Renormalisation Group Analysis of a Spin-1 Kondo Dot out of Equilibrium

Preliminary results of the underscreened Kondo model out of equilibrium in a magnetic field, presented at the march meeting of the Deutsche Physikalische Gesellschaft (DPG) in Dresden, 2011. This foundation for this results was initiated during my diploma thesis at RWTH Aachen University (link to thesis: http://wwwo.physik.rwth-aachen.de/fileadmin/user_upload/www_physik/Institute/theorie_a/publications/Diplomarbeit_Christoph_Hoerig.pdf) and has been finalized as first project of my PhD thesis.

Results have been published in Physical Review B, http://journals.aps.org/prb/abstract/10.1103/PhysRevB.85.054418.

Transcript

1. 14.02.2011 DPG-Tagung Renormalization- Group Analysis of a Spin-1 Kondo Dot out of Equilibrium Christoph Hörig, Sabine Andergassen, and Dirk Schuricht Institute for Theory of Statistical Physics! •! RWTH Aachen! •! Germany [Roch et. al. Nature 453 (2008)] of B and Vb for a constant gate A Zeeman-induced transition -m triplet state j1, 21æ occurs this transition is not directly accessible in our s strate that very specific characteristics of the transition can be observed in transport. The 100 nm 1 µm V b V g b d Even Singlet Triplet E T – E S E S – E T Quantum critical point lc l –1.0 –0.5 0.0 (mV) |B〉 C 60 1 nm
2. • isotropic Spin-1 Kondo dot • fixed charge with S=1 on dot • in magnetic field • only exchange couplings Model h0 J S=1 +V/2 -V/2 J J J Hres = ￿ α=R,L ￿ σ=↑,↓ ￿ dω (ω + µα) a † α1σ1 (ω) aα1￿ σ1￿ (ω), HS = h0 S z , Hint = J 2 ￿ α1,α1￿ ￿ σ1,σ1￿ ￿ dωdω￿ ￿ S · ￿ σσ1σ1￿ a † α1σ1 (ω) aα1￿ σ1￿ (ω￿)
3. • Real-time renormalization group method in frequency space [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] • integrate out the reservoirs & generate decay rates with RG • perturbative expansion up to in the renormalized exchange couplings • solve in weak coupling regime Technique: reduced density matrix Method O(J2) vertex contraction resolvent 12 G 1 1 12 J = 1 2 ln(max{h0, V }/TK) 12 G 21 1 G 12 2 + … ρS(z) = i z − Leff S (z) ρS(t0) dLeff S dΛ =
4. • renormalized magnetic field • up to + logarithmic corrections • leading order: • magnetic field is screened by reservoir electrons • equal to Spin-1/2 due to identical fermionic reservoirs Results: magnetic field O(J2) h = ￿ 1 − 1 2 Tr( ˆ J − ˆ J0) ￿ h0 − J2 2 h0 ￿ j=1,2 ln ￿ Λ2 c h2 0 + Γ2 j + J2 4 (V − h0) ￿ j=1,2 ln ￿ Λ2 c (V − h0)2 + Γ2 j Λc = max{V, h0 }
5. • Decay rates • are generated during the RG flow • cut-off divergencies in the resolvents and logarithms • describe the time-evolution of magnetization and polarization operator • reduced density matrix in Markov-Born-Approximation Results: time evolution Pz ∝ ￿ SzSz − SySy − 1 ￿ Sz (h0 = 0) ρS(t) M = ￿Sz(t)￿ = Tr ￿ Sz ρS(t) ￿ = ￿ ρ11 − ρ−1−1 ￿ e−Γj=1t P = ￿Pz(t)￿ = 1 √ 6 (ρ11 − 2ρ00 + ρ−1−1)e−Γj=2t Γj = π 2 j(j + 1)J2V with j = {0, 1, 2},
6. • Decay rates for finite magnetic field Results: decay rates (h0 ￿= 0) Γ2 = 3πJ2V Γ1 = πJ2V Γ0 = 0 quadrupol tensor spin ￿Pij(t)￿ ￿￿ S(t)￿
7. • magnetization in the stationary state • faster decrease for Spin-1 • qualitatively similar to Spin-1/2 Results: magnetization Spin-1/2 [Schoeller & Reininghaus, Phys. Rev. B 80, 045117 (2009)] vs. Spin-1: ρst S = 1 3 1 + M 2 Sz + P √ 6 Pz
8. • differential conductance • enhancement at due to inelastic cotunneling • qualitatively similar to Spin-1/2 Results: differential conductance V = h0 Spin-1/2: [Schoeller & Reininghaus, Phys. Rev. B 80, 045117 (2009)] Spin-1: G = dI dV G␥ /G0 = ␥ ␲2 2 ͭ ͑Jnd z ͒2 + ͑Jnd Ќ ͒2 ͫ 2 + ͩ 2Jnd Ќ JL Ќ + JR Ќ ͪ2 ͬ + ͑JL z + JR z ͒͑Jnd Ќ ͒2 ͫ 3 + ͩ 2Jnd Ќ JL Ќ + JR Ќ ͪ2 ͬ L2 ͑V − Two interesting features happen at VϷh ˜: There is a the leading-order term in the conductance ͑if Jnd Ќ due to inelastic cotunneling which sets in at this vol jump is superposed by a logarithmic term which largest for V=h ˜. The experimentally accessible p 0 0.1 0.2 0.3 0.4 0 0.5 1 1.5 2 V/h0 G/e2 h REAL-TIME RENORMALIZATION GROUP IN FREQUENCY… PHYSICAL REVIEW B 80, 0451 G0 G ∼ J2 arctan ￿ V − h0 Γj ￿
9. 0 0.1 0.2 0.3 0.4 0 0.5 1 1.5 2 V/h0 G/e2 h G0 Results: differential conductance Spin-1/2: [Schoeller & Reininghaus, Phys. Rev. B 80, 045117 (2009)] Spin-1: • Slope • height-difference of the jump between Spin-1/2 vs Spin 1 ∆(Gγ/G0) = 5 2 π2γJ2 ∆(Gγ/G0) = 3π2γJ2
10. • Anisotropic model • generates term with band width • leading order term dominates Anisotropic model c2 = (Jz)2 − ￿ J⊥ ￿2 ￿= 0 ‣ exchange couplings are strongly relevant c2D > max{h0, V } −πc2DSzSz D c2 > 0 : ￿Sz￿ = ±1 c2 < 0 : ￿Sz￿ = 0
11. • we have derived an analytic solution of a Spin-1 Kondo dot out of equilibrium in the weak-coupling regime • for isotropic couplings we obtained: • the renormalized magnetic field • the time evolution of the spin and polarization operators • analytical solutions for: • magnetization and susceptibility • stationary current and differential conductance • we showed that anisotropic exchange couplings are strongly relevant Outlook: higher orders for logarithmic enhancement Conclusions