Nonequilibrium Kondo impurity: Perturbation about an exactly solvable point
Kingshuk Majumdar
Department of Physics and National High Magnetic Field Laboratory, University of Florida, 215 Williamson Hall,
Gainesville, Florida 32611
Avraham Schiller
Department of Physics and National High Magnetic Field Laboratory, University of Florida, 215 Williamson Hall,
Gainesville, Florida 32611
and Department of Physics, The Ohio State University, Columbus, Ohio 43210-1106
Selman Hershfield
Department of Physics and National High Magnetic Field Laboratory, University of Florida, 215 Williamson Hall,
Gainesville, Florida 32611
Received 31 July 1997; revised manuscript received 14 October 1997
We perturb about an exactly solvable point for the nonequilibrium Kondo problem. In each of the three
independent directions in parameter space, the differential conductance evolves smoothly as one goes away
from the solvable point, and the lowest-order correction contains the logarithm of the band width, or cutoff.
Perturbing towards physically realistic exchange couplings yields differential-conductance curves which more
closely resemble experimental data than at the solvable point. The leading coefficient which describes the
low-temperature and low-voltage scaling changes as one perturbs away from the solvable point, indicating
nonuniversal behavior; however, it is restored to the solvable-point value in the limit of an infinite band width.
S0163-18299800506-2
I. INTRODUCTION
Recently, there has been a resurgence of interest in ex-
actly solvable points for many-body problems in condensed
matter physics. This is due in part to the discovery of new
physical systems and in part to the discovery of new solvable
points. For example, there is strong experimental evidence
for the realization of clean one-dimensional interacting elec-
tron systems in fractional quantum Hall systems
1,2
and quan-
tum wires,
3,4
as well as for tunneling through a single mag-
netic impurity.
5,6
Some of the models used to describe both
of these phenomena have special points in their parameter
space where a simple analytic solution can be found to the
many-body problem. Some of the new solvable points which
have been discovered recently are the Emery-Kivelson line
for the two-channel Kondo model,
7
the g =
1
2
point for static
impurity scattering in a Luttinger liquid,
8
and three new Tou-
louse points for the generalized Anderson impurity model.
9
Besides being exact, one of the main advantages of solv-
able points is that one can easily compute experimentally
observable quantities, e.g., the susceptibility. Historically,
calculations of observables for exact solutions and exactly
solvable points have been done in equilibrium or linear re-
sponse; however, recently there have been some solutions for
nonequilibrium problems.
10,11
This paper concerns one of
those solutions, namely, tunneling though a magnetic impu-
rity connected to two leads.
12
The problem of tunneling through a magnetic impurity
has a long history. Zero-bias anomalies associated with tun-
neling through magnetic impurities were first discovered
13
in
the early 1960’s. These and later experiments
14
showed the
characteristic logarithmic singularities of the Kondo effect.
Shortly after the original experiments, there were perturba-
tive theories
15
which explained all of the qualitative features
of the experiments; however, they were not able to get to the
low-temperature, strong-coupling regime of the Kondo ef-
fect. With the present interest in quantum dots, almost all the
techniques of modern many-body physics have been applied
to this problem.
16
To date, the only exact result on it beyond
linear response is due to an exactly solvable point,
12
which
generalizes the Toulouse
17
and Emery-Kivelson
7
solutions of
the equilibrium Kondo problem.
Using this solvable point for the nonequilibrium Kondo
problem, a host of observables were computed:
12
electrical
current, spin current, current noise, and even time-dependent
response. In the case of the differential conductance, which
is the most widely studied property, the solvable point shows
all the qualitative features of the experiments: there is a reso-
nance at zero bias—this resonance splits in an applied mag-
netic field—the temperature and voltage dependences show
correct Fermi-liquid behavior. Furthermore, assuming uni-
versality, one can actually determine the exact scaling func-
tion for the differential conductance at low temperature and
voltage.
12
Although one can obtain all this information from the
solvable point, it is still only one point in the parameter
space. There is no reason to believe that the experimental
conditions correspond to this point. Thus, a natural question
to ask is what happens as one goes away from the solvable
point? This is the question which we address in this paper. In
particular, we perturb away from the solvable point to lowest
order in all possible directions in parameter space. The ques-
tions which we ask are i is the perturbation away from the
solvable point smooth and nonsingular? ii Does the quali-
PHYSICAL REVIEW B 1 FEBRUARY 1998-I VOLUME 57, NUMBER 5
57 0163-1829/98/575/29919/$15.00 2991 © 1998 The American Physical Society