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Tomonaga-Luttinger Liquid Theory

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Tomonaga-Luttinger Liquid Theory 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 <p k p !′ ≤!′ ′ !′≥ ′ We therefore need to treat nine commutators separately. It is not hard to see that nonzero 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 <p p k<0,k <0' −!′ ′ − ≤! ′ † † + [aRkaRk+p,aRk aRk′ p′ ]. (12) ′ − k p,k 0 ≥−!′≥ Each of the four-fermion operators expanded from this expression can be safely evaluated by exchanging the two operators in the middle. One should finally find that [ρ (p),ρ ( p′)] = R R − pL/(2π)δp,p′ which is the correct value in the space of filled Fermi sea. 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.
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