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Quantum criticality in correlated matter:

a few surprises from a mesoscopic look

Serge Florens

ITKM - Karlsruhe

Recent collaboration with:

Lars Fritz

Rajesh Narayanan

Achim Rosch

Matthias Vojta

Older but related work with:

Antoine Georges

Gabriel Kotliar

Patrice Limelette

Sergei Pankov

Subir Sachdev– p. 1

Motivation: the QPT problem

Classical phase transitions:

• A great challenge of last century

• Solved by K. Wilson in the ’70s

• New ideas (universality) and methods(renormalization)

Quantum phase transitions:

• An exciting problem in the new century

• Puzzling experiments, complex models

• Needs new ideas and methods!

– p. 2

Motivation: mesoscopic look

Environment

Ψ ΨΨΨ ΨΨ

Ψ ΨΨ

Ψ

Idea:go from a collection of quantum objects to anenvironmentally coupled single quantum object

Hope:Simplify life, but preserve crucial aspects of QPT

Bonus:relevant in mesoscopic physics!

– p. 3

Outline

• Reminder on classical phase transitions

• Introduction to quantum phase transitions

• QPT with dissipation: non-classical behavior

• Impurity QPT in superconductors: power ofquantum renormalization

• Criticality in quantum dots: new paradigm

• Conclusion– p. 4

Classical phase transitions

– p. 5

2nd order classical PTIsing magnet:E =

∑

ij JijSiSj

ξ

Associated free energy landscape:

φ φ φ

F F F

DISORDER BY THERMAL FLUCTUATIONS

ORDER BY SPONTANEOUS SYMMETRY BREAKING

– p. 6

Basic conceptsOrder parameter:φ(~x)

Landau energy functional:r ∝ T − Tc

E =∫

dDx[

(~∇xφ)2 + Bφ + rφ2 + uφ4]

Mean field:neglect spatial fluctuationsφ(~x) = M

M(T,B = 0) ∝ (Tc − T )β with β = 1/2

T

M(T, B = 0)

Tc

Universality:critical exponents do not depend onmicroscopic details – p. 7

Beyond the mean-field pictureThe difficulty:

• Spatial fluctuations are singular atD < 4 in thestatistical mechanics ofZ =

∑

φ(~x) e−E[φ(~x)]/kBT

Wilson’s answer:renormalization

• integrate short wavelength fluctuationsa<λ<sa

• iterate untilsa ∼ ξ: perturbation theory works

⇒ for D < 4 non trivial exponentβRG = 12 − 4−D

6 + . . .

D = 2 D = 3 D ≥ 4

βexact 116 0.326 ± 0.001 1

2

βRG 16 + . . . 1

3 + . . . 12 – p. 8

Universal versus non-universalUniversal exponents seen in various systems:(Anti)-ferromagnets, superfluids, liquid-gas transitionand recently metal-insulator transition

V2O3 [P. Limelette Science ’04] κ-BEDT [Kagawa PRB ’04]

The whole thing:full transport in DMFT[P. Limelette, SF, A. Georges, PRL ’04]

0 50 100 150 200 250 3000.0

0.1

0.2

0.3

U=4000 K

ρ (Ω

.cm

)

T (K)

300 bar W=3543 K 600 bar W=3657 K 700 bar W=3691 K 1500 bar W=3943 K 10 kbar W= 5000 K

– p. 9

Quantum phase transitions

– p. 10

First look on QPTWhat is a quantum phase transition?A T = 0 transition driven byquantum fluctuationsbetween two competing ground states

How can it be observed?Change non-thermal parameter to driveTc to zero

Example:LiHoF4 [Bitko et al. PRL ’96]

3D Ising in transverse field

H =∑

ij

JijSzi S

zj +

∑

i

BSxi

|Ferro⟩

=⊗

i| ↑

⟩

i

|Para⟩

=⊗

i(| ↑

⟩

+ | ↓⟩

)i– p. 11

Theory: QPT in insulatorsQuantum partition function:Z = Tr

[

e−(H0+H1)/kBT]

[H0, H1] 6= 0 ⇒ dynamics and statics couple!

New idea[Suzuki 1976]: Z =∑

φ(~x,τ) e−E/∆

with E =∫

dτ∫

dDx[

(∂τφ)2 + (~∇xφ)2 + rφ2 + uφ4]

General statement:(for insulators)Quantum PT in dim. D = Classical PT in dim. D+1

Not so trivial: peculiar time correlations

Many successful applications:

• spin dynamics in undoped cuprates• QPT in coupled ladder magnets

– p. 12

QPT in itinerant magnetsMagnetic critical point in a metal:CeCu6−xAux

criticalQuantum

Competition between RKKYinteraction and magneticcoupling to electrons

Anomalous response:

0 1 2 3 4

T0.75

(K0.75

)

x=0

x=0.05

x=0.1

x=0.15

x=0.2

x=0.3

x=0.5

CeCu6−x

Aux b)

H=0.1T

H II c

0.6

0.4

0.2

0.0

0 1 2 3 4

1/χ

(~q=

~ 0,T

)(m

eV/µ

2 B)

– p. 13

Theoretical puzzleExperiment:CeCu6−xAux

• C/T ∝ log 1/T • χ−1 = χ−10 +aT 0.75 • ρ = ρ0 + AT

Theory:electronic diffusion on spin fluctuations

Landau damping ofφ mode: φφ

e−

e−

E =∫

dω∫

dDq (ω2 + q2 + iω)φ2 ⇒ Deff = D + 2

3D SDW [Hertz; Moriya; Millis]• C/T = γ −

√T • χ−1 = χ−1

0 + aT • ρ = ρ0+AT 3/2

2D SDW [Roschet al.; Pankov, SF, Georges, Kotliar, Sachdev]• C/T = log 1/T • χ−1 = χ−1

0 +a Tln2 T

• ρ = ρ0 + AT

Problem:theory disagrees with experiment!– p. 14

Itinerant criticality: an open problem

Many puzzling compounds:

• Heavy fermions antiferromagnets: CeCu6−xAux,YbRh2(Si1−xGex)2, . . .

• Transition metal ferromagnets: MnSi, ZrZn2

• High temperature superconductors

Some fascinating and fundamental questions:

• Non Fermi Liquids• Exotic Superconductivity

Today:no sound theoretical framework!– p. 15

Mesoscopic QPT

Environment

Ψ ΨΨΨ ΨΨ

Ψ ΨΨ

Ψ

Idea:go from a collection of quantum objectsto a environmentally coupled single one

Hope:Simplify life, but preserve crucial aspects of QPT

Remark:Mesoscopic many-body problem = quantum impurity

– p. 16

Strategy

Practical reason:• Simpler to solve• Develop new concepts and techniques• Test with numerics (now possible)

Physical reason:

• Mesoscopic Effects + QPT = New Physics!• Realization in the lab (quantum dots, STM)

Other important application:

• Basis of quantum mean field theory (DMFT)• Self-consistent quantum impurity⇔ bulk

– p. 17

QPT with dissipation

– p. 18

Simple two-fluids modelSubohmic spin-boson: H = ∆Sx + λSzΦ

ρΦ(ω) ∝ ωs (s ≤ 1)Ohmic cases = 1 standard[Leggettet al. 1987]

Subohmic proposals = 1/2 [Tong and Vojta 2005]

Quantum to classical mapping:Equivalence with long range Ising:E =

∫

dτ∫

dτ ′ 1|τ−τ ′|1+sφ(τ)φ(τ ′) +

∫

dτ[

rφ2 + uφ4]

Quantum phase transition:RG analysis[Fisher-Ma ’72]

• For1/2 < s: u? 6= 0 ⇒ Non trivial exponents• For s < 1/2: u? = 0 ⇒ Mean-field exponents

– p. 19

Non-classical PTRecent NRG simulations:[Bulla, Tong, Vojta 2005]

s>1/2: non-trivial exponents oks<1/2: non mean-field exponents!

0 0.25 0.5 0.75 10

2

4

6

8

F (M )

Fisher-Ma

s

1/β

NRG

New result:[SF & R. Narayanan (unpublished)]

Trick: Majorana~S = −(i/2)~η × ~ηReconsider Landau:u singular ats < 1/2!!

Interpretation:non-analytic free energy

F (M) = hM + rM 2 + vM 2/(1−s)

⇒ β = 1−s2s 6= 1

2 fits numerics!

Future challenges:higher dimension generalization– p. 20

Impurity QPT in superconductors

– p. 21

Pseudogap Kondo modelKondo problem:spin ~S coupled to a Fermi sea

H = Hel. + J ~S · ~Sel.(~x = ~0)

Metals:• finite DoSρc(ε) = const.

• ⇒ Screening of the spinfor all J > 0

εk − µ

kx

ky

HTc superconductors:• linear DoSρc(ε) = |ε|• ⇒ Screening??

Ek = [(εk − µ)2 + ∆2k]1/2

kx

ky

– p. 22

Quantum Phase TransitionQuantum phase transition:[Withoff & Fradkin 1990]

Power-law DoSρc(ε) ∝ |ε|rTrick: r = small parameter

0

T, ω

Free Screened

∞Jc = r J

Quantum critical point:at j = jc

• Fractional spinS′ < 1/2: χimp(T ) = S′(S′+1)3T

• Critical exponents:χloc(ω) = ω−1+ηχ

Universal crossover:for j < jc [Fritz, SF, Vojta 2006]

1e-40 1e-20 10.2

0.25

T/D

Pert. RG

NRG

Tχimp(T )

r = 0.1j < jc

1e-40 1e-20 1

1e-06

0.0001

0.01

ωχ′′(ω, T = 0)

ω/D

loc

Pert. RGNRG

ω2r

ωr2

1e-40 1e-20 10

2

4

6

8

10

1e-40 1e-20 10.01

1

100

ω/D

T ′′(ω, T = 0)

Pert. RGNRG

ωr ω−r

– p. 23

Other progressesGlobal approach:expansions aroundr = 0, 1/2, 1

0 0.2 0.4 0.6 0.8 1r

0

0.2

0.4

0.6

0.8

1

ηχ

0 0.2 0.4 0.6 0.8 1r

0

0.05

0.1

0.15

0.2

0.25

Cim

p =

T χ im

p

CR

ACR

1e-40 1e-20 11e-05

1

1e+05

1e+10

ω/D

χ′′loc(ω, T = 0)

r = 0.47j > jc

Finite temperature dynamics:Still a challenge forthe NRG atω < T

1e-08 0.0001 1 10000 1e+080.0001

0.01

1

ω/T

T (ω, T )

T ? = T

T ? = 0

T ? = 100T

′′

Possible openings:• Clearer experimental studies (STM, nanobridges...)

• Application of our method to more complex models

• Key to understand failure of bosonic approach? – p. 24

Criticality in quantum dots

– p. 25

Two channel Kondo effectModel: spin ~S coupled to two Fermi seas[Nozières 1980]

H = Hel.1 + Hel.2 + J1~S · ~Sel.1 + J2

~S · ~Sel.2

Fermi Sea 2Fermi Sea 1

~S

Critical state:if J1 = J2 ⇒ Simp(T =0) = log(√

2)

imp

1CK

2CK

1CK∞

∞

J2

J1

Conclusion:generic instability!– p. 26

Can one realize 2CK in dots ?Experiments on 1CK:[Goldhaber-Gordon, Kastner 1998]

Mixing ofelectrodes⇒ 1CK!

SPIN

FERMI SEA 1

25 mK1 K

New idea for 2CK:[Oreg & Goldhaber-Gordon 2003]

FERMI SEA 1

FERMI SEA 2

Freeze charge exchangebetween leads 1 and 2by capacitive energy costEc

(Coulomb blockade)

⇒ 2CK possible if largeEc – p. 27

Surprising interplay of correlationsSolution:[SF & A. Rosch PRL 2004]

Phaseθ dual to charge:c†σm = a†σmeiθ [SF & A. Georges ’02]

Results:inversecrossover 1CK→ 2CK if smallEc

0 0.01 0.02 0.03 0.04 0.050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.1 1 10 1000

0.2

0.4

0.6

0.8

S

TEc

S

TT

T/Ec

Sim

p

Ec T 1CK

Ec T 1CK

Sim

p

∼ (T 1CK )2

Ec

T 1CK = De−1/j D

∼ De−2/j

Ec

∼ Ec

T2CK

Implications:

• New paradigm for stability of NFL states

• Optimize experiments: needEc ∼ T 1CK – p. 28

Conclusion

QPT in mesoscopic domain:a promising field!

• QPT areT = 0 quantum-driven transitions

• QPT have fascinating consequences

• New ideas and methods for their understanding

• Mesoscopic QPT are easier to understand, butstill very surprising!

• Cross-fertilization of ideas between traditionalcondensed matter and mesoscopic physics

– p. 29

Future directions of research

Quantum phase transitions:

• Challenge: bulk QPT in itinerant magnets

• Non equilibrium quantum phase transitions

• T = 0 Mott metal-insulator transition

Other problems:

• Transport in novel mesoscopic magnets

• Connection between many-body & ab-initio

– p. 30