Contents - Max-Planck-Institut für Physik komplexer Systeme

2.17 Quantum Criticality out of Equilibrium: Steady States and Effective Temperatures
Quantum phase transitions are of extensive current interest
in a variety of strongly correlated electronic and
atomic systems [1]. A quantum critical point occurs
when such transitions are second order. As a consequence,
there is no intrinsic energy scale in the excitation
spectrum of a quantum critical state. Outof-equilibrium
states near quantum phase transitions
have so far received only limited theoretical attention
despite a long-standing interest in their classical
counterparts. Experimentally, on the other hand,
it seems unavoidable to generate out-of-equilibrium
states during a measurement in the quantum critical
regime as this regime is characterized by the absence
of any intrinsic scale. Any measurement may therefore
potentially perturb the system beyond the linear
response regime where the fluctuation-dissipation theorem
(FDT) no longer holds. To make progress theoretically,
approaches are needed that can capture the
scaling properties of a quantum critical point in equilibrium
and that can be extended to out of equilibrium
settings. This is impeded by the lack of a general understanding
of how to generalize the free energy functional
to stationary nonequilibrium states from which
scaling relations could be obtained.
We have identified a system in which quantum criticality
out of equilibrium can be systematically studied
[2,3]. It was shown earlier that single-electron transistors
(SET) attached to ferromagnetic leads can undergo
a continuous quantum phase transition as their
gate voltage is tuned [4]. The corresponding quantum
critical point separates a Fermi liquid phase from a non-
Fermi liquid one. The key physics is the critical destruction
of the Kondo effect, which underlies a new
class of quantum criticality that has been argued to apply
to heavy fermion metals [5].
(a) (b)
J
Vg
µ 1
Q
µ 2
D
Kondo
Figure 1: (a) Schematic setup of the magnetic SET. The arrows in the
left and right leads indicate the direction of the magnetization in the
leads. For antiparallel alignment, the local magnetic field generated
by the leads vanishes. (b) Phase diagram: the Fermi liquid phase
(“Kondo”) and the critical local moment (“LM”) phase are separated
by a QCP. The voltage Vg allows to tune the system through a quantum
phase transition.
Vg
STEFAN KIRCHNER, QIMIAO SI 4
4 Department of Physics & Astronomy, Rice University, Houston.
QC
g c
LM
g
The couplings of the local degrees of freedom to the
conduction electrons and to the magnons in the leads
allow for a dynamical competition between the Kondo
singlet formation and the magnon drag. As a result, the
low-energy properties are governed by a Bose-Fermi
Kondo model (BFKM) [4, 6].
Hbfk =
Ji,jS · c † σ
i,j
k,k ′ ,σ,σ ′
+
k,i,σ
+ g
β,q,i
k,σ,i
2 c σ ′ ,j
˜ǫ kσi c †
kσi ckσi + hlocSz
Sβ(φβ,q,i + φ †
β,q,i ) + ωq φ
β,q,i
†
β,q,iφβ,q,i. where i,jε{L,R} and hloc = g
i mi, is a local magnetic
field with mL/mR being the ordered moment of the
left/right leads. For antiparallel alignment and equal
couplings one finds mL = −mR. ˜ǫkσi is the Zeemanshifted
conduction electron dispersion, and φβ,i, with
β = x,y, describes the magnon excitations. The spectrum
of the bosonic modes is determined by the density
of states of the magnons,
q [δ(ω − ωq) − δ(ω + ωq)] ∼
|ω| 1/2 sgn(ω) up to some cutoff Λ. A sketch of the magnetic
SET is shown in Fig. 1(a). The resulting equilibrium
phase diagram of the system is displayed in
Fig. 1(b). Nonequilibrium states can be created by
keeping left and right lead at different chemical potentials
but the same temperature T , (eV = µL − µR).
For such a finite bias voltage, we work on the Keldysh
contour. The current-carrying steady state at arbitrary
bias voltage V has been explicitly studied by an extension
of the dynamical large-N limit onto the Keldysh
contour [2]. The dynamical large-N limit in equilibrium
has been shown to correctly capture the dynamical
scaling properties (including, in particular, the
finite-temperature relaxational properties) of the N = 2
case [4, 7]. For a finite bias voltage, we work on the
Keldysh contour. Since we are interested in the steady
state limit, we specify the state of the system at the infinite
past t0 = −∞, so that at finite time all initial correlations
will have washed out.
In this model system, we have shown that the universal
scaling is obeyed by the steady state current,
the pertinent spectral densities, and the associated
fluctuation-dissipation ratios. We have been able to calculate
the entire scaling function for each of these nonequilibrium
quantities. This allows us to elucidate the
concept of effective temperatures. Our theoretical approach
can even be extended to study the transient behavior
and other non-equilibrium probes of quantum
74 Selection of Research Results