Full counting statistics in a disordered free fermion system
Abstract
The Full Counting Statistics (FCS) is studied for a one-dimensional system of non-interacting fermions with and without disorder. For two unbiased
L
site lattices connected at time
t=0
, the charge variance increases as the natural logarithm of
t
, following the universal expression
<δ
N
2
>≈
1
π
2
logt
. Since the static charge variance for a length
l
region is given by
<δ
N
2
>≈
1
π
2
logl
, this result reflects the underlying relativistic or conformal invariance and dynamical exponent
z=1
of the disorder-free lattice. With disorder and strongly localized fermions, we have compared our results to a model with a dynamical exponent
z≠1
, and also a model for entanglement entropy based upon dynamical scaling at the Infinite Disorder Fixed Point (IDFP). The latter scaling, which predicts
<δ
N
2
>∝loglogt
, appears to better describe the charge variance of disordered 1-d fermions. When a bias voltage is introduced, the behavior changes dramatically and the charge and variance become proportional to
(logt
)
1/ψ
and
logt
, respectively. The exponent
ψ
may be related to the critical exponent characterizing spatial/energy fluctuations at the IDFP.