vyacheslavs (slava) kashcheyevs collaboration: christoph karrasch, volker meden (rtwh aachen u.,...
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Vyacheslavs (Slava) Kashcheyevs
Collaboration:Christoph Karrasch, Volker Meden (RTWH Aachen U., Germany)Theresa Hecht, Andreas Weichselbaum (LMU Munich, Germany)Avraham Schiller (Hebrew U., Jerusalem,Israel)
“The Science of Complexity”, Minerva conference, Eilat, March 31st , 2009
Quantum criticality perspective on population fluctuations of a localized electron level
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Quantum criticality perspective on population fluctuations of a localized electron level
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Start non-interactingAverage population <n–>
0
1
Ω
Increase level energy ε–
Critical ε* = EF
V–
ε––
EF
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Add on-site interactions
0
1
Ω
U
V+ε+
V–
b
ε–– +
EF
Average population <n–>
Without V_, b:Increase level energy ε–
Two disconnected, orthogonal ground states, “critical”
at ε–= ε*
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Results in a nutshell
0
1
Ω
U
V+ε+
V–
b
ε–– +
EF
Average population <n–>
For small V_, b:Increase level energy ε–
“narrow”
“broad”
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Motivation
• Population switching in multi-level dots:• is the there room for
abrupt (first order) transitions?
• what determines the transition width for moderate interactions?
• Charge sensing• Qubit dephasing
• A basic (“trivial”) example of criticality
• Connecting limits of different models (Non-) Interacting
resonant level versusanisotropic Anderson
Full weak-to-strong coupling crossover
“Applied” “Fundamental”
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Model Hamiltonian
Strongly anisotropic Anderson model, with local, tilted Zeeman field (b,ε+–ε–)
V– =0 only “+” band interacting resonant level
Caution: definitions of εσ and δU
here are different form those in the paper
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Weaponry
• Analytical mapping to anisotropic Kondo model via bosonisation Pertrubative RG (in tunneling, not U!)
of Yuval-Anderson-Hamann’70
• Numerical Renormalization Group• Functional RG
Fight problems, not people!
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Strategy – renormalization
• Disconnected system at ε–=ε* is RG-invariant a fixed point!
• Tunneling is a relevant perturbation FP is repulsive the system is critical
Fermi liquid(Kondo) FP
Line of critical FP!
D << Ω
D >> Ωvalidity range of perturbative RG
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RG recipe for critical exponents
• Linearize RG equations around the FP:
Bosonization-based mapping:
Reduced to Ω
Started from Γ+
Crossover to strong coupling when ~ 1
Starting (bare) value
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Compare to numerics (alpha)• Numerics done for ε*=0
Consistent with presudo-spin Kondo regime
VK,Schiller,Entin,Aharony ’07Silvestrov,Imry’07
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Compare to numerics (beta)
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Compare to numerics (both!)
A scaling law
Thanks to Amnon Aharony!
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Some open questions
• How does finite voltage dephase/modify the power-laws?
• Will direct measuring of <n-> (e.g., via charge sensing) be destructive for the effect?
• What if both fermionic & bosonic environment are present? Scaling arguments?
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