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 31 st, 2009 Quantum criticality perspective on population fluctuations of a localized electron level

Start non-interacting Average population 0 1 Ω Increase level energy ε – Critical ε * = E F V–V– ε–ε– – EFEF

Add on-site interactions 0 1 Ω U V+V+ ε+ε+ V–V– b ε–ε– – + EFEF Average population 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+V+ ε+ε+ V–V– b ε–ε– – + EFEF Average population 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 versus anisotropic 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 ε σ and δ U 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 >> Ω validity 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 ’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 (e.g., via charge sensing) be destructive for the effect? What if both fermionic & bosonic environment are present? Scaling arguments?