Friedel oscillations and dynamical density of states of an inhomogeneous Luttinger liquid
FFriedel oscillations and dynamical density of states of an inhomogeneous Luttingerliquid
Joy Prakash Das, Chandramouli Chowdhury, and Girish S. Setlur ∗ Department of PhysicsIndian Institute of Technology GuwahatiGuwahati, Assam 781039, India
In this work, the four-point Green functions relevant to the study of Friedel oscillations arecalculated for a Luttinger liquid with a cluster of impurities around an origin using the powerful Nonchiral bosonization technique (NCBT). The two-point functions obtained using the same methodare used to calculate the dynamical density of states (DDOS), which exhibits a power law in energyand closed analytical expressions for the DDOS exponent is calculated. These results interpolatesbetween the weak barrier and weak link cases which are typically studied in the literature. Thedependence of the DDOS on the nature of interactions and the strength of the impurity clustersare highlighted. Finally the special case of the Luttinger parameter g=1/2 is studied and comparedwith existing results.
I. INTRODUCTION
One dimensional systems hold a special position in thestudy of many body physics due to the unusual natureof the mutual interactions between the constituent par-ticles, which forbids the use of Fermi liquid theory andperturbative techniques. An alternative is invoked to de-scribe the state of 1D interacting systems, which goes bythe name ‘Luttinger liquid’ [1] which is based on the lin-earization of the dispersion relations near to the Fermilevel. The study of interactions is quantified by the com-putation of the N-point correlation functions using whicha number of physical phenomena can be described. Themost prevalent method to calculate the correlation func-tions of one dimensional systems is bosonization, whichexpresses a fermionic field as the exponential of a bosonicfield [2]. This field theoretical approach to bosonization,which goes under the name g-ology is well established [3]and can successfully compute the N-point Green func-tions of a homogeneous Luttinger liquid. But the intro-duction of an impurity leads to inadequacy of this methodand to circumvent this, other techniques like renormal-ization group (RG) methods are called for [4].More recently, a new technique has been developedwhich can extract the most singular part of the cor-relation functions of a Luttinger liquid with arbitrarystrength of the external impurities as well as that of mu-tual interactions between the particles [5]. This method,which goes by the name ‘Non chiral bosonization tech-nique’ has been applied successfully to study the one stepfermionic ladder (two 1D wires placed parallel and closeto each other with hopping between a pair of opposingpoints) [6] and slowly moving heavy impurities in a Lut-tinger liquid [7]. Once the Green function of a given sys-tem is obtained, it may be used to predict different phys-ical phenomena occurring in the system. Typical physi-cal phenomena studied in Luttinger liquids with impuri- ∗ [email protected] ties includes Friedel oscillations [8], conductance [9, 10],Kondo effect [11, 12] and so on. In the famous work byKane and Fisher [13], it has been shown how impuritiescan bring drastic effects to the conductance of the parti-cles based on their nature of interaction, viz., attractiveor repulsive. It is also well known that non trivial in-teractions between electrons and impurities leads to theoccurrence of Friedel oscillations in the charge densityprofile of Fermi liquids [14–16].In Luttinger liquids too, it is quite interesting to studythe interplay between mutual interactions and impuritiesby taking into account Friedel oscillations. Egger andGrabert have studied Friedel oscillations in a Luttingerliquid with arbitrary interactions and arbitrary strengthsof impurities [8] by combining the techniques of standardbosonization [1, 13, 17], self-consistent harmonic approx-imation [18, 19] and quantum Monte Carlo simulations.Fernandez et al. provided an alternative approach to thisproblem based on a path-integral bosonization techniquepreviously developed in the context of non local quan-tum field theories and specially suited to consider long-ranged interactions [20]. Grishin et al. studied Friedeloscillations in a Luttinger liquid using the Green func-tions obtained using functional bosonization [21]. Theyalso described the suppression of electron local densityof states (LDOS) at the position of the impurity and ob-tained an analytic expression for the LDOS at any dis-tance from the impurity. For the spinless Luttinger liquidand strong impurity, the density of states has been ex-actly calculated by Von Delft et.al. [2] for the specificcase of K ρ = 0 . K ρ is the Luttinger parameter.With the emergence of 1D materials like carbon nano-tubes [22] and quantum wires [23], the theoretical studiescan be expected to make contact with experiments.In this work, the NCBT is used, which uses a nonstandard harmonic analysis to express the fast parts ofthe density correlation functions in terms of the slowlyvarying parts which is suitable for studying the systemsunder consideration. The oscillating terms in the densitydensity correlation functions (DDCF) are then calculated a r X i v : . [ c ond - m a t . s t r- e l ] N ov as a special case of four point functions. The two-pointcorrelation functions for the same class of systems areused to calculate the DDOS, which for a Luttinger liquidexhibits a power law in energy and closed analytical ex-pressions for the exponent is obtained for the same bothfor points near to and far away from the impurity. Acomparison with existing literature is also made. Thehighlight of this work is that while these physical phe-nomena are studied using combination of different tech-niques besides bosonization, NCBT tackles these issuesin an easier way. II. SYSTEM DESCRIPTION
The system consists of a one dimensional chain of elec-trons with short ranged mutual interactions in the pres-ence of a finite number of barriers and wells clusteredaround an origin. The Hamiltonian of the system is givenas 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 two terms are the kinetic and potential en-ergy term respectively. The potential cluster can con-sist of one delta impurity V δ ( x ), two delta impuritiesplaced close to each other V ( δ ( x + a ) + δ ( x − a )), fi-nite barrier/well ± V θ ( x + a ) θ ( a − x ), etc., where θ ( x ) isthe Heaviside step function. The system is subjected tothe RPA (random phase approximation) which enablesthe calculation of the analytical expressions of the cor-relation functions. In this limit, the Fermi momentumand the mass of the fermion are allowed diverge in sucha way that their ratio is finite (i.e. k F , m → ∞ but k F /m = v F < ∞ ). Units are chosen such that (cid:126) = 1and hence k F is both the Fermi momentum as well as awavenumber [24]. The RPA limit results in linearizingthe energy momentum dispersion near the Fermi surface( E = E F + pv F instead of E = p / (2 m )). Furthermore, if‘2a’ is the width of the impurity cluster (distance betweenthe two delta potentials or width of the finite barrier) it isthen imperative to define how this width ‘2a’ scales in theRPA limit. The assertion made is that in the RPA limit,2 ak F < ∞ as k F → ∞ . In a similar way, 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 differentpotentials consisting the cluster, the only quantities thatwill be used in the calculation of the Green functions isthe reflection (R) and transmission (T) amplitudes whichcan be easily calculated using elementary quantum me-chanics and are provided in an earlier work [5]. Here thegeneralized notion of R and T is used in this work tosignify the reflection and transmission amplitudes of thecluster of impurities in consideration. The third term ineq. (1) is the forward scattering 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 fieldtheoretical approach, 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 as follows. 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 (2)Here θ ν is the local phase which is a function of the cur-rents and densities which is also present in the standardbosonization schemes [3], ideally suited for homogeneoussystems. 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) (3) The new addition in eq. (2) is the ρ s ( − y ) term whichensures the necessary trivial exponents for the singleparticle Green functions for a system of otherwise freefermions with impurities. The single particle Green func-tions of free fermions with impurities can be obtainedusing standard Fermi algebra and they serve as a basisfor comparison for the Green functions obtained usingthe bosonized version of the field operator in eq. (2).The adjustable parameter is the quantity λ which canbe either 0 or 1 as per requirement. Thus the standardbosonization scheme can be easily obtained by setting λ = 0. The factor 2 πi ensures that the fermion com-mutation rules are preserved. The quantities C λ,ν,γ arealso fixed by this comparison and they involve pre-factorswhich do not show any dynamics. The suffix ν signifies aright mover or a left mover and takes values 1 and -1 re-spectively. This field operator (annihilation) as given ineq. (2), to be treated as a mnemonic to obtain the Greenfunctions and not as an operator identity, is clubbed to-gether with another such field operator (creation) to ob-tain the non interacting two point functions after fixingthe C’s and λ ’s. Finally the densities ρ ’s are replaced bytheir interacting versions to obtain the many body Greenfunctions, the details being described in an earlier work[5]. The two point functions obtained using NCBT aregiven in Appendix A. IV. FOUR-POINT FUNCTIONS (FRIEDELOSCILLATIONS)
In the RPA sense, the density ρ ( x, σ, t ) may be “har-monically analysed” as follows. ρ ( x, σ, t ) = ρ s ( x, σ, t )+ e ik F x ρ f ( x, σ, t )+ e − ik F x ρ ∗ f ( x, σ, t )(4)Here ρ s and ρ f are the slowly varying and the rapidlyvarying parts respectively. The auto-correlation functionof the slowly varying part of the density ρ s (the averagedensity is subtracted out, so this is really the deviation)may be written down using Wick’s theorem to give rise tothe density density correlation functions of free fermionsas follows. (cid:104) T ρ s ( x , σ , t ) ρ s ( x , σ , t ) (cid:105) = − δ σ ,σ π (cid:88) ν = ± (cid:18) θ ( x x )[ ν ( x − x ) − v F ( t − t )] + | R | θ ( x x )[ ν ( x + x ) − v F ( t − t )] + (1 − | R | ) θ ( − x x )[ ν ( x − x ) − v F ( t − t )] (cid:19) (5) Here | R | is the reflection coefficient of the cluster of im-purities under consideration and θ ( x ) is the Heavisidestep function. When interactions are taken into account,due to spin charge separation, two different velocities areseen, viz., the holon velocity and the spinon velocity.Thus the density density correlation functions (DDCF)in presence of interactions is given by (cid:104) T ρ s ( x , σ , t ) ρ s ( x , σ , t ) (cid:105) = 14 ( (cid:104) T ρ h ( x , t ) ρ h ( x , t ) (cid:105) + σ σ (cid:104) T ρ n ( x , t ) ρ n ( x , t ) (cid:105) )(6) where ρ h ( x, t ) = ρ s ( x, ↑ , t )+ ρ s ( x, ↓ , t ) is the “holon” den-sity and ρ n ( x, t ) = ρ s ( x, ↑ , t ) − ρ s ( x, ↓ , t ) is the “spinon”density and (cid:104) T ρ a ( x , t ) ρ a ( x , t ) (cid:105) = v F π v a (cid:88) ν = ± (cid:18) − x − x + νv a ( t − t )) − | R | (cid:18) − δ a,h ( v h − v F ) v h | R | (cid:19) v F v a sgn( x )sgn( x )( | x | + | x | + νv a ( t − t )) (cid:19) (7) where a = n (spinon) or h (holon). Here the spinonvelocity is non-different from the Fermi velocity since it isthe total density that couples to the short-range potential( v n = v F ). On the other hand, the holon velocity ismodified by interactions, v h = (cid:113) v F + 2 v F v /π where v isthe strength of interaction between fermions as alreadydescribed in Section 2.The slow part of the DDCF given by eq. (6) can beused to obtain the fast part of the DDCF which cor-responds to Friedel oscillations, which is nothing buta term which oscillates with wavenumber 2 k F such as e ik F ( x − x (cid:48) ) < T ρ f ( x, σ, t ) ρ ∗ f ( x (cid:48) , σ (cid:48) , t (cid:48) ) > . This can bedone using a non standard harmonic analysis suited tostudy inhomogeneous Luttinger liquids like the one understudy. ρ f ( x, σ, t ) ∼ e iπ (cid:82) x −∞ ( ρ s ( y,σ,t )+ λρ s ( − y,σ,t )) dy (8)The above equation is the basis of the NCBT usingwhich eq. (2) is derived. The λ is the same as that ofeq. (2) taking values 0 or 1 and setting λ = 0 yields thestandard harmonic analysis of Haldane. The value of λ is first decided by calculating the non-interacting DDCFusing eq. (8) and comparing with the same DDCF ob-tained using Fermi algebra. After that, similar to thecalculation of the two-point functions, the non interact-ing DDCF in eq. (5) is to be replaced by the interactingDDCF in eq. (6) to obtain the required four-point func-tions in presence of mutual interactions.Define ˜ ρ f ≡ ρ f − < ρ f > . The prescription for choos-ing λ i in eq. (8) as discussed in [5] leads to the unam-biguous conclusion that λ = 1 − λ (where λ and λ corresponds to the points x and x respectively) and thefast parts of the DDCF corresponding to Friedel oscilla-tions are obtained as follows., (cid:68) T ˜ ρ f ( x , σ , t )˜ ρ f ( x , σ , t ) (cid:69) ∼ ( Exp [ (cid:88) ν,ν (cid:48) = ± a = h,n Γ( ν, ν (cid:48) ; a ) Log [( νx − ν (cid:48) x ) − v a ( t − t )]] − (cid:68) T ˜ ρ f ( x , σ , t )˜ ρ ∗ f ( x , σ , t ) (cid:69) ∼ ( Exp [ − (cid:88) ν,ν (cid:48) = ± a = h,n Γ( ν, ν (cid:48) ; a ) Log [( νx − ν (cid:48) x ) − v a ( t − t )]] − One should remember that this really means the timederivative of the logarithms of both sides are equal toeach other. The values of the anomalous scaling expo-nents Γ( ν, ν (cid:48) ; a ) can be obtained from the expression be-low.Γ( ν, ν (cid:48) ; a ) = (cid:18) v F v h δ a,h + 12 δ a,n (cid:19) ( δ ν,ν (cid:48) − δ ν, − ν (cid:48) ) (10) V. DYNAMICAL DENSITY OF STATES
In this section the results for the local dynamical den-sity of states (DDOS), D x ( ω ) at location x is presented.Physically, D x ( ω ) dω is the number of quasiparticle statesper unit length with energy between (cid:126) ω and (cid:126) ( ω + dω )relative to the Fermi energy. In a 1D system of Fermi gas,the density of states is constant for ω small compared tothe Fermi energy and is given by, D ( ω ) = 1 πv F (11)For a Fermi liquid also, the density of states near theFermi energy is constant. But for a Luttinger liquid, theDDOS exhibits a power law in energy: D ( ω ) ∼ ω α where α is the density of states exponent. α depends on the for-ward scattering interaction strength and vanishes whenthis is zero. To prove this, it is important to generalizethe idea of density of states to interacting many bodysystems. The generalization is given below. D x ( ω ) = (cid:90) ∞−∞ dt π e it ( ω + E F ) (cid:104){ ψ ( x , σ, t ) , ψ † ( x , σ, }(cid:105) (12)The above eq. (12) relates the local density of statesto the single particle Green function formulas which areprovided in Appendix A. The DDOS consists of bothslowly and rapidly oscillating (Fridel oscillation) termswhich can be written in terms of the different parts ofthe Green functions as given in eq. (A.1). D x ( ω ) = (cid:90) ∞−∞ dt π e itω (cid:16) (cid:104){ ψ R ( x, σ, t ) , ψ † R ( x, σ, }(cid:105) + (cid:104){ ψ L ( x, σ, t ) , ψ † L ( x, σ, }(cid:105) (cid:17) + (cid:90) ∞−∞ dt π e itω (cid:16) e ik F x (cid:104){ ψ R ( x, σ, t ) , ψ † L ( x, σ, }(cid:105) + e − ik F x (cid:104){ ψ L ( x, σ, t ) , ψ † R ( x, σ, }(cid:105) (cid:17) In the present work we shall be content at evaluatingthe slow contributions to the local DDOS (for fermionswith spin, it is calculated for one spin projection and theanswer is then doubled). For the inhomogeneous systemsunder study (cluster of impurities), there are two inter-esting limits viz. when x is far away from the clusterof barriers and wells and also when x is near or at thelocation of barriers and wells. For this, a dimensionlessparameter proportional to the location x is defined viz. ξ = ω | x | v h and the DDOS at zero temperature is given ingeneral as follows, D ξ ( ω ) ∼ ω α ( ξ ) (13)At the Fermi level, local density of states at finite tem-perature is obtained by simply replacing ω in the aboveequation by k B T viz. the temperature. The exponent α ( ξ ) is given as follows. α ( ξ ) = (cid:40) ( v h − v F ) v F v h + | R | ( v h − v F )( v h + v F )4 v h ( v h −| R | ( v h − v F )) < ξ (cid:28) ( v h − v F ) v F v h ξ (cid:29) A. Far away from the impurity
Consider the Luttinger parameter g (also called K ρ in the literature) which in the present work is equal to v F /v h . From eq. (14), the DDOS exponent for points faraway from the impurity can be written in terms of g as α ( ξ (cid:29)
1) = ( g + g −
2) which is precisely the exponentfound in the textbooks for fermions with spin (Giamarchi[3], eq. (7.27)). From fig. 1. (b), it can be seen that theexponent α is non-negative and its value monotonicallyincreases with an increase in the strength of mutual in-teractions both for attractive and repulsive interactions.It is independent of the strength of the impurity and itvanishes when interactions are switched off giving backthe uniform density of states given by eq. (11). (a)(b)Figure 1. (a) Density of state exponents(0 < ξ (cid:28)
1) as afunction of the interaction parameter v for various values ofreflection amplitudes | R | given in parenthesis. (b) Densityof state exponents( ξ (cid:29)
1) as a function of the interactionparameter v .( v F = 1) B. Near the impurity
The exponent α (0 < ξ (cid:28)
1) is the new result which isthe DDOS exponent for points close to or at the positionof the impurity for arbitrary strength of impurities andmutual interactions. It interpolates between the weakimpurity and weak link results which are typically dis-cussed in the literature. In the case of the half line, Kaneand Fisher [13] have remarked that for spinless fermionsthe density of states is ρ end ( (cid:15) ) ∼ (cid:15) g − . For fermions withspin the exponent may be inferred as half of this [5] viz. D half − line ( ω ) ∼ ω ( g − . Setting | R | = 1 for half-line ineq. (14), α ( ξ ≡
0) = ( v h − v F )2 v F = ( g −
1) since g = v F v h .Thus the exponents are in exact agreement with that ofKane and Fisher. For repulsive interactions, v h > v F theexponent α >
0. This means the density of states nearthe impurity tends to vanish at low energies for repul-sive interactions. This is analogous to the well knownphenomena called ‘cutting the chain’ described by Kaneand Fisher [13]. But for attractive interactions v h < v F and for | R | < | R c | ≡ v F − v h v F − v h the exponent α > | R | > | R c | the ex-ponent α <
0. The exponent near the impurity becomesnegative in some regions signifying that the density ofstates diverges at low energies near the impurity i.e. theimpurity together with attractive interactions effectivelybrings back low-energy quasiparticles which ought not tobe there in a Luttinger liquid. This is analogous to thewell known phenomenon called ‘healing the chain’ [13].
C. Comparison with existing studies (g=1/2 case)
For the spinless case and strong impurity, the density ofstates has been exactly calculated by Von Delft et.al. [2]for the specific case of g = and obtained to be D ( ω ) ∼ ω . Recalculating the exponent in eq. (14) for the spinlesscase (double of this exponent) and strong impurity ( | R | =1), NCBT yields D ( ω ) ∼ ω α where α = g − g = , we have D ( ω ) ∼ ω which is in agreement with VonDelft et.al. [2], Fabrizio & Gogolin[25], Furusaki [26], etc.The main advancement of the present work is beingable to provide simple analytical expressions for expo-nents such as these that interpolate between the no-barrier and strong barrier cases. The novel technicalframework that abandons the g-ology framework in favorof non-chiral bosonization technique with non-standardharmonic analysis of the field operator enables an exacttreatment of free fermions plus impurity problem. VI. CONCLUSIONS
In this work, the non chiral bosonization techniqueis used to calculate the rapidly oscillating parts of thedensity density correlation functions, also called Friedeloscillation terms, which arises because of the presenceof a localized impurity in an otherwise homogeneousLuttinger liquid. The two-point functions obtainedusing the same technique are used to express the localdynamical density of states as a power law and a closed analytical expression of the exponent is obtained as afunction of the strength of impurities and that of mutualinteractions. The interesting limits of far away from theimpurity as well as near or at the position of impurity isalso discussed. A comparison is made with the existingliterature for the dynamical density of states in thespecial case of g=1/2 and an exact match is observed.
VII. 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 ; VIII. FUNDING
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