Transport properties of a Luttinger liquid with a cluster of impurities
TTransport properties of a Luttinger liquid with a cluster of impurities
Joy Prakash Das and Girish S. Setlur ∗ Department of PhysicsIndian Institute of Technology GuwahatiGuwahati, Assam 781039, India
In this work, the correlation functions of a Luttinger liquid with a cluster of impurities aroundan origin obtained using the Non chiral bosonization technique (NCBT) are used to study twoimportant physical phenomena, viz., conductance and resonant tunneling. The latter is studiedwhen the cluster consists of two impurities separated by a distance (measured in units of the Fermiwavelength). Conductance is studied both in the Kubo formalism, which relates it to current-currentcorrelations (four-point functions), as well as the outcome of a tunneling phenomena (two-pointfunctions). In both the cases, closed analytical expressions for conductance are calculated and anumber of interesting physical observations are discussed, besides presenting a favorable comparisonwith the existing literature.
I. INTRODUCTION
One of the most important physical phenomena stud-ied in condensed matter systems is the transport of elec-trons, especially when they are restricted to move in onedimension. This is because of the unique nature of theinter-particle interactions in one dimension which leadsto interesting physics which is substantially different fromthat of the higher dimensions where interactions are tack-led conveniently using the Fermi liquid theory. Secondlythe emergence of advanced technologies has made the re-alization of one dimensional systems possible that haveunusual properties and hold a promising future - carbonnanotubes [1], semiconducting quantum wire [2, 3] andso on. The suitable alternative to the Fermi liquid theoryto capture the many body physics of such 1D systems isthe Luttinger liquid theory [4] which has served as theparadigm for one dimensional systems and is based onlinearization of the dispersion relations of the constituentparticles near the Fermi level.Most of the physical phenomena of such systems can besystematically studied provided one has analytical formsof the correlation functions - to obtain these is the statedgoal in quantum many body physics. In one dimension,this goal is achieved using bosonization methods where afermion field operator is expressed as the exponential ofa bosonic field [5]. This operator approach to bosoniza-tion, which goes under the name g-ology [6], can be usedsuccessfully to compute the N-point Green functions ofa clean Luttinger liquid. But the Fermi-Bose correspon-dence used in the g-ology methods is insufficient to tackleimpurities and to circumvent this, other techniques likerenormalization group (RG) methods are mandatory [7].A novel technique by the name of ‘Non chiral bosoniza-tion technique’ has been developed that uses a basis dif-ferent from the plane wave basis to deal strongly inhomo-geneous Luttinger liquid, without adhering to RG meth-ods [8]. NCBT can extract the most singular part of ∗ [email protected] the correlation functions of a Luttinger liquid with arbi-trary strength of the external impurities as well as thatof mutual interactions between the particles. It has alsobeen applied successfully to study the one step fermionicladder (two 1D wires placed parallel and close to eachother with hopping between a pair of opposing points) [9]and slowly moving heavy impurities in a Luttinger liquid[10]. The Green functions enables one to predict differ-ent physical phenomena occurring in the system such asFriedel oscillations [11], conductance [12, 13], Kondo ef-fect [14, 15], resonant tunneling [16, 17], etc.In the seminal work by Kane and Fisher [18], it hasbeen shown how impurities can bring drastic effects tothe conductance of the particles which can be as severe as‘cutting the chain’ by even a small scatterer. Since thenthe study of transport phenomena in a Luttinger liquidwith impurities has interested a number of researchers[19–22]. The conductance of a narrow quantum wire withnon-interacting electrons moving ballistically is given by e /h . This conductance is renormalized for a Luttingerliquid and is given by ge /h, where g is the Luttinger liq-uid parameter which depends on the mutual interactionstrength of the particles [18, 20, 23]. But no renormal-ization of the universal conductance is required if theelectrons have a free behavior in the source and drainreservoirs [22, 24]. Matveev et al. used a simple renor-malization group method to calculate the conductanceof a weakly interacting electron gas in presence of a sin-gle scatterer [7]. Ogata and Anderson [25] used Green’sfunctions to study conductivity of a Luttinger liquid andshowed that if the spin-charge separation is taken intoaccount, the resistivity has a linear temperature depen-dence. Besides conductance, resonant tunneling is yetanother important phenomena studied in Luttinger liq-uid with double barriers [16, 17, 26, 27]. Kane and Fisherstudied resonant tunneling in a single channel interactingelectron gas through a double barrier and found that thewidth of the resonance vanishes, as a power of temper-ature, in the zero-temperature limit [16, 26]. Furusakiand Nagaosa studied the same for spinless fermions andcalculated the conductance as a function of temperatureand gate voltage [17]. In another work, Furusaki studied a r X i v : . [ c ond - m a t . s t r- e l ] N ov resonant tunneling in a quantum dot weakly coupled toLuttinger liquids [28] and a few years later, this modelwas supported by experimental evidences [2].In this work, the conductance of a Luttinger liquid inpresence of a cluster of impurity is calculated both in theKubo formalism as well as the outcome of a tunneling ex-periment using the correlation functions obtained usingNCBT. All the necessary limiting cases like Launderer’sformula, conductance of a clean Luttinger liquid, half-line, etc. are all obtained. From the tunneling conduc-tance the well known concepts of ‘cutting the chain’ and‘healing the chain’ are elucidated. The condition of reso-nant tunneling for a double impurity system is obtainedand the behavior of the correlation function exponentsnear its vicinity is elucidated. II. SYSTEM DESCRIPTION
The system under study consists of a Luttinger liquidwith short ranged mutual interactions amongst the parti-cles and a cluster of impurities centered around an origin.The Hamiltonian of the system is given as follows. H = (cid:90) ∞−∞ dx ψ † ( x ) (cid:18) − m ∂ x + V ( x ) (cid:19) ψ ( x )+ 12 (cid:90) ∞−∞ dx (cid:90) ∞−∞ dx (cid:48) v ( x − x (cid:48) ) ρ ( x ) ρ ( x (cid:48) ) (1) The first term is the kinetic term followed by the poten-tial energy term which represents the impurity clusterwhich is modeled as a finite sequence of barriers and wellsaround a fixed point. The potential cluster can be as sim-ple as one delta impurity V δ ( x ) or two delta impuritiesplaced close to each other V ( δ ( x + a ) + δ ( x − a )), finitebarrier/well ± V θ ( x + a ) θ ( a − x ) and so on, where θ ( x )is the Heaviside step function. The RPA (random phaseapproximation) is imposed on the system, without whichthe calculation of the analytical expressions of the cor-relation functions is formidable. In this limit, the Fermimomentum and the mass of the fermion are allowed di-verge in such a way that their ratio, viz., the Fermi veloc-ity is finite (i.e. k F , m → ∞ but k F /m = v F < ∞ ). Un-der the choice of units: (cid:126) = 1, k F is both the Fermi mo-mentum as well as a wavenumber [29]. The RPA limit lin-earizes the energy momentum dispersion near the Fermisurface ( E = E F + pv F instead of E = p / (2 m )). It is alsoimperative to define how the width of the impurity clus-ter ‘2a’ scales in the RPA limit and the assertion is that2 ak F < ∞ as k F → ∞ . On the other hand, the heightsand depths of the various barriers/wells are assumed tobe in fixed ratios with the Fermi energy E F = mv F even as m → ∞ with v F < ∞ .In case of the different potentials consisting the cluster,the only quantities that will be used in the calculation ofthe Green functions is the reflection (R) and transmis-sion (T) amplitudes which can be easily calculated usingelementary quantum mechanics and are provided in anearlier work [8]. For instance, in the case of a single delta potential: V δ ( x ), T = 1 (cid:16) V ivF (cid:17) ; R = − iV v F (cid:16) V ivF (cid:17) (2) In the case of a double delta potential separated by adistance 2a between them : V ( δ ( x + a ) + δ ( x − a )), T = 1 (cid:16) V ivF (cid:17) − (cid:16) iV vF e i kF a (cid:17) R = − i V v F sin [2 k F a ] + iV vF cos [2 k F a ] (cid:16) V ivF (cid:17) − (cid:16) iV vF e i kF a (cid:17) (3) In this work the generalized notion of R and T is usedin this work to signify the reflection and transmissionamplitudes of the cluster of impurities in consideration.The third term in equation (1) represents the forwardscattering mutual interaction term such that v ( x − x (cid:48) ) = 1 L (cid:88) q v q e − iq ( x − x (cid:48) ) where v q = 0 if | q | > Λ for some fixed bandwidthΛ (cid:28) k F and v q = v is a constant, otherwise. III. NON CHIRAL BOSONIZATION AND TWOPOINT FUNCTIONS
As in conventional bosonization schemes using the op-erator approach [6], the fermionic field operator is ex-pressed in terms of currents and densities. But in NCBTthe field operator is modified to include the effect of back-scattering by impurities. Hence it is suitable to studytranslationally non invariant systems like the ones con-sidered in this work. ψ ν ( x, σ, t ) ∼ C λ,ν,γ e iθ ν ( x,σ,t )+2 πiλν (cid:82) xsgn ( x ) ∞ ρ s ( − y,σ,t ) dy (4)Here θ ν is the local phase which is a function of the cur-rents and densities which is also present in the conven-tional bosonization schemes [6], ideally suited for homo-geneous systems. θ ν ( x, σ, t ) = π (cid:90) xsgn ( x ) ∞ dy (cid:18) ν ρ s ( y, σ, t ) − (cid:90) ysgn ( y ) ∞ dy (cid:48) ∂ v F t ρ s ( y (cid:48) , σ, t ) (cid:19) (5) The new addition in equation (4) is the optional term ρ s ( − y ) which ensures the necessary trivial exponents forthe single particle Green functions for a system of oth-erwise free fermions with impurities, which are obtainedusing standard Fermi algebra and they serve as a basisfor comparison for the Green functions obtained usingbosonization. The adjustable parameter is the quantity λ which can take values either 0 or 1 as per requirement.Thus NCBT operator reduces to standard bosonizationoperator used in g-ology methods by setting λ = 0. Thefactor 2 πi ensures that the fermion commutation rulesare obeyed. The quantities C λ,ν,γ are pre-factors and arefixed by comparison with the non-interacting Green func-tions obtained using Fermi algebra. The suffix ν signifiesa right mover or a left mover and takes values 1 and -1 respectively. The field operator as given in equation(4) is to be treated as a mnemonic to obtain the Greenfunctions and not as an operator identity, which avoidsthe necessity of the Klein factors that are conventionallyused to conserve the number as the correlation functions,unlike the field operators, are number conserving. Thefield operator (annihilation) is clubbed together with an-other such field operator (creation) to obtain the noninteracting two point functions after fixing the C’s and λ ’s. Finally the densities ρ ’s in the RHS of equation (4)are replaced by their interacting versions to obtain themany body Green functions, the details being describedin an earlier work [8]. The two point functions obtainedusing NCBT are given in Appendix A. IV. CONDUCTANCEA. Kubo conductance
The general formula for the conductance of a quantumwire (obtained from Kubo’s formula that relates it tocurrent-current correlations) without leads but with elec-trons experiencing forward scattering short-range mu-tual interactions and in the presence of a finite numberof barriers and wells clustered around an origin is ob-tained. Consider an electric field E ( x, t ) = V g L between − L < x < L and E ( x, t ) = 0 for | x | > L . Here V g is the voltage between two extreme points. Thus a d.c.situation is being considered right from the start. Thiscorresponds to a vector potential, A ( x, t ) = (cid:26) − V g L ( ct ) , − L < x < L ;0 , otherwise. (6)Here c is the speed of light. This means the averagecurrent can be written as, < j ( x, σ, t ) > = iec (cid:88) σ (cid:48) (cid:90) L/ − L/ dx (cid:48) (cid:90) t −∞ dt (cid:48) V g L ( ct (cid:48) ) < [ j ( x, σ, t ) , j ( x (cid:48) , σ (cid:48) , t (cid:48) )] > LL (7)The current current correlation can be obtained usingthe Green functions derived in the present work (seeAppendix B) to obtain the formula for conductance (inproper units) as follows, G = e h v F v h (cid:18) − v F v h | R | − ( v h − v F ) v h | R | (cid:19) (8) Here v F is the Fermi velocity, v h = (cid:113) v F + 2 v F v /π is theholon velocity and v is the strength of interaction be-tween fermions as already described in Section 2. SeeAppendix B for more details. The Kubo conductance Figure 1. Conductance as a function of the absolute valueof the reflection amplitude as well the interaction parameter( v F = 1) formula obtained in equation (8) is plotted in fig. 1as a function of the reflection coefficient and interactionstrength. It can be seen that when the reflection coef-ficient becomes unity ( | R | = 1), then the conductancevanishes irrespective of the interaction parameter. Onthe other hand, for any fixed value of | R | , the conduc-tance increases as the mutual interaction becomes moreand more attractive (negative v ) and decreases as theinteraction becomes more and more repulsive (positive v ). On the other hand for a fixed value of interactionparameter, the conductance decreases with increase inthe reflection parameter.
1. Limiting cases.
No interaction . In absence of interactions v = 0and hence v h = v F and thus from equation (8), G = e h (1 − | R | ) = e h | T | which is the Landauer’s formula for conductance. No impurity
In this case, there is no reflection andhence | R | = 0 and thus from equation (8), G = e h v F v h = e h g which the renormalized conductance of an infinite Lut-tinger liquid (with parameter g). Infinite barrier
In the case of a half line, | R | = 1 andthus from equation (8), G = 0irrespective of the value of holon velocity v h . B. Tunneling conductance
The Kubo conductance is the linear response to ex-ternal potentials and is therefore related to four-pointcorrelation functions of fermions. Alternatively, conduc-tance may also be thought of the outcome of a tunnelingexperiment [18]. Here fermions are injected from one endand collected from the other end. In this sense the con-ductance is related to the two-point function or the singleparticle Green function. Thus we expect these two no-tions to be qualitatively different from each other. Fromthis point of view, the conductance is ( | T | is the magni-tude of the transmission amplitude for free fermions plusimpurity) , G = e h | T | | v F (cid:90) ∞−∞ dt < { ψ R ( L , σ, t ) , ψ † R ( − L , σ, } > | (9)In this case the results depend on the length of the wire L and a cutoff L ω = v F k B T that may be regarded either asinverse temperature or inverse frequency (in case of a.c.conductance). The result (derived in Appendix B) is G ∼ (cid:18) LL ω (cid:19) − Q (cid:18) LL ω (cid:19) X (10)Here Q and X are obtained from equation (A.4). It is im-portant to stress that the present work has carefully de-fined tunneling conductance and it is not simply relatedto the dynamical density of states of either the bulk orthe half line (Appendix B). Of particular interest is theweak link limit where | R | →
1. The limiting case of theweak link are two semi-infinite wires. In this case, G weak − link ∼ (cid:18) LL ω (cid:19) ( vh + vF )2 − v F vhvF (11)Hence the d.c. conductance scales as G weak − link ∼ ( k B T ) ( vh + vF )2 − v F vhvF . This formula is consistent with theassertions of Kane and Fisher [18] that show that at lowtemperatures k B T → L , the conductancevanishes as a power law in the temperature if the in-teraction between the fermions is repulsive ( v h > v F >
0) and diverges as a power law if the interactions be-tween the fermions is attractive ( v F > v h > G weak − link − nospin ∼ ( k B T ) K − . In order to comparewith the result of the present work, this exponent has tobe halved G weak − link − with − spin ∼ ( k B T ) Kρ − . This ex-ponent is the same as the exponent of the present workso long as | v h − v F | (cid:28) v F ie. ( v h + v F ) − v F v h v F ≈ K ρ − Figure 2. Conductance exponent η as a function of the ab-solute value of the reflection amplitude | R | and the ratio β = v h v F . since K ρ = v F v h . In general, the claim of the presentwork is that the temperature dependence of the tunnel-ing d.c. conductance of a wire with no leads and in thepresence of barriers and wells and mutual interaction be-tween particles (forward scattering, infinite bandwidthie. k F (cid:29) Λ b → ∞ ) is, G ∼ ( k B T ) η ; η = 4 X − Q (12)When η > η < η = 0where the conductance is independent of temperature.This crossover from a conductance that vanishes as apower law at low temperatures to one that diverges as apower law occurs at reflection coefficient | R | = | R c | ≡ v h ( v h − v F )3 v F + v h which is valid only for repulsive interactions v h > v F . For attractive interactions, η < | R | which means the conductance always diverges as a powerlaw at low temperatures. This means attractive interac-tions heal the chain for all reflection coefficients includingin the extreme weak link case. On the other hand for re-pulsive interactions, for | R | > | R c | , η > | R | < | R c | , η <
1. Derivation of RG equation for the tunnelingconductance
In the well-cited work of Matveev et al [7], the RGequation for the tunneling conductance is derived which (a) (b) (c)
Figure 3. Anomalous exponents (L.E) vs impurity strength V for symmetric double barrier: (a) Exponents for (cid:104) ψ R ( X ) ψ † R ( X ) (cid:105) on the same side (b) Exponents for (cid:104) ψ R ( X ) ψ † L ( X ) (cid:105) on the same side (c) Exponents for (cid:104) ψ R ( X ) ψ † R ( X ) (cid:105) on opposite sides. is valid for weak mutual interaction between fermions(they consider both forward scattering as well as back-ward scattering but in the present work we consideronly forward scattering between fermions but of arbi-trary strength and sign subject to the limitation that theholon velocity be real). Both in their work and in thepresent work the transmission amplitude of free fermionscan vary continuously between zero and unity i.e. it isnot constrained in any way. Note that we have chosen aninfinite bandwidth to derive the power-law conductancein equation (12). Had we chosen a finite bandwidth whilecalculating equation (9), the resulting expressions wouldbe considerably more complicated as Matveev et al havealso found. We shall postpone a proper discussion of thisinteresting question to a later publication. For now welook at equation (8) of their paper rather than equation(12) since we are interested in the large bandwidth caseonly for now. Since G ∼ T in their notation, we mayexpand the conductance exponent 4 X − Q in powersof v the forward scattering mutual interaction betweenfermions to leading order (in the notation of Matveev etal this is V (0) and V (2 k F ) ≡ δ TT ≈ X log( ω ) ≈ R v πv F log( ω ) (13)for | v | (cid:28) v F . where R = 1 − T (in the notationof the present work this would be | R | = 1 − | T | and ω → | k − k F | d ∼ k B T . The equation (13) is pre-cisely equation (8) of Matveev et al. Thus mutuallyinteracting fermions renormalize the impurities but iso-lated impurities do not renormalize the homogenous Lut-tinger parameters such as K = v F v h . Note that our re-sults for the conductance equation (12) is the end re-sult of properly taking into account the renormalizationsto all orders in the infinite-bandwidth-forward-scatteringfermion-fermion interactions with no restriction on thebare transmission coefficient of free fermions plus impu-rity. The final answers of equation (12) involve only thebare transmission and reflection coefficients for the samereason why the zero point energy of the harmonic oscilla-tor derived properly using Hermite polynomials (ratherthan using perturbative RG around free particle, say)involves the bare spring constant (ie. (cid:126) (cid:113) km ). Inci-dentally, even the final answers of Matveev et al. such as their equation (13) involve the bare parameters onlysince this formula is the end result of taking into accountall the renormalization properly.It is hard to overstate the importance of these results.They show that it is possible to analytically interpolatebetween the weak barrier and weak link limits withoutinvolving RG techniques. It also shows that NCBT isnothing but non-perturbative RG in disguise. V. RESONANT TUNNELING ACROSS ADOUBLE BARRIER
Resonant tunneling is well-known in elementary quan-tum mechanics. Typically, this phenomenon is studied ina double-barrier system. When the Fermi wavenumberbears a special relation with the inter-barrier separationand height, the reflection coefficient becomes zero andthe Green functions of the system behave as if they arethose of a translationally invariant system. Consider asymmetric double delta-function with strength V andseparation d . Define, ξ = k F d . The resonance conditionin this case is well-known to be, V sin [ ξ ] + v F cos [ ξ ] = 0 (14)Resonant tunneling is studied for a square double barrierpotential in one dimensions by Zhi Xiao et al. [30]. Aftertaking the limiting cases of the square barriers tendingto delta potentials and imposing the RPA limit, equation(14) is obtained.The anomalous exponents of the correlation functionsgiven in Appendix A are plotted in fig. 3 in the vicinityof resonance to see the signatures of resonance tunnelingon the Luttinger liquid Green function. It may be seenthat when the system is at resonance (depicted by thevertical line), all the anomalous exponents take exactlythe same value that they take when there is no barrierat all.For an asymmetric double delta system, V ( x ) = V δ ( x + a ) + V δ ( x − a ), the anomalous exponents can becalculated using NCBT. The form of the exponents arethe same as given in Appendix A but the expression of (a) (b)Figure 4. Anomalous exponents for double barrier: The anomalous exponents (a) X and (b) A as functions of impuritystrength V and V for an asymmetric double delta potential. Near resonance (the point of intersection of the cross lines), thesystem has the same colour it has when both V and V are zero. the reflection amplitude is now different and is given by(here ξ = 2 k F a ) [8]. R = − i V V v F sin [ ξ ] + iv F ( V e iξ + V e − iξ ) (cid:16) i V + V v F + i V V v F (cid:17) + V V v F e iξ (15)For this case also resonance is achieved when both V and V becomes equal ( V = V = V ) and V obeys thesame condition in equation (14). Two of the anomalousexponents X and A (expressions given in equations (A.4)and (15)) for the asymmetric double delta system areplotted in fig. (4). The point of intersection of the crosslines is the condition for resonance and it can easily beseen that the exponent takes the same value (color) atresonance point as it otherwise takes for the no-impuritysystem ( V = V = 0). VI. CONCLUSION
The correlation functions of an inhomogeneous Lut-tinger liquid obtained using the Non chiral bosonizationare successfully used to calculate the conductance in theKubo formalism as well as in a tunneling experiment.The formulas are valid for any strength of the impuri-ties as well as that of the inter-particle interactions andvarious standard results are obtained as limiting cases ofthese formulas. The condition of resonant tunneling isalso obtained and the behavior of the correlation func-tions near resonance is described.
APPENDIX A: TWO POINT FUNCTIONSUSING NCBT
The full Green function is the sum of all the parts.The notion of weak equality is introduced which is de-noted by A [ X , X ] ∼ B [ X , X ] . This really means ∂ t Log [ A [ X , X ]] = ∂ t Log [ B [ X , X ]] assuming that Aand B do not vanish identically. In addition to this,the finite temperature versions of the formulas below canbe obtained by replacing Log [ Z ] by Log [ βv F π Sinh [ πZβv F ]]where Z ∼ ( νx − ν (cid:48) x ) − v a ( t − t ) and singular cutoffsubiquitous in this subject are suppressed in this notationfor brevity - they have to be understood to be present. Notation: X i ≡ ( x i , σ i , t i ) and τ = t − t . (cid:68) T ψ ( X ) ψ † ( X ) (cid:69) = (cid:68) T ψ R ( X ) ψ † R ( X ) (cid:69) + (cid:68) T ψ L ( X ) ψ † L ( X ) (cid:69) + (cid:68) T ψ R ( X ) ψ † L ( X ) (cid:69) + (cid:68) T ψ L ( X ) ψ † R ( X ) (cid:69) (A.1) Case I : x and x on the same side of the origin (cid:68) T ψ R ( X ) ψ † R ( X ) (cid:69) ∼ (4 x x ) γ ( x − x − v h τ ) P ( − x + x − v h τ ) Q × x + x − v h τ ) X ( − x − x − v h τ ) X ( x − x − v F τ ) . (cid:68) T ψ L ( X ) ψ † L ( X ) (cid:69) ∼ (4 x x ) γ ( x − x − v h τ ) Q ( − x + x − v h τ ) P × x + x − v h τ ) X ( − x − x − v h τ ) X ( − x + x − v F τ ) . (cid:68) T ψ R ( X ) ψ † L ( X ) (cid:69) ∼ (2 x ) γ (2 x ) γ + (2 x ) γ (2 x ) γ x − x − v h τ ) S ( − x + x − v h τ ) S × x + x − v h τ ) Y ( − x − x − v h τ ) Z ( x + x − v F τ ) . (cid:68) T ψ L ( X ) ψ † R ( X ) (cid:69) ∼ (2 x ) γ (2 x ) γ + (2 x ) γ (2 x ) γ x − x − v h τ ) S ( − x + x − v h τ ) S × x + x − v h τ ) Z ( − x − x − v h τ ) Y ( − x − x − v F τ ) . (A.2) Case II : x and x on opposite sides of the origin (cid:68) T ψ R ( X ) ψ † R ( X ) (cid:69) ∼ (2 x ) γ (2 x ) γ x − x − v h τ ) A ( − x + x − v h τ ) B × ( x + x ) − ( x + x + v F τ ) . ( x + x − v h τ ) C ( − x − x − v h τ ) D ( x − x − v F τ ) . + (2 x ) γ (2 x ) γ x − x − v h τ ) A ( − x + x − v h τ ) B × ( x + x ) − ( x + x − v F τ ) . ( x + x − v h τ ) D ( − x − x − v h τ ) C ( x − x − v F τ ) . (cid:68) T ψ L ( X ) ψ † L ( X ) (cid:69) ∼ (2 x ) γ (2 x ) γ x − x − v h τ ) B ( − x + x − v h τ ) A × ( x + x ) − ( x + x − v F τ ) . ( x + x − v h τ ) D ( − x − x − v h τ ) C ( − x + x − v F τ ) . + (2 x ) γ (2 x ) γ x − x − v h τ ) B ( − x + x − v h τ ) A × ( x + x ) − ( x + x + v F τ ) . ( x + x − v h τ ) C ( − x − x − v h τ ) D ( − x + x − v F τ ) . (cid:68) T ψ R ( X ) ψ † L ( X ) (cid:69) ∼ (cid:68) T ψ L ( X ) ψ † R ( X ) (cid:69) ∼ where Q = ( v h − v F ) v h v F ; X = | R | ( v h − v F )( v h + v F )8 v h ( v h − | R | ( v h − v F )) ; C = v h − v F v h (A.4) The other exponents can be expressed in terms of theabove exponents. P = 12 + Q ; S = QC ( 12 − C ) ; Y = 12 + X − C ; Z = X − C ; A = 12 + Q − X ; B = Q − X ; D = −
12 + C ; γ = X ; γ = − X + 2 C ; APPENDIX B: CONDUCTANCE OF AQUANTUM WIRE
In this section, the conductance of a quantum wirewith no leads is discussed first using Kubo’s formula andnext using the idea that it is the outcome of a tunnelingexperiment.
A. Kubo formalism
The electric field is E ( x, t ) = V g L between − L < x < L and E ( x, t ) = 0 for | x | > L . Here V g is the Voltage between two extreme points. Thus a d.c. situation isbeing considered right from the start. This correspondsto a vector potential ( c is the velocity of light), A ( x, t ) = (cid:26) − V g L ( ct ) , − L < x < L ;0 , otherwise.This means (since j ≈ j s , the slow part) , < j ( x, σ, t ) > = iec (cid:88) σ (cid:48) (cid:90) L/ − L/ dx (cid:48) (cid:90) t −∞ dt (cid:48) × V g L ( ct (cid:48) ) < [ j ( x, σ, t ) , j ( x (cid:48) , σ (cid:48) , t (cid:48) )] > LL (B.1)
1. Clean wire: | R | = 0 but v (cid:54) = 0 Using the Green function from equation (A.2) and set-ting | R | = 0, the current current commutation relationcan be calculated as, < [ j s ( x, σ, t ) , j s ( x (cid:48) , σ (cid:48) , t (cid:48) )] > = − v F π (cid:88) ν = ± (2 πi ) ∂ v F t (cid:48) (cid:0) δ ( x − x (cid:48) + νv h ( t − t (cid:48) )) + σσ (cid:48) δ ( x − x (cid:48) + νv F ( t − t (cid:48) )) (cid:1) (B.2) Inserting equation (B.2) into equation (B.1), the follow-ing is obtained. < j ( x, σ, t ) > = iec (cid:88) σ (cid:48) (cid:90) L − L dx (cid:48) (cid:90) t −∞ dt (cid:48) V g L ( ct (cid:48) ) (cid:16) − v F π (cid:88) ν = ± (2 πi ) × ∂ v F t (cid:48) (cid:0) δ ( x − x (cid:48) + νv h ( t − t (cid:48) )) + σσ (cid:48) δ ( x − x (cid:48) + νv F ( t − t (cid:48) )) (cid:1) (cid:17) Finally, < j ( x, σ, t ) > = − V g e (2 π ) v F v h or, I = ( − e ) < j ( x, σ, t ) > = V g e (2 π ) v F v h This gives the formula for the conductance (per spin) fora clean quantum wire with interactions, G = e π v F v h or in proper units, G = e π (cid:126) v F v h = e h v F v h A comparison with standard g-ology with the presentchosen model gives the following identifications(Eq.(2.105) of Giamarchi [6]). g , ⊥ = g , (cid:107) = 0 g , ⊥ = g , (cid:107) = g , ⊥ = g , (cid:107) = v g ρ = g , (cid:107) − g , (cid:107) − g , ⊥ = 0 − v − v = − v g σ = g , (cid:107) − g , (cid:107) + g , ⊥ = 0 − v + v = 0 g ,ρ = g , (cid:107) + g , ⊥ = 2 v g ,σ = g , (cid:107) − g , ⊥ = 0 y ρ = g ρ / ( πv F ) = − v πv F y σ = g σ / ( πv F ) = 0 y ,ρ = g ,ρ / ( πv F ) = g ,ρ / ( πv F ) = 2 v / ( πv F ) y ,σ = g ,σ / ( πv F ) = 0 u ρ = v F (cid:113) (1 + y ,ρ / − ( y ρ / = v F (cid:112) v / ( πv F ) ≡ v h K ρ = (cid:115) y ,ρ / y ρ /
21 + y ,ρ / − y ρ / (cid:115)
11 + 2 v / ( πv F ) = v F v h u σ = v F (cid:113) (1 + y ,σ / − ( y σ / = v F K σ = (cid:115) y ,σ / y σ /
21 + y ,σ / − y σ / G = e h v F v h = e h K ρ which is the standard result for a clean quantum wire.
2. The general case: | R | > and v (cid:54) = 0 Again, using the Green function from equation (A.2)for general value of | R | , the current current commutationrelation can be calculated as, < [ j s ( x, σ, t ) , j s ( x (cid:48) , σ (cid:48) , t (cid:48) )] > = − (2 πi ) v F v h π v h ∂ v h t (cid:48) (cid:88) ν = ± (cid:18) δ ( ν ( x − x (cid:48) ) + v h ( t − t (cid:48) )) − v F v h Z h δ ( ν ( | x | + | x (cid:48) | ) + v h ( t − t (cid:48) )) (cid:19) − (2 πi ) σσ (cid:48) v F π ∂ v F t (cid:48) (cid:88) ν = ± (cid:18) δ ( ν ( x − x (cid:48) ) + v F ( t − t (cid:48) )) − | R | δ ( ν ( | x | + | x (cid:48) | ) + v F ( t − t (cid:48) )) (cid:19) where, Z h = | R | (cid:18) − ( v h − v F ) v h | R | (cid:19) Thus, < j ( x, σ, t ) > = ie (cid:88) σ (cid:48) (cid:90) L/ − L/ dx (cid:48) (cid:90) t −∞ dt (cid:48) ∂ v h t (cid:48) V g L (2 πi ) v F π (cid:88) ν = ± (cid:18) θ ( − ν ( x − x (cid:48) ) − v h ( t − t (cid:48) )) − v F v h Z h θ ( − ν ( | x | + | x (cid:48) | ) − v h ( t − t (cid:48) )) (cid:19) therefore, < j ( x, σ, t ) > = 2 iev h V g (2 πi ) v F π (cid:18) − v F v h Z h (cid:19) The conductance of a quantum wire without leads but inthe presence of barriers and wells is, G = e (2 π ) v F v h (cid:18) − v F v h Z h (cid:19) Hence the general formula for the conductance of a quan-tum wire without leads but with electrons experiencingforward scattering short-range mutual interactions andin the presence of a finite number of barriers and wellsclustered around an origin is (in proper units), G = e h v F v h (cid:18) − v F v h Z h (cid:19) (B.3)The above general formula agrees with the three wellknown limiting cases.(i) when v h = v F , Landauer’s formula G = e h | T | isrecovered.(ii) when | R | = 0, the formula G = e h K ρ is alsorecovered.(iii) when | R | = 1, G = 0 regardless of what v h is. B. Conductance from a tunneling experiment
If the conduction process is envisaged as a tunnelingphenomenon as against the usual Kubo formula basedapproach which involves relating conductance to current-current correlation, a qualitatively different formula forthe conductance is obtained.First observe that the quantity | T | and K ρ both serveas a “transmission coefficient” - the former when mutualinteractions are absent but barriers and wells are presentand the latter vice versa. Both these may be relatedto spectral function of the field operator (single particlespectral function) as follows. v F (cid:90) ∞−∞ dt < { ψ ν ( x, σ, t ) , ψ † ν ( x (cid:48) , σ, } > = − (2 πi ) (cid:88) γ,γ (cid:48) = ± θ ( γx ) θ ( γ (cid:48) x (cid:48) ) g γ,γ (cid:48) ( ν, ν ) v F (cid:90) ∞−∞ dt < { ψ ν ( ν L , σ, t ) , ψ † ν ( − ν L , σ, } > = − (2 πi ) g ν, − ν ( ν, ν ) v F (cid:90) ∞−∞ dt < { ψ ν ( ν L , σ, t ) , ψ † ν ( − ν L , σ, } > = T where g γ,γ (cid:48) ( ν, ν ) are functions of the reflection (R) andthe transmission (T) amplitudes of the system and isgiven explicitly as follows. g γ ,γ ( ν , ν ) = i π (cid:104) δ ν ,ν δ γ ,γ + ( T δ ν ,ν + Rδ ν , − ν ) δ γ ,ν δ γ , − ν + ( T ∗ δ ν ,ν + R ∗ δ ν , − ν ) δ γ , − ν δ γ ,ν (cid:105) (B.4) From this point of view, the conductance is related to themagnitude of the above complex number. Choosing it tobe proportional to the magnitude of the complex number(rather than the square of the magnitude) allows perfectagreement with the RG equations of Matveev et al. [7] aswe have seen in the main text ( | T | is the magnitude of thetransmission amplitude of free fermions plus impurity): G = e h | T | | v F (cid:90) ∞−∞ dt < { ψ R ( L , σ, t ) , ψ † R ( − L , σ, } > | (B.5) Note that the above formula is not related to the squareof the dynamical density of states. The dynamical den-sity of states is equal-space and unequal time Green func-tion. For tunneling, an electron is injected at x = − L/ x (cid:48) = + L/ (cid:68) T ψ R ( L , σ, t ) ψ † R ( − L , σ, (cid:69) = i π e − log [ L − v F t ] × e − log [ L − v h t ] e − ( vh − vF )28 vhvF log L − ( vht )2 L ω Hence, (cid:68) { ψ R ( L , σ, t ) , ψ † R ( − L , σ, } (cid:69) = i π e − log [ L − v F ( t − i(cid:15) )] × e − log [ L − v h ( t − i(cid:15) )] e − ( vh − vF )28 vhvF log L − ( vh ( t − i(cid:15) ))2 L ω − i π e − log [ L − v F ( t + i(cid:15) )] × e − log [ L − v h ( t + i(cid:15) )] e − ( vh − vF )28 vhvF log L − ( vh ( t + i(cid:15) ))2 L ω while integrating over t the only regions that contributeare L − v F t ≈ L − v h t ≈
0. When v h (cid:54) = v F these two are different regions. Set L − v F t = y then L − v h t = L − v h v F ( L − y ) and L + v h t = L + v h v F ( L − y ).The implication is, integration over t is now integrationover y and this is important only when y is close to zero.Next set L − v h t = y (cid:48) then L + v h t = 2 L − y (cid:48) and L − v F t = L − v F v h ( L − y (cid:48) ) and the integrals are important only when y (cid:48) is close to zero. This means, v F (cid:90) ∞−∞ dt (cid:68) { ψ R ( L , σ, t ) , ψ † R ( − L , σ, } (cid:69) = (cid:90) ∞−∞ dy i π (cid:16) e − log [ y + v F i(cid:15) ] − e − log [ y − v F i(cid:15) ] (cid:17) e − log [ L (1 − vhvF )+ vhvF y ] e − ( vh − vF )28 vhvF log L − v hv F ( L − y )2 L ω + v F v h (cid:90) ∞−∞ dy (cid:48) i π (cid:18) e − log [ y (cid:48) + v h i(cid:15) ] − e − log [ y (cid:48) − v h i(cid:15) ] (cid:19) e − log [ L (1 − vFvh )+ vFvh y (cid:48) ] e − ( vh − vF )28 vhvF log y (cid:48) (2 L − y (cid:48) ) L ω Only the dependence on L is of interest. Write y = L s and y (cid:48) = L s (cid:48) . Hence, v F (cid:90) ∞−∞ dt (cid:68) { ψ R ( L , σ, t ) , ψ † R ( − L , σ, } (cid:69) ∼ e − ( vh − vF )28 vhvF log L L ω This means the tunneling conductance of a clean (no bar-rier) quantum wire scales as, G clean ∼ e h e − ( vh − vF )24 vhvF log LLω ∼ (cid:18) L ω L (cid:19) ( K ρ + Kρ − where L ω = v F k B T is the length scale associated with tem-perature (or frequency since k B T is interchangeable with ω ). It says that at low temperatures, the tunneling d.c.conductance of a clean quantum wire with no leads butwith interactions ( v h (cid:54) = v F ) diverges as a power law withexponent ( K ρ + K ρ − >
0. Fortuitously, the mag-nitude of this exponent matches with the exponent ofthe dynamical density of states of a clean wire (no im-purity). However when impurities (or a weak link) ispresent, there is no guarantee that this coincidence willpersist. For a clean wire there is nothing for a electron totunnel across so this exercise is pointless. What shouldbe studied is tunneling across a weak link. The gen-eral case involves including a finite number finite bar-riers and wells clustered around the origin. This case0is solved elegantly here where a closed formula for theconductance exponents may be obtained unlike in com-peting approaches found in the literature where a combi-nation of RG and other approaches are needed that fallwell short of providing a closed expression for the expo-nents.
More importantly, the present approach is able toprovide an analytical interpolation from the weak barrierlimit (see above) to the weak link limit to be discussed be-low - something the competing approaches are incapableof doing without solving complicated RG flow equations,often numerically.
In the general case with the barriers and wells, theGreen function for points on opposite sides of the originhas a form that is qualitatively different from the formwhen the points are on the same side of the origin. Thisis the really striking prediction of this work.
C. With the impurities
Consider the general Green function for xx (cid:48) < W = g , − (1 , θ ( x ) θ ( − x (cid:48) ) + g − , (1 , θ ( − x ) θ ( x (cid:48) )), < T ψ R ( L , σ, t ) ψ † R ( − L , σ, > = v F + v h √ v F v h g , − (1 , e (2 X +2 C ) log [ L ] e − log [ L − v F t ] e − log [ L − v h t ] e − ( Q − X ) log [ L − ( v h t ) ] e − C log [ − ( v h t ) ] (B.6)Since G ∼ | v F (cid:82) ∞−∞ dt < { ψ R ( L , σ, t ) , ψ † R ( − L , σ, } > | it is possible to read off the conductance exponent asfollows, G ∼ (cid:18) LL ω (cid:19) − Q (cid:18) LL ω (cid:19) X (B.7)where Q = ( v h − v F ) v h v F and X = | R | ( v h − v F )( v h + v F )8 v h ( v h −| R | ( v h − v F )) .It is easy to see that for a vanishing barrier | R | →
0, theearlier result of the conductance of a clean quantum wireis recovered. The other interesting limit is the weak linklimit where | R | →
1. The limiting case of the weak linkare two semi-infinite wires. In this case, G weak − link ∼ (cid:18) LL ω (cid:19) ( vh + vF )2 − v F vhvF (B.8)Hence the d.c. conductance scales as G weak − link ∼ ( k B T ) ( vh + vF )2 − v F vhvF . This formula is consistent with theassertions of Kane and Fisher ( C. L. Kane and MatthewP. A. Fisher Phys. Rev. Lett. , 1220 (1992) [18]) thatshow that at low temperatures k B T → L ,the conductance vanishes as a power law in the temper-ature if the interaction between the fermions is repulsive( v h > v F >
0) and diverges as a power law if the interac-tions between the fermions is attractive ( v F > v h > G weak − link − nospin ∼ ( k B T ) K − to compare withthe result of the present work this exponent has tobe halved G weak − link − with − spin ∼ ( k B T ) Kρ − . Thisexponent is the same as what we have derived since ( v h + v F ) − v F v h v F ≈ K ρ − v h ≈ v F (weak in-teractions). In general, the claim of the present work isthat the temperature dependence of the tunneling d.c.conductance of a wire with no leads in the presence ofbarriers and wells and mutual interaction between parti-cles is, G ∼ ( k B T ) η ; η = 4 X − Q When η > η < η = 0 where the conductanceis independent of temperature. This crossover from aconductance that vanishes as a power law at low tem-peratures to one that diverges as a power law occurs atreflection coefficient | R | = | R c | ≡ v h ( v h − v F )3 v F + v h which isvalid only for repulsive interactions v h > v F . For at-tractive interactions, η < | R | which meansthe conductance always diverges as a power law at lowtemperatures. This means attractive interactions healthe chain for all reflection coefficients including in theextreme weak link case. On the other hand for repul-sive interactions, for | R | > | R c | , η > | R | < | R c | , η < FUNDING