Contents - Max-Planck-Institut für Physik komplexer Systeme

criticality. The current in the steady state limit is
I(V,T) = e
×
dω ρ(ω)[fL(ω) − fR(ω)]
2
Im T a LL(ω,T,V ) + T a RR(ω,T,V ) .
Here, Tα,β, (α,β = L/R), is the T-matrix of the Bose-
Fermi Kondo model. In the linear response regime of
the quantum critical LM phase, the conductance is proportional
to T 1/2 [4]. In the non-linear regime, V → 0
at T = 0, the conductance goes as V 1/2 . Fig. 2(a) shows
a scaling collapse of I/T 3/2 vs. V/T . Fig. 2(b) demonstrates
scaling of the dynamical spin susceptibiltity in
terms of ω/T and V/T .
(a)
(c) (d)
Figure 2: (a) Scaling of the current-voltage characteristics in the critical
local moment phase and (b) scaling of the imaginary part of the
local dynamical spin susceptibility. (c) The fluctuation-dissipation
ratio for the spin susceptibility in the critical local moment phase
(g = 4 · gc) for V/D = 5.0 · 10−3 = 0.1T0 K , where T0 K is the Kondo
temperature in the absence of magnons (g = 0). (d) An effective
temperature T ∗ χ can be defined such that the FDRχ(ω, T, V ) of (c)
collapses on coth(ω/2T), the equilibrium FDR of a bosonic field.
The ω/T and V/T scaling occurs only in the scaling
regime of the quantum critical LM phase and at
the quantum critical point. As in the equilibrium
relaxational regime (V = 0, ω ≪ kBT/) where
an ω/T scaling implies a linear-in-T spin relaxation
[1] Focus issue: Quantum phase transitions Nature Phys. 4, (2008) 167-204.
[2] S. Kirchner and Q. Si. Phys. Rev. Lett. 103, (2009) 206401.
[3] S. Kirchner and Q. Si. Phys. Status Solidi B 247, (2010) 631.
[4] S. Kirchner, L. Zhu, Q. Si, and D. Natelson. Proc. Natl. Acad. Sci. USA 102, (2005) 18824.
[5] Q. Si, S. Rabello, K. Ingersent and J. L. Smith. Nature 413, (2001) 804.
[6] S. Kirchner and Q. Si. Physica B 403, (2008) pp. 1189.
[7] L. Zhu, S. Kirchner, Q. Si, and A. Georges. Phys. Rev. Lett. 93, (2004) 267201.
[8] P. C. Hohenberg and B. I. Shraiman. Physica D 37, (1989) 109.
[9] L. F. Cugliandolo, J. Kurchan, and L. Peliti. Phys. Rev. E 55, (1997) 3898.
[10] T. Risler, J. Prost, and F. Jülicher. Phys. Rev. Lett. 93, (2004) 175702.
(b)
rate, in the non-equilibrium relaxational regime here
(T = 0, ω ≪ eV/), the ω/V scaling implies a
linear-in-V dependence of the decoherence rate, ΓV ≡
[−i∂ lnχa (ω,T = 0,V )/∂ω] −1
ω=0 = c(e/)V . It is worth
stressing that, in contrast to its counterpart in the highvoltage
(V ≫ kBT 0 K /e) perturbative regime, the decoherence
rate here is universal: c is a number characterizing
the fixed point.
We now turn to the concept of effective temperature
in the non-linear regime. The notion of an effective
temperature for extending the FDT to non-equilibrium
states was first introduced in the context of steady
states in chaotic systems [8], and was later used for
non-stationary states in glassy systems [9] and in the
context of coupled noisy oscillators [10]. Fig. 2(c) displays
the FDRχ(ω,T,V ) for V = 5 · 10−3D = 0.1T 0 K ,
with T 0 K being the Kondo temperature in the absence
of magnons (g = 0). The scaling regime extends up
to energies of the order of T 0 K , so that bias voltages
larger than V = 0.1T 0 K might be affected by sub-leading
contributions. Three important observations underly
the results displayed in Fig. 2(a): (1) for ω >> T ,
FDRχ(ω,T,V ) approaches the value predicted by the
FDT, (coth(ω/2T) → 1), (2) for T >≈ 10−3V , the deviations
from the linear response behavior are hardly
discernible, (3) for T