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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
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