Lectures on Bosonization

C.L. Kane Dept. of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104

This document contains lecture notes from my ﬁrst two blackboard lectures given at the Boulder summer school in July 2005. The second two lectures are available as power point presentations. These lectures are meant to be a pedagogical introduction to interacting electrons in one dimension, with an emphasis on the technique of bosonization and on the Luttinger liquid.

I. A HEURISTIC INTRODUCTION TO E ω BOSONIZATION

E In this lecture we describe the 1D electron gas by exam- F ining the extreme limits of strong and weak interactions v q q in order to develop a intuitive picture of the meaning of k F -k k bosonization. F F 2kF

FIG. 1: Non interacting electrons in one dimension. (a) Dis- A. Non Interacting Electrons persion, (b) Spectrum of particle-hole excitations.

Consider non interacting spinless electrons in one di- θ /π mension, described by the Hamiltonian, a a i Z ¯h2 H † − 2 FIG. 2: A Wigner crystal and the phonon displacement coor- = dxψ (x) ∂x ψ(x) (1) 2m dinate θi.

X 2 2 ¯h k † B. Strong Repulsive Interactions: the 1D Wigner = c c . (2) 2m k k Crystal k

In the ground state the electronic states are filled to the Non interacting electrons are trivial because all of the Fermi energy EF , or equivalently the Fermi momentum energy is kinetic. In a real system, there is a competi- kF , as shown in Fig. 1a. In terms of the 1D electron tion between the kinetic energy and the potential energy density n0 we have of interaction. Here we will consider the opposite limit where the potential energy dominates. In this limit, the kF = πn0. (3) lowest energy configuration of the electrons will be a crystal, with “lattice constant” a =1/n0, as shown in Fig. The velocity of the states at the Fermi energy vF = 2. The low energy excitations are then phonons. To de- (1/¯h)dE/dk(kF )is given by scribe the phonons we introduce a phonon displacement by writing vF =¯hkF /m = π¯hn0/m. (4) 0 a ri = r + θi. (6) The density of states at the Fermi energy (or the com- i π pressibility) is With this definition, the displacement variable θi advances by π when the crystal is displaced by one lattice ∂n/∂µ = N(E )=2/(2π¯hv ). (5) F F constant. To describe the phonons, it is useful to construct the Lagrangian: We will be interested in the low energy, long wavelength properties of this system. Consider the spectrum L = KE − PE, (7) of particle hole excitations arising from states of the form, † | i ck+qck 0 . As shown in Fig. 1b, for small q, ω the particle where the kinetic energy is hole spectrum resembles a sound mode with dispersion Z X 1 ma ω = vF q. This says that the low energy excitations of the KE = mr˙2 = dx θ˙(x)2 (8) electron gas are similar to that of a 1D elastic medium. 2 i 2π2 i This is also the case even in the presence of interactions. To see this, we will now consider the opposite extreme of where we assume that for slow variations θ may be strongly interacting electrons. treated as a continuous variable. The potential energy 2 is given by The second term describes fluctuations at the wavelength Z Z of the Wigner crystal a =1/n0 =2π/2kF , which may 1 V PE = dxdx0 V (x − x0)δn(x)δn(x0)= dx 0 δn(x)2 be characterized by a slowly varying complex number 2 2 n2kF (x) which gives the amplitude and the phase of the (9) oscillation at 2kF , where for the second equalityR we have assumed a short X iqx 2ikF x ranged interaction, withV0 = dxV (x). For a coulomb nqe = n2kF (x)e + c.c. (17) 2 interaction V (x)=e /x screened at large distances by a q∼2kF ground plane at distance R ,wehaveV =2e2 log R /a. s 0 s Since the phase of the 2k density fluctuations advances Note, that here δn(x) is the deviation of the density from F by 2π when θ increases by π we have it’s average value n0. δn can be expressed in terms of θ. θ L − θ −π ∼ 2iθ(x) To see how, note that if ( ) (0) = , then exactly n2kF (x) e . (18) one extra electron resides between 0 and L. Thus, For our present purposes, the constant of proportionality ∂ θ(x) is unimportant. δn(x)=− x (10) π For a perfect crystal h i6 Combining the two terms we now have n2kF =0, (19) Z ma V or equivalently L = dx (∂ θ)2 − 0 (∂ θ)2 . (11) 2 t 2 x h i6 2π 2π lim n2kF (x)n−2kF (0) =0. (20) x→∞ This may be rewritten as This leads to a delta function Bragg peak for scatter- Z ing. For a classical crystal, thermal fluctuations of the ¯h 1 2 2 L = dx (∂tθ) − vρ(∂xθ) , (12) phonons give rise to the Debye Waller factor, which re- 2πg vρ duced the amplitude of the Bragg peak. We will now see that quantum fluctuations are similar, and give rise to where we have introduced the interaction parameter a logarithmically divergent Debye Waller factor, which r π¯hv (barely) destroys the crystaline order. g = F (13) There are many techniques for computing (19) and V0 (20). A powerful method is to use the imaginary time and the phonon velocity path integral. This is done by writing the partition func- r tion as an integral over all trajectories θ(x, τ), where τ = it is imaginary time. For zero temperature (which V0 vF vρ = = . (14) we will focus on here) τ runs from −∞ to ∞.Wethus ma g write, Z This Wigner crystal approach to the problem should be − valid in the strong interaction limit, g 1. However, Z = D[θ(x, τ)]e S[θ(x,τ)]/h¯ (21) note the similarity of the low energy elastic theory for the phonons of a Wigner crystal with the “sound mode” where the action S is given by for non interacting electrons. We shall see that Eq. 12 Z Z 1 1 1 2 2 remains correct for weaker interactions and even for non S = dτL[θ(x, τ)] = dxdτ (∂τ θ) + vρ(∂xθ) . interacting electrons. Eq. 13, however, will need to be ¯h 2πg vρ modified for weaker interactions. (22) It is known that in low dimensions long range order is The action is particularly simple in Fourier transform because the Fourier components decouple, destroyed by fluctuations (either thermal or quantum). It is instructive to see how quantum fluctuations destroy 1 X ω2 S = + v q2 |θ(q, ω)|2. (23) the long range crystaline order at zero temperature. To 2πg v ρ describe the crystaline order, consider a Fourier decom- q,ω ρ position of the electron density, It is now straightforward to compute (19) and (20). First, X X iqx iqx we have n(x) ∼ nqe + nqe + .... (15) Z 2iθ(0,0) 1 2iθ −S q∼0 q∼2kF hn (x, τ =0)i∼he i = D[θ]e e . (24) 2kF Z The first term gives the long wavelength fluctuations of This can be evaluated by doing a simple Gaussian inte- the density: gral over θ(q, ω). However, there is a very useful trick for X ∂ θ(x) evaluating such expectation values, which is to write n eiqx = n − x . (16) q 0 − 1 h 2i π i(2θ) 2 (2θ) q∼0 he i = e . (25) 3 This can be easily remembered by expanding to second 2 order in θ and noting that the term linear in θ has zero n expectation value. For a harmonic theory (quadratic in 0 θ), this relation is exact. -2 The expectation value in the exponent can then be ~x found by noting that for a Gaussian exp[−Cx2/2] we have hx2i =1/C. Thus, Z X dqdω πgv hθ(x = τ =0)2i = h|θ(q, ω)|2i = ρ . (2π)2 ω2 + v2q2 q,ω ρ (26) x/a Changing variables to (q0,q1)=(ω/vρ,q) the integral becomes isotropic and may be written 1 2 3 4 Z ∞ 2πqdq πg g L FIG. 3: Pair correlation function for non interacting electrons. 2 2 = log . (27) 0 (2π) q 2 a The oscillations indicate power law crystaline order. Thus we see that the Debye Waller factor due to quantum fluctuations in one dimension is log divergent. We have C. Bosons: The 1D superfluid cut off the divergence with the system size L and the (electron) lattice constant a. We thus conclude that In our discussion of the Wigner crystal limit we made 2 no use of the fact that electrons are fermions. Since in hn i∼e−2hθ i ∼ (a/L)g → 0 (28) 2kF that limit the electrons never exchange the statistics are A similar calculation for the correlation function (20) in fact irrelevant. Let us now suppose, for the moment, gives that we have bosons instead of fermions. Then, in addition to the Wigner crystal limit, one may consider the − 1 h − 2i h i∼ 2 (2θ(x) 2θ(0)) ∼ 2g limit of Bose condensation. n2kF (x)n−2kF (0) e (a/x) (29) For a Bose condensate, the ground state boson creation Thus, the crystaline correlations decay at long dis- operator acquires a finite expectation value, √ tances, but only as a power law. For strong interactions h †i iφ g 1 the exponent is very close to zero, so that the b = ne (32) system is “almost” a Wigner crystal. Long wavelength fluctuations of the phase of the bose It is instructive to compare this result with the corre- condensate are described by sponding quantity for non interacting electrons. In that Z case it is straightforward to show that g 1 L = dx (∂ ϕ)2 − v (∂ ϕ)2 . (33) 2π v t ρ x h i∼ 2 ρ n2kF (x)n−2kF (0) 1/x . (30) This describes similar elastic distortions to Eq. 12, which In this case the long range correlations are a conse- propagate at velocity vρ. We will see later that the g in quence of Fourier transforming the occupation number Eq. 33 is in fact the same as the g in (12), so that the n(k) which has a sharp step at kF . This leads to the two Lagrangians are dual to each other. familiar Friedel oscillations. These oscillations can also From (33) it can immediately be seen that quantum be seen in the pair correlation function fluctuations in ϕ (barely) destroy the superfluid order, † † h i 2 − 2 2 † ψ (x)ψ (0)ψ(0)ψ(x) = n0(1 sin kF x/x ) (31) hb (x)b(0)i∼1/x1/2g. (34) shown in Fig. 3, which describes the probability of find- Thus, there is power law superfluid order. In the Wigner ing two electrons separated by a distance x. The power crystal limit g<<1 the phase fluctuations are very law decay of the oscillations suggests that non interacting strong and the superfluid correlations die away quickly. electrons are also described by (12) with g = 1. (12) is On the other hand in the opposite limit g →∞the phase indeed more general than the Wigner crystal limit, how- correlations are nearly long range, but the crystaline cor- ever, our formula (13) for g will have to be modified. For relations decay quickly. a Wigner crystal g 1, the peaks at x = na in the pair The duality between θ and ϕ can further be explored correlation function have a similar power law decay, but by noting that the boson creation operator√ increases with a smaller exponent. the density by one unit, so that [n(x), n exp iϕ(x0)] = Before describing a more rigorous approach to δ(x − x0). This implies the canonical commutation rela- bosonization, we will desribe another simple limit tion between number and phase: which will contribute to our intuitive understanding of bosonization. [n(x),ϕ(x0)] = iδ(x − x0) (35) 4

If we write n = n0 + ∂xθ/π, then this may be rewritten 0 0 E [∂xθ(x)/π, ϕ(x )] = iδ(x − x ). (36) In this sense, θ and ϕ are dual variables. The above analysis may be related to fermions by per- forming a Jordan Wigner transformation, which converts bosons to fermions in 1 dimension. We thus write a L R fermion creation operator ψ† as

R x † iπ n(x) † ψ (x)=e −∞ b (x). (37) k The term with the exponent is the “Jordan Wigner string” which counts the number of particles NL to the left of x and mulitiplies the operator by (−1)NL .Itis clear that this converts the boson commutation relation among b(x) into fermion anticommutation relations for ψ(x). Noting that n(x) ∼ n0 + ∂xθ/π we may write the fermion creation operator as ψ†(x) ∼ ei(θ(x)+ϕ(x)). (38)

The fermion operator thus includes both the dual “su- FIG. 4: The Luttinger model, with linear bands extending to perﬂuid” and “crystaline” variables. −∞. Equations (12, 33, 38), which we have “derived” by following our intuition are the hallmarks of bosonization and the Luttinger liquid. In order develop a (slightly) This Hamiltonian is appropriate for describing low energy more rigorous understanding, which will correctly pre- (E EF ) and long wavelength q 1/a properties, dict g and allow for generalizations to include spin and which are independent of the states deep below the Fermi other degrees of freedom it will be necessary to return to level. non interacting electrons, where the exact bosonization It is useful to consider a model for which the linear rep- transformations can be cleanly formulated. resentation (39) is exact. This is known as the Luttinger model, and as described in Fig. 4, the linear dispersion extends between energies ∞. Since the states deep be- II. BOSONIZATION OF NON INTERACTING low the Fermi level are inert, the Luttinger model is ex- ELECTRONS pected to have the same low energy behavior as the real electron gas. The bosonization tranformation is exact for In this lecture we return to non interacting electrons the Luttinger model. and describe the exact bosonization mapping for chiral The Hamiltonian (41) is the 1+1 dimensional analog fermions. of the massless Dirac equation, familiar from particle physics. As in the 3+1 dimensional case, (41) exhibits chiral symmetry, which is manifested by the independent A. The Luttinger Model conservation of the number of right and left moving electrons, NR, NL, given by Now we return to non interacting electrons and fo- Z cus on the low energy excitations. For this purpose we † Na = dxψaψa (42) linearize the dispersion (1) about the Fermi energy EF , which we deﬁne to be 0. X h i Thus in addition to the total charge NR+NL, the “chiral” † † H − − charge N − N is also conserved. = vF q ckF +qckF +q c−kF +qc kF +q . (39) R L q

Defining continuum electron operators B. Bosonization of Chiral Fermions X iqx ψR,L(x)= e ckF +q (40) q We will now consider a single branch of chiral fermions ψR (or equivalently ψL. This will describe “half” of the the Hamiltonian density becomes one dimensional electron gas. In addition to providing a h i building block with which one can construct the one di- H − † − † = ivF ψR∂xψR ψL∂xψL (41) mensional electron gas (also allowing for generalizations 5 to include spin, etc.) chiral fermions are directly relevant to edge states in the quantum Hall effect. We begin by defining the chiral density operator, k+q † n (x)=:ψ (x)ψ (x):. (43) R R R +q Since the electronic states in the Luttinger model extend k to energy −∞, the number of electrons is infinite. There- fore, it only makes sense to talk about deviations in the density from the density of the ground state. This is the purpose of the colons, which indicate that the operator is to be normal ordered, such that the operators -q which annihilate the ground state are always placed on the right side. Equivalently, normal ordering subtracts off the (infinite) vacuum expectation value, FIG. 5: A particle hole excitation created by the chiral density operator. † † −h † i : ψR(x)ψR(x):=ψR(x)ψR(x) ψR(x)ψR(x) 0. (44) It is useful to consider the Fourier transformed density With a bit more work one can convince oneself that operator, this argument works when acting on excited states as Z well as the ground state. In addition, it is clear that X 0 − † [n ,n 0 ] = 0 when q + q =0.6 n = dxe iqxn (x)= : c c : (45) Rq Rq Rq R Rk+q Rk The above properties of the chiral density operators k invite the following transformation to boson creation and where for q =6 0 the normal ordering is unimportant. annihilation operators for q>0: Where necessary, we will assume below that we are in a † 1/2 system of length L with periodic boundary conditions, so bRq =(2π/qL) nRq (48) that the discrete values of q are qn =2πn/L. 1/2 bRq =(2π/qL) nR−q. (49) A fundamental relation obeyed by the chiral density operator is known as the Kac Moody commutation rela- † These obey the Bose commutation relations, [bRq,bRq]= tion 1. Remarkably, the Hilbert space generated by the bo- † identical qL son operators bRq is to the Hilbert space of all [n − ,n ]= . (46) R q Rq 2π excited states of the Luttinger model for fixed particle number. To complete the description we must also in- This can be derived by using (45) and the fermion clude a q = 0 fermion number operator NR which has Pcommutation relations to write the commutator as integer eigevalues between −∞ and ∞. † − † k cRkcRk cRk+qcRk+q. There is a subtlety here, Using the fact that the compressibility of the chiral though, because each of the terms is formally infinite. Fermi gas is ∂n/∂µ =1/(2π¯hvF ) we may write the en- This prevents one from naively changing variables in the ergy as a function of density as first term to get zero. The correct approach is to rewrite Z † † † c c =: c c :+hc c i . The normal ordered 1 2 Rk Rk Rk Rk Rk Rk 0 HR = dxn (x) (50) terms are then perfectly finite, and the variable change 2∂n/∂µ R can be safely done, so the two terms cancel. The difference of the vacuum expectation values, however, gives " # X qL/(2π). π¯hv † = F N 2 + b b (51) An alternative way of seeing this, which emphasizes L R Rq Rq the physical meaning of the chiral density operators is q>0 to consider how the operators act on the ground state. As discussed above the Hilbert spaces are identical in the | i nRq 0 (for q>0) is a superposition of states with a fermion or boson representations. Within this Hilbert single particle hole excitation at k and k + q where the space the boson Hamiltonian (51) is identical to the − hole has momentum k between q and 0 (see Fig. 5). On fermion Hamiltonian, | i the other hand, nR−q 0 = 0 because in the ground state X there are no empty states with momentum less than an H † R = ¯hvF kcRkcRk. (52) − | i occupied state. Moreover, when nR q acts on nRq 0 the k only thing it can do is put the excited particle back into the hole. Thus we conclude that It is often useful to express the chiral density in terms Z of a chiral phase operator, 0 Ldq qL [nR−q,nRq]|0i = |0i = |0i. (47) −q 2π 2π nR(x)=∂xφR(x)/(2π) (53) 6

φ (x) However, this identification is not precise, and it is not π R possible to define the numerical ratio between xc and a. Given this regularization scheme, the appropriate pref- −1/2 x actor is (2πxc) . The second subtlety has to do with carefully treating the q = 0 part of φ. This necessitates the introduction of a Klein factor. The Klein factor κ is the operator which −π connects the ground states of different charge sectors: κ |Ni0 = |N 1i0. (58) iφ (x0) 0 FIG. 6: e R introduces a 2π kink in φR(x)atx = x . In some situations the Klein factors can be ignored, and in the literature you will find many instances where they are not included. However, they are necessary when there In terms of φR the Kac Moody commutation relation is is more than one species of fermion, since they are respon- ∂xφR(x) 0 0 sible for enforcing the anticommutation relation between [ ,φR(x )] = iδ(x − x ). (54) 2π different species of fermion operators κiκj = −κjκi. Thus, our final expression for the fermion operator is This suggests that ∂xφR(x) and φR(x) are canonically + † κ conjugate variables similar to x and p in elementary √ iφR(x) ψR(x)= e . (59) quantum mechanics. This indicates how to write the La- 2πxc grangian, which is usually written as L = pq˙ − H(p, q). In the present case we find (using Eq. (50)) that As an application let us compare the single electron Z Green’s function computed in the fermion and boson representations, H vF 2 = dx(∂xφR) (55) 4π † G(x, τ)=hTτ ψ(x, τ)ψ (0, 0) i (60) and Here Tτ indicates time ordering in imaginary time, in- 1 cluding the appropriate minus sign for the Fermi opera- L = − ∂ φ [∂ φ + v ∂ φ ] (56) 4π x t R F x R tors: † † † Readers will notice that there is a factor of 2 difference Tτ [ψ(τ)ψ (0)] ≡ θ(τ)ψ(τ)ψ (0)−θ(−τ)ψ (0)ψ(τ). (61) in the first term from the expression from elementary In the Fermion representation, this can be easily com- mechanics. This is because unlike p and q, ∂xφR and φR are not independent variables. Below we will transform puted by inserting a complete set of states between the fermion operators. One then finds to sums and differences of φR and φL (See Eq. 74 below). In terms of those new variables the conjugate variables X k(ix−vF τ) − − − are independent, and there is no tricky factor of 2. G(x, τ)= e (θ(k)θ(τ) θ( k)θ( τ)) (62) Finally, we wish to express the fermion creation opera- k tor in terms of the boson fields. This can be accomplished 0 0 by noting that since [φR(x),φR(x )] = iπsgn(x − x ) 1 1 0 0 0 = . (63) we have [φR(x), exp iφR(x )] = πsgn(x − x ) exp iφR(x ). 2π vF τ − ix 0 This means that the operator exp iφ(x ) displaces φR(x) 0 In the boson representation the Klein factor is unim- to introduce a 2π “kink” at x = x as shown in Fig. 6. + − − + 0 portant, since κ κ = κ κ = 1. However, we must This corresponds to an additional charge e at x . explicitly include the minus sign in the definition of the In order to be more precise, we must address two Fermion time ordering. We then have technicalities. The first has to do with the prefactor of † h i exp iφR.√ The prefactor is clearly necessary since ψ has sgn(τ) h i(φR(0,0)−φR(x,τ)) i units 1/ L. However, it depends on the method for reg- G(x, τ)= Tτ e . (64) 2πxc ularizing the theory at large k. The simplest method is to introduce an exponential cutoff, so that sums on k are The expectation of the boson operators can be written replaced by as exp[−C] with X X 1 → e−|k|xc . (57) C = hT (φ(x, τ) − φ(0, 0))2 i. (65) 2 τ k k Using the Lagrangian (56) with ∂t → i∂τ and including The parameter xc should really be interpreted as an in- the exponential regularization this gives finitesimal convergence factor which regularizes formally Z dqdω 2π divergent integrals. xc is often interpreted as a short − − −xc|q| = 2 (1 cos(ωτ qx)) e . (66) distance cutoff a of order the inverse electron density. (2π) q(iω − vF q) 7

The integral over ω can be done by contour integration, and the ϕ representation, and after some manipulation gives Z 1 1 ∞ L[ϕ]= (∂ ϕ)2 − v (∂ ϕ)2 . (76) dq − − − t F x 1 − e q(vF τ ix)sgn(τ) e xcq (67) 2π vF 0 q These equivalent representations in terms of the dual variables θ and ϕ show that non interacting electrons v τ − ix + x sgn(τ) = log F c . (68) are “self dual”. xcsgn(τ) The fermion creation operators have the form

+ Combining the terms in (64) we thus see that the xc and † κ sgn(τ) factors cancel, and the formula (63) is reproduced ψ = √ ei(ϕ+θ), (77) R 2πx exactly for x → 0. + c c † √κ i(ϕ−θ) ψL = e . (78) 2πxc

C. Combine the left and right movers Note the similarity of this description with the description in section 1. The virtue of this bosonized description We now consider both left and right movers, which are is that it is now quit simple to incorporate interactions described by the Lagrangian density into the theory. This will be done in the next lecture.

L = LR + LL (69) with III. REFERENCES

1 The following is a vastly incomplete selection of refer- LR = − ∂xφR (∂tφR + vF ∂xφR) (70) 4π ences for bosonization and the Luttinger liquid. A few orignal articles: 1 L = − ∂ φ (−∂ φ + v ∂ φ ) . (71) L 4π x L t L F x L F.D.M. Haldane, J. Phys. C 14, 2585 (1981). A classic and very readable article where Haldane coins the term We now deﬁne new variables, Luttinger liquid.

φR = ϕ + θ (72) C.L. Kane and M.P.A. Fisher, Phys. Rev. B 46, 15233 φL = ϕ − θ (73) (1992). Long article focusing on the single impurity problem. In terms of these variables the total density is n = Luttinger Liquid and Bosonization Review Arti- (∂xφR − ∂xφL)/(2π)=∂xθ/π. In terms of these new variables we have cles 1 v M.P.A. Fisher and L.I. Glazman, cond-mat/9610037. L = ∂ θ∂ ϕ − F (∂ θ)2 +(∂ ϕ)2 (74) π x t 2π x x Nice review emphasizing physical clarity over mathemat- ical rigor. This has the form pq˙ − H(p, q). The ﬁrst term shows D. Senechal, cond-mat/9908262. More mathematically that ∂xθ/π and ϕ are canonically conjugate variables. “Integrating out” either θ or φ yields two equivalent dual oriented review which draws connections with ﬁeld the- representations of the non interacting electron gas: the θ ory. representation, A.M. Chang, Rev. Mod. Phys 75 No. 4, 1449 (2003). Review which describes Luttinger liquid theory as well 1 1 2 2 L[θ]= (∂tθ) − vF (∂xθ) , (75) as numerous experiments. 2π vF