Tomonaga-Luttinger Liquid Theory

1, Shang Liu ∗

1Department of Physics, Harvard University, Cambridge, MA 02138, United States. (Dated: December 7, 2016)

This is a brief introduction to the Tomonaga-Luttinger liquid theory, including the exactly solvable Tomonaga-Luttinger model and a discussion about generic one- dimensional systems.

I. Introduction 2

II. Tomonaga-Luttinger Model: Exact Diagonalization 2 A. Model Hamiltonian 2 B. *Historical Approach: A Lesson of Infinities 4 C. Bosonization of the Hamiltonian 5 D. Diagonalization 8

III. Tomonaga-Luttinger Model: Physical Properties 9 A. Correlation Functions 9 B. Momentum Distribution 9 C. Tunneling Current 10

IV. The Tomonaga-Luttinger Liquid Concept 10

V. Summary 11

A. Thermo-Average in Free Boson Field Theories 11

B. Bosonized Fermion Operators 12

C. Calculating Correlation Functions in the Fermion Representation 13

Acknowledgements 16

References 16

∗Electronic address: [email protected] 2

I. INTRODUCTION

The one-dimensional interacting electron gas is very different from its higher dimensional counterparts. Perhaps most importantly, the Fermi liquid picture of generic many-particle states breaks down in one dimension and is replaced by the so-called Tomonaga-Luttinger liquid theory. In this note, we will present a brief introduction to this rich theory. We will start by introducing the Tomonaga-Luttinger model [1, 2]whichisaremarkableexactly solvable many-fermion model. This model was first correctly solved by Mattis and Lieb [3]whoinitiatedtheimportantideaofbosonizationandcalculatedthepartitionfunction as well as various physical properties of this model without completely diagonalizing the Hamiltonian. Complete solution of this model with the modern bosonization approach was presented by Haldane [4]andnowcanbefoundinmanyliteratures[5–7]. After exploring the diagonalization and some interesting properties of this model, we will introduce the general Tomonaga-Luttinger liquid concept which, suggested by Haldane [8], is a generalization of this specific model to generic one-dimensional systems.

II. TOMONAGA-LUTTINGER MODEL: EXACT DIAGONALIZATION

A. Model Hamiltonian

Consider a one-dimensional system with two types of spinless fermions which both have linear dispersion relations. More specifically, we consider the following kinetic Hamiltonian.

H = k a† a a† a , (1) 0 Rk Rk − Lk Lk !k " # where aRk and aLk are the annihilation operators for right-moving and left-moving particles, respectively. We confine the configuration space to a finite region of length L and impose periodic boundary condition as we always do, then the allowed momenta k should be of the form 2π integer/L.Ingeneral,thereshouldalsobeanoverallconstantfactor!vF in this × Hamiltonian, but we can always choose our units accordingly to eliminate this factor. This model is not quite physical, because the kinetic energy is not bounded from below. To resolve this problem, we can consider the case of a filled Fermi sea and define the particle and hole operators as follow.

bk (k 0) bk (k<0) aRk = ≥ ,aLk = . (2) $ c† (k<0) $ c† (k 0) k k ≥ 3

εₖ εₖ εₖ

ε k k F

k RL

(a) (b)

FIG. 1: (a) The kinetic energy spectra for right- and left-moving particles. White and black points correspond to b-particles (particles) and c-particles (holes), respectively. (b) Linearization of the spectrum near the Fermi energy.

The Hamiltonian now becomes

H = k (b† b + c† c ), (3) 0 | | k k k k !k where an infinite constant has been neglected. Mathematically, we are converting the Hilbert space of finite R/L particles to the Hilbert space of finite particle/hole excitations. The ki- netic energy spectra of R/L particles as well as the conversion to b/c-particles are illustrated in Fig. 1a. We consider the following interaction term.

1 H = V (p)[ρ (p)+ρ (p)] [ρ ( p)+ρ ( p)] , (4) int 2L R L R − L − p=0 !̸ where ρ (i = L/R)aretheFouriertransformeddensityoperatorsdefinedforp =0as i ̸ ρ (p)= a† a . Note that as long as p =0,thisdensityoperatoriswelldefinedinboth i k ik ik+p ̸ the old and new Hilbert spaces. % One may wonder why at all we are interested in this model with a peculiar linear disper- sion (1). It turns out that, under certain assumptions and approximations, some more prac- tical systems can be described by this simplified model. Consider a generic one-dimensional electron system. If we assume that all physics is restricted to energies much lower than the Fermi energy and that the interactions only modify the occupation near the Fermi surface, it is then plausible to linearize the dispersion relation near the Fermi energy and extend it to infinities. This linearization process is shown in Fig. 1b. The interaction (4)wasfirstconsideredbyMattisandLieb[3]asageneralizationtothat in Luttinger’s paper [2]. This term is very likely to be a realistic model since it includes both inter-branch and intra-branch scattering processes. In fact, it is argued in Ref. [5]thata generic spinless interaction can indeed be approximated by this expression with V (p)being 4 aconstant1. Another possible scattering process (the so-called Umklapp process) where two electrons in the same branch are scattered together to the other branch is only important in ahalf-filledBlochbandduetomomentumconservationandisnotconsideredinthismodel.

B. *Historical Approach: A Lesson of Infinities

In the original paper [2]ofLuttinger,thismodelwasinfactalreadysolvedexactlyinthe case of finite right- and left-moving particles. Luttinger then postulated that his solution also holds in the case of a filled Fermi sea. This intuitive expectation, however, turns out to be incorrect due to a subtle paradox inherent in quantum field theory. We will not discuss here the full story of Luttinger’s solution, but we will instead give an important example to show how things may go wrong. As we will see in Section II C,thecommutatorsbetweendensityoperatorsplayanim- portant role in this problem, so let’s try to calculate one commutator explicitly here, say

[ρ (p),ρ ( p′)] (p p′ > 0). We have R R − ≥

† † [ρR(p),ρR( p′)] = [aRkaRk+p,aRk aRk′ p′ ] (5) − ′ − !k,k′ † † † † = aRkaRk+paRk aRk′ p′ aRk aRk′ p′ aRkaRk+p (6) ′ − − ′ − !k,k′ † † † † = aRk(δk′,k+p aRk aRk+p)aRk′ p′ aRk (δk,k′ p aRkaRk′ p′ )aRk+p (7) − ′ − − ′ − − − !k,k′ † † = aRkaRk+p p′ aRk+p aRk+p =0 (8) − − ′ !k This calculation is perfectly correct in the space of finite R/L particles, but is incorrect in the space of filled Fermi sea. This is because the two series in (8)arebothdivergentinthe case of infinite R/L particles when p = p′.Inotherwords,itisinvalidtoexchangethetwo operators in the middle of any four-fermion operator in (6). To fix this problem, we split the density operator as

ρ (p)= a a† + a† a + a† a , (9) R − Rk+p Rk Rk Rk+p Rk Rk+p k< p p k<0 k 0 !− − !≤ !≥ † † † ρR( p′)= aRk′ p′ aRk + aRk aRk′ p′ + aRk aRk′ p′ . (10) − − − ′ ′ − ′ − k <0 0 k

1 The author is not fully convinced by that argument because the result implies that there is no interaction effect in the case of a δ-function potential. 5 contributions can be grouped into the following two terms.

† † [ρR(p),ρR( p′)] = + [aRk+paRk,aRk′ p′ aRk ] (11) − − ′ &k< p,k

C. Bosonization of the Hamiltonian

The kinetic term of the Hamiltonian is

H = k (b† b + c† c ), (13) 0 | | k k k k !k where bk and ck are annihilation operators for particles (above Fermi sea) and holes (below Fermi sea), respectively. The Fourier transformed density operators satisfy the following commutation relations. pL [ρ (p),ρ ( p′)] = [ρ ( p),ρ (p′)] = δ , for p, p′ > 0, (14) R R − L − L 2π p,p′ [ρ (p),ρ (p′)] = 0, for any p, p′ =0, (15) R L ̸ which inspire the following definition of bosonic operators Ap and Bp (p>0).

2π 2π A = ρ (p),A† = ρ ( p), (16) p pL R p pL R − ( ( 2π 2π B = ρ ( p),B† = ρ (p). (17) p pL L − p pL L ( ( One can easily verify that they satisfy the standard bosonic commutation relations. The interaction in the Hamiltonian can be expressed as quadratic terms of those bosonic operators. Then how about the kinetic term? (If we can do so for the kinetic term, we will be able to exactly diagonalize the whole Hamiltonian!) Note the following commutation relations between the kinetic Hamiltonian and density operators.

[H ,ρ (p)] = pρ (p), (18) 0 R − R [H0,ρL(p)] = pρL(p). (19) 6

They are very similar to the relations in free boson field theories. We therefore expect that the kinetic Hamiltonian can be written in the following form.

H0′ = p(Ap†Ap + Bp†Bp)+C (20) p>0 ! 2π = [ρ (p)ρ ( p)+ρ ( p)ρ (p)] + C. (21) L R R − L − L p>0 !

It turns out that this is indeed true, in other words, we can rigorously prove that H0 = H0′ for some constant C.

Proposition 1. The kinetic Hamiltonian H0 is equivalent to the following operator.

H0′ = p(Ap†Ap + Bp†Bp)+C(QR,QL), (22) p>0 ! where C(Q ,Q )=[Q (Q 1)+Q (Q +1)]π/L, Q and Q are charge operators defined R L R R − L L R L as QR = k 0 bk† bk k<0 ck† ck and QL = k<0 bk† bk k 0 ck† ck. ≥ − − ≥ Proof. The% total Hilbert% space can be decomposed% % as a direct sum of subspaces with H definite charges, i.e. = (m,n) Z Z m,n where m,n is the eigenspace with QR = m and H ∈ × H H Q = n.WeonlyneedtoproveH = H in each of these subspaces. L ) 0 0′ For a given (m, n), we define the vacuum state Ω in as the state where Fermi | m,n⟩ Hm,n sea is filled up to pm 1 =(m 1) (2π/L)forright-movingparticleandp n = n (2π/L) − − × − − × for left-moving particle. We can then generate free bosons from this vacuum by the creation operators A† and B†.Denotethesubspacegeneratedbyallthosefreebosonstatesby ′ , Hm,n more specifically,

′ =subspacegeneratedbyallpossiblestateslikeA† A† B† B† Ωm,n . (23) Hm,n p1 ··· pM q1 ··· qN | ⟩

From the commutation relations between H0 and A†,B†,onecaneasilyshowthatstates like A† A† B† B† Ωm,n are eigenvectors of both H0 and H′ with the same eigen- p1 ··· pM q1 ··· qN | ⟩ 0 values when2 π C(Q ,Q )= Ω H Ω =[Q (Q 1) + Q (Q +1)] . (24) R L ⟨ m,n| 0| m,n⟩ R R − L L L

We therefore conclude that H = H′ is true in ′ . Note that ′ is a subspace of . 0 0 Hm,n Hm,n Hm,n We will now prove that ′ is actually equal to . Hm,n Hm,n The degeneracy of any eigenvalue of H is finite, so we can prove ′ = by 0 Hm,n Hm,n comparing the degeneracies of each eigenvalue of H0 in these two spaces. To do this, let’s

2 These 1 are due to our convention about k = 0 and are never important. One can avoid such kind of ± inconvenience by including a half-integer chemical potential or taking an anti-periodic boundary condition. 7 consider the partition function

Z(β) d exp( βE ), (25) ≡ i − i i ! where Ei are eigenvalues of H0 and di are the corresponding degeneracies. Note that all information about di can be extracted from the partition function. More specifically, suppose that we have known d for i =1, 2, ,k 1, then we can compute d as follows. i ··· − k k 1 − βEk βEi dk =lime Z(β) die− , (26) β − →∞ & i=1 ' ! where we have assumed that E >E when j>i.Defineq exp( πβ/L), then for H in j i ≡ − 0 ,thepartitionfunctionis H 2 ∞ 2 Z = 2 1+q2n . (27) * n=1 . + , - For H in ′,thepartitionfunctionis 0 H

+ 2 ∞ ∞ βC(Q ,Q ) 1 Z′ = e− R L (28) 1 q2n &Q ,Q = '&n=1 ' R !L −∞ + − + 2 ∞ ∞ m(m 1) 1 = q − . (29) 1 q2n *&m= '&n=1 '. !−∞ + − From the Jacobi Triple Product Identity (see, for example, Wikipedia or Theorem 14.6 in Ref. [9]), one can show that

+ ∞ ∞ m(m 1) 2n 2 2n q − =2 1+q 1 q , (30) − m= n=1 !−∞ + , - , - which immediately implies Z = Z′,althoughthosetwoexpressionslookquitedifferent.

If we are interested in the grand canonical ensemble, we may want to include a chemical potential term µ(Q + Q ) in the Hamiltonian. This term only induces a correction to − R L C(QR,QL)andisthusnotimportant. Let us pause and reflect why this beautiful bosonization technique can at all work. Note that the density operator by definition generates or destructs a set of particle-hole pairs. It we want to identify a density operator as a creation or annihilation operator which converts one eigenstate to another, it is necessary that all those particle-hole pairs have the same excitation energy, i.e. the excitation energy of aik† +paik (p>0) should be independent of k. This implies that the dispersion relation of the particles must be linear. If we generalize this argument to higher dimensions, we shall find that identifying density operators as creation 8 and annihilation operators necessarily implies a hyperplain-like dispersion relation which is not physically interesting at all. This is why we only have the nice bosonization property in one dimension.

D. Diagonalization

Let’s assume that V (p)isarealevenfunctionorequivalentlyV (x y)=V (y x)in − − the real space, the full Hamiltonian is bosonized into the following form.

H = H0 + Hint, (31)

H0 = p(Ap†Ap + Bp†Bp)+C(QR,QL), (32) p>0 ! 1 H = pV (p)(A†A + B†B + A†B† + A B +1). (33) int 2π p p p p p p p p p>0 ! This kind of quadratic Hamiltonian can be diagonalized by a Bogoliubov transformation.

More specifically, first define two new sets of boson annihilation operators αp and βp in the following way. A cosh θ sinh θ α p = p − p p , (34) B† sinh θ cosh θ β† & p' &− p p '& p' where θp is determined by the equation

V (p) tanh 2θ = . (35) p V (p)+2π

Then, the total Hamiltonian becomes

H = Ep(αp†αp + βp†βp)+E0(QR,QL), (36) p>0 ! V (p) Ep = p 1+ , (37) ( π E = (E p)+C(Q ,Q ). (38) 0 p − R L p>0 !

Note that αp,βp are connected with Ap,Bp by a unitary transformation (αp,βp†)= S S e (A ,B†)e− where S = θ (A B A†B†), so they indeed satisfy the standard com- p p p p p p − p p mutation relations. The total Hamiltonian H is therefore exactly diagonalized. Ω˜ % | m,n⟩≡ exp(S) Ω is the ground state with Q = m, Q = n,where Ω is the corresponding | m,n⟩ R L | m,n⟩ free theory ground state defined in the proof of Proposition 1. Another form of the ground state is Ω˜ exp( tanh θ A†B†) Ω which can be directly verified by noting that | m,n⟩∝ − p p p p | m,n⟩ annihilation operators serve as derivatives over creation operators. % 9

III. TOMONAGA-LUTTINGER MODEL: PHYSICAL PROPERTIES

A. Correlation Functions

Most interesting physical properties of a solid state system can be extracted from correla- tion functions. In Tomonaga-Luttinger model, correlation functions can be calculated in two different ways: one is to use the original fermion representation, the other one is to somehow bosonize the fermion operators and work in the boson representation. The latter approach is more systematic and is easier once the bosonization is established. Some key results of this bosonization approach are summarized in Appendix B.Calculationsofcorrelationfunctions with this method also involve a trick about thermo-average as summarized in Appendix A. In Appendix C,wegiveonespecificexampleabouthowtocalculatecorrelationfunctions in the fermion representation. In the rest of this section, we will only discuss interesting physical results without going into calculational details.

B. Momentum Distribution

Let’s consider a Tomonaga-Luttinger model system with a proper particle number such that the free ground state is filled up to k (k > 0) for right- and left-moving particles. ± F F After we turned on the interaction, what is the momentum distribution of the new ground state? n b† b is the occupation number operator at momentum k.Theexpectationvalue k ≡ k k of nk can be obtained once we have calculated the equal time correlation function. If one makes a reasonable assumption about V (p), one finds that for k in the vicinity of k , n F ⟨ k⟩ behaves like n n sgn(k k ) k k α. (1) ⟨ k⟩−⟨ kF ⟩∝− − F | − F | The exponent α is

1/2 V (0) 2 − 1 1 α = 1 1= K + 1, (2) − V (0) + 2π − 2 K − * / 0 . / 0 where K = 1+V (0)/π. This result1 shows that the interaction in Tomonaga-Luttinger model removes the discon- tinuity in n at the Fermi surface. Therefore this model realizes a new non-Fermi liquid ⟨ k⟩ phase.

If α<1, the momentum distribution still has an infinite slope at kF .Thereis,soto speak, a residual Fermi surface. If, on the other hand, α>1, then n is perfectly smooth ⟨ k⟩ at kF and all trace of the Fermi surface has been eliminated. 10

C. Tunneling Current

It will be very remarkable if we can experimentally find a Tomonaga-Luttinger liquid, i.e. a system that can be described by the Tomonaga-Luttinger model (see Section IV). According to Ref. [5], some of the most convincing observations of Tomonaga-Luttinger liquid behavior have presumably been the measurements of power laws in the tunneling density of states in carbon nanotubes and the fractional quantum Hall effect.

I V

Tomonaga-Luttinger Liquid

FIG. 2: Tunneling between a one-dimensional wire and a metal. The contact can be made, for example, by a scanning tunneling microscope.

In a tunneling experiment (see Fig. 2 for example) the tunneling current is determined by the local density of states which can be calculated by a suitable correlation function. In Ref. [5], it is shown that if we assume a constant interaction in the momentum space (or equivalently a δ-function interaction in the real space), i.e. V (p)=V0 for all p,fortunneling between two Tomonaga-Luttinger liquids or from a metal to a Tomonaga-Luttinger liquid, the respective zero-temperature I-V characteristics are

K+1/K 1 I V − , (between two Tomonaga-Luttinger liquids), (3) ∼ I V (K+1/K)/2, (between a Tomonaga-Luttinger liquid to a metal), (4) ∼ where K = 1+V0/π is consistent with the definition in Section III B. Note that in a usual Fermi liquid, we expect a simple Ohm’s law I V to be true, but this is not the case for a 1 ∼ Tomonaga-Luttinger liquid where the I-V exponent is always greater than 1.

IV. THE TOMONAGA-LUTTINGER LIQUID CONCEPT

As we have seen, the bosonization technique that we derived so far relies on the linear spectrum of the particles. This linearization is applicable to a real fermion system only when all interesting physics and interaction effects are confined close to the Fermi surface. We may then ask what would happen in more general cases, for example, when the interactions become strong? To answer this question, Haldane [4, 8]suggestedthatthereexistsinone dimension a Tomonaga-Luttinger liquid concept which is similar to the Fermi liquid concept in higher dimensions. Generic one-dimensional systems (there exist some exceptions, of 11 course) can be described by this theory and therefore have many similar properties to those of the Tomonaga-Luttinger model. In Ref. [8]or[6], a phenomenological bosonization approach is introduced to show how fermion operators as well as generic low-energy Hamiltonians can be bosonized and how this Tomonaga-Luttinger liquid theory works, but we will not show those details here.

V. SUMMARY

In this note, we gave a brief introduction to the Tomonaga-Luttinger liquid theory. We started by writing down the Tomonaga-Luttinger model and exactly diagonalized it with the bosonization technique. We then examined some physical properties of this model and showed that it realizes a non-Fermi liquid system. Finally, we talked about the generalization from this specific model to generic one-dimensional systems and introduced the concept of Tomonaga-Luttinger liquids. The purpose of this note is to provide a first sight into this rich and deep topic, so we did not try to be completely general and comprehensive. Nevertheless, we have tried to provide sufficient physical interpretation as well as calculational/mathematical details for the limited content considered in this note.

Appendix A: Thermo-Average in Free Boson Field Theories

We start with the following lemma.

Lemma 1. In a free boson theory with Hamiltonian H = Ea†a, the following identity about thermo-average is true.

exp C1a† exp (C2a) =exp C1C2 a†a . (A1) 2 , - 3 , 2 3- Proof. Using the cyclic property of traces, we have

βH I(C ,C ) exp C a† exp (C a) =Tr e− exp C a† exp (C a) /Z (A2) 1 2 ≡ 1 2 1 2 βH =Tr2 exp, (C2-a)e− exp3 C1a†, /Z , - - (A3) βH βE =Tr,e− exp C2ae− ,exp --C1a† /Z (A4) βE βH βE =exp(, C1C2e− ,)Tr e− - exp,C1a†--exp C2ae− /Z (A5) βE βE =exp(C1C2e− )I(C,1,C2e− ),, - , -- (A6)

βH βH βE where we have used the property e ae− = ae− as well as the well-known rule that

exp(A + B)=exp(A)exp(B)exp( 1/2[A, B]), (A7) − 12 when [A, B]commuteswithbothA and B. Now we can use the above recursion formula repeatedly to obtain (note that β,E > 0)

∞ nβE I(C1,C2)=exp C1C2 e− I(C1, 0) (A8) & n=1 ' ! 1 =exp C C =exp C C a†a . (A9) 1 2 eβE 1 1 2 / − 0 , 2 3-

Lemma 1 can be easily generalized in the following way.

Lemma 2. Suppose that we have a free boson theory with Hamiltonian H = α Eαaα† aα and a set of operators A ,A , A which are all linear combinations of the creation and 1 2 ··· n % annihilation operators (Ai = α λi,αaα +µi,αaα† ). Then the second order cumulant expansion in the thermo-average eA1 eA2 eAn is exact, i.e. %··· 2 3 eA1 eA2 eAn =exp Γ (A ,A , ,A ) , (A10) ··· ⟨ 2 1 2 ··· n ⟩ 2 3 where Γ (A , ,A ) is the second order term in the expansion of eA1 eAn . 2 1 ··· n ··· Proof. By some suitable rearrangement, we can always separate the creation and annihila- tion operators as the form in Lemma 1 and then go through similar calculations. After all those procedures, we will obtain a number exp(ξ)whereξ consists of only quadratic terms of the parameters λi,α and µi,α. Now we can introduce an artificial factor ε to all the λi,α, µi,α, and then expand both exp(ξ)and eA1 eA2 eAn in power series of ε.Thesecondorder ··· term of those two expansions must coincide, so ξ must be equal to Γ (A , ,A ) . 2 3 ⟨ 2 1 ··· n ⟩

Appendix B: Bosonized Fermion Operators

In order to systematically calculate correlation functions, we expect that the fermion field operators ψ (x) (1/√L) eikxa can also be rewritten in terms of bosonic R/L ≡ k R/L,k operators. However, there is no hope to express ψ only with the bosonic creation and % R/L annihilation operators which all preserve QR and QL,soweneedotheroperatorsthatcan increase or decrease charges.

Let’s define operators UR and UL by requiring that they commute with boson operators and obeying the following relations.

U † Ω = Ω , (B1) R| m,n⟩ | m+1,n⟩ U † Ω = Ω . (B2) L| m,n⟩ | m,n+1⟩ 13

It turns out that the explicit expressions for UR/L are

L 1 iπx/L UR/L = dx e− exp iφ† (x) ψ† (x)exp iφR/L(x) , (B3) √L − R/L R/L − 40 " # , - where

πx 2π εpL/(2π) ipx φR(x)= QR +limi e− Ape , (B4) − L ε 0 pL → p>0 ( ! πx 2π εpL/(2π) ipx φL(x)= QL +limi e− Bpe− . (B5) L ε 0 pL → p>0 ( ! We can now easily invert (B3)toobtain

1 iπx/L ψ† (x)= e exp iφ† (x) U † exp iφR/L(x) (B6) R/L √L R/L R/L " # 1 i(π/L π/L)x , - =lim e ± exp iφ† (x)+iφR/L(x) U † . (B7) ε 0 √2εL R/L R/L → " # Proof of these results can be found, for example, in Ref. [4]or[6]. The expressions therein may be slightly different due to different conventions.

Note that, except for the so-called Klein factors UR/L, ψR/L are exponentials of linear terms of boson operators. Given Lemma 2 in Appendix A,wecaneasilycalculatevarious correlation functions with these bosonized fermion operators.

Appendix C: Calculating Correlation Functions in the Fermion Representation

In this section, we sketch the calculation of a single-particle correlation function in the fermion representation. The method used here is a generalization of that in Mattis and Lieb’s paper [3]whereanequal-timecorrelationfunctionwascalculated.Workinginthe fermion representation is straightforward but very tedious. More specifically, we will calculate the following zero-temperature single-fermion correla- tion function.

I(x, y; t ,t) 0′ ψ† (x, t )ψ (y, t ) 0′ . (C1) f i ≡⟨ | R f R i | ⟩

We use the simplified notation 0 and 0′ to denote the free and interacting ground states | ⟩ | ⟩ with a proper particle number such that 0 state is filled up to k (k > 0) for right- | ⟩ ± F F and left-moving particles. ψR is the right-moving fermion field operator defined as ψR(x)= √ ikx (1/ L) k e aRk.Thefollowingcommutationrelationisuseful.

% ipx [ρ (p),ψ (x)] = e− ψ (x). (C2) R R − R 14

Define r x y, t t t ,ourfinalresultofthecorrelationfunctionshouldonly ≡ − ≡ f − i depend on r and t.Wecanusethisfacttoslightlysimplifyourcalculation.

iHt iHt iHt iHt iHt iHt I(r, t)= 0′ e f ψ† (x)e− f e i ψ (y)e− i 0′ = 0′ ψ† (x)e− ψ (y)e 0′ . (C3) ⟨ | R R | ⟩ ⟨ | R R | ⟩

S The two ground states 0 and 0′ are connected by the unitary transformation e where | ⟩ | ⟩ S = θ (A B A†B†), so we have p p p p − p p % S S iHt˜ S S iHt˜ I(r, t)= 0 e− ψ† (x)e e− e− ψ (y)e e 0 , (C4) ⟨ | R R | ⟩ " # , - ˜ where H = p>0 Ep(Ap†Ap + Bp†Bp)+E0(QR,QL)whichisthesameasH except for the replacement (α,β) (A, B). % → S S In order to compute e− ψR(y)e ,definethefollowingquantity.

σS σS f(σ) e− ψ (y)e . (C5) ≡ R One can calculate df(σ)/dσ and solve the differential equation to find

f(σ)=Wσ(y)Rσ(y)ψR(y), (C6) where

2π ipy ipy W (y)=exp (e A e− A†)(cosh(σθ ) 1) , (C7) σ pL p − p p − & p>0 ( ' ! 2π ipy ipy R (y)=exp (e− B e B†)sinh(σθ ) . (C8) σ pL p − p p & p>0 ( ' !

Note that Wσ and Rσ commute with each other and are both unitary operators. From here on, we shall set σ =1anddropitasasubscript. Our correlation function now becomes

iHt˜ iHt˜ I(r, t)= 0 ψ† (x)R†(x)W †(x) e− (W (y)R(y)ψ (y)) e 0 (C9) ⟨ | R R | ⟩ " # What should we do next? Note that the exponents in W and R operators are all linear in the bosonic creation and annihilation operators, so our basic idea is to move creation operators to the left while move annihilation operators to the right in order to hit the vacuum. We will frequently use the well-known rule that

exp(A + B)=exp(A)exp(B)exp( 1/2[A, B]), (C10) − when [A, B]commuteswithbothA and B.Weshouldalsomovetheannoyingtimeevolution 15

iHt˜ iHt˜ factor e− to the right side towards its inverse e for possible simplification. iHt˜ iHt˜ iEpt iHt˜ We can use the fact that e− Ape = Ape to move e− through W (y). After some straightforward calculation, we have

1 iHt˜ iHt˜ W − (x)e− W (y)e = W W+Z1, (C11) − with

2π W =exp A (cosh(θ ) 1)(eipy+iEpt eipx) , (C12) + pL p p − − & p>0 ( ' !

2π ipx ipy iEpt W =exp Ap†(cosh(θp) 1)(e− e− − ) , (C13) − pL − − & p>0 ( ' !

2π 2 ip(x y) iEpt Z =exp (cosh(θ ) 1) (e − − 1) . (C14) 1 pL p − − & p>0 ' ! Likewise, we have 1 iHt˜ iHt˜ R− (x)e− R(y)e = R R+Z2, (C15) − where the expressions for R and Z2 can be obtained from W and Z1 by the replacement ± ± (A ,A†) (B ,B†), (x, y) ( x, y), cosh(θ ) 1 sinh(θ ). p p → p p → − − p − → p The correlation function now becomes

iHt˜ iHt˜ I(r, t)=Z1Z2 0 ψR† (x)W W+R R+e− ψR(y)e 0 (C16) ⟨ | − − | ⟩

iHt˜ iHt˜ and we need to compute e− ψ (y)e .DefineH˜ H˜ E ,thenwecansplitthe R b ≡ − 0 Hamiltonian as H˜ = H˜ + E . With the relation Q ψ = ψ (Q 1), the E part is easily b 0 R R R R − 0 iE0t iE0t ikF t calculated as e− ψRe = ψRe .TheH˜b part can be calculated by solving an operator differential equation as we did for f(σ). We found

itH˜b itH˜b g(t) e− ψR(y)e = Y ψRY+ (C17) ≡ −

2π ipy iEpt 2π ipy iEpt =exp A†e− (e− 1) ψ exp A e (e 1) .(C18) − pL p − R pL p − & p>0 ( ' & p>0 ( ' ! !

Put them together and reduce R , Y+ by hitting on the vacuum, we have ±

ikF t I(r, t)=Z1Z2e 0 ψR† (x)W W+Y ψR(y) 0 . (C19) ⟨ | − − | ⟩

Everything is now quite clear. Exchanging W+ and Y gives a factor Z4.PassingY − − through ψR† (x)givesafactorZ5.PassingW through ψR† (x)andW+ through ψR(y) − altogether give a factor Z0. Also define the free-field equal-time correlation function as 16

ip(y x) Z3 0 ψR† (x)ψR(y) 0 =1/L p k e − .OurfinalresultforI(r, t)is ≡⟨ | | ⟩ ≤ F

% ikF t I(r, t)=Z0Z1Z2Z3Z4Z5e , (C20) where

2π iEpt ipr iEpt Z =exp (cosh(θ ) 1)(1 + e )(1 e − ) , (C21) 0 − pL p − − & p>0 ' !

2π iEpt iEpt ipr Z =exp (cosh(θ ) 1)(e− 1)(e e ) , (C22) 4 − pL p − − − & p>0 ' !

2π ipr iEpt Z =exp e (e− 1) . (C23) 5 pL − & p>0 ' ! We can further simplify the result as

1 I(r, t)=exp( Q(r, t))Z eikF t = exp (ik t ipr Q(r, t)) , (C24) − 3 L F − − p k !≤ F where

2π ipr iEpt iEpt Q(r, t)= (e 1) isin(pr)e− + 1 e− cos(pr) cosh(2θ ) . (C25) pL − − − p p>0 ! , , - - One can easily check that this result reduces to Mattis and Lieb’s [3]whent =0andreduces to the free-field correlation function when θp =0.

Acknowledgements

The author thanks Xiao-Tian Zhang for fruitful discussion and his private note [10]on Giamarchi’s textbook [6].

[1] S. Tomonaga, Progr. Theoret. Phys. 5, 544 (1950). [2] J. M. Luttinger, J. Math. Phys. 4, 1154 (1963). [3] D. C. Mattis and E. H. Lieb, J. Math. Phys. 6, 304 (1965). [4] F. D. M. Haldane, J. Phys. C: Solid State Phys. 14, 2585 (1981). [5] H. Bruus and K. Flensberg, Many-body Quantum Theory in Condensed Matter Physics: An Introduction. (Oxford University Press, 2004) [6] T. Giamarchi, Quantum Physics in One Dimension. (Oxford University Press, 2004) 17

[7] S. Sachdev, “Compressible phases in one dimension: Tomonaga-Luttinger liquids”, http: //qpt.physics.harvard.edu/phys268/Lec3_Luttinger_liquids.pdf [8] F. D. M. Haldane, Phys. Rev. Lett. 47, 1840 (1981). [9] T. M. Apostol, Introduction to Analytic Number Theory (Undergraduate Texts in Mathemat- ics). (Springer-Verlag, 1976) [10] X.-T. Zhang, “Note on bosonization”, private communication.