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Lectures on Bosonization 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- = dx (x) @x (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 crys- tal, 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 ad- vances 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 con- struct the Lagrangian: We will be interested in the low energy, long wave- length 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, ! the particle where the kinetic energy is hole spectrum resembles a sound mode with dispersion Z X 1 ma ! = 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 !2 S = + v q2 |θ(q; !)|2: (23) the long range crystaline order at zero temperature. To 2πg v ρ describe the crystaline order, consider a Fourier decom- q;! 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; !). 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 dqd! πgv hθ(x = τ =0)2i = h|θ(q; !)|2i = ρ : (2π)2 !2 + v2q2 q;! ρ (26) x/a Changing variables to (q0;q1)=(!=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 ad- dition 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.
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