Pumping in Interacting Systems Yuval Oreg Department of Condensed Matter Physics

With Eran Sela

Outline What are pumps? What are electron pumps? Phase Coherent Pumps Non coherent Pumps General formula for interacting systems The two channel Kondo system Topological quantization of spin current Effective magnetic field and Non Fermi liquid behavior at finite T Conclusions

Heat pumps Sadi Nicholas Leonard Carnot Work= Area P V

Single Electron Pumps – Electron Turnstile

N N+1 N N N N N N

Single Electron Pumps

Pumps formulae T Interaction Brouwer BPT ECEC BPT [non interacting] Hartree X1X1 X2X2 δXδX X0X0

Rate equations Pumps formulae T Interaction Brouwer BPT ECEC Non coherent pumps (Sela and YO PRB 2005) Quantum - interacting (Sela and YO PRL 2006)

Non Coherent Pumps With geometric interpretation (Sela and YO PRB2005)  Asymmetry coefficient Q charge on one of the capacitor plates Adiabatic limit τ >RC

x y δXδX X0X0

Pure Spin Pumps

For 2DEG with area A

Prefers polarization A=1μm 2

When non coherent pumps formula applies? L γ δUδU With dephasing (using Buttiker ’ s model) Classical non coherent result With DOS ->1/C and Transmission ->1/R I Class /I Coherent = L k F =L/λ F

Rate equations Pumps formulae T Interaction Brouwer BPT ECEC Non coherent pumps (Sela and YO PRB 2005) ?

Pumps (with interaction) at low temp Kubo Formula Aleiner and Matveev: Open dots (1998) Sharma and Shamon: Lut-Liq (2001, 2003) Citro et al: Lut – Liq (2003) Cohen (2003): Applied to non interacting systems Keldysh J. Splettstoesser et al. (Average time ) approximation

Left Lead Right Lead X 1, X 2 Central area may depend on parameters All parts (including leads) may have interactions X1X1 X2X2 δXδX X0X0 BPT (non interacting)

Adiabatic Limit Relaxation time O(δX 2 ) Curvature X1X1 X2X2 δXδX X0X0

Geometric/Topologic interpretation

Application to Dots Average time Aprox. d d d c c c

εdεd dQ d =Adε d A=#U 2 /(T 2 Γ)+U 2 /T 3  Infinite order in Γ second order in U, Assume: U and Γ «T -#U 2 /(T 2 Γ)

Application to repulsive quantum critical points x y

Spin pumps in the two channel Kondo At x=y=0 NFL point T 1/2 sing. x=Δ=J1-J2, y=h

Emery Kivelson Line hГhГ Δ=(J  1 -J  2 )/Г=Cos( θ) Kondo Temp

Concentrated around r=1 Integral over B=ћ

Δ0Δ0 h0 Δ L2 1

Conclusions Non coherent pumps at high temp. A generic pumping formula for interacting systems, with a geometric interpretation. Application to two channel Kondo physics with anomalous exponents and interesting topology