научная статья по теме ONE-DIMENSIONAL STRONGLY INTERACTING ELECTRONS WITH SINGLE IMPURITY: CONDUCTANCE REEMERGENCE Физика

Текст научной статьи на тему «ONE-DIMENSIONAL STRONGLY INTERACTING ELECTRONS WITH SINGLE IMPURITY: CONDUCTANCE REEMERGENCE»

Pis'ma v ZhETF, vol. 101, iss. 9, pp. 697-702

© 2015 May 10

One-dimensional strongly interacting electrons with single impurity:

conductance reemergence

y. y. Afoniny. Yu. Petrov+

loffe Physical-Technical Institute of the RAS, 194021 St.Petersburg, Russia + Petersburg Nuclear Physics Institute, NRC "Kurchatov Institute", 188300 Gatchina, Russia

Submitted 2 March 2015 Resubmitted 24 March 2015

We show that conductance of ID channel with one point-like impurity critically depends on asymptotic behavior of e—e interaction at small momenta k (about inverse length of a channel). Conductance reemerges (contrary to the case of point-like repulsive potential) if potential V {k = 0) = 0. For example, this happens if the bare e—e interaction is screened by the charges in the bulk. The relation of this phenomena to the long-range order present in the Luttinger model is discussed. We consider spinless electrons but generalization is straightforward.

DOI: 10.7868/S0370274X15090088

Introduction. Theory of one-dimensional interacting electrons is under investigation for a long time [13]. Its relativistic analog, two-dimensional QED, also attracted a lot of attention [4] in the past, since it is a simplest field theory with confinement. During the time it was understood that one-dimensional pure electronic systems (in particular, the Luttinger model [2]) are exactly solvable. To date the properties of the clean systems are very well understood. Situation is different for the ID channels with some impurities that are understood as short-range barriers with transition coefficient K and reflection coefficient R. The simplest system of this kind (with only one impurity) was considered for the first time in Ref. [5]. It turned out that properties of such system depend critically on the sign of the electron-electron (e—e) interaction. Conductance for attractive potentials is equal to the ballistic one and it is not affected by e—e interaction (only the Fermi speed should be renormalized). Conductance for repulsive potentials vanishes. These results were obtained in [5] by the bosonization method.

Another approach with similar results was developed in [6]. The authors returned to the fermion language. Assuming that the interaction is short-range, V = VoS(x) and small Vo <C 1, they summed up the leading infrared logarithms of frequency u> by the renormalization group method. Next-to-leading corrections to the conductivity were also found in [7] by the methods of current algebra.

The approaches of [5] and [6] have different (but overlapping) regions of applicability. The first approach

-'-'e-mail: vasili.afonin@mail.ioffe.ru

employs perturbation theory in reflection (transition) coefficient for an arbitrary attractive (repulsive) potential, while the second one employs perturbation theory in potential for an arbitrary reflection or transition coefficients. A point-like e—e interaction is assumed in both approaches.

We suggested in [8] an alternative approach to the problem based on the path integral formalism. Using the well-known trick [9] we see that Luttinger model can be interpreted as the system of non-interacting electrons in a random external field. Green functions of one-dimensional electrons in any external field can be found exactly. We used this fact to construct pertur-batively a Green function of the system with impurity. At the end we integrate out fermions and arrive at a 0+1-dimensional field theory. This theory describes the evolution with time of the electron phase at the point where the impurity is located. It is completely equivalent to the original Luttinger model with one impurity. For the sake of simplicity we will consider here only electrons without spin.

Using this theory we were able to prove two theorems. First, conductance of the system is zero (for repulsion) or maximal (for attraction) for a wide class of potentials. The arguments in favor of this statement for a point-like potential were given earlier in both approaches of [5] and [6]. We will see below that a necessary condition for such behavior of conductance is that the Fourier transform of the potential V(k) has a non-vanishing limit at k —> 0. The second theorem is a general exact property of the theory which one can call duality. It states that the effective reflection coefficient

17^12 in a theory with an attractive potential is equivalent to the effective transition coefficient \1CU\2 in a theory with repulsion if one exchanges K -f-> R (for a precise formulation of duality transformation of potential, see below). The traces of this property were seen in the perturbation theory in [5] where duality transformation reduces to vc —> v^1 (vc is the renormalized Fermi speed). However, this statement is far more general. It means that it is enough to consider, say, only repulsive potentials.

For a repulsive potential conductance restores if potential vanishes at k —> 0. Such a situation takes place in the systems with a small density of carriers when the screening radius is large. In this case e—e interaction is not point-like from one-dimensional point of view, and it is screened by the image charges on 3-dimensional gates, edges of the channel, etc. Renormalization of the ballistic conductance in such system is finite and will be calculated below. Pay attention that the form and value of conductance is determined not by small k but by the whole region V(k) where the potential is not small. So one needs an approach which is valid for an arbitrary e—e potential, not only for a point-like one as in [5, 6].

The physical reason for critical phenomena of the conductance in the Luttinger model with an impurity is a long-range order which is present in a system of one-dimensional electrons. It is well-known that its analogue — the Schwinger model — exhibits the anomalous breakdown of chiral symmetry. The strength of the interaction in the repulsive Luttinger model is smaller: the system is in the Berezinskii-Kosterlitz-Thouless (BKT) phase [10]. Chiral condensate (consisting of pairs of R electron and L hole with finite density) arises only in the limit of an infinitely large interaction. In the case of an attractive potential there is a charged condensate of Cooper pairs with vanishing density (for the channel with infinite length), i.e. one has a BKT phase as well. The Bose-Einstein principle implies that the chiral condensate increases the probability of reflection, i.e. the effective reflection coefficient at small frequencies, while the charged condensate increases the probability of transition. As a result, IT^J2 = 1 for repulsion and = 1 for attraction. As we mentioned above, this does not happen if V(k —> 0) = 0. We will see that in this case the long- range order in the Luttinger model also disappears. This leads to finite conductance of the channel.

Effective transition/reflection coefficients and conductance. The Fermi surface in one dimension reduces to two isolated points ±pf- The electrons with momenta close to the Fermi surface can be divided in right (R) and (L) movers,

^ = eipFX-iEFt^R + e-ipFX-isFt^

where ^r.l are slowly varying on the scale 1/pf-

By means of the Hubbard [9] trick, the Luttinger model can be reduced to a system of noninteracting electrons in a random external field U(x, t) with a simple Gaussian weight and subsequent integration in all possible realizations of the field. The Schrodinger equation for the non-interaction R, L electrons in the external field reduces to the Dirac equation in d = 1 (in our units h = vp = 1; we will also omit electron charge e to restore it in the final expression for conductivity),

[idt ± idx - U]iprjL = 0.

(1)

The Luttinger liquid is a system which can be solved exactly. The ultimate reason for this is that a one-dimensional fermion Green function in the external field can be found

x ^ x *) — ^^

— r'VTR.LCzMTR.LCz')

TR,l(x) = -fd2x'G^L(x,x')U(x').

(2)

The Green functions Gr^ only by phase differs from the free Green functions

1

2iri(t =F x — idt)

(3)

((5 > 0 is infinitesimal).

A point-like impurity located at x = 0 mixes left and right electrons. Impurity plays the role of a boundary condition, solutions of Eq. (1) should be matched at x = 0. Nevertheless, the general solution in the external field can be found [8]. Solution depends on a new functional variable a(t) which is the difference of phases for R- and L-electrons at the point of impurity

a(t)=7R(0,t)-7L(0,t).

(4)

Construction of the Green function with an impurity is impeded by the Feynman boundary conditions which lead to some integral equation. This equation can be solved perturbatively either in bare reflection or in transition coefficient (for details see [8]).

The Luttinger model has high symmetry: it is invariant both under gauge (vector) and chiral transformations (the latter symmetry is broken by the anomaly). The charge density (p = pb. + Pl) and current (j = = Pr — Pl) can be completely determined from the conservation of the vector and axial currents:

dtp+dxJ= 0, dtj + dxp=--dxU + ®(t)6(x). (5)

71

Here, the first term on the right-hand side is the Adler anomaly [11]. The second term describes the influence

of the impurity, and 35 is the charge jump at x = 0 which depends only on phase a(t). It can be calculated if the Green function is known.

Integrating in fermion degrees of freedom allows to present any quantity clS cl product of Green functions in the external field and fermion determinant describing the sum of the loop diagrams. As it was shown in [8] the effect of impurity is completely determined by the phase a(t): non-trivial part of the Green functions and determinant depends only on a(t). Introducing a as a new variable one can integrate also in U(x, t) and reduce original 1 + 1-dimensional model with impurity to the effective 0+1 field theory (non-local quantum mechanics of the phase a(t)).

The conductance of the channel C(uj) is related to the exact transition c

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