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

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) ψ1,2 = 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 interacting fermions, his analysis was based on the manipulation of the boson-behaved fermionic bilinear operator ψ†ψ. Later the bosonization identity ψ(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 technique 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 ei|k||x| cos φ ∆(x) = = (2π)2 k2 + m2 (2π)2 k2 + m2 1 Z kJ (|k||x|) K (m|x|) = dk 0 = 0 2π k2 + m2 2π

R 2π ix cos φ R ∞ 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(m|x|) = −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 function

Y iβiφ(xi) T < 0, µ| Nme |0, 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 < 0|Πi : e : |0 >, especially < 0| i : e : |0 >. 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

Therefore:

< 0| : xt1X1 : ··· : etnXn : |0 > Y t t [X−,X+] = < 0| e i j i j : et1X1+···+tnXn : |0 > i i i

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 < 0| e i i |0 >= 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

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

This is in agreement with Coleman’s result (4.11).

2.3. Partition function of Sine Gordon model of neutral boson gas. The partition function of the Sine-Gordon model can be written as Z R d2x − 1 (∂µφ)2+2ζ cos βφ Z = Dφ e 2

Using the result from previous section, we can calculate it:

Z ∞ Z n − 1 R d2x (∂ φ)2 X 2 2 ζ iβφ(x ) −iβφ(x ) iβφ(x ) −iβφ(x ) Z = Dφ e 2 µ d x ··· d x (e 1 + e 1 ) ··· (e n + e n ) 1 n n! n=0

Only even n could survive, after we take J → 0. Thus we can let n = 2m, where m is a non-negative integer. Since the gas is neutral, there are C(2m, m) ways of combining 6 SHUCHEN ZHU charges, we need to attach this factor in the partition function: Z ∞ Z 2m 2m − 1 R d2x (∂ φ)2 X X 2 2 ζ (2m)! Y iβq φ(x ) Z = Dφ e 2 µ d x ··· d x e i i 1 2m (2m)! (m!)2 q=±1 m=0 i=1 ∞ 2m X X Z ζ2m Y = d2x ··· d2x h eiβqiφ(xi)i 1 2m (m!)2 qi=±1 m=0 i=1 ∞ Z 2m X X 2 2 ζ P − 1 β2q q ∆(x −x ) = d x ··· d x e i6=j 2 i j i j 1 2m (m!)2 qi=±1 m=0 2.4. Massless free Dirac fermions. The massless Thirring model has Laguangian: 1 L = ψiγ¯ ∂µψ − gjµj µ 2 µ where jµ = ψγ¯ µψ. Since the massless Thirring model is soluble, we can shift the Laguan- gian by −m0σ, where σ = Zψψ¯ , and m0 is just a real parameter and has nothing to do with mass. As indicated by Coleman, the calculation of vacuum expectation values of arbitrary strings of ψ and ψ† is lengthy. The result can be obtained from Klaiber: n Q 2 2 4 (1+b/π) Y 1 i>j[(xi − xj) (yi − yj )M ] (12) hT σ (x )σ (y )i = ( )2n + i − i 2 Q [M 2(x − y )2](1+b/π) i=1 i,j i j g where b = − 1+g/π is just Klaiber’s choice of parameter. 2.5. Equivalence between the two models. Now compare 11 and 12, we can see they 2 2 1 1 β2 are equal if M = cm , −σ± = 2 A± and 1+g/π = 4π . Since the perturbation term in α 0 Laguangin is 2β2 (A+ + A−) for Sine-Gordon model and m (σ+ + σ−) for Thirring model, the two perturbation theories are the same if m0 = α/β. Therefore, we can identity the boson ﬁeld φ in the Sine-Gordon model with the bilinear ψψ¯ in the massless Thirring model as −ψψ¯ ∝ cos βφ For convenience, lets write down the Lagrangian for both theories here again: 1 α α 1 α α L = ∂ φ∂µφ + 0 cos βφ − 0 = ∂ φ∂µφ + 0 (A + A ) − 0 SG 2 µ β2 β2 2 µ 2β2 + − β2 1 1 L = iψ¯∂ψ/ − gjµj − m0Zψψ¯ = iψ¯∂ψ/ − gjµj − m0(σ + σ ) TH 2 µ 2 µ + − As shown in Coleman, not only can we identify the perturbative terms Lagrangians, we can also identify the current in Thirring model as the derivative term in Sine-Gordon model. The commutation relations g [jµ(x), σ (y)] = ∓2(1 + )−1µνD (x − y)σ (y) ± π ν ± [∂νφ(x),A±(y)] = ±βDν(x − y)A±(y)(13) A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 7

µ β µν are identical if j = − 2π ∂νφ.

3. Bosonization of massless fermion theory The first example of bosonization in field theory started from writing the Heisenberg spin chain in the form of spinless fermions by Jordan Wigner transformation [6]. Such spinless fermionic system can be mapped to the sine-Gordon theory by Luther and Peschel [7] in 1975. In the following subsections, we follow the spirit of such motivation, making the Heisenberg model a effective boson model with two subsequent transformations.

3.1. Heisenberg spin chain in fermionic form. We would like to represent the Heisen- berg spin system by a massless fermion system. Define the raising and lowering operators at each site of spin chin n, S±(n): 1 (14) S±(n) = S (n) ± iS (n),S = σ 1 2 i 2 i These operators commute on different sites for m 6= n, and they anticommute on the same site: [S+(n),S+(m)] = [S−(n),S−(m)] = [S+(n),S−(m)] = 0 {S+(n),S−(n)} = I {S+(n),S+(n)} = {S−(n),S−(n)} = 0(15) With such condition the operator creates bosonic excitations on different sites, and there cannot be two excitations on the same site. Define kink operator that rotate sites before n around the z-axis: (iπ Pn−1 S+S−) (16) K(n) = e j=1 j j Define the fermion operator (Jordan-Wigner transformation) as flippling a spin followed by creating a kink: c(n) = K(n)S−(n)(17) The inverse of Jordan-Wigner transformation is:

− −iπ Pn−1 c†(j)c(j) S (n) = e n=1 c(n)(18) These fermion operators satisfy the anticommutation relation, since S− ∗ S− = 0. It can + − be shown (try it out on up and down states) that {Sn ,Kn} = 0, [Sn ,Km] = 0, ∀n 6= m. + − One more identity that relates S3 to S and S is: 1 c†(n)c(n) = S+(n)S−(n) = + S (n) 2 3 1 c(n)c†(n) = S−(n)S+(n) = − S (n)(19) 2 3 8 SHUCHEN ZHU

Hence all the spin operators have the form in terms of raising and lowering operators. The Heisenberg chain can be written as: X x x y y z z H = J(Si Si+1 + Si Si+1) + γJ(Si Si+1) i X 1 1 (20) = J(S†S− + S−S+ ) + γJ(S+S− − )(S+ S− − ) j j+1 j j+1 j j 2 j+1 j+1 2 j Applying the inverse Jordan-Wigner transformation, the spin Hamiltonian becomes fermionic: N N J X X 1 1 H = (c†c + c† c ) + γJ (n − )(n − )(21) 2 j j+1 j+1 j j 2 j+1 2 j j=1 Setting γ to zero yields the X-Y model: N J X Z Z dk H = (c†c + c† c ) = J cos(ka)c† c , where c = eikxj c 0 2 j j+1 j+1 j k k j 2π k j=1 k This system is gapless and the ground is obtained by filling up negative energy states. 3.2. Continuum limit of spin chain. However, the spin chain is a discrete system, and we have to make it to a continuum limit so that fermionic representation becomes possible. 3.2.1. Non-interacting case. We consider: N J X H = (c† c + c† c ), (k) = J cos(ka)(22) 0 2 n n+1 n+1 n n=1 We should expect correlation functions to have a factor of eikx. Since k has to be around ikF n π ikF n n the fermi surface, e . For half filled case, |kF | = 2 ⇒ e = i . This factor varies as we move along the lattice, hence it is neccessary to remove this fluctuation: −n (23) an = i cn The new Hamiltonian becomes: N/2 J X (24) H = i{a† (a − a ) + a† (a − a )} 0 2 2s 2s+1 2s−1 2s+1 2s+2 2s s=1

Define the spinor fields φ1(n) = a2s for even n and φ2(n) = a2s+1 for odd n. Then the Hamiltonian in new spinor fields is: N/2 J X (25) H = i{φ†(2s)[φ (2s + 1) − φ (2s − 1)] + φ†(2s + 1)[φ (2s + 2) − φ (2s)]} 0 2 1 2 2 2 1 1 s=1 Since the spinor φ field is defined by operator c on discrete lattice, its anticommutation relation is discrete: † 0 (26) {φα(n), φα0 (n )} = δαα0 δnn0 A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 9

For a conanical fermion ﬁeld ψα(x) on a continuum, we expect a continous version of anticommutation relation:

† 0 0 {ψα(x), ψα0 (x )} = δαα0 δ(x − x )(27) This can be achieved by deﬁning:

1 δnn0 (28) ψα(x) = √ φα(n), δ(x − x0) = lim 2a a→0 2a By such definition, the difference of discrete fields can be written as the derivative of continuous field: √ ∂ φ (2s + 1) − φ (2s − 1) = 2a 2a ψ (x) 2 2 ∂x 2 √ ∂ φ (2s + 2) − φ (2s) = 2a 2a ψ (x)(29) 1 1 ∂x 1 P R Also, by changing the summation to integration, lima→0 s 2af(s) = dxf(x): J X H = i {ψ†(x)(2a)2∂ ψ (x) + ψ†(2a)2∂ ψ (x)} 0 2 1 x 2 2 x 1 s Z † † = iJa dx{ψ1∂xψ2 + ψ2∂xψ1} Z Z † † 0 1 ∂xψ1 † (30) = iJa dx(ψ1, ψ2) = Jai dxψ σ1∂xψ 1 0 ∂xψ2

H0 Define Hf0 := Ja , hence we find the Hamiltonian for spinor field: Z † (31) Hf0 = dxψ σ1i∂xψ

To put these into chirl fermion ﬁeld representation, where σ1 becomes diagonal. We can deﬁne: 1 ψ1(x) = √ (−R(x) + L(x)) 2 1 ψ2(x) = √ (R(x) + L(x))(32) 2 Then:

† † † † ψ1i∂xψ2 + ψ2i∂x∂1 = −(R i∂xR + L i∂xL) ⇒ Z Z Z † † ∂xL † ¯ (33) Hf0 = (L ,R )iσ3 = ψ iγ0γ1ψ = dxψiγ1∂xψ ∂xR 10 SHUCHEN ZHU

3.2.2. Interacting case. Including the interaction term of Heisenberg Hamiltonian:

N γJ X 1 H = − (a†a − a† a )2 + γJN int 2 j j j+1 j+1 4 j=1 N/2 1 X = − γJ {(φ†(2s)φ (2s) − φ†(2s + 1)φ (2s + 1))2 + (φ†(2s + 1)φ (2s + 1) − φ†(2s + 2)φ (2s + 2))2} 2 1 1 2 2 2 2 1 1 s=1 1 In the second expression of third line, the irrelevant constant 4 γJN is dropped. Taking the continuum limit: N/2 1 X H = − γJ {[ψ†ψ − ψ†ψ ]2 + [ψ†ψ − ψ†ψ ]2}(2a)2 int 2 1 1 2 2 2 2 1 1 s=1 Z † † 2 (34) = −γJa2 dx[−ψ1ψ1 + ψ2ψ2] Working in chiral basis: 0 1 L (35) ψ†ψ − ψ†ψ = R†L + L†R = (L†,R†) = ψψ¯ 2 2 1 1 1 0 R The interacting Hamiltonian in chiral basis is: Z Z ¯ 2 µ ¯ (36) Heint = −2γ dx(ψψ) = γ dxjµj , where jµ = ψγµψ It satisfies the chiral symmetry. 3.3. Bosonization. Now we complete the first step to convert the Heisenberg spin chain to an interactive fermion system. There are a lot of ways to study such system, bosonization is a good non-perturbative method. The U(1) current is jµψγ¯ µψ, in chiral basis it is † † † † j0 = L L + R R and j1 = L L − R R. If the density and current operators are normalized, we obtain a Schwinger term following the calculation from [8], i [j (x), j (x0)] = − ∂ δ(x − x0)(37) 0 1 π x However, for boson field we have such relation: [φ(x), Π(x0)] = iδ(x − x0)(38) This implies that there is a connection between fermion and boson fields, as current jµ is bilinear of ψ. 1 1 1 j (x) = √ ∂ φ, j (x) = −√ ∂ φ(x) := −√ Π(x) 0 π x 1 π t π Since both the left and right mover are conserved, there are two conserved currents. In this case it is the axial current: 5 ¯ 5 ¯ ν 5 ν (39) jµ = ψγµγ ψ = ψµνγ ψ ⇒ jµ = µνj A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 11

The above shows Coleman’s identification between fermion bilinear and single boson field. To obtain the explicit bosonization of single fermion field by Mandelstam [2], consider two operators defined as:

iαφ(x) Oα(x) = e R x1 0 0 R x1 0 0 iβ dx ∂0φ(x0,x ) iβ dx Π(x0,x ) (40) Oβ(x) = e −∞ 1 1 = e −∞ 1 1

iαφ When acting on |{φ(x)}i, Oα simply attaches a phase factor of e , whereas Oβ shifts 0 0 0 φ(x0, x1) to φ(x0, x1) + β, ∀x1 < x. Consider the operator ψαβ(x) of the form:

R x1 0 0 iαφ(x)+iβ dx ∂0φ(x0,x ) (41) ψα,β(x) = Oα(x)Oβ(x) = e −∞ 1 1

To see how this fermion operator satisfy the anticommutation relation, we multiply two of them and apply Baker-Hausduﬀ rule eAeB = eBeAe−[A,B]:

0 0 −iΦ(x,x0) 0 iαβ (42) ψα,β(x)ψα,β(x ) = ψα,β(x )ψα,β(x)e = ψα,β(x )ψα,β(x)e

Where the second and last expression come from:

0 Z x1 Z x1 0 2 0 2 0 0 iΦ(x, x ) = −α [φ(x), φ(x )] − β dy1 dy1[Π(y), Π(y )] −∞ −∞ 0 Z x1 Z x1 0 0 0 (43) − αβ dy1[φ(x), Π(y )] − αβ dy1[Π(y), φ(x )] = −iαβ −∞ −∞

π Clearly, we need αβ = ±π ⇒ α = ± β to have the correct anticommutation relation for fermion ﬁeld. Mandelstam made separate expressions for left and right fermion ﬁelds:

x0 1 −i 2π R 1 dx0 Π(x ,x0 )+i β φ(x) R(x) = √ : e β −∞ 1 0 1 2 : 2πa x0 1 −i 2π R 1 dx0 Π(x ,x0 )−i β φ(x) L(x) = √ : e β −∞ 1 0 1 2 :(44) 2πa

The U(1) current jµ [2] is:

β (45) j = ∂νφ µ 2π µν √ Compare the above current with the Coleman identity, we require that β = 4π. However, we should expect that the bosonized chiral fermion field could be represented by chiral boson field, it is true if we write boson field in mode expansion and separate it into chiral 12 SHUCHEN ZHU components: Z ∞ dk 1 φ(x) = (a(k)ei(|k|x0−kx1) + a†(k)e−i(|k|x0−kx1)) ⇒ −∞ 2π 2|k| Z ∞ dk 1 Π(x) = (i|k|a(k)ei(|k|a(k)x0−kx1) − i|k|a†(k)e−i(|k|x0−kx1)) ⇒ −∞ 2π 2|k| Z ∞ dk 1 ik(x0−x1) † −ik(x0−x1) φR(x) = φR(x0 − x1) = (a(k)e + a (k)e ) 0 2π 2k Z ∞ dk 1 −ik(x0+x1) † ik(x0+x1) (46) φL(x) = φR(x0 + x1) = (−a(k)e − a (k)e ) 0 2π 2k Introduce the duel field θ such that the field operator φ and it obey the Cauchy-Riemann equation:

(47) ∂0φ = Π := ∂1θ The ﬁelds ﬁnally decompose as chiral components as:

φ(x0, x1) = φR(x0 − x1) + φL(x0 + x1)

θ(x0, x1) = −φR(x0 − x1) + φL(x0 + x1)(48) Put the above expression into Mandelstam’s identity, the chiral fermion is identical to the exponential of the chiral boson field: 1 √ R(x) = √ : ei2 πφR : 2πa 1 √ L(x) = √ : e−i2 πφL :(49) 2πa It is shown [8] that by explicit calculation, the operator product yields: √ † 1 −i2 πφ(x) lim R (x0, x1)L(x0, y1) = : e : y1→x1 2πa √ † 1 +i2 πφ(x) lim L (x0, x1)R(x0, y1) = : e : y1→x1 2πa Now we have formulae to bosonize any fermion field that appears in Lagrangian, and then we put them together: ¯ ¯ 2 (50) LF = H0 − Hint = iψ∂ψ/ − γ(ψγµψ) 1 γ (51) = (∂ φ)2 − ∂νφµλ∂ φ 2 µ π µν λ The first term of the last expression comes from the conservation of axial current j5: 1 1 ∂ j5µ = µν∂ j = √ µν ∂ ∂λφ = √ ∂2φ µ µ ν π νλ µ π 1 (52) ∂ j5µ = 0 ⇒ √ ∂2φ = 0, ∂2 = ∂2 − ∂2 µ π 0 1 A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 13

If we rescale the boson ﬁeld and coupling constant:

r 2γ (53) ϕ := 1 + φ π The bosonized Lagrangian becomes: 1 (54) L = (∂ ϕ)2 B 2 µ We now understand that the massless fermion theory is equivalent to free boson theory. Moreover, the interacting massive fermion system is equivalent to sine-Gordon model.

4. Luttinger model Luttinger liquid is a general model of 1D electron gas at low energy regime. The shape of 1D system can be thought as a rectangle, with its length L way longer than its width 2π d. By such assumption, the momentum is dense, because ∆p ∼ L , and the gap between 1 different energy states is large: ∆E ∼ d . Thus we only need to think it is a one band system with all electrons filling up to the fermi surface. To the nature of 1D system, the excited particles’ momentum are either parallel or antiparallel to the direction of fermi energy (posivie for negative). This indicates that the excited particle and the hole it lefts behind are correlated at long distance - they form a long-lived bound state (collective mode). Therefore, one would suspect that the low energy excitation of fermions is effective theory of boson. The retarded Green’s function recognizes the probability of excitation for given set of energy and momentum. In Landau’s fermi gas theory, the spectral function of free fermion is maximized but smears out for a wide range of momentum. However it is quite different for Luttinger liquid, where the spectral function has two poles at qvf and −qvf . This tells us the excitation consists of left and right movers as bosons.

4.1. Spinless fermion. Consider the Hamiltonian:

† 2 2 H = H0 + Hint = ψ (x)(αivF ∂x)ψ(x) + 2g2ρR(x)ρL(x) + g4(ρR(x) + ρL(x)) (55) where g2 measures the strength of backscattering and g4 that of the forward scattering. At precisely half-ﬁlling, the Umklapp interaction has to be incoorperated into the Hamiltonian. This allows scattering process to be momentum conserved up to a one reciproval lattice vector: † † HUmklapp = g3 lim(ψ (x)ψ (y)ψL(x)ψL(y) + R ↔ L)(56) y→x R R The Umklapp term cannot be written as the product of density operators, hence it breaks the continuous chiral symmetry of the Hamiltonian. Using the identities derived before, 14 SHUCHEN ZHU the free fermion Hamiltonian is: v v H = ψ†(x)(αiv ∂ )ψ(x) = F (Π2 + (∂ φ)2) = F ((∂ θ)2 + (∂ φ)2) together with previous identities: 0 F x 2 x 2 x x 1 1 1 1 ρ = √ (∂ φ − π) = √ ∂ (φ − θ), ρ = √ (∂ φ + π) = √ ∂ (φ + θ) R 2 π x 2 π x L 2 π x 2 π x The new Hamiltonian without Umklapp term is: v 1 (57) H = (πv + g )(ρ2 + ρ2 ) + 2g ρ ρ = ( (∂ θ)2 + K(∂ φ)2) where F 4 R L 2 R L 2 K x x r s g4 2 g2 2 vF + g4/π + g2/π v = (vF + ) − ( ) ,K = π π vF + g4/π − g2/π The bosonized Hamiltonian is invariant with duality transformation: 1 (58) φ ↔ θ, K ↔ K

The forward scattering factor g4 renormalizes the speed and the backward scattering factor g2 renormalizes the stiﬀness constant K. Clearly, for repulsive interaction g2 > 0, K > 0; and for attractive interaction g2 < 0, K < 0. For the case of half ﬁlling that includes Umklapp scattering term, the bosonized Hamiltonian has an additional term gµ cos(4πφ) that breaks the chiral symmetry of the system.

4.2. Spin fermion. Consider the spin Hamiltonian: X X † X † † X † † H = −ivF sψs,σ∂xψs,σ + g4 ψs,σψs,−σψs,−σψs,σ + g2 ψ1,σψ−1,σ0 ψ−1,σ0 ψ1,σ σ=↑,↓ s=± σ,s σ,σ0 (59) X † † X † † + g1,k ψ1,σψ−1,σψ1,σψ−1,σ + g1,⊥ ψ1,σψ−1,−σψ1,−σψ−1,σ σ σ The ﬁrst line has the kinetic term, the forward scattering of the same branch and opposite branch. The second line includes the backscattering process. Also the Umklappp term should be considered at half ﬁlling case:

i(4pF −G)x † † (60) Hu = g3e ψ−1,↑ψ−1,↓ψ1,↓ψ1,↑ + h.c.

The H0 term can be easilly expressed as: vF X H = (Π2 + (∂ φ )2)(61) 0 2 σ x σ σ Deﬁne the charge and spin boson ﬁelds and transform the free Hamiltonian: 1 1 φc = √ (φ↑ + φ↓), φs = √ (φ↑ − φ↓) ⇒ 2 2 v v H = F (π2 + (∂ φ )2) + F (Π2 + (∂ φ )2)(62) 0 2 c x c 2 s x s A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 15

The bosonized free Hamiltonian describes the charge and spin ﬁelds moving at the same velocity. Adding interaction terms will renormalize the speed of the propagating bosonic waves: (63) √ √ vc 1 2 2 vs 1 2 2 H = ( Πc + Kc(∂xφc) ) + ( Πs + Ks(∂xφs) ) + Vc cos(2 2πφc) + Vs cos(2 2πφs) 2 Kc 2 Ks r g g g r g g v = (v + 4 )2 − ( 1,k − 2 )2, v = (v − 4 )2 − ( 1,k )2 c F 2π 2π π s F 2π 2π s s 2πvF + g4 + 2g2 − g1,k 2πvF − g4 − g1,k Kc = ,K2 = 2πvF + g4 − 2g2 + g1,k 2πvF − g4 + g1,k g g V = 3 ,V = 1,⊥ c 2(πa)2 s 2(πa)2 Clearly the ineractions modify the spin and charge velocity, thus it leads to the spin charge separation.

4.3. Scaling. Bosonization is a method that maps a system to another system, it has no use to solve the problem. We only hope the mapped system is easier to analysis so we could see the behavior of the original system that is hard to study. For the sine-Gordon model, we would like to ﬁnd the limit that the interacting cosine term is useful, in other word if it is an important term adding to the free boson Langrangian. Suppose we are interested in the correlation function of operator φn(r).If the system is at ﬁxed point such that it is translation, rotation and scale invariant; we would expect the correlation function have a power law dependence on the distance:

0 1 (64) hφn(r)φn(r )i = |r − r0|2∆n where ∆n is the scaling dimension of φn. For sine-Gordon model, we are interested in the 1 iϕ −iϕ correlation of cos(ϕ) = 2 (e + e ), the vertex operator. For sine-Gordon Langrangian K 2 u in Euclidean space of the generic form L = 2 (∂ϕ) + a2 cos ϕ, the correlation of vertex operator is:

1 2 1 2 (65) heinϕ(0)e−inϕ(x)i = ( )n /(4πK)( )n /(4πK) z z¯ n2 n2β2 Thus the scaling dimension of the vertex operator is ∆1 = 4πK = 4π . A similar analysis shows that the scaling dimension of (∂ϕ)2 is 2, which is independent of any parameter. Thus if ∆1 > 2, the interaction is irrelavant, marginal if ∆1 = 2, and relavant if ∆1 < 2. For Luttinger model, we may write it as the summation of the charge and spin sections H = Hc + Hs. Simillar to the above analysis, Kc = 1 ⇒ ∆c = 2, the Umklapp process is marginal. Kc > 1 makes it relevant, and it is associated with the repulsive interaction. For spin section, the scaling dimension behaves exactly the same with respect to Ks. 16 SHUCHEN ZHU

5. Bosonization identity from condensed matter physics We may view bosonization as a tool that can be constructed from few simple assumptions of physics system. This method of building bosonization identity is called constructive bosonization. The starting point only requires the definition of fermion field in terms of fermion creation and annihilation operators which statisfy the usual anticommutation relation: 2π X (66) ψ (x) = ( )1/2 e−ikxc , η L kη k where η represents different fermion species. The fermion operators are defined as: † (67) ckη |0i ≡ 0, ∀k > 0 ckη |0i ≡ 0, ∀k < 0 . Then the N-particle ground state can be defined from vacuum sate and fermion operators:

¯ N1 NM (68) |Ni0 ≡ C1 ··· CM |0i  c† ··· c† N > 0 Nηη 1η η Nη  (69) Cη ≡ 1 Nη = 0  c(Nη+1)η ··· c0η Nη < 0 Now define the boson operators as: ∞ ∞ i X −i X (70) b† ≡ √ c† c b ≡ √ c† c . qη n k+q η kη qη n k−q η kη q k=−∞ q k=−∞ The boson operator shifts the momentum of all the particles in N-particle ground state ¯ by q or −q. Acting b on the N-particle ground state yields 0, so that |Ni0 can be treated as a vacuum state for boson operators. Moreover, boson operators satisfy the expected commutation relation: † † † (71) [bqη, bq0η0 ] = [bqη, bq0η0 ] = 0, [Nqη, bq0η0 ] = [Nqη, bq0η0 ] = 0 , ˆ P∞ † where the number operator is defined as Nη ≡ k=−∞ : ckηckη :. The normal ordering is introduced to make the number operator counting the fermions lying above the fermion sea. The last relation tells that the boson operator does not change the number of fermions in the system, hence it is neccessary to introduce one operator that is able to change the fermion numbers. It is called Kelin factor and defined as: † † † † [bqη,Fη0 ] = [bqη,Fη0 ] = [bqη,Fη0 ] = [bqη,Fη0 ] = 0(72) (73) F † |Ni ≡ f(b†)c† |N ,...,N ,...,N i ≡ f(b†)Tˆ |N ,...,N ,...,N i η Nη+1 1 η M 0 η 1 η+1 M 0 † † ˆ (74) Fη |Ni ≡ f(b )cNη+1 |N1,...,Nη,...,NM i0 ≡ f(b )Tη |N1,...,Nη−1,...,NM i0 . † The effect of Fη is clear: it creates one more η-fermion on the lowest possible level of the ground state. From the definition above, we can deduce that the Klein factors are unitary A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 17 operator which are denoted by U in the literatures. With all the necessary operators in hand, the boson field can be defined as: X 1 X 1 (75) ϕ (x) ≡ − √ e−iqxb e−aq/2 ϕ† (x) ≡ − √ eiqxb† e−aq/2 η n qη η n qη q>0 q q>0 q † (76) φη(x) ≡ φη + φη . The boson fields can be checked (appendix A.5) that they satisfy the appropriate commutation relations: 0 † † 0 [ϕη(x), ϕη0 (x )] = [ϕη(x), ϕη0 (x )] = 0(77) † 0 2π 0 (78) [ϕ (x), ϕ (x )] = −δ 0 ln [i (x − x − ia)],L → ∞ . η η0 ηη L The fermion density operator can now be expressed in terms of boson fields: 2π (79) ρ (x) ≡: ψ†(x)ψ (x) := ∂ φ (x) + Nˆ , a → 0 . η η η x η L η This is the same form that Coleman claimed in (5). The explicit bosonization identity can be obtained by acting the fermion field on an arbitrary N-particle state, applying the available identities and we are able to find, for L → ∞:

−1/2 −iφη(x) (80) ψη(x) = Fηa e .

6. Justification and experiment of Luttinger liquid In this section we would like to give some handwaving arguments toward the analysis of Luttinger liquid.

6.1. Why 1D?. The Luttinger model is special because the dispersion relation is linear around the fermi point . Since we are only interested in the low energy excitation, we can linearize the dispersion relation as = vf (k − kf ), ∀k ∈ (−∞, ∞). It is not the case for larger dimensions, as in two dimension the fermi points turn into a continuous circle.

Figure 1. The despersion relation of 1D model can be linearized at fermi point[12]. 18 SHUCHEN ZHU

Figure 2. In 2D it is not possible to linearize the dispersion relation[12].

In order to visualize a given interacting Hamiltonian, it is better to break the fermion ﬁeld in Hamiltonian as left and right movers. The fermion ﬁeld in 1-d wire can be written as:

−ikF x ˜ ikF x ˜ (81) ψ(x) = e ψL(x) + e ψR(x) . where the tilde means we extend the range of k from (kF , ∞) to (−∞, ∞).

Figure 3. single particle spectrum dispersion relation. Occupied states are shown in grey, the dark area shows the unphysical state. From [13].

The state lies way below the fermion sea is unaﬀected by low energy excitations. The left/right movers have momentum and energy unbounded. This modiﬁcation makes the 1-D model solvable, in a sense that the Hamiltonian can be written as a free Hamiltonian, even the interactions exist. The electron interaction Hamiltonian is:

(82) H = Hkin + Hint . A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 19

Assuming linear dispersion relation, the kinetic part is (explicit in appendix B.2): ∞ X X Z L/2 dx 1 (83) H = k : c† c := : [˜ρ2 +ρ ˜2 ](x): . kin kη kη 2π 2 L R η=L,R k=−∞ −L/2 For the interaction part (B.2): 1 Z (84) H = dxdx0 ψ†(x)ψ†(x0)V (x − x0)ψ(x0)ψ(x) . int 2 ee

Since each ψ has right and left two components, the Hint should have 16 terms in total. We drop the 8 terms that do not conserve the number of right and left movers. There are 8 terms left that can be catagorized into two groups. One group is interbranch interaction where the interaction happens between the same fermion species, and the other group is intrabranch interaction where the interaction occurs between diﬀerent fermion species. We assume the strengh is diﬀerent for these two kinds of interactions, so eventually we are left with: Z L/2 1 2 2 (85) Hint = : g2ρ˜L(x)˜ρR(x) + g4[˜ρL(x) +ρ ˜R(x)]] : . −L/2 2 The full Hamiltonian can be expressed as: Z L/2 v dx 1 2 2 (86) H0 = Hkin + Hint = :[ (˜ρL +ρ ˜R) + g(˜ρL − ρ˜R) ](x): 4 −L/2 2π g q r 2 2 1 + g4 − g2 v ≡ (1 + g4) − g2 g ≡ . 1 + g4 + g2 6.2. What is bizarre? There are some strange behaviors that belong only to 1D electron gas. Tunneling is a quantity that can be measured experimentally for 1D quantum wires. It occurs between the wire and the tip of a scanning tunneling microscopy. Another example is a 1D wire connected weakly to two leads at ends. One interesting property that determines

Figure 4. two tunneling examples. From [14] the tunneling current is the local density of state, at zero temerature reads: Z ∞ dt iωt † (87) ρdos(ω) ≡ e hG| Ψ(t, x = 0)Ψ (0, x = 0) |Gi . −∞ 2π With the diagonalized free Hamiltonian and boson ﬁeld Φν ready from last section, the correlation function can be separated as product of two independent correlation functions. 20 SHUCHEN ZHU

Each one relates to one of the boson species φν. the correlation function is found to be proportional to (appendix B.3): − 1 ( 1 +g) (88) hG| Ψ(t, x = 0)Ψ†(0, x = 0) |Gi ∼ t 2 g Thus the local density of state is proportional to: 1 ρ ∼ ων−1, ν = (1/g + g) . dos 2 For g 6= 1, ν > 1, so that the tunneling density vanishes at low frequencies. The problem becomes more complex if impurity is added to the site x = 0, as it leads to forward and backward scattering. In the presence of impurity, the Hamiltonian has to include two more terms HF + FB. HF is for forward scattering, and HB for backward scattering. The full Hamiltonian can then be separated into two H− + HF and H+ + HB, which can be diagonalized using refermionization. After the refermionization, the full Hamiltonian is the sum of two independent free Hamiltonians. Up to this point, the tunneling density can be treated the same way as this section does for non impurity situtation. It is found [9] that for g = 1/2, the correlation function is proportional to t−2, so that ρ(ω) ∼ ω. The tunneling density of states of Luttinger liquid would lead to an interesting phenomenon that is known as orthogonality catastrophe, or power-law suppression of the tunneling diﬀerential conductance. The diﬀerential tunneling conductance G(V,T ) is proportional to ν−1 ν−1 ρdos. Thus for T = 0, G ∼ ω ∼ V . For ν 6= 1, we must have vanishing conductance at low bias. When the bias becomes larger compared with the temperature, the normal Ohmic behavior of condutance appears again.

Figure 5. The diﬀerential tunneling conductance as a function of bias. From [8]

Its momentum distribution function has a jump at the fermi surface. Unlike the normal fermi liquid where the distribution function vanishes completely at fermi surface. The width of the jump is related to the weight of quasiparticle Z, in this case Z < 1. The stronger the interaction, the smaller the weight of quasiparticle in excitation. For Luttinger liquid, Z ∼ 0 indicates there is actually no quasiparticle excitations. The excitation takes the form of particle hole pair that are strongly correlated. A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 21

Figure 6. The momentum distribution function of Luttinger liquid. There is a small discontinuity at the fermi surface. From [8]

For Luttinger liquid invovles spin, the Hamiltonian becomes more complicated to bosonize and diagonalize. However, we can give a short argument for spin-charge separation in terms of Hamiltonian represented in momentum space: X Hkin ∼ [ρRσ(q)ρRσ(−q) + ρLσ(q)ρLσ(−q)](89) σ,q X (90) Hint ∼ [ρRσ(q) + ρLσ(q)][ρRσ0 (−q) + ρLσ0 (−q)]] . q,σ,σ0 Now if we deﬁne the spin and charge densities as

(91) ρ = ρ↑ + ρ↓ σ = ρ↑ − ρ↓ . Then the Hamiltonian becomes: (92) X Hkin ∼ [ρR(−q)ρR(q) + ρL(q)ρL(−q) + σR(−q)σR(q) + σL(q)σR(q) + σL(q)σL(−q)] q (93) X Hint ∼ [ρR(q) + ρL(q)][ρR(−q) + ρL(−q)] . q The charge density and spin density operators have decoupled from each other in the kinetic term, and the interaction Hamiltonian only contains charge density terms. If we continue to bosonize this Hamiltonian with new boson ﬁelds, it is not suprising that we can get two decoupled Hamiltonian Hσ and Hρ such that charge and spin oscillations propagate with diﬀerent velocities. This is the spin-charge separation, this refers that two excitations are moving independently of one another. To visualize the spin-charge separation, consider the cartoon: After the system undergoes a particle-hole excitation, the point labeled by holon has a net charge but zero net spin, and the point labeled by spinon has a net spin but zero net charge. The holon and spinon are independent of each other, and their motion are not 22 SHUCHEN ZHU

Figure 7. (a) has a particle-hole excitation (b) spin and charge are separated (c) in higher dimension such phenomenon breaks down [15] constrained to each other. In higher dimension, in order to create a spinon, we have to align electron with neighbors of the same spin.

7. Conclusion In this note we reviewed the development of bosonization technique both in particle and solid state physics. However, there are more can be reviewed under this catalogue: the non abelian bosonization, the scaling of operator product expansion, the universality classes of 1D electron liquid, the non-linear spectrum Luttinger model, the application of Luttinger liquid to 1D Hubbard model, the Kondo model, and possible comformal ﬁeld theory. Moreover, bosonization is only one of the many famous methods that solve the 1D system. It might be true that we have only seen an iceberg of the one dimensional world, and this is a branch of physics in which both theorists and experimentalists could found amazing physics that is diﬀerent from what we usully see in higher dimensions.

References [1] S. Coleman, Phy. Rev. D 11 (1975) 2088 [2] S. Mandelstam, phys. rev. D11 (1975) 3026 [3] E. D. Witten, Commun. Math. Phys. 92 (1984) 455 [4] A. Belavin, A. Polyakov, A. Zamolodchikov, Nuclear Phys. B, 241 (2) (1984) [5] S. Tomonaga, Prog. Theor. Phys. (Kyoto) 5 (1950) 544 [6] P. Jordan and E. Wigner, Z. Physik 47, 631 (1928) [7] A. Luther and I. Peschel, Phys. Rev. B 12, 3908 (1975) [8] E. Frandkin, ”Field Theories of Condensed Matter Physics ”, Cambridge University Press, 2013 A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 23

[9] Jan Von Delft, Herbert Schoeller, Bosonization for Beginners - Refermionization for Experts. cond-mat/9805275v3 [10] P. D. Francesco, “Conformal Field Theory”, Springer, 1996 [11] M. Srednicki, “An Introduction to Quantum Field Theory”, Cambridge University Press, 2012 [12] D. Senechal, An introduction to bosonization. cond-mat/9908262v1 [13] H.J. Schulz, G. Cuniberti, P. Pieri, Fermi liquids and Luttinger liquids. cond-mat/9807366v2 [14] S. Eggert, One-dimensional quantum wires: A pedestrian approach to bosonization. arXiv:0708.0003v3 [15] T. Giamarchi, Quantum Physics in One Dimension. Clarendon Press, 2003 24 SHUCHEN ZHU

Appendix A. Bosonization identity in condensed matter In this appendix we followed Delft’s [9] constructive approach to construct the bosonization identity. For new comers this approach is preferred since the bozonization identity, −iφη(x) ψη(x) ∼ Fηe , is better understood from its origin. The beginning point is to set 2π 1/2 P −ikx a fermion ﬁeld ψη(x) = ( L ) k e ckη within a system of length L. Then all other operators can be constructed in terms of ckη and have their physical interpretation.

A.1. Fermion operator and ﬁeld. Let the fermion creation and annihilation operators to have the canonical anticommutation relations: † (94) {ckη, ck0η0 } = δηη0 δkk0 , k ∈ [−∞, ∞], η = 1,...,M η is a species index, it can denote spin or left/right-moving electrons. The momntum index k is 2π 1 k = (n − δ ), n ∈ Z, δ ∈ [0, 2)(95) L k 2 b k b

δb is useful in determining the boundary conditions of the fermion fields, where the fermion field is defined as:

∞ 2π X (96) ψ (x) ≡ ( )1/2 e−ikxc η L kη k=−∞ Z L/2 −1/2 ikx ⇒ ckη = (2πL) dx e ψη(x)(97) −L/2 From (95) and (96) it is easy to see:

iπδb ψη(x + L/2) = e ψη(x − L/2)(98)

Thus δb = 1(0) it gives anti-periodic(periodic) condition. To conﬁrm the anticommutation relation of the constructed fermion ﬁelds, using (94) and (95) we have:

† 0 2π X −i 2π (n − 1 δ )(x−x0) (99) {ψ (x), ψ (x )} = δ 0 e L k 2 b η η0 ηη L n∈Z π 0 i δb(x−x ) 0 = δηη0 e L δ((x − x) − nL) P iny P In the second line we used the identity n∈Z e = 2π n∈Z δ(y − 2πn). Thus we see that the expected anticommutation relation is satisﬁed.

A.2. Vacuum and N-particle ground state. The vacuum state |0i is deﬁned to be: † ckη |0i ≡ 0, ∀k > 0 ckη |0i ≡ 0, ∀k < 0(100) It says at vacuum state no electron or positron can be annihilated. The vacuum state is the fermi sea, and is used as a reference state. The fermion-normal-ordering is deﬁned with A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 25 respect to the fermi sea:

† (101) : ABC ··· := ABC − h0| ABC . . . |0i , A, B, C · · · ∈ {ckη, ckη} The fermion-normal-ordering is not the same as the timer-normal-ordering. It moves fermion creation(annihilation) operator with negative(positive) momentum to the right of other operators. To give it a check, let k > 0 ∧ k0 < 0,

† † † † † : ckck0 := −ck0 ck = ckck0 − δkk0 = ckck0 − hckck0 i0 Let Nˆ ≡ P∞ : c† c := P∞ [c† c − hc† c i ] be the number operator that η k=−∞ kη kη k=−∞ kη kη kη kη 0 counts the number of η electron with respect to the fermion sea. The eigenvalue of Nˆη M running through all η should be a vector N = (N1, ··· ,NM ) ∈ Z . This makes sense since the eigenvalue of c† c is either 1 or 0, and by sutbstracting hc† c i we only count the kη kη kη kη 0 number of electrons above the fermion sea or positrons under the fermion sea. Denote |Ni0 as the N-particle ground state relative to the fermion sea, and no particle-hole excitation is allowed:

¯ N1 NM (102) |Ni0 ≡ C1 ··· CM |0i  c† ··· c† N > 0 Nηη 1η η Nη  (103) Cη ≡ 1 Nη = 0  c(Nη+1)η ··· c0η Nη < 0 Based on our construction, all N-particle excited state can be built from N-particle ground state |Ni0, since they have the same number of particles for each species.

A.3. Bosonic operator. Deﬁne the bosonc creation and annihilation operators as: ∞ ∞ i X −i X (104) b† ≡ √ c† c b ≡ √ c† c qη n k+q η kη qη n k−q η kη q k=−∞ q k=−∞

2π + † where q ≡ L nq, nq ∈ Z , so that q is positive. It can be seen that bqη shift all the η electrons’ momentum by q, if it acts on |Ni. Similarly for bqη which lowers the momentum by q. The normalization constant gives the expected commutation relation (suppressing the η indices):

† 1 X † † [bqη, bq0η0 ] = [ck−qck−q0 − ck−q+q0 ck](105) nq k If q 6= q0, it is normal-ordered because the vacuum state does not come into play. We can let k + q0 → k so that it yields 0. If q = q0, we have to normal-roder it ﬁrst:

† X 1 † † † † (106) [bqη, bq0η0 ] = δqq0 {[: ckck : − : ck+qck+q :] + [hckcki0 − hck+qck+qi ]} nq 0 k 26 SHUCHEN ZHU

The ﬁrst term yields 0 because it is normal ordered. The second term, based on our deﬁnition of fermion creation/annihilation operator acting on the vacuum state (100), gives:

0 −nq 1 X X 1 ( − ) = nq = 1(107) nq nq nk=−∞ nk=−∞

† Therefore, we obtained [bqη, bq0η0 ] = δqq0 δηη0 . Also the remaining relations are easy to check:

† † † [bqη, bq0η0 ] = [bqη, bq0η0 ] = 0, [Nqη, bq0η0 ] = [Nqη, bq0η0 ] = 0(108) Based on (102) and the anticommutation relation of fermion creation/annihilation operators, we can deduce bqη |Ni0 = 0 for all q, η. This makes sense since for ground state there is no excitation, down shifting the momentum of every particle/hole by a positive value q gives 0. Therefore, we can treat |Ni0 as the vacuum state for bosonic operator as to |0i for fermion operator, and the normal ordering for bosonic operator can be similarly deﬁned as:

(109) : ABC ··· := ABC · · · − 0 hN| ABC . . . |Ni0

Clearly as |Ni0 is interchangeable with |0i, the operator product is boson-normal-ordered if and only if it is fermion-normal-ordered. It is obvious that every excited state |Ni is a linear combination of bilinear fermion operators acting on the ground state:

† (110) |Ni = f(ckηck0η) |Ni0 Remarkably this statement is also nontrivially true for a linear combination of bosonic † operators bqη, as was proved by Haldane [?]. Here we take the proof, and a profound consequence can be seen from this statement. If we write the bilinear fermion operator as bilinear fermion ﬁeld operator, by (97) Z Z † 1 0 i(k0x0−kx) † 0 c c 0 = dx dx e ψ (x)ψ (x )(111) kη k η 2πL η η

† 0 This indicates that ψη(x)ψη(x ) can be written as bosonic operators, and shows the sign of possibility of building a bosonization rule.

A.4. Klein factors. From (108) since the bosonic operator commute with the number operator, the bosonic operator does not change the number of fermions of a given state. Hence for the completeness of bosonization, it is necessary to include operators that could † change the number of fermions. The Klein factor Fη and Fη are introduced for this purpose, where the notation comes from Kotliar and Si [?]. First it is required that they commute with bosonic operators

† † † † [bqη,Fη0 ] = [bqη,Fη0 ] = [bqη,Fη0 ] = [bqη,Fη0 ] = 0(112) A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 27

Then we require (113) F † |Ni ≡ f(b†)c† |N ,...,N ,...,N i ≡ f(b†)Tˆ |N ,...,N ,...,N i η Nη+1 1 η M 0 η 1 η+1 M 0 † † ˆ (114) Fη |Ni ≡ f(b )cNη+1 |N1,...,Nη,...,NM i0 ≡ f(b )Tη |N1,...,Nη−1,...,NM i0 † † It can be seen that Fη |Ni has the same excitation as |Ni, because it is the same f(b ) † acting on ground states. They differ by the ground state in which Fη |Ni has one more Pη−1 Nˆ η-electron in the lowest empty level of |Ni. Tˆη = (−1) i=1 i is a phase factor we need to take into account when c† passes by all the fermion creation operators in front of Nη+1 Nη † Cη . From (113) and (114), Fη Fη |Ni = |Ni, hence the Klein factor is unitary. Other commutation relations are (easy to check): (115) † ˆ † † ˆ ˆ † † ˆ {Fη,Fη } = 2δηη0 , NηFη = Fη (Nη + 1) ⇒ [Nη0 ,Fη ] = δη0ηFη , [Nη0 ,Fη] = −δη0ηFη A.5. Boson field. With all the operators properly defined for the bosonic ground state |Ni, we are now able to give definition to the boson field: X 1 X 1 (116) ϕ (x) ≡ − √ e−iqxb e−aq/2 ϕ† (x) ≡ − √ eiqxb† e−aq/2 η n qη η n qη q>0 q q>0 q X 1 1 (117) φ (x) ≡ − [√ e−iqxb + √ eiqxb† ]e−aq/2 η n qη n qη q>0 q q Then the electron density operator can be straightforwardly expressed as boson fields: 2π X X ρ (x) ≡: ψ†(x)ψ (x) := e−iqx : c† c :(118) η η η L k−q,η kη q k 2π X √ 2π X = i n (e−iqxb − eiqxb† ) + : c† c :(119) L q qη qη L k,η kη q>0 k 2π = ∂ φ (x) + Nˆ , a → 0(120) x η L η The commutation relations of boson fields can be checked: (121) 0 † † 0 [ϕη(x), ϕη0 (x )] = [ϕη(x), ϕη0 (x )] = 0 (122) † 0 X 1 −q[i(x−x0)+a] −i 2π (x−x0−ia) 2π 0 [ϕ (x), ϕ (x )] = δ 0 e = −δ 0 ln [1 − e L ] = −δ 0 ln [i (x − x − ia)],L → ∞ η η0 ηη n ηη ηη L q>0 q Notice the importance of a in (122) when x = x0, where a is treated as a UV cut off to regularize the result. With (122) and Weyl’s identity, we can easily show:

† L † L (123) eiϕη(x)eiϕη(x) = ( )1/2eiφη(x) e−iϕη(x)e−iϕη(x) = ( )1/2e−iφη(x) 2πa 2πa 28 SHUCHEN ZHU

Using (117) and (122), the commutator of φη(x) with its derivative can be ﬁnd (to order 1 of L ), as taking the limit L → ∞ ﬁrst and a → 0 second:

0 0 1 [φ (x), ∂ 0 φ 0 (x )] = δ 0 2πi[δ(x − x ) − ](124) η x η ηη L

A.6. Bosonization identity. We are now able to ﬁnd the bosonization identity by applying the fermion ﬁeld on both boson vacuum state and arbitrary boson states. Let i iqx αq(x) ≡ √ e , and write both ψη(x) and bqη in terms of fermion operators. The com- nq mutation relation can be seen as:

† ∗ (125) [bqη0 , ψη(x)] = δηη0 αq(x)ψη(x)[bqη0 , ψη(x)] = δηη0 αq(x)ψη(x)

Since it yields 0 for boson annihilation operator acting on the boson vacuum state, with (125) we immediately have:

(126) bqη0 ψη(x) |Ni0 = δηη0 αq(x)ψη(x) |Ni0

Equation (126) tells that ψη(x) |Ni0 is an eigenstate of bqη0 , therefore ψη(x) |Ni0 must can be expressed in the form:

P † † q>0 αq(x)bqη ˆ −iϕη(x) ˆ (127) ψη(x) |Ni0 = e Fηλη(x) |Ni0 = e Fηλη(x) |Ni0

The validity of (127) can be checked by inserting it into (126), with the identity C = B B † ˆ [A, B] ⇒ [A, e ] = Ce ,A = bqη0 ,B = −iφη(x),C = δηη0 αq(x). The value of λη(x) can † be determined as follow, as noticing ﬁrst that 0 hN| Fη ψη(x) |Ni0 = λη(x). Then we can easily ﬁnd, by writing ψη(x) as combination of fermion operators, that

† 2π 1/2 −i 2π (N − 1 δ )x (128) hN| F ψ (x) |Ni = ( ) e L η 2 b 0 η η 0 L

† Two more identities are needed, let |Ni = f({bqη0 }) |Ni0, then

† † ∗ f({bqη0 }) |Ni0 = f({bqη0 − δηη0 αq(x)})ψη(x)(129)

† ∗ −iϕη(x) † iϕη(x) (130) f({bqη0 − δηη0 αq(x)}) = e f({bqη0 })e

Equation (129) uses the identity [A, B] = DB ∧[A, D] = [B,D] = 0 ⇒ f(A)B = Bf(A+ D), and (130) uses the identity c = [A, B]∧[A, C] = [B,C] = 0 ⇒ e−Bf(A)eB = f(A+C). A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 29

The ﬁnal step is to evaluate ψη(x) |Ni:

(131) † † ∗ ψη(x) |Ni = ψη(x)f({bqη0 }) |Ni0 = f({bqη0 − δηη0 αq(x)})ψη(x) |Ni0 used (129) † † ∗ −iϕη(x) ˆ (132) = f({bqη0 − δηη0 αq(x)})e Fηλη(x) |Ni0 used the relation (127) ψη |Ni0 † ˆ −iϕη(x) † ∗ † (133) = Fηλη(x)e f({bqη0 − δηη0 αq(x)}) |Ni0 since Fη commutes with bqη0 † ˆ −iϕη(x) −iϕη(x) † iϕη(x) = Fηλη(x)e [e f({bqη0 })e ] |Ni0 used (130)(134) † ˆ −iϕη(x) −iϕη(x) † = Fηλη(x)e e f({bqη0 }) |Ni0 since bqη0 |Ni0 = 0(135) † −iϕη(x) −iϕη(x) (136) = Fηλˆη(x)e e |Ni

Since the choice of |Ni is arbitrary, we conclude that

† −iϕη(x) −iϕη(x) (137) ψη(x) = Fηλˆη(x)e e

2π 1/2 −i 2π (N − 1 δ )x −iϕ† (x) −iϕ (x) (138) = F ( ) e L η 2 b e η e η η L 2π 1/2 −i 2π (N − 1 δ )x −iφ (x) = F ( ) e L η 2 b e η used (123)(139) η L

This is the ﬁnal result of bosonization identity, and cearly for L → ∞, we have a better 2π 1/2 −iφη(x) looking result Fη( L ) e .

Appendix B. Luttinger model (heuristic view) In this section we go through the tunneling density of states at in a Luttinger liquid, the discussion will not be explicit due to its length and the time constraint I have.

B.1. Fermion and boson left and right movers. The physical fermion ﬁeld in a 1-D wire of length L can be deﬁned as:

∞ 2π X (140) ψ (x) ≡ ( )1/2 eipxc phy L p p=−∞ 30 SHUCHEN ZHU

If we let p = ∓(k + kF ), the above ﬁeld can be written as: ∞ 2π X (141) ψ (x) = ( )1/2 (e−i(kF +k)xc + ei(kF +k)xc ) phy L −(kF +k) kF +k k=−kF ∞ 2π X ≡ ( )1/2 (e−i(kF +k)xc + ei(kF +k)xc )(142) L k,L k,R k=−kF ∞ 2π X → ( )1/2 (e−i(kF +k)xc + ei(kF +k)xc )(143) L k,L k,R k=−∞ −ikF x ˜ ikF x ˜ ≡ e ψL(x) + e ψR(x)(144)

In the last line we extended the range of k from [kF , ∞) to (−∞, ∞). The inclusion of unphysical state is possible, they are unaﬀected by the low energy excitation. The tilde means it is a mathematical left/right mover, and they shall only be used in low energy sit- uations. Clearly we have to quantize the momentum k by imposing boundary conditions, we can take δb = 1 in (95) to have antiperiodic condition.

Having the fermion left/right operators properly defined, we can follow exactly what we did in appendix A.3 to define the bosonic left/right operators bq L/R in terms of ck L/R. Then the Klein operator, number operator could all be written in terms of left/right movers. Here we have: ˜ X 1 −aq/2 ∓iqx ±iqx † 2π (145) φL/R(x) ≡ − √ e [e bq L/R + e bq L/R] from (117), q = L nq nq nq>0 2π ˆ 1 ˜ ˜ −1/2 ∓i (NL/R− δb)x −iφL/R(x) (146) ψL/R(x) = a FL/Re L 2 e compare with (123) 2π (147) ρ˜ (x) ≡: ψ˜† (x)ψ˜ (x) := ±∂ φ˜ (x) + Nˆ compare with (120) L/R L/R L/R x L/R L L/R Notice the plus minus sign in the above equations, the reason is that the left and right movers in (144) are differed by a sign of x in their phase factors.

B.2. Diagonalizing the electron interaction Hamiltonian by bosonization. One of the important usages of bosonization is to diagnonalize the interaction Hamiltonian and result in a free Hamiltonian. Here we start with a local electron-electron interaction, bosonize it and in the end make use of a Bogoliubov transformation to get a free Hamiltonian. Assuming linear dispersion relation, the kinetic term of the Hamiltonian is: ∞ Z L/2 X X † ˜† ˜ ˜† ˜ (148) Hkin = k : ckηckη := :[ψL(x)(i∂x)ψL(x) + ψR(x)(−i∂x)ψR(x)] : η=L,R k=−∞ −L/2 X 2π 1 Z L/2 dx 1 Z L/2 dx 1 (149) = [ Nˆ 2 + : (∂ φ (x))2 :] = : [˜ρ2 +ρ ˜2 ](x): L 2 ν 2π 2 x ν 2π 2 L R ν=L,R −L/2 −L/2 A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 31

In (149) there are some subtleties for the ﬁrst equality and we do not discuss here, and the second equality comes from (120). The interaction term can be expressed as:

(150) 1 X † † H = V c c c 0 c int 2L q k−q k0+q k k kk0q (151) 1 Z = dxdx0 ψ†(x)ψ†(x0)V (x − x0)ψ(x0)ψ(x), let V (x − x0) = δ(x − x0) 2 ee ee (152) 1 Z → dxdx0 ψ† (x)ψ† (x0)V (x − x0)ψ (x0)ψ (x), substitute (144) into it 2 phy phy ee phy phy (153)

1 Z dx dx0 g = δ(x − x0){ 4 [ψ† (x)ψ† (x0)ψ (x0)ψ (x) + R ↔ L] 2 2π 2π 2 R R R R † † 0 0 + g2[ψR(x)ψL(x )ψL(x )ψR(x) + R ↔ L

−2ikF x † 0 † 0 + e [ψR(x)ψR(x )ψR(x )ψL(x) + R ↔ L]

−4ikF x † † 0 0 (154) + e [ψR(x)ψR(x )ψL(x )ψL(x) + R ↔ L]}

Only the ﬁrst two terms conserve the number of right and left movers, and the last two terms can be dropped since their phase factors are in ‘high-energy’ region. Thus we have:

Z L/2 1 2 2 Hint = : g2ρ˜L(x)˜ρR(x) + g4[˜ρL(x) +ρ ˜R(x)]] :(155) −L/2 2

The Hamiltonian then is:

Z L/2 v dx 1 2 2 (156) H0 = Hkin + Hint = :[ (˜ρL +ρ ˜R) + g(˜ρL − ρ˜R) ](x): 4 −L/2 2π g q r 2 2 1 + g4 − g2 v ≡ (1 + g4) − g2 g ≡ 1 + g4 + g2 32 SHUCHEN ZHU

Substitute ρ in terms of boson operators using (147): (157) 2π v 1 X 1 X 1 X H = {( + g) [ Nˆ 2 + n b† b ] + ( − g)[Nˆ Nˆ − n (b b + b† b† )]} 0 L 2 g 2 ν q qν qν g L R q qR qL qR qL ν=L,R q q (158) 2π X X = v [gνNˆ 2 + n B† B ] by Bogoliubov transformation L ν q qν qν ν=± q (159) X 2π Z L/2 dx 1 = v [ gνNˆ 2 + : (∂ Φ (x))2 :] reproduce the previous line by subsituting in Φ (x) L ν 2π 2 x ν ν ν=± −L/2 (160) ≡ H0+ + H0− where we have the new transformed boson operators:

1 1 √ 1 √ † † Bq± ≡ √ {(√ + g)(bqL ∓ bqR) ± (√ − g)(b ± b )} 8 g g qL qR 1 1 Nˆ ≡ (Nˆ − Nˆ ), Nˆ ≡ (Nˆ + Nˆ ) + 2 L R − 2 L R X 1 Φ (x) ≡ − √ e−aq/2[e−iqxB + eiqxB† ] ± n q± q± q>0 q 1 1 √ ˜ ˜ 1 √ ˜ ˜ (161) = √ {(√ + g)[φL(x) ∓ φR(−x)] ± (√ − g)[φL(−x) ∓ φR(x)]} 8 g g 2π √ ρ (x) ≡ ∂Φ (x) + 2g±1/2Nˆ ± ± L ± B.3. Tunneling density at site x = 0 without impurity. In appendix B.2 we diagonalize the fermion interaction Hamiltonian, so that the Hamiltonian is free of interaction expressed in terms of transformed boson operators. We are interested in the tunneling density: Z ∞ dt iωt † (162) ρdos(ω) ≡ e hG| Ψphy(t, x = 0)Ψphy(0, x = 0) |Gi −∞ 2π In our case we set x = 0, from (144) and (146): ˜ ˜ −1/2 −iφL −iφR Ψphy(x = 0)a (FLe + FRe )(x = 0)(163) The left and right boson ﬁeld can be expressed as the transformed boson ﬁeld through (161): 1 1 √ φL/R(x = 0) = √ (√ Φ−(x = 0) ± gΦ+(x = 0))(164) 2 g A NOTE ON BOSONIZATION OF 1 DIMENSIONAL ELECTRON GAS 33

Therefore, the fermion ﬁeld can be completely expressed as boson ﬁeld in free Hamiltonian: √ √ √i −1/2 − Φ− −i g/2Φ+ i g/2Φ+ Ψphy(x = 0) = a e 2g (FLe + FRe )(165) † Since H = H++H−, and [H+,H−] = 0, hG| Ψphy(t, x = 0)Ψphy(0, x = 0) |Gi = D−(t)D+(t). Where

(166) √i √i iH−t − Φ− −iH−t Φ− D−(t) ≡ h0−| e (e 2g )e (e 2g ) |0−i it only has Φ− part (167) √ √ √ √ −1 iH+t −i g/2Φ+ i g/2Φ+ −iH+t † i g/2Φ+ † −i g/2Φ+ D+(t) ≡ a h0+| e (FLe + FRe )e (FLe + FRe ) |0+i (168) = DLL(t) + DRR(t) + DLR(t) + DRL(t)

Without impurity, H− and H+ are just free Hamiltonian, so that the evaluation up to a constant is: 1 1 h0−|Φ−(t)Φ−(0)−Φ−(0)Φ(0)|0−i − (169) D−(t) ∼ e 2g = (1 + ivt/a) 2g −1 − g (170) D+(t) ∼ a (1 + ivt/a) 2 Therefore, the tunnelign density is proportional to: 1 ρ ∼ ων−1, ν = (1/g + g) 2 For g = 1, the tunneling density is a constant and does not vanish at Fermi energy. But it is diﬀerent for g 6= 1, the tunneling density vanishes at Fermi energy.