Transcript Quantum criticality perspective on population fluctuations

Quantum criticality perspective
on population fluctuations of a
localized electron level
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
Start non-interacting
Average population <n–>
1
V–
EF
–
ε–
Ω
0
Increase level energy ε–
Critical ε* = EF
Add on-site interactions
Average population <n–>
1
V–
EF
–
Ω
ε–
+
b
U
ε+
Without V_, b:
0
V+
Increase level energy ε–
Two disconnected, orthogonal ground states, “critical” at ε–= ε*
Results in a nutshell
Average population <n–>
1
“narrow”
V–
EF
–
ε–
“broad”
Ω
+
b
U
ε+
For small V_, b:
0
V+
Increase level energy ε–
Motivation
“Applied”
“Fundamental”
• Population switching in • A basic (“trivial”)
multi-level dots:
example of criticality
• is the there room for
• Connecting limits of
abrupt (first order)
different models
•
transitions?
what determines the
transition width for
moderate interactions?
• Charge sensing
• Qubit dephasing


(Non-) Interacting
resonant level versus
anisotropic Anderson
Full weak-to-strong
coupling crossover
Model Hamiltonian
Caution: definitions of εσ and δU
here are different form
those in the paper
Strongly anisotropic Anderson model,
with local, tilted Zeeman field (b,ε+–ε–)
V– =0  only “+” band 
 interacting resonant level
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!
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
D << Ω
D >> Ω
Line of critical FP!
validity range of
perturbative RG
RG recipe for critical exponents
• Linearize RG equations around the FP:
Bosonization-based mapping:
Crossover to strong
coupling when ~ 1
Reduced to Ω
Starting
(bare) value
Started from Γ+
Compare to numerics (alpha)
• Numerics done for ε*=0
Consistent with
presudo-spin
Kondo regime
VK,Schiller,Entin,
Aharony ’07
Silvestrov,Imry’07
Compare to numerics (beta)
Compare to numerics (both!)
A scaling law
Thanks to
Amnon Aharony!
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?