arXiv:cond-mat/9909295v2 [cond-mat.str-el] 21 Mar 2000 oe.Freege mle hntebandwidth the than smaller Tomonaga-Luttinger same energies the the For of have that would the model. as con- for models physics LL, and gapless of low-energy spin 1D name with the that LL coined jectured who the Haldane, (this for LL). pole both spinless quasiparticle true elec- no is the shows consequence property a function as Green’s and LL tron the of excitations tary em iud:Lna quasiparticles Landau liquids: Fermi iesoa oessc steHiebr hi,the chain, Heisenberg one the gapless and as for Hubbard such liquids class universality models Luttinger a so-called dimensional constitute of spin-charge LL generic as (LL): known is is and model separation Hubbard the of property ayectto ftefatoa unu alfli at fluid Hall quantum elemen- fractional the the filling charge of is the excitation which probably tary is quasiparticle, example Laughlin famous sys- one-third most the The of particles the constituent tem. of the part exis- of only numbers the carrying quantum is excitations elementary that of fractionalization, tence is systems correlated nabatflsto htnieexperiments noise shot of set beautiful a in rtn otnu o ∆ gen- spinons for (the two continuum magnon by a the replaced is erating ex- model: boson) elementary Heisenberg Goldstone the 1D usual also the is later of spinon Takhtajan citation same and Faddeev the as that spinon. known showed the into dynamics, and split holon independent electron the the with of excitations spin and two charge the that vealed hsc shwvrfudi n ieso:teexact the model dimension: Hubbard one the matter in of condensed found solution in however is fractionalization physics of example liest n ftems tiigpoet fsm strongly some of property striking most the of One ν 1 = ASNmes 11 m,71.27+a , th Pm and 71.10 holon Numbers: describe. the PACS to LL: able a are in we longer which any states fractional realized not is separation o h utne iudwt pni lodrvd ntepre the In quas derived. 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Luttinger the in Excitations Fractional 8 S utne iud r non- are liquids Luttinger . excitations 1 = 3 aoaor ePyiu e oie,asc´ uCNRS associ´e au Solides, des Physique de Laboratoire yBteAnsatz Bethe by 9 .V hm .GbyadP Lederer P. and Gabay M. Pham, K.-V. r o elemen- not are 2 h ear- The . 6 , 7 Spebr1,2018) 18, (September Universit´e Paris–Sud 10 ra,France Orsay, 91405 This . 10 4 – re- 12 , 1 eoiyfrcletv oe and modes collective for velocity iain naL a enlmtdt h pnni the in spinon the to ex- limited fractional been of has of study LL case the a to in spectroscopic citations application the alterna- stress yet, an does is state- aspects: which theory this bosonization field below to Conformal show just tive we is incorrect. As is hamiltonian ment phonons. effective acoustic a the of of since that excitations the phonons, relevant in are physically only LL more, the sometimes What’s is that it the stated formalism, knowledge. to bosonization the LL authors’ of the framework the in the of excitations in fractional best sys- of no states exists study there fractional tematic liquids, of Luttinger several (Bethe existence of solutions spectrum the exact that show fact the Ansatz) bosonization despite of and formalisms CFT, or the through established well system. the of compressibility the to eslt ls stestof set the uni- invari- liquid as conformal Luttinger class the with identify versality to dimen- theories allowed critical has two this 1) describes ance; + 1 which con- (or by (CFT) provided bosonic sional theory is free LL field a the of formal on be- that perspective straightforward Another is are field. hamiltonian the computations effective from the the stems cause all bosonization that of fact popularity and success h atrmdli xdpito h renormalization the of point fixed a (RG) is group model latter the aitna eciigfe phonons acoustic free gaussian describing a hamiltonian into model Tomonaga-Luttinger the nt-ieaayi fLtigrlqisfis nrdcdby introduced first Haldane liquids Luttinger of analysis finite-size free gaussian the hamiltonian towards RG boson under flow which models all hrceie ytefloigparameters: following theory the phenomenological a by as characterized described be can theory LL e,atog h Ldsrpini upsdyquite supposedly is description LL the although Yet, o egv natraierepresentation alternative an give we ion SU hBteAst hc rdcssimilar predicts which Ansatz Bethe th gt:i h o-nrylmt efind we limit, low-energy the in ights: osrcino h e spectroscopy new the of construction r 17 ec famgei ed spin-charge field, magnetic a of sence , 18 2 ymty hc stestainrelevant situation the is which symmetry, (2) l nesodi em fphonons of terms in understood lly nteasneo pnrotational spin of absence the in r sntasmo ngnrl trial a general; in semion a not is pnnaete elcdb new by replaced then are spinon e 13 , 8 n1,bsnzto lost transform to allows bosonization 1D, In . ntrso h asinhmloin the hamiltonian, gaussian the of terms In . prils h oo operator holon The iparticles. 16 a quasiparticle-quasihole nal F a loe ofraiethe formalize to allowed has CFT . c CFTs 1 = K hc sproportional is which 14 h considerable the ; 15 ..testof set the i.e. , u hc sa is which 19 for the Heisenberg chain . Fractional excitations must and the ∆Sz = 1 continuum, is therefore incidental. The exist in a Luttinger liquid if Bethe Ansatz is correct but important lesson to be learned is that in the presence of as far as the authors are aware, the characterization of an Ising anisotropy, the ∆Sz = 1 and ∆Sz = 0 continua these very unconventional fractional states, through ei- may involve different fractional spin states: in the first ther bosonization or CFT, is mostly terra incognita as case we have spinons20, but in the second case the spinon the following list of issues may show: identification is not always correct. What happens in the 1) ∆S = 1 excitations for the Heisenberg chain form a case of an arbitrary anisotropy will be dealt with in this continuum of pairs of spinons. When an Ising anisotropy paper. is introduced, in the massless regime (with obvious no- 2) The holon appearing in the exact solution of the 5 tations: Jz Jx (= Jy) ) the continuum still exists Hubbard model is a spinless charge one excitation . and evolves| | smoothly ≤ as a function of the anisotropy The issues raised for the spinon (operator, wavefunction, 7 ∆= Jz/Jx . The continuum is again ascribed to pairs of statistics) extend naturally to the holon. spinons20. It is intuitively clear that these spinons should 3) Spin-charge separation is an asymptotic property of be in some sense deformations of the SU(2) spinon. We the Hubbard model valid in the low-energy limit. When will derive in this paper creation operators for these non- a magnetic field is applied Frahm and Korepin found that SU(2) spinons using the bosonization method. spin-charge decoupling was not realized even in the low- In the case of SU(2) symmetry, trial wavefunctions energy limit25. In this paper we derive the new excita- for these spinons can be found by making use of the tions replacing the holon and the spinon. exact solvability of the Haldane-Shastry (HS) chain21. 4) In the context of the Calogero-Sutherland (CS) The HS model shares the properties of the Heisenberg model26 the existence of fractional excitations similar to chain: it is a gapless SU(2) symmetric spin chain with Laughlin quasiparticles was suggested27. In a variant of a continuum of spinon excitations. Its ground state and the standard LL known as the chiral LL used to describe spinon wavefunctions22 are remarkably similar to those edges of a FQHE sample, Laughlin quasiparticles do ap- one would write for bosonic Laughlin states at filling pear but the existence of such states follows from that of ν =1/2: the same excitations in the bulk28. In the CS model the proposal was triggered by the similarity of the ground 2 Ψgs(x1, .., xN )= (zi zj ) , (1.1) state with that of the 2D Laughlin wavefunctions and by − ifermions and bosons): we will show formalism similar to that of Bethe Ansatz can be intro- that the statistics is affected when an anisotropy is intro- duced and for the restricted case of rational couplings duced. λ = p/q special selection rules are found for the dynam- A continuum is also found by Bethe Ansatz for ∆Sz = ical structure factor: p pseudo-holes (a pseudo-hole is 0 transitions24. Low-lying excitations are described in a hole in the Fermi sea of pseudo-momenta) must be that approach as two-strings states in the string for- accompanied in any excitation by q pseudo-particles29. malism customary to Bethe Ansatz. This description For a charge ( 1) pseudo-particle, this means one has is similar to that given for the ∆S = 1 continuum of a charge 1/λ =−K for the pseudo-hole. In the interpre- the isotropic chain. In the latter case it is quite clear tation of those selection rules it is proposed to view the that a spin one-half should be ascribed to each of the CS model as a gas of non-interacting pseudo-particles (pseudo) ”hole” states in the string, which leads to the with anyonic statistics πλ and one rewrites the ground λ spinon interpretation since each state should contribute state as an anyonic wavefunction (zi zj) . The 6 i

2 There are however several limitations to those views. per. In subsection IB we give a short review of the LL Firstly, these considerations are only valid for rational physics in order to set the notations used throughout the couplings (the pseudo-particle selection rules can not be paper; the issues discussed in the present subsection IA extended to irrational λ): the physics of the CS model will be amplified in subsection I B 2 in which we present is by contrast completely continuous with the coupling the standard view on excitations of the LL. In the next and does not discriminate between rational and irrational section II we will show that an alternative eigenstate ba- couplings. The impossibility to describe rational and ir- sis can be built: that quasiparticle basis allows a natural rational couplings on the same footing means the repre- discussion of fractional states. In section III, we will gen- sentation is not adequate. Secondly a disymmetry be- eralize our analysis to the LL with spin. When a mag- tween particle and hole excitations is introduced. netic field is added on to the Hubbard model, spin-charge In a parallel strand of ideas, Laughlin quasiparticles separation no longer occurs25. The standard spin-charge were also proposed in studies of transport in a LL. The separated Luttinger liquid theory is not applicable any basis of the argument is that for a LL with an impurity more. We will introduce in subsection III B a general potential a charge K and not a charge unity is backscat- framework related to the K topological matrix of Wen’s tered at the impurity location (where K is the LL pa- chiral Luttinger liquids28, which yields simple criteria of 30 rameter, i.e. the conductance of the LL) . The impurity spin-charge separation in terms of a Z2 symmetry: we potential can be rewritten as a hopping potential for a will be able then in subsection III C to derive the frac- charge K state whose exchange statistics is πK as seen tional excitations which replace the holon and the spinon. from the commutation relations (i.e. anyonic). But for The general LL theory we have introduced will then be K =1/(2n + 1) these states are just those given by Wen applied to the Hubbard model in a magnetic field in sub- for the Laughlin quasiparticles of his chiral LL: this sug- section III D in which we explain the relation between gests to identify these states as Laughlin quasiparticles. our approach and the formalism of the dressed charge The main difficulty in that argument is that it relies on matrix due to Frahm and Korepin. Let us mention that the introduction of an impurity potential in the LL: this section II of this paper expands on a short version which obscures the question of the existence or not of a Laughlin contained results in the case of a spinless LL32, whereas quasiparticle in the pure non-chiral Luttinger liquid. In part III presents totally new material. summary what is missing is a proof that states similar to Laughlin quasiparticles might be exact eigenstates of the LL boson hamiltonian (i.e. of the RG fixed point in the B. The Luttinger liquid. low-energy limit). The existence of Laughlin quasipar- ticles for the non-chiral LL for arbitrary couplings must 1. Notations. then be considered at this point as an unproved conjec- ture. This section will define the notations employed The long list of issues we have brought up in points throughout the paper. We exclusively deal with Lut- (1 4) above should convince the reader that a thorough − tinger liquids and therefore when considering some spe- discussion of fractional excitations for Luttinger liquids cific models such as the Heisenberg spin chain or the Hub- within the formalisms of bosonization or CFT remains bard model we implicitly assume that we are working in to be done. This is what motivated us to re-examine in the LL part of their phase diagrams. The whole physics that paper the spectrum of Luttinger liquids. We want of the LL is embodied in the following hamiltonian: to stress that although the previous examples concern in- tegrable models, the detailed physics of such integrable L u −1 2 2 models is not really our main interest: what matters for HB = dx K ( Φ(x)) + K( Θ(x)) (1.6) 2 ∇ ∇ us is the universal low-energy content of these theories Z0 and of course we will be unable to tell anything through supplemented by the so-called bosonization formulas. We bosonization on the high-energy physics. Although it work on a ring of length L. u and K are the LL param- seems to be taken for granted that the excitations of a eters. Φ can be interpreted as a displacement field for LL are the holon and the spinon on account of Bethe phonons, while Θ is a superfluid phase; indeed the parti- Ansatz studies of the Hubbard model, we are not aware cle and current densities are defined as: of any existence proof of such fractional excitations when- ever the model is non-integrable: this is so because any 1 ρ(x) ρ0 = Φ(x), (1.7) proof must resort to the universality hypothesis, that is − −√π ∇ to the LL and bosonization frameworks. The theoreti- 1 j(x)= Θ(x). (1.8) cal formalism we wish to introduce aims at bridging that √π ∇ gap by focusing on the universal structure of fractional excitations of Luttinger liquids through the bosonization Renormalized current: Actually j(x) is a bare current method. density which corresponds to the correct one only in the The structure of the paper will be as follows: section non-interacting case K = 1: the continuity equation I is an introduction to the topics considered in the pa-

3 shows that the correct current density is renormalized Bosons : J even integer, (1.18) and is Fermions : Q J even integer . (1.19) − jR(x)= uKj(x) (1.9) Both Q and J are integers. The zero modes are some- times extracted from the definition of the fermion opera- (the Fermi velocity has been set to unity). We will discuss tor which defines the U± operators, first built by Heiden- in section II the meaning of such a renormalization. reich and Haldane for the Tomonaga-Luttinger model31,8: The particle operators for bosons and right and left moving fermions respectively are given as: U = exp i√π (Θ Φ ) . (1.20) ± 0 ± 0

ΨB(x)=:exp i√πΘ(x):, (1.10) It will be useful to consider the commutation properties Ψ (x)=:exp i√π (Θ(x) Φ(x)) : exp ik x, (1.11) of the following operators: F,R − F ΨF,L(x)=:exp i√π (Θ(x)+Φ(x)) : exp ikF x (1.12) V (x) =: exp i√π (αΘ(x) βΦ(x)) : . (1.21) − α,β − − (in the following we will assume that these operators are Using Campbell-Haussdorf formula, one finds: normal ordered). kF = πN0/L is the Fermi momentum where N0 is the number of particles which is fixed by Vα,β(x)Vα,β (y) −iπαβ sgn(y−x) the chemical potential. Θ and Π = Φ are canonical = Vα,β(y)Vα,β (x)e (1.22) conjugate boson fields: ∇ where sgn(x) is the sign function, which shows in partic- [Θ(x), Π(y)] = iδ(x y). (1.13) ular that Ψ (x) is a fermionic operator. We define the − F exchange statistics of an operator per: The zero modes of the charge and current density are respectively: O(x)O(y) = O(y)O(x)exp iθ sgn(y x). (1.23) L L 1 − − N = N + Q = ρ(x)dx = N Φdx, (1.14) 0 0 − √π ∇ For instance θ = π for fermions. Z0 Z0 L L 1 Jb = j(xb)dx = Θ(x)dx. (1.15) √π ∇ Z0 Z0 2. Excitations. b Q has integral eigenvalues as befits a charge operator; Until the work of Heidenreich31 and subsequently of in the bosonization mapping, the charge quantization is Haldane8, the only excitations considered in the gaussian takenb into account by the topological quantization of the 14 L model were the bosonic phonon (or plasmon ) modes. phase field Φ. Similarly, since 0 j(x)dx is a closed line But the hamiltonian contains a second part correspond- integral (around the LL), it is a quantized number: this is R ing to the energies of states with non-zero charge or cur- just the topological quantization of the superfluid phase; rent with respect to the ground state. In reciprocal space, the normalization of the fields have been chosen so that the gaussian hamiltonian becomes: J is an integer. For fermions, Q = N+ + N− and J = u N+ N− where N+ and N− are respectively the (integral) −1 2 HB = K ΠqΠ−q + Kq ΘqΘ−q number− of (bare) electrons added to the ground state at 2 b b b q6=0 the right and left Fermi points. The construction we have X reviewed above is due to Haldane34. πu Q2 + + KJ 2 . (1.24) Integrating the Fourier expansions of the charge and 2L K ! current density gives: b We have split the hamiltonian intob the phonon part and √π 1 2πn the non-bosonic zero mode part. The first term can in- Θ(x)=Θ0 + Jx + Θn exp i x, (1.16) L √L L deed be rewritten as: nX6=0 b 1 √π 1 2πn H = u q b+b + (1.25) Φ(x)=Φ0 Qx + Φn exp i x. (1.17) phonon | | q q 2 − L √L L q6=0   nX6=0 X b Note that these fields are not periodic: this allows for with the phonon operators: the above mentioned topological excitations. We demand K q q that the boson or fermion operators are physical objects b = | | Θ Φ , (1.26) q 2 q − K q q and be periodic on the ring: ΨB/F (x)=ΨB/F (x + L); r  | |  this then implies the following selection rules on the K q q b+ = | | Θ Φ . (1.27) eigenvalues Q and J of the zero modes Q and J: q 2 −q − K q −q r  | |  b b 4 The second term in the hamiltonian is standard in confor- tions brought by the low-energy limit will allow a com- mal field theory; it corresponds to finite size corrections plete characterization, giving for instance easy access to to the energy when one adds particles or creates persis- statistical phases. tent currents in the Luttinger liquid. The correspond- ing states are built by means of Haldane’s U± operators which act as ladder operators in Fock space8. This (Q,J) II. FRACTIONAL EXCITATIONS OF THE part of the hamiltonian is often called in CFT a zero SPINLESS LUTTINGER LIQUID. mode part. The corresponding excitations may however carry momentum. A non zero J excitation creates indeed This section is divided as follows: first, we discuss a persistent current with momentum JkF . These states the property of chiral separation which is central to the are therefore non-dispersive since their momentum may physics of fractionalization; then, we exhibit fractional only assume the discrete values JkF . quasiparticles for the bosonic LL before turning to the The spectrum of the hamiltonian results from a convo- fermionic LL for which we will find a different set of ele- lution of plasmon excitations and of these (Q,J) excita- mentary excitations. tions as is apparent in figures (1,2): two linear plasmon branches rise from each local minimum of the energy ob- tained for the zero-mode states (Q,J). It is important A. Chiral separation. to note that there are selection rules on the allowed val- ues of (Q,J), which refer back to the quantum statistics 1. Chiral vertex operators and fractionalization. of the particles: as reviewed in the previous section, the gaussian model can be considered either for bosons or The gaussian model is endowed with a very basic prop- fermions, which results in different bosonization formu- erty which is that of chiral separation, i.e. we can split it las. For bosons, J is constrained to be an even integer into two commuting parts corresponding to right or left while for fermions, Q and J must have the same par- propagation of the fields. This is a property which is sys- ity. This then leads to two different spectra as can be tematically used by CFT in the analysis of conformally seen from figures (1) and (2): for instance, for bosons invariant systems. Indeed: the state (Q =1,J = 0) is available while it is forbidden for fermions; conversely (Q = 1,J = 1) is available to u L fermions but not to bosons. Thus we have two different H = dx K−1Π(x)2 + K( Θ(x))2, (2.1) B 2 ∇ theories: the same hamiltonian leads to different proper- Z0 ties depending on whether we consider a Fock space of 34 bosons or a Fock space of fermions . We will call the 1 LL with bosonic (resp. fermionic) selection rules: the ∂2 ∂2 Θ(x, t) = 0 (2.2) ⇒ x − u2 t bosonic (resp. fermionic) LL. For the bosonic LL, as de-   picted in figure (1) the spectrum in arbitrary charge sec- More precisely we introduce the following chiral fields: tors has the same form but for a shift in energies: in the charge sector Q one must add the constant πuQ2/(2L) to Φ(x) Θ (x)=Θ(x) (2.3) the energy. The same energies are found for the fermionic ± ∓ K LL in charge sectors for which Q is an even integer, but if Q is an odd integer there is a new spectrum with local which are related to the phonon operators by: minima at momenta kF and not k = 0 (figure 2). In the rest of the paper± we refer to this parametrization K q q> 0 : bq = | |Θ+,q, (2.4) of the spectrum in terms of phonons and zero modes as r 2 the zero mode basis; this is to be distinguished from the K q quasiparticle basis which we will build later. A property q< 0 : bq = | |Θ−,q. (2.5) 2 which will prove crucial for the rest of the discussion is r the fact that in the free-fermion case a quasiparticle basis In terms of these fields the hamiltonian becomes: exists as an alternative to the zero mode basis: instead of the zero modes basis, it is indeed possible to parametrize HB = H+ + H−, (2.6) the spectrum in terms of the usual Landau quasiparti- L uK 2 cles. Below we show that a similar quasiparticle basis H± = dx : (∂xΘ±(x)) : (2.7) 4 can be built in the interacting case. While fractional 0 Z 2 quasiparticles do occur in exactly solvable models (the πu Q KJ = u q : b+b :+ ± . (2.8) holon, the spinon), scant contact had been made with | | q q LK 2 ±q>0 ! the bosonization approach as mentioned earlier. In the X b b low-energy limit, using the bosonization formalism, we H only contains right-moving phonons and similarly will directly recover the fractional excitations predicted + for H with left-moving phonons. It is clear also that in Bethe Ansatz, with the advantage that the simplifica- −

5 [H+,H−] = 0. Let us show now that these fields Θ± are We now define the chiral vertex operators which ap- chiral; they obey the equal-time commutation relations: peared in the previous expression as:

± K V (x) = exp i√πQ±Θ±(x), (2.17) Θ (x), ∂ Θ (y) = iδ(x y), (2.9) Q± − ± ∓ 2 y ± −   where the upperscript refers to the direction of propa- ± which implies that the momentum canonically conjugate gation. They obey the following commutation rules: K to Θ± is ΠΘ± = 2 ∂xΘ±. The equations of motions for ± ± ∓ ρ(x), V (y) = Q±δ(x y) V (x), (2.18) these fields are: Q± − Q± h i Q, V ± (x) = Q V ± (x), (2.19) u∂ Θ = ∂ Θ . (2.10) Q± ± Q± x ± ∓ t ± h ± i Q± ± Thus: Θ±(x, t) = Θ±(x ut) which means that J,b V (x) = V (x), (2.20) ∓ Q± K Q± we have chiral fields indeed. The superfluid phase h i Q±KJ has therefore been parametrized as: Θ(x, t) = which showsb they carry charges Q± = which are 1 2 [Θ+(x ut)+Θ−(x + ut)]. non-integral in general. The above operator identity 2 − One may define chiral density operators as well as the means that the charge is ’sharp’: by ’sharp’ we mean that corresponding chiral charges as: the charge found is not a quantum average ( is not necessarily quantized of course). This is a point we want 1 δρ(x) Kj(x) ρ±(x)= ∂xΦ±(x)= ± , (2.11) to stress because this means that these quantum states 2√π 2 are genuinely fractional. This shows then that if one in- Q KJ jects Q particles with current J in a LL, one should ob- Q± = ± . (2.12) serve a charge Q = Q+KJ state propagating to the right 2 + 2 Q−KJ b b at velocity u and a charge Q− = 2 going to the left Thoseb chiral densities obey the anomalous (Kac-Moody) with velocity u. For instance, let us inject an electron commutation relations: exactly at the− right Fermi point: this is a (Q =1,J = 1) iK excitation (with no plasmon excited); there would then [ρ (x),ρ (y)] = ∂ δ(x y). (2.13) 1+K ± ± x be fractionalization into a charge 2 state going to the ∓ 2π − 1−K right and a charge 2 going to the left. Let us now consider the injection of Q particles with a The most important property of these fractional states momentum q and current J. In that case, the plasmon is that they are exact eigenstates of the gaussian hamil- πQ total momentum is equal to q J(kF + ). In the tonian. The proof requires a proper definition of their − L bosonization formalism, the operator creating this state Fourier transform because they are anyons, as will be is: shown shortly: from equation (1.22) it is clear indeed L that the commutation relations are anyonic with an any- 1 i(q−JkF )x VQ,J (q)= dxe : exp i√π(QΘ JΦ) : . onic phase √L 0 − − Z Q2 (2.14) θ = π ± . (2.21) ± K This can also be rewritten as: ± Due to its anyonic character VQ± (x) does not obey pe- L riodic boundary conditions; if we use the expressions of 1 i(q−JkF )x VQ,J (q)= dxe exp i√πQ+Θ+(x) the fields Φ and Θ (equations 1.17, 1.16), we immediately √L − Z0 find that: exp i√πQ−Θ−(x). (2.15) × − Q2 V ± (x + L) = exp i2π ± V ± (x). (2.22) As a function of time: Q± ± K Q± L The Fourier transform is then defined as: 1 i(q−JkF )x VQ,J (q,t)= dxe exp i√πQ+Θ+(x ut) √L − − 1 L 2π 2π Q2 Z0 ± ± ± VQ± (qn)= dx exp i n x VQ± (x), exp i√πQ Θ (x + ut). (2.16) √L 0 − L ± L K × − − − Z   (2.23) There is therefore a splitting into two counter- propagating states. For non-interacting electrons the chi- with a pseudo-momentum qn quantized as: ral charges Q± are integers since K = 1 and the opera- 2 tors exp i√πQ±Θ±(x ut) are just those of Q± Landau 2π 2π Q± − ∓ qn = n (2.24) quasiparticless. But in the general case this is not true L ± L K anymore: we will therefore have states carrying fractional 2π Q2 = q ± , (2.25) charges. n ± L K

6 (where we have defined a phonon part qn of the momen- to the system. In sharp contrast, in the zero mode basis, tum). the spectroscopy is not useful because the states used in ± The operators VQ± (qn) are such that: that basis are the phonons (which have no charge) and ± the U operators (which have charge but no dynamics). 1) V (q) Ψ0 > is an exact eigenstate of the chiral ± Q± | We will give such an argument in the next section: this hamiltonian H± with energy: will prove in an independent manner the fractionalization 2 of the LL spectrum (in a way which does not depend on πu Q± E(Q±, qn)= u qn + . (2.26) the explicit construction of the fractional states opera- | | 2L K   tors). where Ψ0 > is the interacting ground state (see ap- pendix).| It has a linear dispersion. ± 3. Selection rules and fractionalization. 2) The states created by the VQ± (qn) to which one adds the phonon excitations form a complete set. This The fractional charges carried by the fractional excita- is obvious because the states VQ,J (x) span the full Fock space35. tions considered above are not arbitrary: they must take on the values Q KJ Q = ± , (2.27) 2. The LL spectrum in terms of fractional quasiparticles. ± 2 where both Q and J are integers. We may view these Let us consider figures (1,2) which show the spectrum of the LL hamiltonian in various charge sectors and ask constraints on the allowed spectrum of fractional charges as selection rules. These selection rules have however a the following question: what happens when one adds Q particles to the system (i.e. in the charge sector Q)? clear physical meaning which we discuss now. Although these excitations do not carry the electron In the standard view of the LL spectrum based on the phonon and zero modes basis, the dynamics of the charge quantum numbers because of the fractionalization of the spectrum, nevertheless the elementary constituents of our added to the LL is unclear because it is concealed in system are electrons (they are the high energy elemen- the zero modes. The parametrization of the spectrum in tary particles of our systems): this means that they alone terms of the zero modes and the phonons does not allow define the structure of Fock space, with the implication to find what happens once the charge Q is added to the that all physical states must consist of an integrer num- system because that choice of basis involves the use of ber of electrons. Despite the fact that there are fractional non-dynamical states (Haldane’s U operators, which pm states, the previous remark implies that these fractional describe the zero modes). By contrast the quasiparticle states will be created in appropriate combinations so that basis only involves states which have a dynamics (the phonons and the fractional states) and we are therefore the total charge is always an integer. This is the expla- nation of the previous selection rules we found, which in able to tell what happens to the charge, how much of it will move to the right, and so forth: if we consider the fine enforce the basic constraint that we started out with electrons. We may view these selection rules as being two branches starting from k = JkF in the charge sector Q, the right branch corresponds to a right-moving frac- topological since they are directly related to the struc- ture of the Fock space. tional excitation with linear dispersion and with charge Q = Q+KJ , while the left branch is due to a left-moving It is easy to show that eq. (2.27) immediately follows + 2 from the requirement that all states are electronic. We fractional state with charge Q = Q−KJ . The contin- − 2 consider two counterpropagating states with arbitrary uum in between the branches simply results from the cre- charge Q+ and Q−; we make no hypothesis on the val- ation of the two fractional excitations with both non-zero ues of the charges, nor on the nature of the chiral states momentum q+,n and q−,n (on the right branch, a charge + − (we do not assume they correspond to VQ and VQ ). Q = Q−KJ is also created but it has zero momentum + − − 2 The only assumptions we make are the following: (a) q−,n = 0, and conversely on the left branch). the one-dimensionality which means that the eigenstates The direct way to find out how the charge Q will be- have momenta in one of either two directions and (b) have, is to exhibit the quantum states which will describe that the current density operator is renormalized. We the propagation of the charge. This is what the quasi- then have two constraints on the values that the charges particle basis does because it directly considers the states Q+ and Q− may assume: since our Fock space is that involved in the dynamics of the charge. Of course, the of electrons, all the states contain an integer number of two bases (the quasiparticle basis and the zero mode ba- electrons i.e. Q+ + Q− = Q is an integer. The second sis) are mathematically equivalent and therefore lead to constraint stems from the renormalization of the current identical physics: therefore the charge dynamics can also density operator: in principle be determined in the zero mode basis, but in the quasiparticle basis, we have the benefit that the spec- 1 j (x)= uKj(x)= uK ∂ Θ(x) (2.28) troscopy immediately tells us the fate of the charge added R √π x  

7 Q+KJ where j(x) is the current density in the non-interacting Q+ = 2 Q Q−KJ case. This expression can be derived from the continuity − 2 8     equation . Going around a ring of length L in the LL we 1 K = Q 2 + n . (2.33) get a (persistent) current which must be quantized: 1 K  2  −  L This implies that in real space the fractional charge JR = dxjR(x)= uKJ (2.29) Z0 excitation is: 36 Q n where J is an integer ; but the current carried by the V ± (x)= V ± (x) V ± (x) , (2.34) states with charges Q and Q is J = u(Q Q ). Q± 1/2 ±K + − R + − − Therefore: h i   where Q and n are now independent integers of arbitrary sign: (Q,n) Z2. In reciprocal space, one has a convo- (Q+ Q−)= KJ, J integer (2.30) − lution for the∈ exact fractional eigenstate: while: Q n ± ± ± V (q)= .. dqi V (qi) dpj V (pj ) (Q+ + Q−)= Q,Q integer (2.31) Q± 1/2 ±K Z Z i=1 j=1 Y h i Y   Solving for these constraints, one recovers the selection Q n rules (2.27), i.e the spectrum of fractional charges. We δ( q + p q) (2.35) × i j − observe in passing that this argument does not depend on i=1 j=1 X X our formal algebraic derivation of subsection (II A 1) and provides an alternative proof of the existence of fractional (the momenta in that expression are the phonon parts ± states as well as it yields the allowed charge spectrum. In qn of the momentum of the operator: for VQ± (qn), qn = 2 that argument, fractionalization follows from the renor- 2π Q± 2πn 37 qn L K and qn = L ) . malization of the current in the presence of interactions. The± above equation demonstrates clearly that the ex- ± citation VQ± (q) can be built from Q charge 1/2 states V ± and n charge K states V ± . The whole spectrum B. Elementary excitations of the bosonic LL. 1/2 ± ±K of fractional excitations is thus built by repeated creation ± ± 1. Elementary excitations. of V1/2 and V±K which means that they are the elemen- tary excitations we were seeking. These two elementary excitations will be identified in the following as respec- We now establish a series of new results concerning tively the spinon (for spin systems) and a (1D) Laughlin the elementary chiral excitations of a non-chiral LL. We quasiparticle. would like to find a basis of elementary excitations, i.e. identify objects from which all the other excitations can be built. It will be useful to use a spinor notation to 2. Wavefunctions of the fractional excitations. represent the fractional states:

Q+KJ To be complete, we compute the wavefunctions of all Q+ 2 = Q−KJ . (2.32) the chiral excitations. We will first need the ground state Q−    2  wavefunction which is simply a Jastrow wavefunction: (2.32) should be understood as follows: the fractional this is of course expected since the gaussian hamiltonian state V + which is an anyon propagating with velocity is the 1D version of the acoustic hamiltonian of Chester Q+ and Reatto’s Jastrow theory of He433 and is also iden- u is created along with the fractional state V − which Q− tical to Bohm-Pines RPA plasmon hamiltonian adapted propagates in the opposite direction with the velocity u. 38 − to 1D . Since the gaussian hamiltonian is a sum of os- The selection rules are encoded in the second spinor: the cillators, the ground state is a gaussian function of the equation is then read as meaning that addition of Q par- densities: ticles with (persistent) current J will result in a splitting into the two counterpropagating fractional states V + Q+ Ψ0,B( ρq ) and V − . { } π Q− = exp( ρ ρ ) (2.36) We must carefully distinguish between Bose and Fermi − 2K q q −q q6=0 statistics because of the constraints on Q and J. Let us X | | 1 π consider bosons first: since J is even we can rewrite it as = exp dxdx′ ρ(x) ln sin (x x′) ρ(x′). (2.37) J = 2n where n is now an arbitrary integer. But then 2K L − Z Z for bosons this implies that the spinor can be written in This expression is validb for the bosonic LL;b for the terms of two other independent spinors: fermionic LL, antisymmetry is recovered by observing

8 ∓Q±/K that the fermionic LL simply derives from the bosonic LL (zi z) Q± i xi = − exp ikF + x , through a singular gauge transformation on the bosonic z z ± K N0 i i LL (the Jordan-Wigner transformation)39: this is ex- Y  | − |  P  actly as in the composite boson Chern-Simons theory (2.42) for which the hamiltonian is plasmon-like at the one- loop level (RPA) and whose ground state is of course where we use the definitions of ρ and z introduced above symmetric (the modulus of Laughlin wavefunction); in and where that theory, the Laughlin state is then found after undo- 1 x ing the Chern-Simons gauge transformation40. Similarly θ(x)= ln − , (2.43) iπ x undoing the Jordan-Wigner transformation amounts to  | |  multiplying the bosonic ground state by the phase fac- is the Heaviside step function. (xi−xj ) tor sgn(xi xj )= (that phase fac- i

1/K Q+/K 1/K ψ ( x )= z z . (2.38) = C (zi z) zi zj 0,B { i} | i − j | − | − | i − | + V1/2(x)Ψ0,B (x1, .., xN ) δ 1/2K 1/K = exp α = C (zi z) zi zj − | − | δρ(x0) i i

9 3. The spinon. Laughlin quasi-hole and the Luttinger liquid Laughlin quasiparticles are indeed very strong. For instance as in ± 2D one can use the plasma analogy to find the fractional We found an elementary excitation V1/2(x) for the bosonic LL carrying a charge 1/2. When we consider charge: spins, which are hard-core bosons, this result translates 2 into having a state carrying a spin ∆Sz = 1/2 with re- Ψ (z ) 2 = (z z ) z z 1/K spect to the ground state. In spin language, adding a | Laughlin−qp 0 | i − 0 | i − j | i i

10 1 The velocity u has been normalized to the Fermi velocity H [J, ∆] = J S+S− + S−S+ + ∆SzSz . so that u = 1 in the absence of interactions for fermions −2 i i+1 i i+1 i i+1 i   (K = 1). We have found that current excitations were X
due to Laughlin quasiparticles. The natural explanation (2.54) of the renormalization is therefore that the current is no longer carried by Landau quasiparticles but by Laughlin As ∆ is varied, one finds three phases: i) for ∆ > 1 quasiparticles with velocity u and charge K. one gets an Ising antiferromagnet the twofold degenerate groundstate of which leads to solitonic excitations with spin one-half 1/2 domain walls; ii) for ∆ < 1 one has an Ising ferromagnet; iii) for -1 ∆ 1 we have the so- 5. The bosonic LL spectrum in terms of fractional ≤ ≤ elementary excitations. called XY phase: this is the Luttinger liquid phase we are interested in. The isotropic Heisenberg chain with SU(2) invariance corresponds to ∆ = 1 . The Luttinger For the bosonic LL we can now add the following pre- liquid parameter was determined exactly by Luther and cisions to the description of the spectrum. Peschel on the basis of a comparison with the Baxter For Q = 0 excitations (see figure (1)), the continuum model43: is due to multiple Laughlin quasiparticle-quasihole pairs: π the right branch starting at k =2kF corresponds to the K(∆) = . (2.55) propagation of a 1D Laughlin quasielectron while the left 2arccos( ∆) branch is due to a Laughlin quasihole; in between the − two lines, we have a continuum generated by these two The spectrum in the sector ∆Sz = 1 is shown in figure (3) for the Heisenberg model; its linearization through excitations. More generally at k = 2nkF where n is an arbitrary integer, the two branches create a continuum bosonization is also shown in figure. of n Laughlin quasiparticle-quasihole pairs. (Note that Given that a spin one-half can be mapped onto a hard- 44 for n = 0, which is an exceptional case, we have multi- core boson: through the Holstein-Primakov transfor- phonon processes.) Therefore the spectrum in the zero mation, we can transpose the results we found for the charge sector is not a Landau quasiparticle-quasihole pair bosonic LL to the XXZ spin chain. continuum except at the special value K = 1 which de- If we want to compare the bosonization linearized spec- scribes indeed in the low energy limit a gas of hard-core trum to the exact one there are however two provisos: (a) bosons. we have to shift the bosonization spectrum by a momen- In the charge sector Q = 1, pairs of charge one-half ex- tum π : this is due to the bipartite transformation one citations are created: they correspond to the ”spinons” of makes in the bosonization of the XXZ spin chain (in or- spin chains; the pairs are superimposed on the previous der to change the sign of the XY term) and (b), there is a Brillouin zone: therefore we have to identify momenta Laughlin quasiparticle continuum: for instance a 2kF ex- citation generates a Laughlin quasiparticle-quasihole pair modulo 2π and since the Fermi vector is kF = π/2, ex- in addition to the ”spinon” pair. citations with J/2 odd (resp. even) correspond to the same harmonics k = π + Jk 0 (resp. π). Taking The Laughlin quasiparticle and the spinon are dual F ≡ states for the bosonic LL; by duality we mean electro- (a) and (b) into account , we can use the results of the magnetic duality which exchanges charge and current previous section pertaining to the bosonic LL, to recover processes. Indeed the ”spinon” is associated with charge the linearized spectrum of the XXZ chain. processes while the Laughlin quasiparticles is due to cur- We first consider the spin sector ∆Sz = 0. In figure rent excitations. The duality operation which maps a (3) starting from momentum π we have two straight lines bosonic LL onto another bosonic LL is:K =1/(4K′) with corresponding to left and right moving phonons, bound- Θ = 2Φ′ and Φ = Θ′/2; zero modes then transform as ing a continuum; due to the folding of the continuum J = 2Q′ et Q = J ′/2. With these relations, the selec- spectrum of the bosonic LL, one superimposes on these tion rule remains bosonic (J ′ even) while the hamiltonian lines the lines due to the creation of any even number of ′ ′ ′ Laughlin qp-qh pairs (the qp dispersion being given by HB [K, Θ, Φ] = HB [K , Θ , Φ ] retains a gaussian form. It is clear then that K = 1/2 is a self-dual point while one line, and that of the qh by the other; if the qp is right- handed, its dispersion is that of the right line, etc...). VK and V1/2 create dual quasiparticles. This is not true for the fermionic LL. Similarly the lines starting from momentum zero or 2π correspond to the creation of an odd number of Laugh- lin qp-qh pairs. The continuum is therefore seen to be 6. The XXZ spin chain. parametrized entirely in terms of the phonons and Laugh- lin quasiparticle-quasihole pairs while the zero mode ba- Let us illustrate these results on the specific example of sis relies on phonons and zero modes. The ∆Sz = 1 the anisotropic Heisenberg XXZ spin chain. The hamil- continuum is described in a similar manner but for the tonian of the XXZ spin chain with anisotropy ∆, after substitution of the phonons by a pair of counterpropagat- a bipartite rotation is: ing spinons. In the special case of SU(2) symmetry, the Laughlin quasiparticle and the spinon become identical

11 operators. The previous parametrization reduces then to indeed generates Laughlin qp-qh pairs as the above equa- one involving only pairs of spinons because a pair of coun- tion shows. The continuum for zero charge excitations terpropagating spinon plus a pair of counterpropagating (Q = 0) is often depicted as a (Landau) particle-hole con- spinon-antispinon is equivalent to a pair of spinons prop- tinuum as in the non-interacting system (K = 1): this agating in arbitrary directions. One then recovers the is incorrect; we have instead a Laughlin quasiparticle- Bethe Ansatz result. quasihole continuum. The latter does reduce to the In the low-energy limit, we can now answer the various standard Landau quasiparticle continuum when K = 1. questions raised in the introduction about the spectrum of For k = JkF there is a local minimum of the energy the XXZ chain: from which two linear branches rise corresponding to -what is the nature of the ∆Sz = 1 continuum? It J/2 = n pairs of Laughlin quasiparticles and quasi- is indeed a spinon pair continuum; but superimposed holes. For− the fermionic LL, the Laughlin quasiparticle on them Laughlin quasiparticle-quasihole pairs can ex- is not the only state which may have an irrational charge: ist. The spinon changes when the anisotropy is varied: this is also possible for the hybrid state. it acquires a statistical phase π/4K = arccos( ∆)/2. The hybrid state is created in mixed charge and current Therefore the spinons at different anisotropy are not− adi- processes: this is the main difference with the bosonic abatically connected: they are orthogonal states;45 this LL for which the decoupling between charge and current is consistent with numerical computations of the spectral processes is complete. As reviewed in the introduction, density of the SU(2) spinon, where it is found that the there are even-odd effects in the fermionic spectrum: the SU(2) spinon has a zero quasiparticles weight for the XY spectra obtained by adding an even or an odd number of chain46. particles are qualitatively different for the fermionic LL. -what is the nature of the ∆Sz =0 continuum? It is a The Q = 1 continuum is understood as follows: the two Laughlin qp-qh pair continuum with an unquantized spin branches at kF correspond to a pair of hybrid excitations 1−K 1+K Sz = K = π/(2 arccos( ∆)); in the SU(2) symmet- carrying a charge 2 and 2 , and propagating with ric case,± they± are identical− to the spinons. In the XY velocities respectively u and u . At k the correspon- − − F limit, one recovers the standard spin one continuum pre- dence is reversed. More generally at JkF (J is an odd dicted through a Jordan-Wigner transformation (K =1, integer if Q = 1), in addition to the hybrid quasiparticles, Sz = 1). But in between these two points, the elemen- one also creates J/2 Q pairs of Laughlin quasiparticle tary excitation± is neither a spinon nor a Jordan-Wigner and quasihole. When− K = 1 the hybrid states reduce fermion. to Landau quasiparticles. It is interesting to note an evidence for these states in the work of Safi and Schulz who considered the evolution of a charge 1 wavepacket in- C. The fermionic Luttinger Liquid. jected at kF in a LL: they found that there was a splitting with an average charge = (1+K)/2 propagating to 1. Elementary excitations : the Laughlin quasiparticles and the right and an average charge = (1 K)/2 going the ”hybrid state”. to the left47. This is exactly what we predict.− Note how- ever a crucial difference: the charge they find is a quan- We now turn to fermions; the analysis of the elemen- tum average while we deal with elementary excitations tary excitations will differ from that found for the bosonic (exact eigenstates); this has an important consequence: LL because the allowed (Q,J) states obey different selec- while it is clear that on average a charge may assume ir- tion rules, namely J is not constrained any more to be rational values, our result goes beyond that observation an even integer, but must have the same parity as Q. We since it proves that there may exist in condensed mat- may therefore write Q J =2n. Then for fermions using ter systems a genuine good quantum state with sharp eq.2.27: − irrational charge. In this section, we have found that for the fermionic LL 1+ K 1 K there are two elementary excitations. One is the Laugh- (Q ,Q )= Q , − n (K, K) . (2.56) + − 2 2 − − lin quasiparticle already found for the bosonic LL. The   second one is a hybrid state intermediate between the The most general excitation once again, is built by ap- ± ± spinon and the Laughlin quasiparticle. The excitations plying Q times V 1±K and/or n times V±K to the ground 2 corresponding to Q = 0 transitions (they are particle-hole state ; this means that we have identified a set of elemen- excitations in the non-interacting case), form a Laughlin tary excitations for the fermionic LL. Here too we find quasiparticle-quasihole pair continuum when K = 1. ± 6 Laughlin quasiparticles V±K , but instead of the spinon we get a ”hybrid state”: this is a consequence of statis- tics; as we will show below, that hybrid state is self-dual 2. Dual basis and the dual quasiparticles. and is intermediate between the Laughlin quasiparticle and its dual state. The elementary excitations we have derived form a ba- The Laughlin quasiparticle is created by current exci- sis from which all the LL spectrum is recovered; by no tations: for a pure current process (Q = 0,J = 0) one 6

12 means is this choice of basis unique: other bases of el- kF harmonics: this is exactly the composite fermion con- ementary excitations are generated with matrices asso- struction and it may then be more fitting to speak of a ciated with a basis change having integer entries whose composite fermion (indeed, the statistics of the operator inverses are also integer-valued: this ensures that all ex- is (2n + 1)π and not π). Because of the similar long dis- citations are integral linear combinations of the elemen- tance behavior of their Green functions, Stone proposed tary excitations. (The matrices belong to SL(2,Z).) For to identify such a sub-dominant operator -which he calls instance for fermions, another basis of elementary exci- a hyperfermion- with Wen’s electron operator introduced ± ± 48 tations consists of states V 1±K and V1 : in the chiral LL . This hyperfermion is identical to the 2 dual state for K =1/(2n + 1). In general the dual state 1+ K 1 K and the electron are however orthogonal: this is quite (Q ,Q )= J , − + n (1, 1) . (2.57) + − 2 2 clear when one considers the LL with spin. The dual   state is then generalized to a state with the same quan- It is actually a dual basis to the previous one: for tum numbers as the electron (carrying a unit charge and fermions the electro-magnetic duality which exchanges a spin one-half), but with again anyonic statistics. But charge and current excitations is expressed by K due to spin-charge separation, that state is not stable 1/K and Φ Θ. This is a canonical transforma-←→ and decays into a spinless charge one quasiparticle which ←→ tion; it results in: HB [K, Θ, Φ] = HB [1/K, Φ, Θ]. The is none other than the holon, and a spin one-half ex- fermionic selection rule Q J even is obviously preserved: citation, which is just the spinon. The dual excitation the transformation is therefore− a duality operation for we have found is therefore the analog of the spinon and the fermionic LL. Observe that the transformations dif- the holon for the spinless LL and has nothing to do (in fer from those for the bosonic LL. general) with an electron. In the following in accordance ± with the previous remarks we will call these states holons What is the nature of the elementary excitation V1 ? ± ± (for the spinless LL) or dual states. Under the duality transformation VK V−1 and ± ± ± −→ These dual holon states also occur in Haldane’s in- V−K V1 . Therefore V1 is an excitation dual to the Laughlin−→ quasiparticle. It carries a charge unity and terpretation of the Calogero-Sutherland model: he pro- its wavefunction is: posed that a natural interpretation of such a model was not in terms of electrons or bosons but as a gas of non- (z z ) interacting anyons22,27. The basis for that interpreta- V +(z )Ψ = (z z )1/K z z 1/K i − j 1 0 F i − 0 | i − j | z z tion is the finding that for rational values of the coupling i i

13 particle-hole duality: there exist both a charge one anyon the holon and the spinon cannot be the elementary exci- with statistics π/K (the dual holon) and a similar charge tations of the system anymore. To answer the question 1 anyon; the same applies to Laughlin quasiparticle for we have to turn to the low-energy effective theory. Frahm which− we have both quasiholes and quasielectrons. Our and Korepin’s results imply that an effective description selection rules involve the hybrid state and the Laughlin in terms of the gaussian model should be possible since quasiparticles, and the holon only appears in the dual conformal invariance is realized. We will find a gener- basis where it is accompanied again by the hybrid quasi- alization of the spin-charge separated gaussian hamilto- particle. nian suitable for a description of the Hubbard model in Finally, a symmetric basis can be associated with the a magnetic field. Our formalism is very similar to Wen’s hybrid excitation: K matrix approach to edge states of the FQHE. This will enable us to characterize very precisely, in the low- 1+K 1−K Q+ 2 2 energy limit, the properties of 1D gapless models with = m 1−K + n 1+K , (2.61) Q− or without spin-charge separation such as the Hubbard    2   2  model in a magnetic field: we will find that in the latter Q+J Q−J where m = 2 and n = 2 ; m and n are again in- case, although there is no spin-charge separation, there is dependent integers. They physically correspond to the still a generalized decoupling. The excitations are again number of electrons added to the system at the right fractional; as expected the holon and the spinon are no and left Fermi points respectively. That self-dual basis longer present in the spectrum and we will give the gen- reduce to Landau quasiparticles when K = 1. The phys- eral framework allowing the description of the fractional ical processes generated in that symmetric basis are not states which replace them. charge or current excitations but addition of electrons at the Fermi surface. Note that the arbitrariness in the choice of a basis simply reflects the possibility to stress A. Spin-charge separated Luttinger liquid. various specific physical processes as elementary. But ex- periments probe Q+ and Q−; for a given set of Q and We start with the standard case when spin-charge sep- J, Q± assume the same value irrespective of the basis aration exists. Although fractional excitations are clearly choice. present in Bethe Ansatz, no description of these special states was attempted in the low-energy limit through bosonization. In the following we answer several ques- III. THE LUTTINGER LIQUID WITH SPIN. tions: how does the holon evolve with interaction? What would be an effective wavefunction for it? Is it a semion? In this section we generalize the construction of frac- First, we consider the ground state of the two-component tional excitations developed in section (II) to the full Lut- gaussian model, because it will suggest to us a possible tinger liquid with spin. One of the main properties ex- generalization of the gaussian model which will prove to hibited by the effective theory is spin-charge separation, be the correct one for the description of gapless mod- the complete decoupling of spin and charge dynamics. els without spin-charge separation such as the Hubbard In the exact solution of the Hubbard model by Bethe model in a magnetic field. Ansatz, excitations display such a spin-charge separa- tion: one state, the holon is a spinless particle carrying the charge of the electron, while the other, the spinon, 1. Ground state of the gaussian hamiltonian. is a neutral spin one-half state5. This is an asymptotic property only valid in the low-energy limit (the Hub- We consider a two component model by introducing bard model in the large U limit is an exception because an internal quantum number such as the SU(2) spin. spin-charge separation is realized at all energy scales). We consider the charge and spin densities as well as their However this property is not obtained for all the gapless associated phase fields: itinerant 1D models, in the low energy limit. According to the universality hypothesis they should be described ρ = ρ + ρ ; ρ = ρ ρ , (3.1) c ↑ ↓ s ↑ − ↓ by the Luttinger liquid framework if the interaction is 1 not too long-ranged. One such example is the Hub- ρ = ∂ Φ , σ = , (3.2) σ √π x σ bard model in a magnetic field which does not display − ↑ ↓ [Θ (x), ∂ Φ ′ (y)] = iδ ′ δ(x y), (3.3) spin-charge separation even in the low-energy limit, al- σ x σ σσ − though it is a short range gapless model. This model Φ↑ Φ↓ Φc/s = ± , (3.4) was analyzed by Frahm and Korepin in the framework √2 of Bethe Ansatz plus conformal field theory25: they were [Θτ (x), ∂xΦτ ′ (y)] = iδττ ′ δ(x y); τ = c, s. (3.5) able to compute the anomalous exponents for the corre- − lation functions. Several issues remain unclear for such The effective hamiltonian derived for instance from the models in a magnetic field: in particular what the exci- Hubbard model in the absence of a magnetic field is: tations are. Since spin-charge separation does not occur,

14 2π H = Hc + Hs, (3.6) to particles with spin σ) and if one sets z = exp i L x, L one easily finds: uτ −1 2 2 Hτ = dx Kτ ( Φτ ) + Kτ ( Θτ ) ; τ = c, s. (3.7) 2 0 ∇ ∇ gσiσj Z Ψ0( xi, σi )= zi zj . (3.14) { } | − | i

We define the charge and spin parts of the ground state is omitted): per: Ψ ( u , v )= u u λ v v λ u v µ , 1 0 { i i} | i − j| | i − j | | i − j | Ψc/s = exp [ i

The previous ground state may be rewritten in terms of Ψ = u u v v u v the densities of each species: c  | i − j| | i − j | | i − j | i

16 1/2 a single charge state, which has charge Q = 1. In the 2kF i xi (xi xj ) exp i + x0 − , (3.32) non-interacting case, the charge velocity uc and the spin × − Kc N xi xj  P  x0,i

17 where again Q↑ J↑ = 2n↑ and Q↓ J↓ = 2n↓. In B. Luttinger liquid without spin-charge separation. − −± 1±Kc addition to the hybrid quasiparticles Vc (Q = 2 ) ± 1±Ks 1. The general LL and the charge matrix. and Vs (Sz = 4 ) which already existed in the pre- vious basis, we have two excitations associated with pure charge current or spin current processes: Vc(Qc = Kc) We now generalize the standard LL theory to include and Vs(Sz = Ks/2). Under electromagnetic duality the situations with no spin-charge separation. We start from latter are conjugate to the holon and spinon respectively. the ground state of the gaussian hamiltonian: Actually they are obtained by spin-charge separation of ′ the two-component Laughlin quasiparticle, and we may 1 ′ π(x x ) ′ Ψ0 [g] = exp dxdx ρσ(x)gσσ′ ln sin − ρσ′ (x ), call them a Laughlin holon and a Laughlin spinon. Let 2 L Z Z us compute the wavefunctions of these two excitations; (3.41) one finds: b

Vc(Qc = Kc,z0)Ψc λ µ g ′ = , (3.42) = (z z )1/2 σσ µ λ i − 0   i −1 Y Kc = λ + µ, (3.43) xi −1 exp i2k + x Ψ , (3.37) Ks = λ µ. (3.44) × − F N 0 c −  P  and relax the constraint g↑↑ = g↓↓ (while g is kept sym- z Vs(S = σ0Ks/2,z0)Ψs λ µ metric). We consider the charge matrix: g ′ = σσ µ λ′ σiσ0/2 b = (zi z0)   − and the associated wavefunction Ψ0 [g]. We introduce for i Y convenience the eigenvalues of the charge matrix and the xi↑ xj↓ unitary matrix P : exp i2σ0(k↑ k↓) − + x0 Ψs (3.38) b × − − M  P  1/K 0 (3.39) P −1gP = D = 1 (3.45) 0 1/K2   In the previous expression σ0 takes on the values 1 ± δττ ′ and k , k are the Fermi vectors associated with particles Pστ Pσ′τ ′ gσσb ′ = b ; τ =1, 2 (3.46) ↑ ↓ K of spin up and down: k = π N and N = N N , τ σ L σ ↑ − ↓ M = N↑ N↓.Ψc and Ψs are given for the bosonic The normal modes of the charge matrix are simply: LL in eq.(3.20)− and (3.21); for the fermionic LL, there ρ = P ρ ρ = P ρ τ =1, 2 (3.47) are additional phase factors given in the text follow- τ στ σ ⇔ σ στ τ ing eq.(3.20) and (3.21). Plasma analogy allows to get the charge and spin of the Laughlin holon eq.(3.37) and ′ 1 ′ ρ (x)g ′ ρ ′ (x )= ρ (x) ρ (x ). (3.48) spinon eq.(3.39)). The Laughlin holon and spinon are σ σσ σ τ K τ ′ τ τ the stable excitations into which the Laughlin quasipar- Xσσ X ticle decays as a result of spin-charge separation; in the If the charge matrix obeys a Z2 symmetry then the nor- localized Wannier basis we have considered throughout, mal modes are just the charge and spin density (up to a the product of the wavefunctions of the two excitations normalization factor) ρ1 = ρc/√2 and ρ2 = ρs/√2). The yields indeed: wavefunction now reads:

δσ σ gσiσj Ψqp(z0, σ0)= (zi z0) 0 i zi zj , (3.40) − | − | Ψ0[g] i i

18 1 1 Anomalous exponents are easily computed as functions ρτ = ∂xΦτ , jτ = ∂xΘτ , (3.50) −√π √π of the charge matrix g which leads to compact expressions valid both for the LL with spin-charge separation or for the more general LL; one introduces the Fermi vectors: ′ ′ b [Φτ (x), ∂xΘτ (y)] = iδττ δ(x y). (3.51) πNσ − kF σ = L . For instance density-density correlators are It is then clear that Ψ0[g] is the exact ground state of the obtained with H[g] and with the bosonization formulas, following family of two component gaussian hamiltonians yielding the static structure factor; the dominant Fourier components are k = 0, k = 2k , k = 2k and k = for arbitrary velocities ub1,u2: b F ↑ F ↓ 2kF ↑ +2kF ↓. Near k = 0: H[g,u1,u2] −1 = HB[u1,K1]+ HB[u2,K2] g σσ′ < δρσ(0)δρσ′ (x) >k=0= , (3.57) b L 2(πx)2 uτ −1 2 2 = dx Kτ (∂xΦτ ) + Kτ (∂xΘτ ) . (3.52) d −1 2 A g ′ τ=1,2 Z0 < δρ(0)δρ(x) >= k=0 ; A = σσ . (3.58) X h i (πx)2 k=0 2 σσ′ The two hamiltonians HB[u1,K1] and HB [u2,K2] com- X d mute by construction. The next section will be devoted For the higher harmonics one includes a mode at to the properties of that generalized LL theory. Let us 2kF ↑ +2kF ↓ =2πρ (which appears in the Hubbard model stress here the main property of this general LL now: in a magnetic field): by construction that theory corresponds to a generalized separation: the normal modes will not be charge and spin Ak=0 cos(2kF ↑x) cos2kF ↓x < δρ(0)δρ(x) >= + a↑ + a↓ modes but mix charge and spin in a proportion fixed in (πx)2 x2+α(2kF ↑) x2+α(2kF ↓) time. This will translate for the fractional excitations to cos(2kF ↑ +2kF ↓)x states with both fractional charge and fractional spin. +b , (3.59) x2+α(2kF ↑+2kF ↓)

2. Main properties. −1 2+ α(2kF σ)=2gσσ , (3.60) −1 2+ α(2k +2k )=2 g ′ . (3.61) The compressibility and spin susceptibility are easily F ↑ F ↓ σσ b ′ computed and one finds: Xσσ d 2 Ak=0 is fixed in the low-energy limit but the other 1 ∂2E π u constants a ,a ,b are non-universal and depend on high- κ−1 = 0 = τ P , (3.53) ↑ ↓ L ∂ρ2 4 K στ energy processes. When there is spin-charge separation, 0 τ τ σ ! X X the exponent for 4kF oscillations and the constant Ak=0 are related by the equation 2+ α(4kF )=4Ak=0; in the 2 general case, we find: 1 ∂2E π u χ−1 = 0 = τ σP . (3.54) s L ∂ρ2 4 K στ s τ τ σ ! 2+ α(2kF ↑ +2kF ↓)=4Ak=0 X X ρ0 and ρs are the charge and spin mean densities, while The derivation of the exponents is done in exactly the E0 is the ground state energy. The Drude peak is: same manner as in the spin-charge separated LL.

2 Spin-spin correlation functions are: 2 1 ∂ E0 D = 2 = uτ Kτ Pστ , (3.55) ′ −1 L ∂φ ′ σσ g ′ τ σ ! σσ σσ X X = 2(πx)2  where φ is a flux threading the LL ring of length L. P d This expression can also be recovered using the Kubo for- cos2kσx + −1 , (3.62) 2gσσ mula; one then needs the expression of the current den- σ x sity which one finds with the continuity equation. The X | | current is renormalized as for the spinless LL and the LL b with spin-charge separation. In contrast to the case of cos(kF ↑ + kF ↓)x = , (3.63) the LL with spin-charge separation, here the expression x γ involves both K and K because both modes one and | | 1 2 1 two involve charge: γ = g−1 g + (g−1 + g) . (3.64) ↑↓ − ↑↓ 2 σσ " σ # X ∂xΘσ b b j (x)= u K P P ′ . (3.56) Electronic Green functions decay as: R  τ τ στ σ τ  √π σ ′ X Xσ ,τ   19 exp ik x 1 < Ψ (0)Ψ+(x) >= σ ; 1+ α (σ)= (g + g−1) . V (x) = exp i√π (QσΘσ(x) JσΦσ(x)) . (3.69) σ σ 1+α(σ) F σσ − − x 2 σ | | X b (3.65)b Fractionalization stems from two decouplings: chiral sep- aration and a separation for the internal quantum num- For bosons the exponent is modified as: 1 + αB(σ) = ber generalizing spin-charge separation. In terms of the 1 2 gσσ. These exponents are derived with the bosonization normal modes fields Θτ and Φτ (τ =1, 2): formulas but can also be found by plasma analogy. b V = exp i√π ( P Q )Θ ( P J )Φ . − στ σ τ − στ σ τ 3. The charge matrix: a summary. τ σ σ ! X X X (3.70) For the two-component LL there are three interesting situations which we summarize below: The chiral fields are: i) in the general case, the (symmetric) g matrix has arbitrary entries; there is no spin-charge separation but Θτ,±(x)=Θτ (x) Φτ (x)/Kτ , (3.71) ∓ a more general two-mode separation: b and therefore: λ µ g = ′ . (3.66) V (x)= exp i√πQ Θ (x). (3.72) µ λ − τ,± τ,±   τ,± Y As will be shownb below, the Hubbard model in a mag- This expression explicitly shows a decoupling into four netic field can be described by such a theory. components. We have defined in the above the chiral ii) the g matrix has a Z2 symmetry; this case pertains to spin-charge separation: charges: b λ µ 1 1 1 g = ,K = ,K = (3.67) Qτ,± = ( Pστ Qσ) Kτ ( Pστ Jσ) . (3.73) ρ σ 2 ± µ λ λ + µ λ µ " σ σ #   − X X Indeedb the symmetry under the exchange of up and down The following operators are exact eigenstates of each chi- spins implies that the normal modes of the charge matrix ral hamiltonian H±,τ τ =1, 2: are just the charge and spin modes. The LL parameters are then the eigenvalues of the inverse of the charge ma- V ±(Q , q)= dx exp iqx exp i√πQ Θ (x), (3.74) trix. This situation describes models with spin-charge τ τ,± − τ,± τ,± separation but with a spin anisotropy, for instance a Z 2 2πn 2π Qτ,± Hubbard model to which one would add some Ising term q = . (3.75) L ∓ L Kτ n Sz(n)Sz(n + 1). iii) the g matrix corresponds to a SU(2) symmetric The chiral charges correspond to the charge and spin car- P case (Kσ = 1): ried by each of these excitations up to a normalization b factor: µ +1 µ 1 g = ,K = ,K = 1 (3.68) µ µ +1 ρ 2µ +1 σ ± ±   Qσ, Vτ (Qτ,±, q) = Qτ,±Pστ Vτ (Qτ,±, q), (3.76) Thisb situation describes the low-energy limit of the Hub- h i b ± bard model. It is noteworthy that the wavefunctions Ψ [g] which implies that the charge and spin of Vτ (Qτ,±, q) for that sub-case were used in a variational approach of are: the 1D t J model giving very good results although − b it was not realized they were the exact ground states of Q = Qτ,± Pστ , (3.77) 54 the gaussian model . The reason why it is so is now σ ! X transparent. 1 S = Q σP . (3.78) z τ,± 2 στ σ ! C. Elementary excitations for the generalized LL. X Thus for an arbitrary charge matrix, fractional excita- tions carry both charge and spin. However the ratio of We now consider the excitations of the general bosonic charge to spin is constant for each given mode τ = 1, 2: and fermionic LL with or without spin-charge separation. 1 σP Q = S ( P ); of course the phonons Let us inject particles in the Luttinger liquid. In real 2 σ στ z σ στ associated to each mode mix charge and spin in exactly space this is described by the operator: theP same proportions
since:P

20 1 1 Once again we find a charge 1/2 particle and a charge ρτ (x)= Pστ ρc(x)+ σPστ ρs(x) 2 2 Kc Laughlin quasiparticle as for the bosonic spinless σ ! σ ! X X LL. But in addition we find new states resulting from 1 a fractionalization of ”pseudo-spin” for the bosons (half- = P ρ (x)+ P s (x) (3.79) 2 στ c στ z spinons for instance). σ ! σ ! X X For the fermionic LL the elementary excitations are where sz(x) is a spin density. obtained by the equation: It is convenient to define the charge to spin ratio: 1 K 1 K Q = Q P ± τ + Q P ± τ Q τ,± ↑ ↑τ 2 ↓ ↓τ 2 r = , (3.80)     2Sz +n ( P K )+ n ( P K ) , (3.86) ↑ ∓ ↑τ τ ↓ ∓ ↓τ τ for each mode. Unitary implies that if for the first mode: where we have resolved the constraint: Qσ Jσ = 2nσ. Q This fully characterizes the low-energy elementary− exci- r = = p (3.81) 2Sz tations of a LL in a magnetic field (see below). then for the second mode: Q 1 D. Application to the Hubbard model. r = = . (3.82) 2Sz −p To illustrate the previous results we discuss the Hub- where p is arbitrary. Note that for a Fermi liquid these bard model in one dimension. The model was solved ex- ratios are r = 1 (we are characterizing Landau quasi- ± actly by Bethe Ansatz by Lieb and Wu. In zero magnetic particles (or holes) of either spin) and when spin-charge field, for repulsive (U > 0) interactions, a LL metallic separation is realized the ratio is either r =0or r = . ±∞ phase exists both for weak and strong coupling, except Let us give the elementary excitations. The simplest at half-filling. For very large U the spin-charge decou- case is that of bosons: pling is valid at all energy scales. This was shown by P P Ogata and Shiba who also found that the Bethe Ansatz ↑τ ↓τ 55 Qτ,± = Q↑ + Q↓ ground state then took a remarkable factorized form : 2 2     it is the product of a charge part (a Slater determinant for free fermions involving all electrons ) and a Bethe J↑ J↓ wavefunction similar to that of the Heisenberg model on + ( P K )+ ( P K ) . (3.83) 2 ± ↑τ τ 2 ± ↓τ τ a reduced lattice from which one has removed the holes To simplify the notation, we have only written a sin- ΨHubbard(xi, σi) gle line, but Qτ,± and the other entries should be read Jσ = det(exp ikj ri, kj kF )ΨHeisenberg (yi, σi) (3.87) as four-vectors. Qσ and 2 are arbitrary independent | |≤ ± integers, which shows that the states Vτ (Qτ,±, q) where (yi is the coordinate in the reduced lattice of particle i P↑τ P↓τ Qτ,± = 2 , 2 , P↑τ Kτ or P↓τ Kτ are elementary ex- whose real position is xi). citations. As an± illustration± let us considere the simple It is instructive to compare it to the two-component casee of a Z2 symmetric charge matrix for bosons which Jastrow wavefunctions which are also explicitly spin- have a pseudo-spin index. The unitary matrix P is: charge decoupled. The Slater determinant is rewritten as (in terms of the circular coordinates z): 1/√2 1/√2 Pστ = (3.84) 1/√2 1/√2 Ψ (x , σ )= (z z )Ψ (y , σ )  −  Hubbard i i i − j Heisenberg i i i

1/2Kc σiσj /2Ks Q+ 1/2 1/2 Ψ= z z z z | i − j| | i − j | Q− 1/2 1/2 i

vτ = vc/s. (3.99) b g−1 g−1 ↓↓ − ↑↓ . (3.105) The zero modes can be identified term by term; it is −1 −1 × g ↑↓ g ↑↑ ! sufficient to consider the current terms to uniquely de- −d d termine the charge matrix. The charge zero modes yield Term by term identification of the charge zero modes extra relations which lead to the very same expression d d Qσ would lead to exactly the same expression for g; in- for g. Indeed expanding the squares gives: deed: K P 2 = (Z Z )2, 2 b b 1 ↑1 cc − sc 1 2 (Zss) 2 2 P↑1 = 2 , K1P↓1 = (Zsc) , K1 (det Z) 2 K1P↑1P↓1 = (Zcc Zsc)Zsc, 1 2 (Zcs Zss) 2 − 2 P↓1 = − 2 , K2P = (Zcs Zss) , K1 (det Z) ↑2 − 2 2 K2P↓2 = (Zss) , 1 (Zcs Zss)Zss P↑1P↓1 = − − 2 , K P P = (Z Z )Z . (3.100) K1 (det Z) 2 ↑2 ↓2 cs − ss ss 1 (Z )2 2 Since: 2 sc P↑2 = 2 , K2 (det Z) −1 g σσ′ = Kτ Pστ Pσ′τ (3.101) 2 1 2 (Zcc Zsc) τ P↓2 = − 2 , X K2 (det Z) d it follows that the inverse of the charge matrix is: 1 (Z Z )Z P P = − cc − sc sc , (3.106) K ↑2 ↓2 2 g−1 = (Z Z )2 + (Z Z )2 2 (det Z) ↑↑ cc − sc cs − ss −1 −1 2 2 and since g ′ = K P P ′ one recovers g ↓↓ = (Zsc) + (Zss) σσ τ τ στ σ τ d equ.(3.105). As it should be, one can check that the g−1 = g−1 = (Z Z )Z + (Z Z )Z (3.102) P d↑↓ ↓↑ cc − sc sc cs − ss ss anomalous exponentsb predicted for H [uτ , g] agree then completely with Frahm and Korepin’s results. Wed defined the matrix Z obtained from the dressed charge Let us illustrate these results in two situations,b one matrix Z by subtracting the second line from the first: with spin-charge separation, the other without. From e these, we can exhibit the criterion for spin-charge sepa- Z Z Z Z Z = cc − sc cs − ss (3.103) ration within the matrix formalism. Zsc Zss   In the presence of spin-charge separation, Frahm and e Korepin find that the dressed charge matrix Z is Z = Then: Z = ξ Z =0 cc cs . This implies that g and its −1 T Z = ξ/2 Z =1/√2 g = ZZ (3.104)  sc ss  inverse g−1 are: This is the most important result of the present sec- b b e e ξ2 1 ξ2 1 tion: the low-energy properties of the Hubbard model b −1 4 + 2 4 2 g = ξ2 1 ξ2 − 1 (3.107) are expressed in terms of quantities which can be com- + ! 4 − 2 4 2 puted from the microscopic parameters and spin-charge 1 1 1 1 2 + 2 separation simply follows from the Z2 symmetry of our b ξ 2 ξ 2 g = 1 1 1 − 1 (3.108) charge matrix. In the framework of the dressed charge 2 2 +  ξ − 2 ξ 2  matrix Z approach, the second feature is not easily de- coded from the structure of Z which is then triangular The charge matrixb explicitly exhibits spin-charge separa- tion and takes the form characteristic of SU(2) symme- with some relations between its matrix elements whose 2 60 −1 ξ physical interpretation is quite unclear. We can use the try. The eigenvalues of g are Kc = 2 and Ks = 1.

23 b The Z matrix can also be explicitly computed in the An important test, of course, would be to observe ex- limit of infinite repulsion with a magnetic field close to perimentally (or numerically) all these fractional states. the critical field hc for which all the spins are polarized Although the existence of the holon and the spinon was (i.e. close to the ferromagnetic phase). In terms of the ascertained theoretically quite a long time ago5 no ex- parameter periment has yet allowed their detection: in fact, the property of spin-charge separation itself is not yet es- h h δ = c − (3.109) tablished experimentally. The observation of two of the h r c fractional states discussed in this paper would be partic- ularly important: the LL Laughlin quasiparticle and the the dressed charge matrix is: hybrid state. Indeed they may assume irrational charges. 1 0 The precise spectroscopy of fractional excitations we have Z = 2 1 (3.110) done in this paper allows us to determine which processes π δ 1 π δ  −  are involved in their creation: for the Laughlin states, which implies that the (inverse of the) charge matrix g current probes are needed, while the hybrid particle is is: created by addition of an odd number of electrons. For −1 b Laughlin quasiparticles shot noise is likely an adequate g = probe: the shot noise coefficient for Luttinger liquids can be computed exactly and is predicted to be equal to K;63 1 2 2 2 b 2 2 1 2 in the two-dimensional electron gas at filling ν =1/3 this 1 π δ + 1 π δ 1 π δ π δ 1 π δ 2 − 2 2 − 1 2 − 1 2 2− 2 − . yields a charge 1/3 . The latter situation involves Wen’s 1 δ δ 1 δ 1 δ + δ ! − π
π − −
π − π

π
chiral Luttinger liquid. The identification of the shot




(3.111) noise coefficient with the charge of a carrier has been de- bated because the coefficient one measures might actually This expression shows explicitly the breakdown of spin- be the conductance rather than a quasiparticle charge (at charge separation (except for δ = π/4). ν =1/3 the conductance also assumes the value 1/3). For the non-chiral Luttinger liquid our spectroscopy of frac- tional states allows to resolve that ambiguity: the shot IV. CONCLUSIONS AND PERSPECTIVES. noise coefficient is indeed identical to the conductance K of the LL but the (backscattering) current-current cor- The main goal of our paper was to establish and relation function measured in shot noise involves charge describe fractional excitations for the Luttinger liquid K excitations, because charge K LL Laughlin quasiparti- within the bosonization scheme. bf Bethe Ansatz gives cles are precisely generated by current excitations. These exact eigenstates and shows the existence of some frac- might thus be detected in any physical realization of the tional excitations: however their description is quite com- Luttinger liquid: quantum wires, or possibly nanotubes. plex in that framework and it is unclear how to generate An intriguing possibility is also suggested by recent ex- systematically a complete set of excitations. In the low- periments of tunneling at the edge of a two-dimensional energy limit, the Luttinger liquid approach allows a very gas in a magnetic field64. I V characteristics measured precise characterization of the fractional states already at the edge showed very surprising− non-Fermi liquid be- known from exact solutions but what’s more allows us haviour compatible with a chiral LL with unquantized to discover novel fractional excitations which may carry LL parameter K; the I V curves evolve smoothly when irrational charges (the 1D Laughlin quasiparticle, the hy- one varies the filling fraction− and do not show a plateau brid state). In section II the low-energy spectrum of Lut- structure. There seems to be a continuum of Luttinger tinger liquids can be reinterpreted in terms of fractional liquids living at the edge: this caused quite a stir be- states: for instance, the particle-hole continuum consists cause the chiral LL theory can presumably be derived of a Laughlin quasiparticles-quasihole continuum. The only for incompressible filling fractions. These puzzling quasiparticle perspective clarifies many properties of the results are sofar unexplained, but if an interpretation in LL: the renormalization of the current operator is a di- terms of a single-boson mode chiral LL with unquantized rect consequence of fractionalization; for spin chains, we parameter K can be in some manner justified, according present the correct description of the spinon excitation in to the results given in the present paper this would im- the generic case of a violation of SU(2) invariance. We ply that there exists charge K Laughlin quasiparticles in also show that the Sz = 0 continuum of spin chains in- that experimental setting: such a chiral LL is identical volves the analogs of Laughlin quasiparticles. In section to the chiral half of the non-chiral gaussian hamiltonian III we describe fractional excitations such as the holon considered throughout our paper. or the spinon for the Luttinger liquid with spin; we also The authors wish to acknowledge the late Heinz Schulz present a generalization of the gaussian theory valid for for insightful remarks on the results of this paper. We Luttinger liquids without spin-charge separation and dis- also thank Vincent Pasquier for a discussion on the play in that situation the new fractional states replacing Calogero-Sutherland model. Bernard Jancovici gave us the holon and spinon (see III C). information on one dimensional plasmas for which we are

24 grateful. eigenstates with the same eigenvalue of the appropriate chiral hamiltonian (H+ or H−). This is enough to prove ± that VQ± (qn) Ψ0 > is an exact eigenstate of H±: APPENDIX A: DISPERSION OF THE | FRACTIONAL STATES. 2 ± + πu Q± H±V (qn) Ψ0 > = u q a aq + (A5) Q± | | | q 2L K We show here that V ± (q) Ψ > (where Ψ > is the "q>0 # Q± 0 0 X b | | ± interacting ground state) is an exact eigenstate of the V (qn) Ψ0 > (A6) × Q± | chiral hamiltonian H± with energy: 2 2πn πu Q± ± = u + V (qn) Ψ0 >, (A7) 2 L 2L K Q± | πu Q±   E(Q±, qn)= u qn + . (A1) | | 2L K 2πn   where qn = L is the momentum due to phonons. ± Let us rewrite the state considered V (qn) Ψ0 > in Q± | terms of zero modes and phonon operators (we use equ. (1.16), (1.17), (1.26) and (1.27)):

2π Φ ± √ 0 VQ± (x) Ψ0 >= exp iQ± Q±x π Θ0 | ∓ K − ∓ K 1 R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983)    2 b L. Saminadayar, D. C. Glattli, Y. Jin, B. Etienne, Phys. 2πn Rev. Lett. 79 2526(1997); R. de Picciotto, M. Reznikov, exp i√πQ Θ exp i x Ψ > (A2) M. Heiblum, V. Umansky, G. Bunin, D. Mahalu, Nature × − ± ±,n L | 0 n6=0 389, 162 (1997) X 3 J. Hubbard, Proc. Roy. Soc. A 240, 539 (1957); 243, 336 Taking into account the fact that the operators bq an- (1958); 276, 238 (1963). nihilate the ground state, it follows that: 4 H. A. Bethe, Z. Phys. 71, 205 (1931). 5 E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968). 6 + 2π Φ0 L. D. Fadeev, L. A. Takhtajan, Phys. Lett. A 85, 375 V (x) Ψ0 >= exp iQ+ Q+x √π Θ0 Q+ | − K − − K (1981)    7 J. D. Johnson, S. Krinsky, B. M. McCoy, Phys.Rev. A 8, b 2526 (1972) 8 L 2πn F. D. M. Haldane, Phys. Rev. Lett. 45, 1358 (1980); ibid, exp i√πQ b+ exp i x Ψ >, (A3) × − + Kπ n n L | 0 J. Phys. C 14, 2585 (1981) n>0 s 9 X | | L. D. Landau, Sov. Phys. JETP 3, 920 (1957); 5, 101 8 − (1957); , 70 (1959). with a similar expression for V (x) Ψ0 > (the sum is 10 Q− | J. M. Luttinger, J. Math. Phys. 4, 1154 (1963). then over negative momentum phonons). Going back to 11 S. Tomonaga, Prog. Theor. Phys. 5, 544 (1950). reciprocal space: 12 D. C. Mattis and E. H. Lieb, J. Math. Phys. 6, 304 (1963). 13 C. Bourbonnais, L.G. Caron, Int. J. Mod.Phys.B 5, 1033 L ± 1 2π (1991); R.Shankar, Rev. Mod. Phys. 66, 129 (1994) V (qn) Ψ0 > = dx exp i nx 14 Q± We call these excitations describing collective density fluc- | √L 0 − L Z tuations interchangeably phonons, or plasmons. Φ0 15 exp iQ+ √π Θ0 c is a real number known as the conformal anomaly. × − ∓ K 16    A. A. Belavin, A. M. Polyakov, and A. B. Zamolod- chikov, Nucl. Phys. B 241, 333 (1984); C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Cambridge Uni- L 2πp verstity Press, Cambridge, 1989, vol. 2; P. Ginsparg, in exp i√πQ b+ exp i x Ψ >, (A4) × − + Kπ p p L | 0 Fields,Strings and Critical Phenomena, Les Houches Sum- ±p>0 s X | | mer School 1988, Eds. E.Brezin, J.Zinn-Justin (North- ± Holland, Amsterdam, 1990). which shows that V (qn) Ψ0 > only spans chiral 17 Q± | H. W. J. Bl¨ote, J. L. Cardy, and M. P. Nightingale, Phys. phonons with momentum n > 0 ; in other words this ± Rev. Lett. 56, 742 (1986) state is obtained by the action of the zero mode of the 18 N. Kawakami, S. K. Yang, Phys.Rev.Lett. 67, 2493 (1991) chiral field plus the creation of phonons with momenta of 19 P. Bouwknegt, A. Ludwig, K. Scoutens, Phys. Lett. B 338, the same sign. When one expands the phonon exponen- 448 (1994); K. Schoutens, Phys. Rev. Lett. 79, 2608 (1997) tial, the integral over position will select configuration of 20 O. Babelon, H. de Vega, and C.M Viallet, Nucl. Phys. B 2πn phonons with identical total momentum qn = L . All 220, 13 (1983); K. Sogo, Phys. Lett. A 104, 51 (1984). these configurations consist of phonons of identical chi- 21 F. D. M.Haldane, Phys. Rev. Lett. 60, 635 (1988); B. S. rality and total momentum which means that they are Shastry, Phys. Rev. Lett. 60, 639 (1988)

25 22 F. D. M. Haldane, in Proceedings of the 16th Taniguchi 41 Here and in the remaining of the paper for convenience Symposium on Condensed Matter Physics, Kagoshima, we will consider the operators in real space and the wave- Japan, 1993, Eds A.Ojiki, N.Kawakami (Springer, Berlin- functions will be localized: the exact eigenfunctions are Heidelberg-New-York, 1994) of course recovered by taking the Fourier transform of the 23 We omitted a phase factor corresponding to a momentum localized (Wannier) states. π boost. It can be ignored after a bipartite transformation. 42 B. Horowitz, Phys. Rev. Lett 48, 1416 (1982) 24 F. Woynarowich, J. Phys. A 15, 2985 (1982) 43 A. Luther, I. Peschel, Phys. Rev. B 12, 3908 (1975) 25 H. Frahm, V. E. Korepin, Phys. Rev. B 42, 10553 (1990); 44 The Jordan-Wigner transformation which maps spins onto ibid, Phys. Rev. B 43, 5653 (1991) fermions might lead one to think that the spectrum of 26 F. Calogero, J. Math. Phys. 10, 2191(1969); B. Sutherland, the spin chain is identical to that of the Jordan-Wigner Phys. Rev. A 4, 2019 (1971) fermions. This is correct only for the spectrum of an even 27 M. R. Zirnbauer, F. D. M. Haldane, Phys. Rev. B 52, 8729 number of fermions added in (this corresponds to spin tran- (1995); sitions ∆Sz = 2p where p is an integer): imposing periodic 28 X. G. Wen, Phys. Rev. B 41, 12838 (1990); X. G. Wen, (PBC) or anti-periodic (ABC) boundary conditions forces Adv. Phys. 44, 405 (1995) the number of fermions to keep a given parity (even for 29 Z. N. C. Ha, Phys. Rev. Lett 73, 1574 (1994) PBC and odd for ABC), thus the mapping of spins onto 30 M. P. A. Fisher, L. I. Glazman, in Mesoscopic electron fermions is not valid when one adds an odd number of transport, ed. by L. Kovenhoven, G. Schoen, L. Sohn fermions to the ground state. The spin flip operator is not (Kluwer, Dordrecht). a Jordan-Wigner fermion since one also creates a flux tube 31 R. Heidenreich, R. Seiler, and A. Uhlenbrock, J. Stat. Phys. (the Jordan-Wigner string). This has led to some confu- 22, 27 (1980). sion in the literature. The ∆Sz = 0 continuum is correctly 32 K-V Pham, M. Gabay, P. Lederer, Eur. Phys. Journ. B 9, described by the excitations of the Jordan-Wigner fermion 573 (1999) system since this corresponds to the addition of an even 33 G. V. Chester, L. Reatto, Phys. Lett. 22, 3 (1966); R. P. number of fermions. For the XX spin chain this means Feynman, Statistical Physics, (Addison-Wesley, New-York, that the ∆Sz = 0 continuum is due to spin one states which 1972) are Jordan-Wigner fermions. But if the mapping were also 34 F. D. M. Haldane, Phys. Rev. Lett. 47, 1840 (1981) valid for ∆Sz = 1 excitations, this would mean that the 35 The zero mode basis (phonons and U operators) forms a continuum is also due to spin one states. This is not cor- ± complete family of states: this was proved by Haldane in his rect because as we know this latter continuum is due to computation of the grand canonical partition function of pairs of spinons. 8 45 the Tomonaga-Luttinger model . Therefore VQ,J operators By adiabaticity, we mean a finite quasiparticles weight, as span the Fock space since VQ,J operators are products of for the Landau quasiparticles which is an adiabatic defor- Haldane’s U operators and of an exponential of phonons. mation of the screened free electron. ± 46 Since the fractional states VQ± come from the decay of the J. C. Talstra, S. P. Strong, Phys. Rev. B 56, 6094 (1997) ± 47 VQ,J operators, they, along with the phonons span the Fock I. Safi, PhD Thesis, Universit´ede Paris, Orsay, unpub- space. lished; ibid, Annales de Physique 22, 463 (1997) 36 This is just the quantization of the superfluid phase: 48 M. Stone, M. P A. Fisher, Int. Journ. Mod. Phys. B 8, the boson operator is ΨB exp i√πΘ and therefore 2539 (1994) ∝ 49 √π (Θ(L) Θ(0)) must be an integer multiple of π. The K.-V. Pham, PhD thesis, Universit´ede Paris, Orsay, un- − total current around the ring is defined as the integral published. L uK 50 B. I. Halperin, Helv. Phys. Acta 56, 75 (1983) 0 dx jR(x) = JR; therefore JR = √π (Θ(L) Θ(0)) and − 51 The g matrix is distinguished from the K matrix to avoid we may then write JR = uKJ where J is an integer. 37 R a possible confusion because its eigenvalues are 1/K and The previous equationb assumesb that both Q and n are pos- c 1/Ks. itive; if that is not so, it suffices to take their absolute value 52 b b It would be more correct to call this operator an anti-holon and replace the operators by their conjugate: for instance since its electrical charge is Qe = ( e)Q = e; but apart − − Q Q and W1±/2(x) W ±1/2(x) . from the sign of the charge all fractional particles consid- −→ | | −→ − 38 D. Bohm, D. Pines,h Phys.i Rev.h 85, 338 (1952)i ered in this paper and their particle-hole conjugates have 39 Recall that bosonization formulae for the electron opera- the same properties. So we call this state a holon. 53 tors are obtained by making a Jordan-Wigner transforma- A naive decoupling of the electron operator would lead to: tion on the boson operators; likewise the fermionic (Q, J) Ψσ = exp i√π (Θσ Φσ ) selection rules (Q+J even) derive directly from the change − − in statistics: a (Q, J) excitation of the bosonic LL becomes = exp i π/2(Θc Φc) − − a (Q, J ′ = J + Q) states due to the Jordan-Wigner trans- exp i π/2(Θs Φs) formation which attaches one flux-tube to each particle (Q − p − tubes to Q particles, which increases J by Q units). Since J The first term is sometimesp identified as the holon and it is even for the bosonic LL, Q + J ′ is even for the fermionic has a semionic statistics. The second term is likewise sup- LL. 40 posed to be the spinon. These identifications are in general S. C. Zhang, H. Hansson, S. Kivelson, Phys. Rev. Lett. 62, incorrect because these states are unstable because of chi- 82 (1989); S. C. Zhang, Int. J. Mod. Phys. B 6, 25 (1992) ral separation (see the anisotropic spin chain). In the spe-

26 cial case of SU(2) symmetry, the above expression for the spinon is accidentally correct. 54 C. S. Hellberg and E. J. Mele, Phys. Rev. Lett. 67, 2080 (1991). 55 M. Ogata, H. Shiba, Phys. Rev. B 41, 2326 (1990) 56 F. Woynarovich, J. Phys. A 22, 2115 (1990) 57 K. Penc and J. S´olyom, Phys. Rev. B 47, 6273 (1993) 58 The ambiguity which is rather technical is the following: there are four kinds of anomalous exponents (two chiralities and two modes). For a given excitation in a given mode, the sum of anomalous exponents for the two chiralities is fixed but their difference is not. The dimensions are therefore only known up to a constant; to fix their values, Frahm and Korepin make the unproven assumption that they are perfect squares if the excitation corresponds to a primary field. 59 Note that in Frahm and Korepin’ notations Ns = N is not ↓ the spin density. 60 The charge matrix and its inverse have the same properties, Z2 symmetry or SU(2) symmetry in the relevant cases. 61 P. W. Anderson, Physics Today, 50, n◦ : 10, pp 42-47 (1997) 62 W. P. Su, J. R. Schrieffer, A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979) 63 P. Fendley, A. Ludwig, H. Saleur, Phys. Rev. Lett. 74, 3005 (1994) 64 M. Grayson et al, Phys.Rev.Lett. 80, 1062 (1998)

27 FIG. 1. Spectrum of the gaussian model with a bosonic Fock space in the charge Q sector. The energy at zero mo- mentum is ∆(Q)= πuQ2/(2L) as a function of Q the number of particles added to the system. The spectrum for fermionic Fock spaces is identical if Q is an even integer. The contin- uum is enclosed within the straight lines which are supported 2 by the parabolic enveloppe πu(k/kF ) /(2L).

28 FIG. 2. Spectrum of the gaussian model for fermionic Fock spaces in the case of charge sectors for which Q is an odd integer. Notice that the energy has now local minima for momenta k = 1, 3, 5, ... (in units of kF ) instead of ± ± ± k = 0, 2, 4,... ± ±

29 FIG. 3. Spectrum of the XXZ spin chain for a trans- verse exchange J and anisotropy ∆; the continuum is en- closed within the curves E(k) = (πα/2) sin(k) and | | E(k) = πα sin(k/2). The parameter α is related to the anisotropy ∆ = cos θ by α = sin θ/θ. The linearized spec- trum found by bosonization is also shown.

30