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A Note on Bosonization of 1 Dimensional Electron Gas 11 A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS SHUCHEN ZHU Abstract. We study the original development of bosonization chronologically, and in- vestigate in the one dimensional electron gas - Luttinger liquid. Many important results are reproduced with help from various sources. 1. A brief introduction of the historical development More than two hundred years ago, a French mathematician Augustin-Louis Cauchy found the identity which named after him: Q 1 i<j(zi − zj)(wi − wj) (1) det = Q : zi − wj i;j(zi − wj) In the 1970s it was interpreted in the context of two-dimensional (2D) quantum field i y theory. Suppose we have a chiral fermionic Lagrangian, L = 2π R@z R, where @z = 1 2 (@x + @t). Then, as it follows from Wick's theorem, the l.h.s. of eq.(1) coincides with y y 2n-point correlation functions h R(z1) ··· R(zN ) R(w1) ··· R(wN )i : At the same time its r.h.s. can be understood as the correlation function of the expo- nential fields built from the chiral boson heiφR(z1) ··· eiφR(zN )e−φR(w1) ··· e−iφR(wN )i. iφ (z) In fact, Cauchy's identity suggests the remarkable relation R(z) = e R between the Fermi and Bose fields. All the above can be repeated to the left movers and formulated as an equivalence between non-chiral free Fermi and Bose theories described by the Lagrangians1 ¯ = 1 2 (2) LF = iΨ@Ψ ∼ LB = 2 (@µΦ) : In the seminal work [1] in 1975, Coleman extended this equivalence and \bosonized" the interacting theory: (g;M) ¯ = g ¯ µ ¯ ¯ (3) LF = Ψi@Ψ − 2 Ψγ ΨΨγµΨ + MΨΨ; by means of the so-called sine-Gordon model 1 2 LSG = 2 (@µΦ) + cos Φ(4) Among others the bosonization implies that U(1) current in the fermionic theory coincides with the topological current in the sine-Gordon model: 1 (5) Ψ¯ γµΨ = − µν@ Φ : 2π ν p 1Here Ψ = p1 ( ; )T , whereas Φ = 2 π(φ + φ ). 2π R L R L 1 2 SHUCHEN ZHU The above identification between the boson operator and fermion bilinear indicates that boson is a bound state of fermion-antifermion pair. It starts from the comparison of the correlation functions of fermionic fields and exponential of bosonic fields. He found them are equal if the coupling constants satisfy certain condition. This leads to the identity of currents in (5). In the same year of 1975, explicit formulae for single fermion field were obtained by Mandelstam [2] : 2πi R x dζ φ_(ζ)∓ 1 iφ(x) (6) 1;2 = C1;2 e −∞ 2 : Historically, there is a parallel development of bosonization in condensed matter physics. In the 1950s Tomonaga [5] first identified the boson-behaved density operator for interact- ing fermions, his analysis was based on the manipulation of the boson-behaved fermionic bilinear operator y . Later the bosonization identity (x) ∼ eiφ(x) was used by various studies in 1D electron systems. Luttinger model is a model which describes interacting electrons in 1D, and it is the massless version of eqn (3). The motivation to study such system theoretically is that it actually describes systems in the reality. Although our world is 3D, there exists systems that can be described effectively by 1D models, such as quantum wires. Bosonization tech- nique can be introduced to study such model with interaction at low energy excitation ( ∼ F ), where linear dispersion relation can be assumed. Recently it appears that nonlin- ear Luttinger liquid is under investigation. The most interesting phenomenon of Luttinger liquid is spin charge separation, where spin and charge density are treated as separated particles propagating independent of each other. In this note, we will first discuss the original discovery of Bosonization identity by Coleman. Then we will bosonize the xzz spin chain for the non interactive case in continuum limit. Finally we will go through constructive bosonization that is frequently used in condensed matter physics. In the last section, we will briefly motion some experimental phenomenon of Luttinger liquid. 2. Coleman's identity In this section we derive explicitly the n point correlation function of the free boson field, and make comparison to the n point correlation function of the free fermion field. The identities in Coleman's paper [1] follow directly from the comparison of the correlation functions. 2.1. 2 point correlation function. The euclidean action for free boson is, by Francesco (2.96) [10]: 1 Z S = d2x @ φ∂µφ + m2φ2 2 µ A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 3 1 As worked out in Srednicki [11] chapter 8, the propagator is k2+m2 (in Euclidean space). And in 2 dimension we can integrate it out: Z d2k eikx Z k dkdφ eijkjjxj cos φ ∆(x) = = (2π)2 k2 + m2 (2π)2 k2 + m2 1 Z kJ (jkjjxj) K (mjxj) = dk 0 = 0 2π k2 + m2 2π R 2π ix cos φ R 1 xJ0(x) Here we used the identity for the Bessel functions: 0 dφe = 2πJ0(x) and 0 dx x2+m2 = 1 K0(m). Expand around x = 0, K0(mjxj) = −γ +ln 2−ln mx. So that ∆(x) = − 2π ln cmx. 2.2. N point corollation function. 2.2.1. Normal ordering method. From Coleman (4.2), he first considered a correlation func- tion Y iβiφ(xi) T < 0; µj Nme j0; µ > i 1 2 2 Knowing the form of free propagator of boson field is ∆F = − 4π ln cµ x . He concluded iβφ µ2 β2=8π iβφ that with Nme replaced by ( m2 ) e through the identity (2.11), we could have the result of the correlation function: 2 µ (P β2=8π) Y 2 2 β β =4π ( ) i [cµ (x − x ) ] i j m2 i j i>j Ignore the constant in front of the exponential operators, we want to show the result of Xi Q tiXi < 0jΠi : e : j0 >, especially < 0j i : e : j0 >. The colon is used to indicate normal ordering. Let us begin with: : xX1 : ··· : eXn :(7) + − where Xi = Xi + Xi , plus and minus Xi are associated with creating and annilation operators, repectively. Using the Weyl's identity in the third equality: X X++X− X+ X− 1 [X+;X−] X++X− 1 [X+;X−] X : e :=: e := e e = e 2 e = e 2 e Then 7 becomes: Y − 1 [X−;X+] X X (8) e 2 i i e 1 ··· e n i 4 SHUCHEN ZHU Applying the general Weyl's identity on 8, we have: Y − 1 [X−;X+] Y 1 [X ;X ] X +···+X (9) e 2 i i e 2 i j e 1 n i i<j Y − 1 [X−;X+] X Y 1 [X ;X ] (X++···+X+)+(X−+···+X−) = e 2 i i e 1 e 2 i j e 1 n 1 n i i<j Y − 1 [X−;X+] X Y 1 [X ;X ] (X++···+X+)+(X−+···+X−) = e 2 i i e 1 e 2 i j e 1 n 1 n i i<j Y − 1 [X−;X+] X Y 1 [X ;X ] 1 [X−+···+X−;X++···+X+] X +···+X = e 2 i i e 1 e 2 i j e 2 1 n 1 n : e 1 n : i i<j Y [X−;X+] = e i j : eX1+···+Xn : i<j Therefore: < 0j : xt1X1 : ··· : etnXn : j0 > Y t t [X−;X+] = < 0j e i j i j : et1X1+···+tnXn : j0 > i<j Y t t [X−;X+] = e i j i j < 0j : et1X1+···+tnXn : j0 > i<j Y t t [X−;X+] = e i j i j i<j Y (10) = etitj <0jXi;Xj j0> i<j Then Coleman (4.5) [1] follows immediately, and (4.11) as a more specific case can be checked using (4.5) with small n. 2.2.2. Path integral method. When making calculation, we can add a source Jφ in the Lagrangian, and take J ! 0 after the calculation. The path integral can be written, as shown in Srednicki chapter 7 and 9, and also Francesco (2.107): Z − 1 R d2x[@µφ∂ φ+m2φ2−J(x)φ(x)] 1 R d2x d2y J(x)∆(x−y)J(y) Dφ e 2 µ = e 2 Therefore the n point exponential corollation function can be written as: Y iβ φ(x ) Y iβ δ 1 R d2x d2y J(x)∆(x−y)J(y) T < 0j e i i j0 >= e i i e 2 i i A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 5 2 Where δiJ(x) = δ (xi − x). Taking J ! 0, we immediately get P − 1 ∆(x −x )β β e i6=j 2 i j i j β β P i j ln cm2(x −x )2 =e i6=j 8π i j β β P i j ln cm2(x −x )2 =e i<j 4π i j β β Y 2 2 i j = [cm (xi − xj) ] 4π i<j This is in agreement with Coleman (4.5). Now we can specialize our computation to get the result of Coleman (4.11). Y T h eiβ(φ(xi)−φ(yi))i i Y iβ(δ −δ ) 1 R d2xd2yJ(x)∆(x−y)J(y) = e xi yi e 2 i P − 1 β2∆(x −x ) P − 1 β2∆(y −y ) P 1 β2∆(x −y ) =e i6=j 2 i j e i6=j 2 i j e i;j 2 i j 2 2 Q 2 2 β Q 2 2 β [(xi − xj) m c] 4π [(yi − yj) m c] 4π (11) = i<j i<j Q β2 i;j[(xi − yi)m] 4π This is in agreement with Coleman's result (4.11).
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