+3

4N+4

 Parity effect with 4-fold degeneracy
 Low-T conductance reaches 2 / (unitary limit) Zero-bias anomaly = Kondo resonance23

Shot noise
As Kondo state is broken,
noise is enhanced.

2G (2e /h)

1.00.5

Kondo regime
16 mK

1.023

222Vg (V)

20.8
0.6

10.4
0.2

20.8

Conventional shotnoise2 | |1.0

SI (100.60.20.0-20I (nA)

2A /Hz)0.50.4

0.0-10-50I (nA)

5100-27 2dI/dV (2e /h)

1.0A /Hz)

Coulomb blockade (non-Kondo)

TK=2K

-27 2

dI/dV (2e /h)

21SI (100.020No noise = perfect Kondo.

0.024

Effective charge
20.8

0.20.0-10-50I (nA)510

enhanced shot noisereflected currentA /Hz)

10.4

-27 2

0.602-27 2

2TK=2KSI (10dI/dV (2e /h)

1.0

SK (10 A /Hz)
Noisereflection (Back scattered)

2.

10-4

∗

-2.

0Current(nA)
IK(nA)

shot noisereflected current

242

Kondo correlation appears in nonlinear non-eq. regime.
∗ / is close to 5/3 as theoretically predicted.
Theory：Sela, Oreg, von Oppen, and Koch, PRL 97, 086601 (2006); Golub, PRB 73, 233310 (2006);
Gogolin and Komnik, PRL 97, 016602 (2006); Mora, Leyronas, and Regnault PRL 100, 036604(2008); Vitushinsky, Clerk, and Le Hur, PRL 100, 036603 (2008). Fujii, JPSJ 79, 044714 (2010);25
Sela and Malecki, PRB 80, 233103 (2010); Sakano, Fujii, and Oguri, PRB 83, 075440 (2011).

Fermi Liquid26

Fermi liquid theory Landau, 1957

L. Landau

K. G. Wilson

Describes the low-energy physics of interacting Fermion systems.
Applicable to liquid He, usual metals, heavy fermion systems, neutron stars...

Single particle excitations are quasi-particles with energies ̃ , inone-to-one correspondence with those of non-interacting system
̃ ,,,

1
2,,,,

: momentum distribution, : deviation from the groundstate,,, ,,Residual interaction: characterized by Wilson ratio R

You can treat the system as if it is a non-interacting one.
Kondo state = locally formed Fermi liquid.
Yamada, Prog. Theor. Phys. 53, 970 (1975);
Yosida and K. Yamada, ibid. 53, 1286 (1975).
Nozières, and, Blandin, J. Physique 41, 193 (1980)27

Wilson ratio

Free electron gas = simplest quantum liquid
Specific heat
C

Magnetic
susceptibility

S

N

3

Wilson ratio→ Charge and spin “united”

28Wilson ratio Free electron gas at Magnetic (Pauli para.): Thermal (Sp. heat): Wilson ratio

Nozières (1974)→ 0: charge frozen
with spin fluctuatedspin susceptibility

charge susceptibility

Wilson ratio = measure of quantum fluctuation
29

Meaning of
Two scattering processes at Kondo state
Double charge
Single charge
a free particle is reflected.

two particles interact via “residual”
interaction.

Probability

Probability

Kondo state

Kondo state→ In the Kondo limit (),holds.

∗30

Wilson ratio determined

Costi et al, J. Phys.: Condens. Matter 6 (1994); Oguri, PRB 64, 153305 (2001)

Conductance scales to Temperature, Magnetic Field, andBias Voltage.
≅̅

112

̅
13Γ

̅

̅

14Γ

̅

̅Γ
1514

 Wilson ratio is determined 1.9∼ 2.
 Strict quantitative test of quantum many bodytheory in the nonequilibrium regime.
31Symmetry Crossover32SU(4) Kondo effectKondo effect of even # of electrons
Spin-orbital Kondo effectCoulomb oscillation

cf. Conventional: SU(2) Kondo effect
N=1

∼

N=2

N=3

Low T

High TFerrier et al., Nature Physics 12, 230 (2016)33

Symmetry and Noise

M. Ferrier, et al., Phys. Rev. Lett. 118, 196803 (2017);
Y. Teratani et al., J. Phys. Soc. Jpn. 85, 094718 (2016).

SU(4) Kondo

32
Noise

∗

Reflected current (nA)

SU(2) Kondo∗53
NoiseSpin & Orbital

Reflected current (nA)34

SU(N) Kondo effect: exp. vs theory
Link between interaction andquantum fluctuations acrosssymmetry crossover

effective charge
Wilson ratio∗/ : interaction: quantum fluctuations

PRB 83,
Theory predicts… Sakano-Fujii-Oguri,075440 (2011).

Symmetry

effectivecharge ∗ /Wilson ratio

SU 2

5/32SU 4

3/2

4/3

…

…

…

SU ∞11 (free particle)∗→119511111(Kondo limit:→ ∞)35

Conclusion

Ferrier et al., Nature Physics 12, 230 (2016).
Teratani et al., J. Phys. Soc. Jpn. 85, 094718 (2016).
Ferrier et al., Phys. Rev. Lett. 118, 196803 (2017).
Hata et al., Phys. Rev. Lett. (in press).

 Realize ideal Kondo state Strict test of non-eq. quantum

many body theory
 Fluctuation increases as
symmetry lowers: SU(4) → SU(2) Connect different symmetries
cf: BEC-BCS crossover, symmetry in SC

On-chip collision experiment toprobe non-equilibrium quantum liquids

36