One-dimensional strongly interacting electrons with single impurity: conductance reemergence
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r One-dimensional strongly interacting electrons with single impurity: conductancereemergence
V.V. Afonin and V.Yu. Petrov ( A.F.Ioffe Physical-Technical Institute of the RAS, St.Petersburg, Russia) (Petersburg Nuclear Physics Institute, Gatchina 188300, St. Petersburg, Russia) We show that conductance of 1D channel with one point-like impurity critically dependson asymptotic behavior of e − e interaction at small momenta k (about inverse length of achannel). 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 chargesin the bulk. The relation of this phenomena to the long-range order present in the Luttinger modelis discussed. We consider spinless electrons but generalization is straightforward. INTRODUCTION
Theory of one-dimensional interacting electrons isunder investigation for a long time [1]-[3]. Its relativisticanalog, two-dimensional QED, also attracted a lotof attention [4] in the past, since it is a simplestfield theory with confinement. During the time it wasunderstood that one-dimensional pure electronic systems(in particular, the Luttinger model [2]) are exactlysolvable. To date the properties of the clean systemsare very well understood. Situation is different for the1D channels with some impurities that are understoodas short-range barriers with transition coefficient K andreflection coefficient R . The simplest system of this kind(with only one impurity) was considered for the firsttime in Ref.[5]. It turned out that properties of suchsystem depend critically on the sign of the electron-electron ( e − e ) interaction. Conductance for attractivepotentials is equal to the ballistic one and it is notaffected by e − e interaction (only the Fermi speed shouldbe renormalized). Conductance for repulsive potentialsvanishes. These results were obtained in [5] by thebosonization method.Another approach with similar results was developedin [6]. The authors returned to the fermion language.Assuming that the interaction is short-range, V = V δ ( x ) and small V ≪ , they summedup the leading infrared logarithms of frequency ( ω ) by the renormalization group method. Next-to-leadingcorrections to the conductivity were also found in [7] bythe methods of current algebra.The approaches of [5] and [6] have different (butoverlapping) regions of applicability. The first approachemploys perturbation theory in reflection (transition)coefficient for an arbitrary attractive (repulsive)potential, while the second one employs perturbationtheory in potential for an arbitrary reflection ortransition coefficients. A point-like e − e interaction isassumed in both approaches.We suggested in [8] an alternative approach to theproblem based on the path integral formalism. Usingthe well-known trick [9] we see that Luttinger model can be interpreted as the system of non-interacting electronsin a random external field. Green functions of one-dimensional electrons in any external field can be foundexactly. We used this fact to construct perturbativelya Green function of the system with impurity. Atthe end we integrate out fermions and arrive at a0+1-dimensional field theory. This theory describes theevolution with time of the electron phase at the pointwhere the impurity is located. It is completely equivalentto the original Luttinger model with one impurity. Forthe sake of simplicity we will consider here only electronswithout 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 conditionfor such behavior of conductance is that the Fouriertransform of the potential V ( k ) has a non-vanishinglimit at k → . The second theorem is a general exactproperty of the theory which one can call duality . It statesthat the effective reflection coefficient |R ω | in a theorywith an attractive potential is equivalent to the effectivetransition coefficient |K ω | in a theory with repulsionif one exchanges K ↔ R (for a precise formulationof duality transformation of potential, see below). Thetraces of this property were seen in the perturbationtheory in [5] where duality transformation reduces to v c → v − c ( v c is the renormalized Fermi speed). However,this statement is far more general. It means that it isenough to consider, say, only repulsive potentials.For a repulsive potential conductance restores ifpotential vanishes at k → . Such a situation takes placein the systems with a small density of carriers when thescreening radius is large. In this case e − e interaction isnot point-like from one-dimensional point of view, andit is screened by the image charges on 3-dimensionalgates, edges of the channel, etc. Renormalization of theballistic conductance in such system is finite and will becalculated below. Pay attention that the form and valueof conductance is determined not by small k but by thewhole region V ( k ) where the potential is not small. Soone 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 theconductance in the Luttinger model with an impurityis 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 anomalousbreakdown of chiral symmetry. The strength of theinteraction in the repulsive Luttinger model is smaller:the system is in the Berezinskii-Kosterlitz-Thouless(BKT) phase[10]. Chiral condensate (consisting of pairsof R electron and L hole with finite density) arises onlyin the limit of an infinitely large interaction. In the caseof an attractive potential there is a charged condensateof Cooper pairs with vanishing density (for the channelwith infinite length), i.e. one has a BKT phase aswell. The Bose-Einstein principle implies that the chiralcondensate increases the probability of reflection, i.e. theeffective reflection coefficient |R ω | at small frequencies,while the charged condensate increases the probabilityof transition. As a result, |R ω | = 1 for repulsion and |K ω | = 1 for attraction. As we mentioned above, thisdoes not happen if V ( k →
0) = 0 . We will see thatin this case the long-range order in the Luttinger modelalso disappears. This leads to finite conductance of thechannel.
EFFECTIVE TRANSITION/REFLECTIONCOEFFICIENTS AND CONDUCTANCE
The Fermi surface in one dimension reduces to twoisolated points ± p F . The electrons with momenta closeto the Fermi surface can be divided in right (R) and (L)movers, Ψ = e ip F x − iε F t ψ R + e − ip F x − iε F t ψ L , where ψ R,L are slowly varying on the scale /p F .By means of the Hubbard [9] trick, the Luttingermodel can be reduced to a system of noninteractingelectrons in a random external field U ( x, t ) with asimple Gaussian weight and subsequent integration inall possible realizations of the field. The Schr¨odingerequation for the non-interaction R,L electrons in theexternal field reduces to the Dirac equation in d = 1 (inour units ~ = v F = 1 ; we will also omit electron charge e to restore it in the final expression for conductivity). [ i∂ t ± i∂ x − U ] ψ R,L = 0 . (1)The Luttinger liquid is a system which can be solvedexactly. The ultimate reason for this is that a one-dimensional fermion Green function in the external field can be found G R,L ( x, x ′ ) = G (0) R,L ( x − x ′ ) e iγ R,L ( x ) − iγ R,L ( x ′ ) γ R,L ( x ) = − Z d x ′ G (0) R,L ( x, x ′ ) U ( x ′ ) , (2)The Green functions G R,L only by phase differs from thefree Green functions G (0) R,L ( x, t ) = 12 πi ( t ∓ x − iδt ) (3)( δ > is infinitesimal.)A point-like impurity located at x = 0 mixes left andright electrons. Impurity plays the role of a boundarycondition, solutions of eq. (1) should be matched at x =0 . Nevertheless, the general solution in the external fieldcan be found [8]. Solution depends on a new functionalvariable α ( t ) which is the difference of phases for R - and L -electrons at the point of impurity α ( t ) = γ R (0 , t ) − γ L (0 , t ) . (4)Construction of the Green function with an impurity isimpeded by the Feynman boundary conditions which leadto some integral equation. This equation can be solvedperturbatively either in bare reflection or in transitioncoefficient (for details see [8]).The Luttinger model has high symmetry: it is invariantboth under gauge (vector) and chiral transformations(the latter symmetry is broken by the anomaly). Thecharge density ( ρ = ρ R + ρ L ) and current ( j = ρ R − ρ L )can be completely determined from the conservation ofthe vector and axial currents: ∂ t ρ + ∂ x j = 0 , ∂ t j + ∂ x ρ = − π ∂ x U + D ( t ) δ ( x ) . (5)Here the first term on the right-hand side is the Adleranomaly [11]. The second term describes the influence ofthe impurity, and D is the charge jump at x = 0 whichdepends only on phase α ( t ) . It can be calculated if theGreen function is known.Integrating in fermion degrees of freedom allows topresent any quantity as a product of Green functionsin the external field and fermion determinant describingthe sum of the loop diagrams. As it was shown in [8]the effect of impurity is completely determined by thephase α ( t ) : non-trivial part of the Green functions anddeterminant depends only on α ( t ) . Introducing α as anew variable one can integrate also in U ( x, t ) and reduceoriginal -dimensional model with impurity to theeffective field theory (non-local quantum mechanicsof the phase α ( t ) ).The conductance of the channel C ( ω ) is related to theexact transition coefficient C ( ω ) = e |K ω | πv r ( ω ) . (6)Here v r ( ω ) is the renormalized speed of an electron v r ( ω ) = r V ( ω ) π , v c = v r (0) , (7) |R ω | πδ ( ω − ω ′ ) = iπ | ω | W ( ω ) v r ( ω ) hh α ( − ω ′ ) D ( ω ) ii . (8)Here the average is understood as an integral with theeffective action: hh . . . ii = 1 Z Z Dα . . . D et imp exp (cid:20) − Z dω π α ( − ω ) α ( ω )2 W ( ω ) (cid:21) , (9)and W ( ω ) is W ( ω ) = − Z dk πi k V ( k )( ω − k + iδ )( ω − v r ( k ) k + iδ ) . (10)The "kinetic energy"in Eq.(9) is a well-knowncontribution of the Adler anomaly [4] to the effectiveaction rewritten in terms of phase α ( t ) (see [8] fordetails).At last, D et imp takes care about the loop diagramsdescribing the multiple rescattering on the impurity. Itcan be expressed in terms of the charge jump D log D et imp = − i Z dλ Z dω π α ( − ω ) D [ λα ]( ω ) , D ( ω ) = 2 i δδα ( − ω ) log D et imp [ α ] . (11)Expression (8) is only one of the possiblerepresentations for the effective reflection coefficient.Another useful representation |R ω | = 2 π | ω | W ( ω ) v r ( ω ) [ g ( ω ) − g ( ω )] , (12)relates R ω to the Green function of the electron phase g ( τ − τ ′ ) = hh α ( τ ) α ( τ ′ ) ii ( g is a Green function withoutimpurity determinant). Expression (12) can be obtainedfrom eq. (8) taking functional integral by parts, it is oneof the Ward identities in the effective theory.The determinant D et imp can be built as a series in barereflection coefficient log D et imp [ α ] = ∞ X n =1 ( − n +1 n (cid:18) | R || K | (cid:19) n T n − [ α ] , (13)where T n = Z dτ . . . dτ n (2 πi ) n +1 − cos[ α ( τ ) − α ( τ ) + . . . α ( τ n )]( τ − τ − iδ ) . . . ( τ n − τ − iδ ) . (14)In fact, we derived in [8] expression not for D et imp but forthe charge jump D ( ω ) which is a variational derivativeof the determinant in α according to eq. (11). Formulae (8)-(14) allow to calculate conductance forattractive e-e interaction as in this case W ( ω ) is positive.For the repulsive interaction one should use a differentform of D et imp . As it was proved in [8] eq. (13) can bealso presented in a dual form as a series in the inverseparameter log D et imp [ α ] = ∞ X n =1 ( − n +1 n (cid:18) | K || R | (cid:19) n T n − [ e α ] ++ Z dω π | ω | π e α ( − ω ) e α ( ω ) , (15)where e α ( ω ) = sign( ω ) α ( ω ) . It is natural to call the firstterm a dual determinant and combine the second onewith the "kinetic energy"eq. (10). Introducing e α as a newvariable we obtain a dual theory with the kinetic energy f W − ( ω ) = − W − ( ω ) − | ω | π . (16)Effective coefficients |R ω | and |K ω | can be writteneither as functional integrals eq. (9) in the original theoryor as the functional integrals in the dual theory after thetransformation: | R | ↔ | K | , |R ω | ↔ |K ω | , W ↔ f W . (17)
CONDUCTANCE REEMERGENCE
It is well-known [5] that the coefficient |R ω | → foran attractive potential and |K ω | → for a repulsive onewhen ω → . This happens only if V ( k = 0) = 0 dueto the infrared divergency. We reproduced this result forour effective 0+1 dimensional theory in [8].An example of a different behavior (for a repulsivepotential with V ( k = 0) = 0 ) is given by a system ofone-dimensional electrons with small concentration. Wewill see that conductance remains finite in this case.Indeed, if the concentration is small then the 3Dscreening radius can be much larger than the width d of the channel (which is considered to be zero in our one-dimensional theory). The bare e − e interaction is screenedby the "image charges" that arise on the split gates. Thescreened interaction is Coulomb one at distances smallerthan the distance to the gate ( l ) and it is dipole-dipolein the opposite limit [12]: V ( k ) = − ζ (cid:26) log | k | d, kl ≫ kl ) log | k | d, kl ≪ . (18)Here ζ ≡ /πa B p F ≫ , a B is the Bohr radius. Weassume that d ≪ l ≪ L where L is the length of theone-dimensional channel and ζ is a largest parameterof the problem. Moreover, we will consider the casewhen transition coefficient is small: | K | ≪ . Thenone can leave only first term in the expansion (15) andcalculate the charge jump D according to eq. (11). TakingGaussian integral in α we arrive at |K ω | = | K | π Z dτ − cos ωτ | ω | τ e − σ ( τ ) , (19)where σ ( τ ) = Z dω ′ π f W ( ω ′ ) (1 − cos ω ′ τ ) . (20)Behavior of transition coefficient at small ω is related tothe asymptotic σ ( τ ) at large τ . The kinetic energy f W ( ω ) in eq. (16) is nonsingular for repulsive potential (18) at ω → . For this reason σ ( τ ) has a finite limit σ ∞ at τ → ∞ . This limit determines the conductance C ( ω = 0) = e | K | π e − σ ∞ , σ ∞ = Z dω π f W ( ω ) (21)The renormalized Fermi speed v c = 1 here.At l √ ζ/d ≫ the main contribution to the integral ineq. (21) comes from kl ≫ , where the potential is notscreened. The kinetic energy f W ( ω ) is determined by thepole | ω | = v r ( k ) | k | ≈ | k | p ζ log (1 /k d ) in eq. (10). Itis equal to f W ( ω ′ ) = 2 π | ω ′ | q ζ log ( p ζ/ | ω ′ | d ) (22)at ω ′ l ≥ . Finally σ ∞ ≈ p ζ (cid:18) log p ζ ld (cid:19) ≫ (23)Hence, conductance of channel in this limit is small.Let us note that conductance is determined not by the k → but by k ∼ /l where interaction of electrons isimportant.Consider now also the opposite limit σ ∞ ≪ while K is not necessarily small. It can be implemented atintermediate concentrations if a 3-dimensional screeningradius is of the order of the channel thickness. In thiscase we are dealing with a weak interaction and theconductance is determined by an expansion in powersof α T n − = n Z dω (2 π ) | ω | α ( ω ) α ( − ω ) − n Z dω . . . dω (2 π ) ×× δ X i =1 ω i ! Γ ( ω i ) α ( ω ) . . . α ( ω ) + . . . , (24)where the vertex Γ is Γ ( ω i ) = X i | ω i | − X i
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