The Low-Energy Limiting Behavior of the Pseudofermion Dynamical Theory

Nuclear Physics B 737 [FS] (2006) 237–260 The low-energy limiting behavior of the pseudofermion dynamical theory J.M.P. Carmeloa,b, L.M. Martelob,c,K.Pencd,∗ aDepartment of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA bGCEP, Center of Physics, University of Minho, Campus Gualtar, P-4710-057 Braga, Portugal cPhysics Department, Engineering Faculty of University of Porto, P-4200-465 Porto, Portugal dResearch Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary Received 1 September 2005; accepted 19 December 2005 Available online 18 January 2006 Abstract In this paper we show that the general finite-energy spectral-function expressions provided by the pseudofermion dynamical theory for the one-dimensional Hubbard model lead to the expected low-energy Tomonaga–Luttinger liquid correlation function expressions. Moreover, we use the former general expressions to derive correlation-function asymptotic expansions in space and time which go beyond those obtained by conformal-field theory and bosonization: we derive explicit expressions for the pre-factors of all terms of such expansions and find that they have an universal form, as the corresponding critical exponents. Our results refer to all finite values of the on-site repulsion Uand to a chain of length Lvery large and with periodic boundary conditions for the above model, but are of general nature for many integrable interacting models. The studies of this paper clarify the relation of the low-energy Tomonaga–Luttinger liquid behavior to the scattering mechanisms which control the spectral properties at all energy scales and provide a broader understanding of the unusual properties of quasi-one-dimensional nanostructures, organic conductors, and optical lattices of ultracold fermionic atoms. Furthermore, our results reveal the microscopic mechanisms which are behind the similarities and differences of the low-energy and finite-energy spectral properties of the model metallic phase. 2006 Elsevier B.V. All rights reserved. PACS: 70. *Corresponding author. E-mail address: [email protected] (K. Penc). 0550-3213/$ – see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.12.016 238 J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 1. Introduction Over the past twenty five years it has been shown that the low-energy physics of a variety of models of one-dimensional (1D) correlated electrons can be described by the Tomonaga– Luttinger liquid (TLL) theory [1]. Indeed, the low-energy physics of such interacting quantum problems displays universal properties which are also found in the simple and exactly solvable Tomonaga [2] and Luttinger [3] models. Importantly, the low-energy TLL universal behavior was observed in different real materials and systems, as for instance in carbon nanotubes [4,5], ballistic wires [6], quasi-1D organic conductors [7], 1D metallic chains [8], and quasi-1D quantum gases of ultracold fermionic atoms [9]. On the other hand, the low-energy phases of some quasi-1D compounds are not metallic and correspond to broken-symmetry states [10]. Recently, the resolution of photoemission experiments has improved, and the normal state of these compounds was found to display exotic spectral properties [11]. However, such a metallic phase refers to finite energies and is not described by the TLL theory. The 1D Hubbard model is one of the few realistic models for correlated electrons in a discrete lattice for which one can exactly calculate all the energy eigenstates and their energies [12, 13]. It includes a first-neighbor transfer-integral t1, for electron hopping along the chain, and an effective on-site Coulomb repulsion U. For finite-energy values, the metallic phase of this model is not a TLL and thus the study of spectral functions is a very involved many-electron problem. Fortunately, the recently introduced pseudofermion dynamical theory (PDT) provides explicit expressions for these functions [14,15]. Moreover, the theory describes successfully the unusual spectral features of quasi-1D compounds for the whole finite-energy band width [16]. More recently, consistent results were obtained by numerical techniques, involving the use of the dynamical density matrix renormalization group method [17]. Furthermore, when combined with the renormalization group, the use of the PDT reveals that a system of weakly coupled Hubbard chains is suitable for the successful description of the phase diagram observed in quasi-1D doped Mott–Hubbard insulators [18]. The PDT is a generalization for all values of U/t1of the method introduced in Ref. [19] for U/t1→∞. Such an extension was fulfilled by means of the relation of the original electrons to the exotic objects whose occupancy configurations describe all energy eigenstates of the model [20]. The electron–rotated-electron unitary transformation [20], defined in the whole Hilbert space, and the pseudoparticle–pseudofermion unitary transformation [14, 15,21], defined in the subspace where the oneand two-electron excitations are contained, play a major role in the construction of the PDT. In turn, the low-energy physics of the model corresponds to the universal TLL behavior and was studied by different techniques, such as bosonization [22] and conformal-field theory [23, 24]. There are many investigations where the low-energy conformal invariance was combined with the model exact Bethe-ansatz solution in the study of the asymptotics of correlation functions and related quantities [25–34]. The connection of the low-energy TLL behavior to the microscopic scattering mechanisms which control the unusual spectral properties of the model at all energy scales [14,21] remains an interesting open problem, which we study in this paper. Indeed, while conformal-field theory and bosonization techniques do not provide correlation-function expressions for finite energy, we show here that the general finite-energy PDT introduced in Refs. [14,15] reproduces the expected correlation-function expressions in the limit of low energy. Moreover, we derive the corresponding correlation-function asymptotic expansions in space and time. Such expansions go beyond those obtained by conformal-field theory and bosonization: we derive explicit expressions for the pre-factors of all terms of such expansions and find that they have an universal form, as J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 239 the corresponding critical exponents. We also find the relation of the low-energy pseudofermion description to the conformal-field theory primary fields and Virasoro-algebra generators [23,24]. In this paper the emergence of the TLL low-energy physics is described in terms of the general non-perturbative microscopic scattering mechanisms of the model at all energy scales [14,15,21]. Thus, our results provide further information about the microscopic mechanisms and scattering processes behind both the low-energy and finite-energy properties of one-dimensional fermionic interacting problems. For instance, we clarify why there occurs a different type of momentum and energy dependence for the low-energy and finite-energy parts of important singular spectral features of the model metallic phase. Furthermore, our findings lead to a broader understanding of the unusual properties observed in low-dimensional materials and nanostructures [4–8] and systems of interacting ultracold fermionic atoms in 1D optical lattices [35]. Following the investigations on quasi-1D quantum gases of ultracold fermionic atoms [9], studies about twoatom correlation functions of interacting ultracold fermionic atoms in 1D optical lattices are in progress [36]. Recently, the model was used in preliminary theoretical investigations of the density profiles and collective models of 1D ultracold fermionic atoms confined in an optical lattice with harmonic trapping potential [37]. Our study provides the details of the preliminary results on the universal form of the prefactors of the correlation-function asymptotic expansions presented in short form in Ref. [38]. The paper is organized as follows: in Section 2we introduce the model and summarize the basic information about the PDT which is needed for our studies. The general finite-energy PDT spectral-function expressions are shown in Section 3to recover in the limit of low energy the correct correlation-function expressions and corresponding asymptotic expansions in space and time. Moreover, we are able to obtain expressions for the pre-factors of all terms of such expansions. In Section 4we discuss the universal form of the pre-factors of all terms of the correlation-function asymptotic expansions and the emergence of the TLL low-energy physics in terms of the general scattering mechanisms which control the model spectral properties at all energy scales. Furthermore, in that section we discuss the qualitative difference between the low-energy and finite-energy parts of the singular charge and spin spectral features of the metallic phase and the relation of the low-energy pseudofermion description to the conformal-field theory primary fields and Virasoro-algebra generators. Finally, the concluding remarks are presented in Section 5. 2. The model, the general correlation functions, and the pseudofermions The 1D Hubbard model reads (1) ˆ H=−t1 j,σ c† j,σ cj+1,σ +h.c.+U jˆnj,↑ˆnj,↓, where c† j,σ and cj,σ are spin-projection σ=↑,↓electron operators at site j=1,2,...,N aand ˆnj,σ =c† j,σ cj,σ . The model (1) describes N↑spin-up electrons and N↓spin-down electrons in a chain of Nasites. We denote the electronic number by N=N↑+N↓. The number of lattice sites Nais even and very large. For simplicity, we use units such that both the lattice spacing a and the Planck constant are one. In these units the chain length reads L=Na. Our results refer to periodic boundary conditions. We consider an electronic density n=n↑+n↓in the range 0<n<1 and a spin density m=n↑−n↓such that 0 <m<n, where nσ=Nσ/L and σ=↑,↓. We introduce the Fermi momenta which except for 1/L corrections are given by ±kFσ =±πnσ 240 J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 and ±kF=±[kF↑+kF↓]/2=±πn/2. The Hamiltonian (1) commutes with the generators of the η-spin and spin SU(2)algebras [20,39,40]. We call the η-spin and spin eigenvalues of the energy eigenstates ηand S, respectively, and the corresponding projections ηzand Sz. We consider the following general N-electron correlation function (2)χl N(k, ω) =l +∞  −∞ dωBl N(k, ω) ω−ω+il0, where Bl N(k, ω) is the corresponding N-electron spectral function given by (3)Bl N(k, ω) = ff|ˆ Ol N(k)|GS 2δω−l[Ef−EGS],lω>0,l=±1. Here the general N-electron operators ˆ O+1 N(k) ≡ˆ O† N(k) and ˆ O−1 N(k) ≡ˆ ON(k) carry momentum k,thefsummation runs over the excited energy eigenstates, the energy Efcorresponds to these states, and EGS is the ground-state energy. Most common examples are the operator ˆ O1(k) =ck,σ and different choices of charge, spin, and Cooper-pair N=2 operators. For simplicity, we use in expression (3) a momentum extended scheme such that k∈(−∞,+∞), yet it is a simple exercise to obtain the corresponding spectral function expressions for the first Brillouin zone. The double Fourier transform ˜χl N(x, t) of the general correlation function (2) relative to the momentum kand energy ωcan be expressed in terms of the corresponding Fourier transform ˜ Bl N(x, t) of the spectral function (3) as (4)˜χl N(x, t) =−i2πθ(lt) ˜ Bl N(x, t), where here and in other expressions provided below θ(y)=1fory>0 and θ(y)=0fory⩽0. One of the goals of this paper is the evaluation of a general asymptotic expansion for the correlation function (4). To reach such a goal, in Section 3we use the finite-energy expressions derived in Ref. [14] for the general spectral function (3) by means of the PDT. The pseudofermion description is related to the holon and spinon representation for the model: all its energy eigenstates can be described in terms of occupancy configurations of η-spin 1/2 holons, spin 1/2, spinons, and η-spin-less and spin-less cpseudoparticles [20]. We use the notation ±1/2 holons and ±1/2 spinons according to the values of the η-spin and spin projections, respectively. For large values of U/t1,the+1/2 holons and −1/2 holons become the holons and doublons, respectively, used in the studies of Ref. [41]. The electron–rotated-electron unitary transformation [20] maps the electrons onto rotated electrons such that rotated-electron double occupation, unoccupation, and spin-up and spin-down single occupation are good quantum numbers for all values of U.The ±1/2 holons of charge ±2eand zero spin and the charge-less ±1/2 spinons are generated from the electrons by that unitary transformation, where −edenotes the electronic charge. The corresponding holon and spinon number operators ˆ Mc,±1/2and ˆ Ms,±1/2, respectively, are of the form given in Eq. (24) of Ref. [20] and involve the electron–rotated-electron unitary operator. While the −1/2 and +1/2 holons refer to the rotated-electron doubly occupied and unoccupied sites, respectively, the −1/2 and +1/2 spinons correspond to the spin degrees of freedom of the spin-down and spin-up rotated-electron singly occupied sites, respectively. The charge degrees of freedom of the latter sites are described by the spin-less and η-spin-less cpseudoparticles, which are composite objects of a chargeon and a antichargeon, and thus carry charge −e or +efor the description of the transport of charge in terms of electrons and electronic holes, J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 241 respectively [20].Thecν pseudoparticles (and sν pseudoparticles) such that ν=1,2,... are ηspin singlet (and spin singlet) 2ν-holon (and 2ν-spinon) composite objects. Thus, the numbers of ±1/2 holons (α=c) and ±1/2 spinons (α=s) read (5)Mα,±1/2=Lα,±1/2+∞  ν=1 νNαν,α=c,s, where Nαν denotes the number of αν pseudoparticles and Lc,±1/2=[η∓ηz]and Ls,±1/2= [S∓Sz]gives the number of ±1/2 Yang holons and ±1/2 HL spinons, respectively. Those are the holons and spinons that are not part of composite pseudoparticles. The total number of holons (α=c) and spinons (α=s) is given by (6)Mα=[Mα,+1/2+Mα,−1/2],α=c,s. All energy eigenstates can be described by occupancy configurations of cpseudoparticles, αν pseudoparticles, −1/2 Yang holons, and −1/2 HL spinons [20]. For the ground state, Nc=N, Ns1=N↓, and Ncν =Nsν=Lα,−1/2=0forα=c,s,ν>0, and ν>0. The construction of the PDT involves a second unitary transformation, which maps the c pseudoparticles (and composite αν pseudoparticles) onto cpseudofermions (and composite αν pseudofermions) [14,15,21]. Such a transformation introduces shifts of order 1/L in the pseudoparticle discrete momentum values and leaves all other pseudoparticle properties invariant. As a result of such momentum shifts and in contrast to the cpseudoparticles and composite αν pseudoparticles, the corresponding pseudofermions have no residual-interaction energy terms. A concept widely used in the PDT is that of a CPHS ensemble subspace [14,42]. (Here CPHS stands for cpseudofermion, holon, and spinon.) Such a subspace is spanned by all energy eigenstates with fixed values for the −1/2 Yang holon number Lc,−1/2,−1/2 HL spinon number Ls,−1/2,cpseudofermion number Nc, and for the sets of αν pseudofermion numbers {Nαν} corresponding to the composite pseudofermion branches. 3. General asymptotic expressions of correlation functions Here we derive the pre-factors of all terms of the general asymptotic expansion for the correlation function (4) by use the finite-energy spectral-function expressions derived in Refs. [14,15] by means of the PDT. To reach such a goal, we start by defining the low-energy subspace for the electronic densities and spin densities considered in this paper [43] and providing further information about the pseudofermion description when defined in such a subspace. 3.1. Pseudofermion description in the low-energy subspace For each correlation function, the electronic number deviations N↑and N↓have welldefined values and forelectronic densities 0 <n<1 and spindensities 0 <m<n, all low-energy excited energy eigenstates belong to a single CPHS ensemble subspace such that Nc=N, Ns1=N↓, (7){Ncν}={Nsν}={Lc,−1/2}={Ls,−1/2}=0,ν=1,2,3,..., ν=2,3,.... The results of this section refer to such a correlation-function low-energy subspace. 242 J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 For simplicity, in this paper we denote the s1 branch by sbranch.The cand sindices used here correspond to the c0 and s1 indices in all quantities of Refs. [14,15,21,42]. In our study we use the index ι=±1, which refers to the right (ι=+1) and left (ι=−1) α, ι Fermi points.Forthe ground state and except for 1/L corrections [20] such points read ιq0 Fc =ι2kFand ιq0 Fs =ιkF↓. All the α(and ι) sums and products appearing in the expressions provided throughout this paper run over the values α=c,s (and ι=+1,−1). The αpseudofermion number deviation Nαand current deviation J F αof the excited energy eigenstates relative to the initial ground state are given by (8)Nα= ι NF α,ι,J F α=1 2 ι ιNF α,ι,α=c,s, where (9)NF α,ι =N0,F α,ι +ιQ0 α/2π, α =c,s, ι =±1. Here N0,F α,ι stands for the number of αpseudofermions created (N0,F α,ι >0) or annihilated (N0,F α,ι <0) as a result of the ground-state–excited-energy-eigenstate transition and Q0 α/2isa scattering-less phase shift that has a single and well-defined value for the correlation-function excitation CPHS subspace such that Q0 c/2=0,N seven,Q 0 c/2=±π/2,N sodd, (10)Q0 s/2=0,N c+Nseven,Q 0 s/2=±π/2,N c+Nsodd. It is useful for our study to consider the pseudofermion subspace (PS). It is spanned by an initial ground state |GSand all excited energy eigenstates contained in the onetwo-electron excitations [14,15]. The pseudoparticle–pseudofermion unitary transformation which maps the αpseudoparticle ontothe αpseudofermion isdefined in thePS. The αpseudoparticlehasdiscrete bare-momentum values qj=[2π/L]Iα jsuch that Iα jare consecutive integers or half-odd integers [20]. These values are good quantum numbers whose allowed occupancies are one and zero only [20]. Due tothe values of such quantumnumbers, the currentdeviations J F αare integersor halfodd integers depending on the parities of the number deviations N =Ncand N↓=Ns as follows J F c=Nc+Ns 2mod 1 =N +N↓ 2mod 1, (11)J F s=Nc 2mod 1 =N 2mod 1. On the other hand, the αpseudofermion has discrete canonical-momentum values given by [14,21], (12)¯qj=¯q(qj)=qj+QΦ α(qj)/L, α =c,s, where j=1,2,...,N∗ αand the number N∗ αis such that N∗ α=Nα+Nh α.HereNh αdenotes the number of αpseudofermion (and αpseudoparticle) holes [14,15,20]. For the PS low-energy sector such numbers are given by N∗ c=Na,N ∗ s=N0 ↑+N↑, (13)Nh c=Na−N0−N, Nh s=N0 ↑+N↑−N0 ↓−N↓. J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 243 Thus, the bare-momentum values are defined in the range −q0 α⩽q⩽+q0 αwhere the limiting bare-momentum q0 αreads (14)q0 c=π, q0 s=kF↑. In these expressions we have neglected 1/L corrections [20]. When below we refer to the α pseudofermion bare-momentum q, we mean that qis the bare-momentum value that corresponds to the pseudofermion canonical momentum ¯q=q+QΦ α(q)/L. Except for the discrete momentum values, the above pseudoparticle and pseudofermion have the same properties. Thus, all the energy eigenstates that span the low-energy sector of the PS can described by occupancy configurations of αpseudofermions. The functional (15)QΦ α(qj)/2=π α N∗ α  j=1 Φαα(qj,q j)Nα(qj), α =c,s, of Eq. (12) is the scattering part of the overall pseudofermion or pseudofermion hole phase shift [21] (16)Qα(q)/2=Q0 α/2+QΦ α(q)/2,α=c,s, where Q0 α/2 is the scattering-less phase shift given in Eq. (10). Such an overall phase shift plays an important role in the αpseudofermion scattering theory and related spectral properties. On the right-hand side of Eq. (15),Nα(qj)≡Nα(qj)−N0 α(qj)is the excited-state α branch bare-momentum distribution-function deviation relative to the initial ground state value and πΦαα(q, q)is a two-pseudofermion phase shift such that the α(and α) pseudofermion or hole of momentum q∈[−q0 α,+q0 α](and q∈[−q0 α,+q0 α]) is the scattering center created under the ground-state–excited-state transition (and the scatterer) [21]. The two-pseudofermion phase shifts πΦαα(q, q)are defined in Appendix A. The low-energy correlation-function CPHS subspace contains several J-CPHS subspaces. The current deviation values of the energy eigenstates which span each of the latter subspaces differ in at least one of the two current deviation values {J F c,JF s}. At low-energy, the reduced JCPHS subspaces considered in Ref. [14] are spanned by a single energy eigenstate. Since such a state is the lowest-energy eigenstate of the corresponding J-CPHS subspace, we call it J-ground state. It corresponds to a cand spseudofermion bare-momentum densely packed occupancy configuration such that qFα,−1⩽q⩽qFα,+1. Here the J-ground-state Fermi point qFα,ι reads qFα,ι =ιq0 Fα +qFα,ι =q0 Fα,ι +Q0 α L,q 0 Fc =2kF,q 0 Fs =kF↓, (17)α=c,s, ι =±1, where we have neglected 1/L corrections to the value of q0 Fα [20] and qFα,ι denotes the α, ι bare-momentum Fermi-point deviation relative to the corresponding ground-state value given by (18)qFα,ι =q0 Fα,ι +Q0 α L=ι2π LNF α,ι =ι2π LNα 2+ιJ F α,α=c,s, ι =±1. In these expressions the bare-momentum q0 Fα,ι and corresponding deviation q0 Fα,ι read (19)q0 Fα,ι =ιq0 Fα +q0 Fα,ι,q 0 Fα,ι =ι2π LN0,F α,ι ,α=c,s, ι =±1. 244 J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 A J-ground state has excitation momentum (20)kF 0= α P F α, where (21)P F α= ι ιq0 FαNF α,ι =2q0 FαJ F α,α=c,s. Such a state is generated from the ground state by zero-energy and finite-momentum elementary processes (B), which create or annihilate αpseudofermions at or from the four α, ι Fermi points, respectively. In turn, the PDT processes (A) do not exist at low energy [14]. The corresponding J-CPHS subspace is spanned by energy eigenstates generated from the J-ground state by smallmomentum and low-energy processes in the vicinity of the α, ι Fermi points, which we call elementary processes (C). Such processes conservethe set of {Nc,N s,JF c,JF s}deviation values. For each low-energy J-CPHS subspace, the general momentum spectrum provided in Eq. (29) of Ref. [14] simplifies and is single valued and given in Eq. (20). A crucial point for the low-energy scattering properties and corresponding correlationfunction asymptotic expansions studied in this paper is that the αpseudofermions and holes created by the above processes (C) are not active scattering centers, once the phase shifts generated by the created pseudofermions exactly cancel those originated by creation of the corresponding holes [14]. It follows that the overall scattering phase shift (15) has for each αpseudofermion or hole scatterer of momentum qthe same value QΦ α(q)/2=π α ι Φααq,ιq0 FαNF α,ι (22)=π α ι Φααq,ιq0 FαNα 2+ιJ F α,α=c,s, for all excited states spanning given J-CPHS subspace. Note that the scattering part of the overall phase shift, Eq. (15), vanishes and is finite for the initial ground state and excited states, respectively. Thus, the ground-state–excited-energyeigenstate transition leads to a shift (23)¯qFα,ι =qFα,ι +QΦ αιq0 Fα/L =q0 Fα,ι +Qαιq0 Fα/L, α =c,s, ι =±1, in the value of the four α, ι canonical-momentum Fermi-points. Such a shift is the excited-state deviation in the value of ¯qcorresponding to ¯q=q=ιq0 Fα for the initial ground state. The square of these shifts in units of 2π/L plays a key-role in the spectral properties at all energy scales and is denoted by 2ι α. It can be written as follows (24)2ι α≡¯qFα,ι [2π/L]2 =ιN0,F α+Qα(ιq0 Fα) 2π2 ,α=c,s, ι =±1. The general expressionin terms of two-pseudofermion phase shifts of the functional Qα(ιq0 Fα)/2 appearing in the second expression of Eq. (24) is given in Eqs. (35)–(37) of Ref. [14]. However, for the excited energy eigenstates that span the low-energy subspace the finite-energy deviation given in Eq. (14) of that reference vanishes. This property together with the values of the numbers giveninEq.(7) implies that at low energy the general expression of the general functional (24) J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 245 simplifies to 2ι α=2ι αNc,N s,JF c,JF s= αιξ0 αα NF α 2+ξ1 ααJ F α2 , (25)α=c,s, ι =±1, where the parameters ξj ααcan be expressed in terms of the two-pseudofermion phase shifts as follows (26)ξj αα=δα,α+ ι=±1ιjΦααq0 Fα,ιq0 Fα,j=0,1,α,α =c,s. Expressions (25) and (26) are consistent with the low-energy form of the scattering phase shift given in Eq. (22). The overall phase shift (22) controls the unusual spectral properties of the model through the pseudofermions anticommutators [14,21]. To illustrate the dependence of the latter anticommutators on the overall phase shifts for the two pseudofermion branches used in our study, let us consider pseudofermion creation and annihilation operators f† ¯q,α and f¯q,α, respectively. When the canonical momentum values ¯qand ¯q=qcorrespond to an excited-energy-eigenstate and the initial ground-state J-CPHS ensemble subspaces, respectively, the pseudofermion anticommutation relations read [14,15], (27) f† ¯q,α,f q,α=δα,α1 N∗ α e−i(¯q−q)/2eiQα(q)/2sin(Qα(q)/2) sin([¯q−q]/2),α,α =c,s, and {f† ¯q,α,f† q,α}={f¯q,α,f q,α}=0. Here N∗ αis the number whose value is given in Eq. (13). The anticommutation relations (27) are indeed controlled by the value of the overall phase shift (16), which in our case has the same value for all excited energy eigenstates spanning a given J-CPHS subspace. In addition to the overall phase shift (16), the group velocities (28)vα(q) =∂α(q) ∂q ,v α≡vαq0 Fα,α=c,s, play an important role in our studies. Here c(q) and s(q) are the cand spseudofermions energy dispersions defined by Eqs. (C.15) and (C.16) of Ref. [20], respectively. These energy bands are plotted in Figs. 6 and 7 of Ref. [42], respectively, as a function qfor several values of U/t1and nfor m=0. 3.2. The asymptotic expressions of correlation functions Our starting point for the study of low-energy correlation functions and associated correlationfunction asymptotic expansions in space xand time tis the general expression for the N-electron spectral function (3) given in Eq. (41) of Ref. [14]. Fortunately, such a general expression simplifies for the low-energy problem considered here. Indeed, the numbers NphNF c0≡NphNF cand NphNF s1≡NphNF sof the summation on the right-hand side of the aboveequation vanish in our case because the corresponding bare-momentum distribution function deviation given in Eq. (14) of the same reference vanishes in the low-energy limit considered here. Also the numbers NF αν,ι of the summation of the former equation vanish. This follows from the number values of Eq. (7), 252 J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 Due to the convolutions in Eqs. (38) and (39), the asymptotic expansion (52) was obtained by use in expression (46) of the α, ι spectral function (40). To study the above issue it is useful to perform the inverse Fourier transform of the asymptotic expansion (52) relative to both xand t. This provides the behavior of χl N(k, ω) near the singularities corresponding to the branch lines whose linear shape is defined by the following parametric equations (53)ω=ιvαk−lkF 0,α=c,s, ι =±1, where the momentum kF 0and velocity vαare given in Eqs. (20) and (28), respectively. The obtained expression corresponds to a range of small values of ωand (k −lkF 0)such that ω≈ιvα(k −lkF 0). By performing the double inverse Fourier transform relative to xand tof the leading-order term of the general asymptotic expansion (52), one finds that this behavior is associated with the following kand ωdependence of the correlation function (2) (54)χl N(k, ω) ∝lω −ιvαk−lkF 0ζα,ι ,α=c,s, ι =±1, where the exponent reads (55)ζα,ι =−1+2ι α+2+1 ¯α+2−1 ¯α,α=c,s, ι =±1, and 2ι αis the functional given in Eq. (25),¯c=s, and ¯s=c. For the values of kand ωthat these expressions refer to, the real and imaginary parts of χl N(k, ω) have the same kand ωdependence, but differ in the pre-factors. Thus, one also finds (56)Bl N(k, ω) ∝lω −ιvαk−lkF 0ζα,ι ,α=c,s, ι =±1, for the general spectral function given in Eq. (3). When applied to specific N-electron spectral functions, expression (56) with the power-law exponent given in Eq. (55) provides the universal and well, known low-energy TLL behavior for the 1D Hubbard model [25–29,34], Tomonaga–Luttinger model [45–47], and many other models whose low-energy physics corresponds to the same universality class. When ζα,ι <0, such an expression refers to a linear singular spectral feature. The PDT studies of Ref. [14] reveal that the spectral feature whose shape is defined by Eq. (53) is the low-energy part of a spectral-function αbranch line which also exists for finite energy values. The parametric equations which define the (k, ω)-plane points belonging to such a α pseudofermion (or αpseudofermion hole) branch line is of the general form (57)k=lkF 0−c1ιq0 Fα +c1q,ω=lEα(k) =lc1α(q), αν =c,s, ι =±1, where (58)q∈     [−q0 Fα,+q0 Fα],α=c,s, ι =±1,c 1=−1, [+q0 Fα,+q0 α],α=c,s, ι =+1,c 1=+1, [−q0 α,−q0 Fα],α=c,s, ι =−1,c 1=+1, and the constant c1is such that c1=+1 (and c1=−1)forcreationofaαpseudofermion (and a αpseudofermion hole), as discussed below. Note that for (59)q=ιq0 Fα +lc1k−lkF 0,α=c,s, ι =±1, with (k −lkF 0)small one finds (60)lEα(k) =lc1αιq0 Fα +lc1k−lkF 0≈ιvαk−lkF 0,α=c,s, ι =±1. J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 253 Here we used the property that α(ιq0 Fα)=0[14,20,42,48]. This confirms that for baremomentum values qin the vicinity of ιq0 Fα the energy ωis small and the line defined by the parametric equation (57) becomes indeed the line (53). However, although the latter line is continuously reached from the general line (57) as qapproaches ιq0 Fα,thekand ωdependence of the corresponding spectral feature has two regimens, for small and finite values of ω, respectively. Indeed, use of the general PDT reveals that the line defined by the parametric equation (57) corresponds to a spectral feature called αpseudofermion (c1=+1) or αpseudofermion hole (c1=−1) branch line [14]. Such spectral features were observed for the one-electron removal case by photoemission experiments in quasi-1D compounds [11,16]. A spectral-function α branch line is produced by creation for the values of the momentum and energy given in Eq. (57) of a αpseudofermion or αpseudofermion hole, as a result of ground-state–excited-energyeigenstate transitions with such values for the excitation momentum and energy. Therefore, the branch lines are named according to the corresponding pseudofermion or pseudofermion hole, once the shape of the branch line in the (k, ω)-plane coincides with that of that object energy dispersion. The use of the spectral-function expressions derived in Ref. [14] reveals that for (k, ω)-plane points located just above (l=+1) or below (l=−1) the branch line whose shape is defined in Eq. (57), the weight distribution has the following form for finite values of ω, (61)Bl N(k, ω) ∝lω −Eα(k)ζα(k),α=c,s, where the exponent reads (62)ζα(k) =−1+2+1 c(k) +2−1 s(k) +2+1 c(k) +2−1 s(k), α =c,s. In this expression the parameters 2ι α(k) correspond to the general functional given in Eq. (24). However, they are not given by expression (25), which corresponds to the low-energy limit of such functionals. In the present general case the phase-shift dependence is that provided in Eq. (40) of Ref. [14]. The dependence on the momentum koccurs through the corresponding dependence on the scattering center bare-momentum of the phase-shift scattering component given in Eq. (36) of that reference. In contrast to the low-energy limit studied here, the general PDT expressions derived in Ref. [14] include contributions from pseudofermion and/or hole scattering centers created off the Fermi points for finite values of the excitation energy. The above α pseudofermion or hole which generates the spectral feature (61) is an example of such scattering centers. Note that when the exponent (62) is such that ζα(k) < 0, expression (61) refers to a singular spectral feature. As k→lk0 F(and q→ιq0 Fα) and ω→0, the parameters 2ι α(k) of the exponent expression (62) become those of Eqs. (25) and (55), with 2ι α(lkF 0)=2ι α. This result together with comparison of the 2ι αdependence of the exponents (55) and (62) confirms that the latter exponent does not evolve continuously onto the former exponent as q→ιq0 Fα and ω→0. The origin of such two different behaviors of the spectral function in the vicinity of the branch line for small and finite values of ω, respectively, can be explained by an effect which is as a particular case of a general PDT mechanism studied in Ref. [14].Asq→ιq0 Fα and thus ω→0 the spectral function corresponds to the vicinity of a αν =c,s branch line end point, (k =lkF 0,ω=0). That for this low-energy TLL limit the expression of the spectral function in the vicinity of the cor s branch-line is not that of Eq. (61) results from a resonance effect: the branch line group velocity vα(q) equals the velocity vα(ιq0 Fα)=ιvαassociated with the α, ι pseudofermion particle–hole excitation sub-branch generated by the elementary processes (C). Due to such a resonance effect, which also occurs for finite energies corresponding to the lower limits of the first, second, 254 J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 and higher-order upper Hubbard bands, it is shown in Ref. [14] that the momentum and energy dependence of the spectral function in the vicinity of the αpseudofermion or pseudofermion hole branch line is instead given by Eq. (73) of that reference. The above low-energy expression (54) corresponds to a particular case of the general expression given in that equation. In turn, the finite-ωexpression (61) is a particular case of the general expression (70) of that reference for the spectral function in the vicinity of cand spseudofermion branches lines considered here. The latter expression corresponds to the same spectral function in the vicinity of any pseudofermion branch line, including those corresponding to the cν and sνpseudofermion branches such that ν>0 and ν>1, respectively. We note that there is an intermediate regimen in the vicinity of the αbranch where the spectral function is neither given by the low-energy TLL expression (54) nor by the finite-energy expression (61). These expressions correspond to vα(q) ≈ιvαand vα(q) = ιvα, respectively. The energy and momentum widths of the crossover regimen are infinitesimal. In turn, the energy and momentum widths of the low-energy linear regimen of Eqs. (53) and (56) are controlled by the value of |vα(q) −ιvα|. The low-energy TLL behavior emerges when such difference can be written as (63) vα(q) −ιvα≈aαq0 Fαk−lkF 0,a α(q) =∂vα(q) ∂q ,α=c,s, ι =±1, where the qvalues are in the ranges given in Eq. (58) and the relation between kand qis defined by the first expressionof Eq. (57).Asthevalueofqapproaches ιq0 Fα the behavior (63) is reached. For smaller values of |aα(q0 Fα)|the value of |vα(q) −ιvα|can remain small for larger values of |(k −lkF 0)|and thus of ω≈ιvα(k −lkF 0). It follows that the momentum and energy widths of the (k, ω)-plane region in the vicinity of (lkF 0,0)where the TLL liquid behavior (54) is valid increase for decreasing values of |aα(q0 Fα)|, provided that vαis finite. For instance, in the limit of zero spin density, m→0, the value of |as(q)|is small in two relatively large qregions in the vicinity of q=−kFand q=+kF, respectively, and thus the domain of the corresponding spin sbranch lines where the low-energy TLL expression (56) is valid increases in that limit. 4.3. Relation to conformal-field theory primary fields and Virasoro algebras The relation of the low-energy conformal-field theory [23,24] to bosonization [1,22] is well established. Thus, here we briefly discuss the connection of the general pseudofermion description to the conformal-field theory primary fields and Virasoro-algebra generators [23,24,43]. Implicitly, that also provides information about the relation of that description to bosonization. In the limit of low-energy considered here the reduced J-CPHS subspaces of the general PDT [14] are spanned by a single energy eigenstate. We have called it J-ground state: it is the lowestenergy state of a J-CPHS subspace. Within the pseudofermion description a J-ground state can be written as (64)|J−GS= αˆ U† α ι Fα,ι|GS, where the initial ground state reads (65)|GS= α +q0 Fα  qj=−q0 Fα f† qj,α|0, J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 255 and |0is the pseudofermion vacuum such that fqj,α|0=0. The operator F† α,ι and the transpose of the operator ˆ U† αappearing in expression (64) are given by Fα,−1=θN0,F α,−1−q0 Fα  qj=q0 F α,−1 f† qj,α +θ−N0,F α,−1 q0 F α,−1  qj=−q0 Fα fqj,α,α=c,s, (66)Fα,+1=θN0,F α,+1 q0 F α,+1  qj=q0 Fα f† qj,α +θ−N0,F α,+1 q0 Fα  qj=q0 F α,+1 fqj,α,α=c,s, and (67) ˆ Uα=expN∗ α  j=1 f† qj,α[f¯qj,α −fqj,α],α=c,s, respectively. In the pseudofermion operator f¯qj,α of Eq. (67) ¯qj=qex j+QΦ α(qj)/L =qj+ Qα(qj)/L where qex j=qj+Q0 α(qj)/L denotes the excited-state discrete bare-momentum values. Moreover, qjstands for the ground-state discrete bare-momentum values and in the four pseudofermion operators of Eq. (66) and two remaining pseudofermion operators of Eq. (67) the discrete canonical-momentum values are those of the initial ground state (65) such that QΦ α(qj)/2=0 and, therefore, ¯qj=qj. The operator (67) is unitary and leaves the pseudofermion vacuum invariant and thus ˆ U† α|0=|0. Once the functional 2ι αgiven in Eq. (25) is shown in Appendix A to be the conformal dimension of the α, ι primary field, it is straightforward to show by analysis of the corresponding finite-size energy and momentum spectra that the J-ground state (64) is a highest-weight state (HWS) of the model cand sVirasoro algebras [24]. Thus, the α, ι operator (68)Gα,ι =ˆ U† αFα,ι ˆ Uα,α=c,s, ι =±1, where Fα,ι and ˆ Uαare expressed in terms of αpseudofermion operators in Eqs. (66) and (67), respectively, refers to the pseudofermion representation of the corresponding α, ι primary field. It follows that the initial ground state (65) plays the role of the vacuum of conformal-field theory and the zero-energy and finite-momentum processes (B) generate the HWSs of the cand s Virasoro algebras from such a vacuum. For the pseudofermion description, application onto the ground state of the operator Gα,ι creates |N0,F α,ι |αpseudofermion scattering centers (N0,F α,ι >0) or αpseudofermion-hole scattering centers (N0,F α,ι <0) at the α,ι Fermi point. This leads to an overall phase shift Qα(q)/2 for all αpseudofermions (Nα(q) =1) or αpseudofermion holes (Nα(q) =0) of baremomentum q∈[−q0 α,+q0 α]. In particular, this shifts the α, ι canonical-momentum Fermi point by ¯qFα,ι =[q0 Fα,ι +Qα(ιq0 Fα)/L]. The square of such a shift in units of 2π/L is denoted by 2ι αin Eq. (24). In the present low-energy limit, the latter quantity has the form given in Eq. (25) and for the conformal-field theory it is the conformal dimension of the α, ι primary field. On the other hand, the generators of the small-momentum and low-energy αpseudofermion particle–hole processes (C) in the vicinity of the α, ι Fermi point, correspond in the present lowenergy limit to the generators of two α=c,s Virasoro algebras [24]. These generators have a much simpler form in terms of the pseudofermion creation and annihilation operators than that of 256 J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 those given in Eqs. (66)–(68). Thus, the excited energy eigenstates generated from the J-ground state by the elementary processes (C) correspond to the tower of states of conformal-field theory. A crucial point of the pseudofermion scattering theory is that the αpseudofermions and holes created by the latter processes are not active scattering centers. As discussed above, the overall phase shifts generated by the created pseudofermions exactly cancel those originated by creation of the corresponding holes. This implies that the overall scattering phase shift (15) has for each α pseudofermion or hole scatterer of momentum q∈[−q0 α,+q0 α]the same value given in Eq. (16) for all excited states generated by the elementary processes (C) from a given J-ground state. For the conformal-field theory, this means that all tower states obtained from application of the generators of each of the two αVirasoro algebras onto a given HWS correspond to the same value of the conformal dimension 2±1 αof the two corresponding α, ±1 primary fields. Thus, while the pseudofermion scattering controls the model spectral properties at all energy scales [14], in the limit of low energy considered in this paper the pseudofermion operators are closely related to the conformal-field theory operators and fields. This reveals that rather than corresponding to the original electrons, the conformal-field theory spectrum and operators correspond to the low-energy limit of the general pseudofermion description. 5. Concluding remarks In this paper we have shown that in the limit of low energy the general finite-energy spectralfunction expressions derived in Refs. [14,15] by means of the PDT fully recover the TLL universal expressions of correlation and spectral functions. Importantly, we were able to derive explicit expressions for the pre-factors χ0,Eq.(50), of all terms of the asymptotic expansion (52) for the correlation functions of the 1D Hubbard model. Furthermore, we have shown that the form of these pre-factors is universal for all correlation functions. Our results have also clarified the relation of the low-energy TLL behavior to the general scattering mechanisms which control the model exotic spectral properties at all energy scales. Such a relation was used in the description of the effects behind the qualitative difference in the momentum and energy dependence of the low-energy and finite-energy parts of important singular features of the general spectral functions given in Eq. (3). The low-energy connection of the conformal-field primary fields and Virasoro algebra generators to the pseudofermion description was also clarified. While the studies of this paper considered the 1D Hubbard model, which describes successfully some of the exotic properties observed in low-dimensional materials [11,16–18,49], our results are of general nature for many integrable interacting problems [1,50] and therefore have wide applicability. Such results provide a broader understanding of the low-energy properties of carbon nanotubes [4,5], ballistic wires [6], quasi-1D conductors [7,8], and interacting ultracold fermionic atoms in 1D optical lattices [35,37]. Indeed, our results relate these properties to the general scattering processes of the objects whose occupancy configurations describe the exotic quantum phases of matter corresponding to different energy scales of quasi-1D materials and systems. This is confirmed for finite energies in Refs. [11,16], where the general PDT weight distributions [14,15] are shown to describe the photoemission features of quasi-1D compounds for the whole finite-energy band width, whereas the TLL universal behavior was observed in quasi-1D materials and systems whose low-energy phase is metallic [4–9], as mentioned in Section 1. J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 257 Acknowledgements We thank Patrick A. Lee, Sung-Sik Lee, João M.B. Lopes dos Santos, and Tiago C. Ribeiro for stimulating discussions. J.M.P.C. and L.M.M. thank the hospitality and support of MIT and the financial support of FCT under the grant POCTI/FIS/58133/2004, J.M.P.C. and K.P. thank the support of the ESF Science Programme INSTANS 2005–2010, J.M.P.C. thanks the financial support of the Gulbenkian Foundation and Fulbright Commission, and K.P. thanks the financial support of the OTKA grant T049607. Appendix A. The cand stwo-pseudofermion phase shifts and Fermi point shifts Here we define the two-pseudofermion phase shifts πΦαα(q, q)on the right-hand side of Eqs. (15) and (22) for the scattering part of the αoverall phase shift at bare-momentum q, Eq. (16). Furthermore, we show that the low-energy expression (25) of the square of the four α, ι canonical-momentum Fermi points in units of 2π/L equals that of the conformal dimension of the conformal-field theory four α, ι primary fields used in the studies of Refs. [25–29,34].We start by the definition of the two-pseudofermion phase shifts. These quantities can be expressed as, (A.1)πΦαα(q, q)=π¯ Φα,α4t1Λ0 α(q) U,4t1Λ0 α(q) U,α,α =c,s, where π¯ Φα,α(r, r)is the corresponding rapidity two-pseudofermion phase shift defined below and (A.2)Λ0 c(q) =sink0(q), k0(q), and Λ0 s(q) are ground-state rapidity functions [20]. Those are single-valued functions of the bare-momentum q. Thus, they can be given in terms of their inverse functions, which are the functions q0 c(k) and q0 s(Λ) ≡q0 s1(Λ), respectively, defined in Eq. (A.1) of Ref. [14]. The rapidity two-pseudofermion phase shifts π¯ Φαα(r, r)on the right-hand side of Eq. (A.1) are particular cases of the corresponding general PDT rapidity two-pseudofermion phase shifts. In spite of a different notation for the cν and sνbranches of Refs. [14,15,21], such that ν= γand ν=γ+1, respectively, the general integral equations which define the rapidity twopseudofermion phase shifts ¯ Φαν,αν(r, r)are those given in Eqs. (B30)–(B40) of Ref. [48]. While the phase shifts π¯ Φαα(r, r)considered here refer to the two α=c,s pseudofermion branches whose occupancy configurations describe the low-energy eigenstates, the phase shifts ¯ Φαν,αν(r, r)refer to all the pseudofermion branches. From direct use of the general system of coupled integral equations which defines the PDT two-pseudofermion phase shifts, we find that the phase shifts π¯ Φαα(r, r)involved in our low-energy study are uniquely defined by the following integral equations (A.3) π¯ Φss(r, r)=arctanr−r 2−1 π r0 c  −r0 c dr arctan(r −r) 1+(r −r)2 + r0 s  −r0 s drG(r, r)π ¯ Φss(r,r), 258 J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 (A.4)π¯ Φcs(r, r)=−arctan(r −r)+1 π r0 s  −r0 s dr π¯ Φss(r,r) 1+(r −r)2, (A.5)π¯ Φsc(r, r)=−arctan(r −r)+ r0 s  −r0 s drG(r, r)π ¯ Φsc(r,r), and (A.6)π¯ Φcc(r, r)=1 π r0 s  −r0 s dr π¯ Φsc(r,r) 1+(r −r)2. In the above equations the function arctan(y) corresponds to the branch such that −π/2⩽ arctan(y) ⩽+π/2, the kernel G(r, r)is given by (A.7)G(r, r)=− 1 2π1 1+[(r −r)/2]2−2 π r0 c  −r0 c dr 1 [1+(r −r)2][1+(r−r)2], and the integration limiting values read (A.8)r0 c=4t1sinQ U,r 0 s=4t1B U, where Q=k0(2kF)and B=Λ0 s(kF↓)are the parameters appearing in the expressions of Ref. [12]. They are such that q0 c(±Q) =±2kFand q0 s(±B) =±kF↓, their value being selfconsistently defined by the solution of the relations given in Eq. (A.5) of Ref. [14]. Finally, let us confirm that the low-energy limit of the square of the shift in the value of the α, ι canonical-momentum Fermi point given in Eq. (25) is indeed the conformal dimension of the α, ι primary field. To reach such a goal, we start by noting that combination of Eqs. (26) and (A.1) reveals that the parameters defined in Eq. (26) can for j=1 be expressed as (A.9)ξ1 αα=Ωααr0 α,α,α =c,s, where the function Ωαα(r) is given by (A.10)Ωαα(r) =δα,α+ ι=±1 ι¯ Φα,αr, ιr0 α,α,α =c,s. Based on Eqs. (A.3)–(A.7), it is straightforward to confirm that the functions defined by Eq. (A.10) obey the following integral equations: (A.11)Ωss(r) =1+ r0 s  −r0 s drG(r, r)Ωss(r), (A.12)Ωcs(r) =1 π r0 s  −r0 s dr Ωss(r) 1+(r −r)2, J.M.P. Carmelo et al. / Nuclear Physics B 737 [FS] (2006) 237–260 259 (A.13)Ωsc(r) =1 πarctanr+r0 c−arctanr−r0 c+ r0 s  −r0 s drG(r, r)Ωsc(r), and (A.14)Ωcc(r) =1+1 π r0 s  −r0 s dr Ωsc(r) 1+(r −r)2. From analysis of the form of the kernel function given in Eqs. (A.7), one straightforwardly finds that Eqs. (A.10)–(A.14) are equivalent to those that define the entries of the conformal-field theory dressed charge matrix of Ref. [25] and the transposition of that of Eq. [28]. Thus, this shows that the four parameters ξ1 ααgiven in (A.9) are indeed the entries of these matrices. Finally, if one uses similar procedures for the four parameters ξ0 αα, one finds that they are the entries of the transposition of the inverse of the conformal-field theory dressed charge matrix of Ref. [25] and the inverse of that of Eq. [28]. The quantity 2ι αgiven in Eq. (25) has the same expression as the conformal dimension of the α, ι primary field, provided that the four parameters ξ1 αα(and four parameters ξ0 αα)arethe entries of the conformal-field theory dressed charge matrix (and the entries of the transposition of the inverse of the conformal-field theory dressed charge matrix). Thus, we conclude that such a quantity, which was obtained by considering the low-energy limit of the general functional given in Eq. (24), is indeed the conformal dimension of the α, ι primary field of the 1D Hubbard model αconformal-field theory. References [1] J. Voit, Rep. Prog. 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