0.5 setgray0 0.5 setgray1 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
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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