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

Quantum criticality perspective on population fluctuations of a localized electron level

Start non-interactingAverage population <n–>

0

1

Ω

Increase level energy ε–

Critical ε* = EF

V–

ε––

EF

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*

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”

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”

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

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

Line of critical FP!

D << Ω

D >> Ωvalidity range of perturbative RG

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

Compare to numerics (alpha)• Numerics done for ε*=0

Consistent with presudo-spin Kondo regime

VK,Schiller,Entin,Aharony ’07Silvestrov,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?