CITE(2009arXiv0902.4170A):
For the reasons noted above we think it worthwhile
to develop an alternative formulation of the transport
theory of Luttinger liquids, formulated entirely in the
fermionic language. The fermionic representation oï¬€ers
the possibility to connect the fermionic degrees of freedom in the (non-interacting) leads smoothly with the
degrees of freedom in the interacting system. In other
words, it allows to use the scattering states of the system in the non-interacting limit as a basis of description.
On the simplest level, in lowest order in the interaction,
this program has been carried out in a seminal paper by
Yue, Matveev and Glazman.13 The physics of this problem lies in the scale-dependent build-up of a polarization
potential around the bare barrier, induced by the Friedel
oscillations of the density. For repulsive interaction, the
polarization potential is found to extend farther and far-
2
ther out as the temperature is lowered, until at T = 0
it is inï¬nitely extended, leading to a vanishing transmission probability across the barrier.... The process of gradual growth of the eï¬€ective potential barrier may be described in the language of renormalization group theory
applied to the transmission probability as a function of
the temperature. A generalization of the approach of Yue
et al. to the case of two barriers has been given in Ref.
[14].... These predictions, based on the continuum version
of the theory, were thoroughly checked and conï¬rmed by
the fermionic functional renormalization group method,
starting from the Hubbard models on a lattice.15,16
In this paper we report on a signiï¬cant extension of
the work of Yue et al. to the case of systems of spinless fermions in 1d interacting via arbitrarily strong forward scattering (parameter g2 ) and subject to a short
range barrier potential (width a). The principal tool we
will be using is perturbation theory in the interaction,
summed to inï¬nite order.... A solution to this problem, in the lowest
order of u2 was presented in the early work.4,5 Focussing
on the small-energy sector of the problem, one uses the
4
renormalization group approach, removing the higher energy states of the problem while simultaneously rescaling
the parameters of the action. It turns out that under this
procedure the amplitude of the BSG term is renormalized
as
u2 â†’ u2 e(1âˆ’K)Î›
(7)
where the RG scaling variable Î› = ln EF /Ç« > 0 . As a
result, the conductance at small u2 , given by G = |t|2 â‰ƒ
1 âˆ’ u2 , undergoes a similar renormalization
2
G â‰ƒ 1 âˆ’ |u2 |2 e2(1âˆ’K)Î›
âˆ’1
âˆ’1)Î›
Fermionic approach, RG for S-matrix
Yue, Matveev and Glazman13 have developed a theory,
starting from the formalism of scattering states and taking into account the fermionic interaction in lowest order
of perturbation. They arrived at the RG equation for the
transmission amplitude t in the form
dt
= âˆ’gt(1 âˆ’ |t|2 )
dÎ›
(10)
(8)
with the lower cutoï¬€ in Î› deï¬ned by temperature or voltage bias, Î› = ln EF / max[T, V ]. We shall conï¬ne ourselves to the the linear response regime, V â‰ª T in the
following....
Lett. 74, 3005 (1995).
D. Yue, L. I. Glazman, and K.