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General spectral function expressions of a 1D correlated model

We introduce a method that allows the evaluation of general expressions for the spectral functions of the one-dimensional Hubbard model for all values of the on-site electronic repulsion U. The spectral weights are expressed in terms of pseudofermion operators such that the spectral functions can be written as a convolution of pseudofermion dynamical correlation functions. Our results are valid for all finite energy and momentum values and are used elsewhere in the study of the unusual finite-energy properties of quasi-one-dimensional compounds and the new quantum systems of ultra-cold fermionic atoms on an optical lattice.


I. INTRODUCTION
The main goal of this paper is to provide a general method for the evaluation of matrix elements of one-, twoelectron, or N -electron operators such that N is finite, between the ground state and excited energy eigenstates of the one-dimensional (1D) Hubbard model. Our results correspond to an important part of the derivation of the one-electron and two-electron spectral-weight distributions used in Refs. [1,2] in the study of the phase diagram and unusual one-electron spectral properties of quasi-1D compounds. Indeed, the matrix-element and general spectralfunction expressions derived here are used in Ref. [3] in the evaluation of closed-form expressions for the one-electron and two-electron spectral-weight distributions of the model metallic phase. The 1D Hubbard model is one of the few realistic models for which one can exactly calculate all the energy eigenstates and their energies [4,5]. In addition to the applications to the study of the unusual properties of the quasi-1D compounds presented in Refs. [1,2], our results are also of interest for the understanding of the spectral properties of the new quantum systems described by ultra-cold fermionic atoms in optical lattices with on-site repulsion [6].
The electron -rotated-electron unitary transformation [7] used in Ref. [8] for all values of the on-site repulsion U and the pseudofermion scattering theory introduced in Ref. [9] play a central role in the construction of the pseudofermion dynamical theory introduced here. For finite values of U very little is known about the finite-energy spectral properties of the model. This is in contrast to simpler models [10]. Unfortunately, combination of the model Bethe-ansatz solution [4,5] with bosonization, conformal-field theory, or g-ology and Renormalization Group [11,12] only allows the derivation of low-energy correlation-function expressions. In the limit of infinite U the spectral functions can be evaluated by the method presented in Ref. [13] and there are recent numerical results for finite values of U [14], but it is difficult to extract from them information about the microscopic processes that control the unusual spectral properties of the model. The paper is organized as follows: In Sec. II we introduce the model and the spectral-function problem and summarize the pseudofermion operational description used in our study. In Sec. III we write the general spectral functions in terms of rotated-electron operators. The description of the rotated-electron elementary processes in terms of pseudofermion operators is the problem addressed in Sec. IV. In Sec. V we express the spectral functions as a convolution of pseudofermion spectral functions and study the pseudofermion determinants involved in the expressions of these functions. Finally, the concluding remarks are presented in Sec. VI. Here the operators c † j, σ and c j, σ are the spin-projection σ electronic creation and annihilation operators at site j and n j, σ = c † j, σ c j, σ where j = 1, 2, ..., N a . The number of lattice sites N a is even and very large. We consider periodic boundary conditions. In the first expression of Eq. (1), µ c = 2µ, µ s = 2µ 0 H, µ 0 is the Bohr magneton, and the diagonal generators of the η-spin and spin SU (2) algebras [15,16]Ŝ c z andŜ s z , respectively, are given in Eq. (2) of Ref. [8]. The HamiltonianĤ SO(4) of Eq. (1) commutes with the six generators of these two algebras, their off-diagonal generators being given in Eqs. (7) and (8), respectively, of Ref. [8]. The electron number operator readsN = σN σ , whereN σ = Na j=1n j, σ . For simplicity, we use units such that the Planck constant and electronic lattice constant are one. The model (1) describes N ↑ spin-up electrons and N ↓ spin-down electrons in a chain of N a sites, whose length in the units used here reads L = N a . We introduce the Fermi momenta which, in the thermodynamic limit L → ∞, are given by ±k F σ = ±πn σ and ±k F = ±[k F ↑ + k F ↓ ]/2 = ±πn/2, where n σ = N σ /L and n = N/L. The electronic density can be written as n = n ↑ + n ↓ and the spin density is given by m = n ↑ − n ↓ . We denote the η-spin value η and projection η z (and spin value S and projection S z ) of an energy eigenstate by S c and S z c (and S s and S z s ), respectively. The momentum operator reads, and commutes with the Hamiltonians given in Eq. (1). The spin-projection σ momentum distribution operator appearing in Eq. (2) is given byN σ (k) = c † k, σ c k, σ . Here c † k, σ and c k, σ are the spin-projection σ electron creation and annihilation operators of momentum k. These operators are related to the above local operators as follows, The Bethe-ansatz solvability of the 1D Hubbard model is restricted to the Hilbert subspace spanned by regular states, i.e. the lowest-weight states (LWSs) of the η-spin and spin algebras such that S α = −S α z , where α = c, s [8]. For simplicity, in this paper we restrict our considerations to values of the electronic density n and spin density m such that 0 ≤ n ≤ 1 and 0 ≤ m ≤ n, respectively. Often our expressions are different for the n = 1 Mott-Hubbard insulator phase and 0 < n < 1 metallic phase (and for m = 0 zero spin density and 0 < m < n finite spin densities).
The main aim of this paper is the evaluation of expressions for finite-ω N -electron spectral functions B l N (k, ω), such that l = ±1, of the general form, where the operators in the matrix elements are such that, Here the f summation runs over the excited energy eigenstates, the energies E f correspond to these states, E GS is the ground-state energy, and we use a momentum extended scheme such that k ∈ (−∞, +∞). The operatorsÔ † N (k) andÔ N (k) carry momentum k and are denoted in Eq. (5) byÔ l N (k) where l = +1 and l = −1, respectively. They are related to the local operatorsÔ † N , j ≡Ô +1 N , j andÔ N , j ≡Ô −1 N , j , respectively, by a Fourier transform. The local operatorsÔ l N , j can be written as a product of N = lc, ls=±1 N l lc, ls ; l = ±1 , local electronic creation and annihilation operators. Here N l lc, ls is the number of local electronic creation and annihilation operators of the operatorÔ l N , j for l c = −1 and l c = +1, respectively, and with spin down and spin up for l s = −1 and l s = +1, respectively. It is assumed that the ratio N /N a vanishes in the thermodynamic limit. Note that, by construction, N +1 lc, ls and N −1 lc, ls are such that N +1 lc, ls = N −1 −lc, ls , N +1 −1, ls ≥ N +1 +1, ls , and N −1 −1, ls ≤ N −1 +1, ls . For N > 1 the operatorÔ l N , j has a well defined local structure involving the N −1, ls electronic creation operators of spin projection l s /2, and N l +1, ls electronic annihilation operators of spin projection l s /2 located in neighboring lattice sites. The more usual cases for the description of experimental studies correspond to N = 1 and N = 2. Examples of N -electron operatorsÔ N (k) ≡Ô −1 N (k) are the one-electron operator O 1 (k) = c k, σ (measured in the angle-resolved photoelectron spectroscopy), the spin-projection σ density operator O σsd 2 (k) = 1 √ Na k ′ c † k+k ′ , σ c k ′ , σ , the transverse spin-density operatorÔ sdw 2 (k) = 1 √ Na k ′ c † k+k ′ , ↑ c k ′ , ↓ , the on-site s-wave singlet superconductivity operatorÔ oss 2 (k) = 1 √ Na k ′ c k−k ′ , ↑ c k ′ , ↓ , and the spin-projection σ triplet superconductivity operator O σts 2 (k) = 1 √ Na k ′ cos(k ′ ) c k−k ′ , σ c k ′ , σ . The corresponding local operatorsÔ −1 N , j ≡Ô N , j arê O 1, j = c j, σ ,Ô σsd 2, j = c † j, σ c j, σ ,Ô sdw 2, j = c † j, ↑ c j, ↓ ,Ô oss 2, j = c j, ↑ c j, ↓ , andÔ σts 2, j = c j, σ c j+1, σ , respectively. The charge density operator (measured in density-density electron energy loss spectroscopy and inelastic X-ray scattering) is written in terms of the above spin-up and spin-down density operators. The operatorsÔ l N (k) of physical interest, correspond in general to operatorsÔ l N , j whose N elementary electronic operators create or annihilate electrons in a compact domain of lattice sites. For instance, ifÔ +1 2, j = c † j, ↓ c † j+i, ↑ and thusÔ −1 2, j = c j+i, ↑ c j, ↓ , the interesting cases correspond to i = 0 (on-site s-wave singlet superconductivity) and i = 1 (extended s-wave singlet superconductivity).
The k dependence of the spectral functions (4) can be transferred from the N -electron operatorsÔ l N (k) to the excited energy eigenstates as follows, Here,Ô l N , 0 is the j = 0 local operatorÔ l N , j considered above, k f is the momentum of the excited energy eigenstates, and k GS denotes the ground-state momentum. In this expression, we have chosen j = 0 for the local operatorÔ l N , j . Due to translational invariance, the value of the functions (7) is independent of this special choice.
Let us summarize the basic information about the holon, spinon, pseudoparticle, and pseudofermion descriptions needed for our studies. (For further information, see Refs. [3,8,17,18].) These studies involve the electronrotated-electron unitary transformation, such that rotated-electron double occupancy is a good quantum number for all U/t values [8]. As the Fermi-liquid quasiparticles, the rotated electrons have the same charge and spin as the electrons, but refer to all energies and reorganize in terms of [N a − N c ] η-spin 1/2 holons, N c spin 1/2 spinons, and N c spinless and η-spinless c pseudoparticles, where N c is the number of rotated-electron singly occupied sites [8]. We use the notation ±1/2 holons and ±1/2 spinons, which refers to the η-spin and spin projections, respectively. The ±1/2 holons of charge ±2e correspond to rotated-electron unoccupied (+) and doubly-occupied (−) sites. The complex behavior occurs for the spin-projection σ-rotated electrons occupying singly occupied sites: their spin degrees of freedom originate chargeless spin-projection σ spinons, whereas their charge part gives rise to η-spinless and spinless c pseudoparticles of charge −e.
The local αν pseudofermion creation (and annihilation) operator f † xj , αν (and f xj, αν ) creates (and annihilates) a αν pseudofermion at the effective αν lattice site of spatial coordinate x j = j a 0 αν . Here j = 1, 2, ..., N * αν and a 0 αν = L/N * αν = N a /N * αν is the effective αν lattice constant given in Eq. (55) of Ref. [17] in units of the electronic lattice constant. The general expression of the number of effective αν lattice sites N * αν is given in Eq. (B6) of Ref. [8], where the number of αν pseudofermion holes N h αν is provided in Eq. (B.11) of the same reference. (The number of pseudofermion and pseudofermion holes equals that of the corresponding pseudoparticle and pseudoparticle holes [3,17].) The operator f † xj, αν (and f xj, αν ) is related to the operator f † qj , αν (and fq j , αν ), which refers to αν pseudofermions of canonical-momentumq j , by a Fourier transform. The discrete canonical-momentum values of the αν pseudofermions have a functional character and read [3,9], where q j = [2πI αν j ]/L [8] is the bare-momentum carried by the αν pseudoparticles. Here I αν j are the actual quantum numbers provided by the Bethe-ansatz solution [8]. Although the αν pseudoparticles carry bare-momentum q j , one can also label the corresponding αν pseudofermions by such a bare-momentum. When one refers to the pseudofermion bare-momentum q j , one means that q j is the bare-momentum value that corresponds to the canonical momentum q j = q j + Q Φ αν (q j )/L. Here and in Eq. (8) Q Φ αν (q j )/2 is a αν pseudofermion overall scattering phase shift given by [9], where ∆N αν (q j ) = ∆N αν (q j ) is the distribution function deviation ∆N αν (q j ) = N αν (q j ) − N 0 αν (q j ). The canonicalmomentum distribution function N αν (q j ) (and bare-momentum distribution function N αν (q j )) is given by N αν (q j ) = 1 and N αν (q j ) = 0 (and N αν (q j ) = 1 and N αν (q j ) = 0) for pseudofermions and pseudofermion holes (and pseudoparticles and pseudoparticle holes), respectively [3]. The ground-state densely-packed bare-momentum distribution function 3) of Ref. [8]. The αν = c0, s1 pseudofermion limiting canonical-momentum values play an important role in the theory and read, where q 0 αν is the ground-state limiting bare-momentum value given in Eqs. (C.13) and (C.14) of Ref. [8] and q αν the excited-energy-eigenstate limiting bare-momentum value provided in Eq. (B.14) of the same reference. In contrast to the αν pseudoparticles, the αν pseudofermions have no residual-interaction energy terms [3]. Instead, under the ground-state -excited-energy-eigenstate transitions the αν pseudofermions and αν pseudofermion holes undergo elementary scattering events with the α ′ ν ′ pseudofermions and α ′ ν ′ pseudofermion holes created in these transitions [9]. This leads to the elementary two-pseudofermion phase shifts π Φ αν, α ′ ν ′ (q j , q ′ j ) on the right-hand side of the overall scattering phase shift (9), which are defined by a set of integral equations [3,9]. The overall αν pseudofermion or hole phase shift, plays an important role in the pseudofermion theory [3,9]. Here Q 0 αν /2 can have the values Q 0 αν /2 = 0, ±π/2 [3,9]. The pseudofermion description refers to a Hilbert subspace called pseudofermion subspace (PS) where the Nelectron excitations are contained [3,9]. The PS is spanned by the initial ground state and the excited energy eigenstates generated from it by the following types of processes (A)-(C), which are defined in more detail in Ref. [3]: (A) finite-energy and finite-momentum elementary c0 and s1 pseudofermion processes plus creation of αν = c0, s1 pseudofermions with bare-momentum values q = ±q αν , (B) zero-energy and finite-momentum processes that change the number of c0 and s1 pseudofermions at their Fermi points, which for the ground state and L → ∞ read, plus creation of independent −1/2 holons and/or −1/2 spinons, and (C) low-energy and small-momentum elementary c0 and s1 pseudofermion particle-hole processes in the vicinity of the Fermi points. The PS contains subspaces of several CPHS ensemble subspaces. (Here CPHS stands for c0 pseudofermion, holon, and spinon.) A CPHS ensemble subspace is spanned by all energy eigenstates with fixed values for the −1/2 Yang holon number L c, −1/2 , −1/2 HL spinon number L s, −1/2 , c0 pseudofermion number N c0 , and for the sets of composite αν pseudofermion numbers {N cν } and {N sν } corresponding to the ν = 1, 2, ... branches. The pseudofermion bare-momentum dependent energy dispersions ǫ c0 (q), ǫ s1 (q), ǫ cν (q) = 2νµ + ǫ 0 cν (q) for ν > 0, and ǫ sν (q) = 2νµ 0 H + ǫ 0 sν (q) for ν > 1, where µ = µ(n) and H = H(m) correspond to the density and magnetization curves, respectively, are defined and studied in Refs. [3,8,18]. Such energy dispersions play a crucial role in the expressions of the N -electron spectral functions. For m = 0, the energy 2µ is an increasing function of U and a decreasing function of the density n with the following limiting values, where E MH is the half-filling Mott-Hubbard gap [4]. The evaluation of matrix elements between energy eigenstates considered in Sec. V involves pseudofermion operators f † q, αν and fq′ , αν such that the canonical momentum valuesq andq ′ = q ′ correspond to an excited-energy-eigenstate and initial ground-state CPHS ensemble subspaces, respectively. In that case the unusual pseudofermion anticommutation relations read [3,9], and

III. SPECTRAL FUNCTIONS IN TERMS OF ROTATED-ELECTRON OPERATORS
Here we express the general N -electron spectral functions (7) in terms of rotated-electron creation and annihilation operators and evaluate the spectral-weight contributions from the Yang holons and HL spinons. Our first goal is identifying the set of CPHS ensemble subspaces which are spanned by the excited energy eigenstates generated by application onto the initial ground state of the operatorÔ l N , 0 of Eq. (7). For clarification of this problem, we must find the set of deviation numbers ∆N c0 , ∆N s1 , {∆L α, −1/2 }, and {∆N αν } for αν = c0, s1 that are generated by application onto the ground state of that operator. According to the results of Refs. [3,8], for the ground state M c, −1/2 = L α, −1/2 = N αν = 0 for the αν = c0, s1 branches and thus ∆M c, −1/2 = M c, −1/2 , ∆L α, −1/2 = L α, −1/2 , and ∆N αν = N αν for the latter branches.
First, we note that the values of the +1/2 holon and +1/2 spinon number deviations are such that, and thus are dependent on the values of the −1/2 holon and −1/2 spinon numbers and c0 pseudofermion number deviations. Also the occupancy configurations of the −1/2 holons and −1/2 spinons determine those of the +1/2 holons and +1/2 spinons. Indeed, the −1/2 holons and +1/2 holons correspond to the rotated-electron doubly-occupied sites and unoccupied sites, respectively, of a charge sequence [17]. The point is that the spatial position of the unoccupied sites corresponds to the sites left over by the rotated-electron doubly occupied sites of a charge sequence. The same applies to the −1/2 spinons and +1/2 spinons, provided that we replace the rotated-electron doubly-occupied sites and unoccupied sites by sites singly occupied by spin-down and spin-up rotated electrons, respectively, and the charge sequence by the spin sequence [17]. Moreover, the values of the corresponding +1/2 Yang holon and +1/2 HL spinon number deviations read, and respectively, and thus are not independent. One does not need to provide these values in order to specify a CPHS ensemble subspace. Therefore, often we do not consider in the expressions below the values of the holon numbers M c, +1/2 and L c, +1/2 and of the spinon numbers M s, +1/2 and L s, +1/2 . The values of the deviations ∆N ↑ and ∆N ↓ specific to a given N -electron operator, lead to sum rules for the values of the number deviations of pseudofermions, −1/2 Yang holons, and −1/2 HL spinons as follows, and Only transitions to excited energy eigenstates associated with deviations obeying the sum rules (18) and (19) are permitted. The same deviations are associated with sum rules obeyed by the numbers N l lc, ls of Eq. (6) for the operatorÔ l N , j appearing in the general spectral-function expressions of Eq. (7). Such sum rules read, Furthermore, it is straightforward to show that the following selection rule is valid for initial ground states corresponding to the density values considered in this paper: The values of the numbers of −1/2 Yang holons and −1/2 HL spinons generated by application onto the ground state of the N -electron operatorÔ l N , j , Eq. (7), are restricted to the following ranges, respectively. Here the numbers N l lc, ls are those of Eq. (6) specific to that operator. Further selection rules in terms of the rotated-electron expressions for the operatorÔ l N , 0 of the general spectralfunction (7) are given in the ensuing section.
Let us label the excited energy eigenstates of the state summations of the general N -electron spectral function (7) according to their CPHS ensemble subspace. (We recall that all excited energy eigenstates belonging to a given CPHS ensemble subspace have the same values for the set of deviation numbers ∆N c0 and ∆N s1 and numbers {L α, −1/2 } for α = c, s, and {N αν } for the αν = c0, s1 branches.) This procedure leads to the following expression for the spectral function (7), where l = ±1 and the summation {∆Nαν }, {L α, −1/2 } runs over the CPHS ensemble subspaces whose deviation values obey the sum rules (18) and (19) and selection rules (21). Moreover, the summation f runs over the excited energy eigenstates |f ; C of a given CPHS ensemble subspace, ∆E cphs is the excitation energy, and ∆k cphs the corresponding excitation momentum. A general energy eigenstate |f with finite values for the numbers L c, −1/2 and/or L s, −1/2 , can be expressed as follows, Here, and the η-spin flip Yang holon (α = c) and spin flip HL spinon (α = s) operatorsŜ † α are the off-diagonal generators of the corresponding SU (2) algebras given in Eqs. (7) and (8), respectively, of Ref. [8]. These operators remain invariant under the electron -rotated-electron unitary transformation and thus have the same expression in terms of electronic and rotated-electron creation and annihilation operators. Moreover, in Eq. (23) |f.L is the LWS that corresponds to the state |f . For a state |f ; C belonging to a given CPHS the corresponding LWS is denoted by |f.L ; C . However, note that a non-LWS |f ; C and the corresponding LWS |f.L ; C belong to different CPHSs, once they correspond to different values of the numbers L c, −1/2 and/or L s, −1/2 .
It is useful to reexpress the spectral-function expression (22) in terms of matrix elements between regular states only. The ground state is a LWS of both the η-spin and spin SU (2) algebras and thus has the following property, Let us introduce the operatorsΘ l N , j andΘ l N , k such that, By suitable use of Eq. (25), it is straightforward to show that the operatorsΘ l N , j andΘ l N , k are given by the following commutators,Θ l N , j = α=c, s Thus, Note that when the operatorΘ l N , 0 is given by Eq. These relations provide information about the numbers of electronic creation and annihilation operators of the Nelectron operatorΘ l N , j expression relative to the corresponding numbers of theÔ l N , j expression. While the number of electronic creation (and annihilation) operators decreases (and increases) by 2L c, −1/2 , the number of electronic spin-down creation and spin-up annihilation (and spin-down annihilation and spin-up creation) operators decreases (and increases) by 2L s, −1/2 .
We note that the numbers N l lc, ls of Eq. (6) for the operatorÔ l N , j on the right-hand-side of Eq. (27) obey the sum rules (20). Thus, following the relations of Eq. (30), the numbersN l lc, ls of the corresponding operatorΘ l N , j are such that, The first relation of Eq. (20) just states that the difference in the number of electronic creation and annihilation operators of the original N -electron operatorÔ l N , j equals the value ∆N of the electron number deviation generated by such operator. Similarly, the first relation of Eq. (31) states that the difference in the number of rotated-electron annihilation and creation operators ofΘ l N , j equals twice the value ∆S c = ∆η of the η-spin value deviation generated by that N -rotated-electron operator. Similar considerations apply to the second relations of Eqs. (20) and (31).
We emphasize that all matrix elements of the general spectral-function expression (29) refer to regular energy eigenstates. Indeed, by changing from the spectral function representation (7) to (29) we have eliminated the explicit presence of −1/2 Yang holons and −1/2 HL spinons. This was done by evaluation of the contribution of these quantum objects to the N -electron spectral weight. Such a procedure corresponds to the computation of the commutator [ α=c, s (Ŝ α ) L α, −1/2 ,Ô l N , j ] on the right-hand side of Eq. (27). Our next step is the expression of the operatorΘ l N , j for the general spectral-function expression (29) in terms of rotated-electron creation and annihilation operators. Here we use the results of Refs. [3,8,17] concerning the expression of the rotated electrons in terms of ±1/2 holons, ±1/2 spinons, and c0 pseudofermions. It is this direct relation that makes convenient the rotated-electron expression for the N -electron spectral functions. The expression of the local N -electron operatorΘ l N , j in terms of rotated-electron creation and annihilation operators is obtained by use of the following relation, HereŜ is the operator defined by Eqs. (21)-(23) of Ref. [8] andΘ l N0, j has the same expression in terms of rotatedelectron creation and annihilation operators asΘ l N , j in terms of electronic creation and annihilation operators and thus N 0 = N . It is given by, The operatorsΘ l Ni, j on the right-hand side of Eq. (32) such that i = 1, 2, ... can be written as a product of N i rotated-electron creation and annihilation operators and the value of the coefficient c l i is a function of n, m, and U/t such that c l i → 0 as U/t → ∞. For instance, for i = 1 and i = 2 we find, and respectively, and the i > 2 operator terms are easily generated and involve similar commutators. For simplicity, here we omit the longer expressions of the latter terms. It is useful for the study of the spectral-function expressions to divide each CPHS ensemble subspace in a set of well-defined subspaces. The number deviation ∆N αν for the αν = c0, s1 branches and the number N αν = ∆N αν for the αν = c0, s1 branches can be expressed in terms of other related numbers as follows, Here ∆N F αν, ±1 is the deviation in the number of αν = c0, s1 pseudofermions at the right (+1) and left right (−1) Fermi points, ∆N F αν and ∆J F αν are the corresponding number and current number deviations, respectively, and ∆N N F αν gives the deviation in the number of αν = c0, s1 pseudofermions away from these points. Moreover, ∆N 0,F αν, ±1 is the actual number of αν pseudofermions created or annihilated at the right (+1) and left right (−1) Fermi points and Q 0 αν /2 is the scattering-less phase shift on the right-hand side of Eq. (11). For the αν = c0, s1 branches, N F αν, ι is the number of αν pseudofermions with limiting bare-momentum value q = ι q 0 αν such that ι = ±1, J F αν is the corresponding current number, and N N F αν is the number of αν pseudofermions whose bare-momentum values obey the inequality |q| < q 0 αν . Let us also consider the number N phN F αν of finite-momentum and finite-energy αν pseudofermion particle-hole processes (A), which refers to the αν = c0, s1 branches only [3]. N phN F αν is zero or a positive integer such that Here N αν gives the number of αν pseudofermion creation and annihilation operators involved in the expression of the generators of the elementary processes (A).
The J-CPHS ensemble subspaces are the subspaces of a CPHS ensemble subspace spanned by the excited energy eigenstates with the same values for the numbers Use of Eq. (32) for j = 0 in the general spectral-function expression (29) leads to, where the summations {∆Nαν }, αν, ι } run over CPHS ensemble subspaces and the corresponding J-CPHS ensemble subspaces of each of these spaces, respectively. The function B l,i (k, ω) on the right-hand side of Eq. (36) reads, where the summation f runs over the excited energy eigenstates |f.L; JC which span each J-CPHS ensemble subspace. Thus, there is a function B l,i (k, ω) for each of these subspaces. (We recall that |f.L; JC is the LWS of a state |f ; JC related to it by the general equation (23).) Finally, let us use a notation for the number of spin-down and spin-up rotated-electron creation and annihilation operators of the operatorΘ l Ni, j such that i = 0, 1, 2, ... similar to that associated with the numbersN l lc, ls of Eq. (30). Thus, we introduce the numbers, which refer to the operatorΘ l Ni, j . HereN l,i lc, ls is the number of rotated-electron creation and annihilation operators for l c = −1 and l c = +1, respectively, and with spin down and spin up for l s = −1 and l s = +1, respectively. The operatorΘ l N0, j of Eq. (32) has the same four rotated-electron numbers {N 0,l lc, ls } = {N l lc, ls } as the corresponding operatorΘ l N , j in terms of electrons.

IV. ROTATED-ELECTRON SUM RULES, SELECTION RULES, AND ELEMENTARY PROCESSES IN TERMS OF PSEUDOFERMION OPERATORS
In this section we provide sum rules and selection rules which for the PS arise from the direct relation between rotated electrons and the holons, spinons, and pseudofermions. Furthermore, we use such a relation to express the elementary rotated-electron processes in terms of the pseudofermion creation and annihilation operators.
An important symmetry is that all six generators of the η-spin and spin SU (2) algebras are invariant under the electron -rotated-electron unitary transformation [8]. Thus, the number of spin-projection σ electrons equals the number of spin-projection σ rotated electrons. This also applies to the deviations ∆S c and ∆S s in the η-spin and spin values, respectively, generated by application onto the ground state of a N -electron operator. This symmetry implies that all i = 0, 1, 2, ... operatorsΘ l Ni, j on the right-hand side of Eq. (32) generate the same deviations ∆S c and ∆S s , as the operatorΘ l N , j on the left-hand side of the same equation. It follows that the valuesN l,i lc, ls for all these i = 0, 1, 2, ... operators with the same value of l = ±1 obey the following sum rules, These rules provide useful information about the expression of all i = 1, 2, ... operatorsΘ l Ni, j . In addition to the rotated-electron creation and annihilation operators ofΘ l N0, j , such an expression includes pairs of rotated-electron creation and annihilation operators with the same spin projection σ. Thus, such additional creation and annihilation operators only generate rotated-electron particle-hole excitations and do not change the net number of spin-projection σ rotated electrons created or annihilated by application of the operatorsΘ l Ni, j onto the ground state. The general situation refers to N -electron operators that are not invariant under the electron -rotated-electron unitary transformation. (The problem is trivial for those that are invariant, onceΘ l Ni, j = 0 for i > 0 in that case.) The precise form ofΘ l Ni, j for i = 1, 2, ... depends on the specific N -electron operator under consideration. However, a general property that follows from the relations (39) is that for increasing values of i = 1, 2, ... the operatorsΘ l Ni, j are constructed by adding toΘ l N0, j an increasing number of particle-hole elementary spin-projection σ rotated-electron pairs. In equation (32) the N -electron operatorΘ l N , j is expressed in terms of rotated-electron creation and annihilation operators. From the direct relation between the rotated electrons and the holons, spinons, and pseudofermions it is straightforward to find useful selection rules. Such rules refer to restrictions in the values of the number of −1/2 holons and thus of 2ν-holon composite cν pseudofermions generated by application onto the ground state of each of the i = 0, 1, 2, ... operatorsΘ l Ni, j of expression (32). [The expression of these operators determines the value of the N -electron spectral function of Eq. (36), as confirmed by the form of the related functions (37).] For the PS excited energy eigenstates which have finite overlap with the N -electron excitations, the values of the −1/2-holon number, M c, −1/2 , and number of finite-energy and finite-momentum c0 pseudofermion particle-hole processes, N phN F c0 , of the elementary processes (A) [3] are restricted to the following ranges, Here the numbersN l,i lc, ls are those of Eq. (38) for the operatorΘ l Ni, j . The i = 0 operatorΘ l N0, j has the same expression in terms of rotated-electron creation and annihilation operators as the corresponding operatorΘ l N , j of Eq. (32) in terms of electronic creation and annihilation operators. Therefore, for i = 0 the selection rules given in Eq. (40) read, where the numbersN l −1, ls =N l,0 −1, ls are those of Eq. (30) specific to the operatorΘ l N , j . The first exact ground-state charge selection rule of Eq. (40) concerning the number of −1/2 holons, M c, −1/2 , is equivalent to the following selection rule involving the number deviation −∆M c = ∆M s = ∆N c0 , Indeed, the combination of the inequalities (42) with the relations (27)-(29) of Ref. [8] and that of Eq. (39), readily confirms the equivalence of the first selection rule given in Eq. (40) and that of Eq. (42). Moreover, the −1/2 spinon number deviation ∆M s, −1/2 is fully determined by the value of the −1/2 holon number of the first selection rule of Eq. (40) and reads, Equations (18) While the above rules are exact, direct evaluation of the weights by the method introduced in Sec. V and further developed in Ref. [3] reveals that 94% to 98% of the N -electron weight corresponds to excited energy eigenstates with numbers in the following range, For i = 0 the relation (44) can be written as, where the numbersN l lc, ls =N l,0 lc, ls are those of Eq. (30) specific for the operatorΘ l N , j of Eq. (32). Local −1/2 holons (and −1/2 spinons) correspond to local 2ν-holon composite cν pseudofermions (and 2ν-spinon composite sν pseudofermions). Local αν pseudofermions are associated with the operators f † xj, αν and f xj , αν on the right-hand side of Eq. (34) of Ref. [3]. Let us denote the rotated-electron spin projections σ =↑, ↓ by σ = −1/2, +1/2, respectively, and consider the elementary processes of the N -electron excitations in terms of occupancy configurations of local ±1/2 holons, ±1/2 spinons, and c0 pseudofermions [17]: (i) To create one spin-projection σ = ±1/2 rotated electron at the unoccupied site j, we need to annihilate a local +1/2 holon and create a local c0 pseudofermion and a local ±1/2 spinon at the same site. Annihilation of a spinprojection σ = ±1/2 rotated electron at a spin-projection σ = ±1/2 rotated-electron singly occupied site j, involves the opposite processes.
(ii) To create one spin-projection σ = ±1/2 rotated electron at a spin-projection σ = ∓1/2 rotated-electron singly occupied site j, we need to annihilate a local ∓1/2 spinon and a local c0 pseudofermion and to create a local −1/2 holon at such a site. Again, to annihilate a spin-projection σ = ±1/2 rotated electron at a rotated-electron doubly occupied site j, involves the opposite processes.
(iii) The creation of two rotated electrons of opposite spin projection onto the unoccupied site j involves the annihilation of a local +1/2 holon and the creation of a local −1/2 holon at such a site. Annihilation of two rotated electrons of opposite spin projection onto the doubly-occupied site j, involves the opposite processes.
(iv) The annihilation of one spin-projection σ = ±1/2 rotated electron and creation of one spin-projection σ = ∓1/2 rotated electron at the singly-occupied site j, involves the annihilation of one local ±1/2 spinon and the creation of one local ∓1/2 spinon.
Other processes can be expressed as suitable combinations of the above elementary processes. The local rotatedelectron operator terms which transfer spectral weight from the ground state to each of the J-CPHS ensemble subspaces appearing in the state summation of the spectral-function expression (36) have a specific and uniquely defined form in terms of c0 pseudofermion and composite αν pseudofermion creation and annihilation operators. In order to find the pseudofermion form of these operator terms it is crucial to take into account the initial ground-state pseudofermion occupancies, given in Eqs. (C.24)-(C.25) of Ref. [8]. (We recall that the pseudoparticle-number values of the latter equations equal those of the corresponding pseudofermion numbers.) Before illustrating how the elementary processes (i)-(iv) are generated by the pseudofermion creation and annihilation operators, it is convenient to provide some basic rules for the use of the latter operators. Since following the use of the relations of Eq. (26) all matrix elements are between the ground state and regular excited states, in the processes considered below, the deviations in the numbers of Yang holons (α = c) and HL spinons (α = s) are such that ∆L α = ∆L α ,+1/2 . Some of these processes involve creation or annihilation of +1/2 Yang holons and/or +1/2 HL spinons. However, we recall that within the pseudofermion representation, the +1/2 Yang holons and +1/2 HL spinons do not appear explicitly. Such processes are taken into account by the deviations in the number of discrete bare-momentum (and canonical-momentum) values and effective αν lattice sites of the cν = c0 branches and/or sν branches, respectively [17]. Given the values of the corresponding pseudofermion number deviations, this is readily confirmed if one compares the number (B.6) of Ref. [8] of discrete bare-momentum values and of effective lattice sites of the excited energy eigenstate and ground state CPHS ensemble subspaces. Since ∆L α = ∆L α ,+1/2 , note that following Eqs. (B.6) and (B.7) of Ref. [8], the value of the number N * αν changes when the value of the number L c = L c ,+1/2 of +1/2 Yang holons and/or L s = L s ,+1/2 of +1/2 HL spinons also changes. Thus, creation and annihilation of +1/2 Yang holons (and +1/2 HL spinons) are processes that are taken into account in the definition of the effective cν = c0 pseudofermion lattices of the initial ground state and excited energy eigenstates.
In the following we provide different examples of local rotated-electron operator expressions in terms of pseudofermion creation and annihilation operators. For simplicity, each of such pseudofermion expressions corresponds to the term of the local rotated-electron operator which transfers spectral weight from the initial ground state onto a single excitation J-CPHS ensemble subspace. Such a pseudofermion term includes a coefficient factor 1/C J whose value is well defined for each subspace. The full local rotated-electron operator term which transfers spectral weight from the ground state to a given J-CPHS ensemble subspace is the product of that studied here by another pseudofermion operator term given in the ensuing section. (The general form of 1/C J is also given in that section.) The latter operator transfers from the ground state to the J-CPHS ensemble subspace the part of the spectral weight which corresponds to the processes (B) and (C), whereas the pseudofermion terms studied here transfer the part of that weight associated with the processes (A).
A i = 0 local rotated-electron operatorΘ l N0, j always has one or a few dominant CPHS ensemble subspaces which correspond to the whole spectral weight transferred from the ground state in well defined limits. For one-electron problems such that N 0 = N = 1 this refers to the limits where the spectral-weight distribution is δ-function like, as in Eq. (77) of Ref. [3]. For N 0 = N = 2, to the limits where such a distribution can be expressed as a simple integral whose integrand is a δ function, as in Eq. (78) of the same reference. In the general N 0 = N > 2 case, to the limits where the spectral-weight distribution can be written as an integral whose integrand is a product of [N − 1] δ functions. For instance, for N = 1 this occurs for U/t → 0. For the one-electron problem the amount of spectral weight transferred from the ground state to the set of J-CPHS ensemble subspaces contained in the dominant CPHS ensemble subspaces is weakly dependent on U/t: while for U/t << 1 it corresponds to the whole spectral weight transferred from the ground state by the local rotated-electron operator, for U/t >> 1 it corresponds typically to more than 0.94% of that weight. (Comparison of the amount of transferred weight for U/t << 1 with that obtained by use of the methods of Ref. [13] for U/t >> 1 confirms such a weak U/t dependence.) For U/t finite there arise an infinite number of pseudofermion terms, each corresponding to a J-CPHS ensemble subspace compatible with the local rotated-electron operator. Another example is the N = 2 charge dynamical structure factor, where there are different dominant CPHS ensemble subspaces for U/t → 0 and U/t → ∞, respectively. In this case the amount of spectral weight transferred from the ground state to the set of J-CPHS ensemble subspaces contained in the dominant CPHS ensemble subspaces is a decreasing (and increasing) function of U/t for the U/t → 0 (and U/t → ∞) dominant subspaces and vanishes as U/t → ∞ (and U/t → 0). Again, for intermediate finite values of U/t there arise an infinite number of pseudofermion terms, each corresponding to a J-CPHS ensemble subspace compatible with the local rotated-electron operator. However, for all i = 0 local rotated-electron operatorsΘ l N0, j the pseudofermion terms associated with the dominant subspaces together with a small number of other terms correspond to more than 99% of the spectral weight. It follows that in applications of the pseudofermion dynamical theory introduced here and in Ref. [3] only a finite number of pseudofermion terms should be considered.
The αν = c0, s1 pseudofermion number deviations and αν = c0, s1 pseudofermion numbers are related to the rotated-electron number deviations by Eqs. (18) and (19). Given the values of the αν = c0, s1 pseudofermion number deviations and αν = c0, s1 pseudofermion numbers of the specific J-CPHS ensemble subspace under consideration, the expression of the local rotated-electron operator in terms of pseudofermion creation and annihilation operators is always uniquely defined. Let us start by providing some of the simplest pseudofermion operator terms of local rotated-electron operators. For local one-and two-rotated-electron operators these operator terms involve in general αν pseudofermion creation and/or annihilation operators belonging to branches such that ν < 2. The case of other terms associated with excitation J-CPHS ensemble subspaces generated from the ground state by processes involving creation of composite αν pseudofermions for ν > 1 is discussed later.
In the following expressions the αν effective lattice integer site index j ′ is such that j ′ = 1, 2, ..., N * αν [17]. An important property is that an operator whose expression in terms of rotated-electron operators is local at x j = ja = j can be written as a product of local αν pseudofermion operators at x j ′ ≈ x j where x j ′ = j ′ a 0 αν . Here and in all expressions given below j ′ is defined for the αν = c0 branches as the closest integer number to jn * αν , whereas j ′ = j for αν = c0. We note that for the former branches the site j ′ occupied by one αν pseudofermion corresponds to 2ν sites of the rotated-electron lattice. Thus, |x j ′ − x j | is always smaller than the very small intrinsic uncertainty which corresponds to the 2ν rotated-electron lattice sites occupied by the local αν pseudofermion. Moreover, we emphasize that the rotated-electron lattice site j associated with the effective αν = c0 lattice site j ′ ≈ jn * αν defined above always belongs to the domain of 2ν rotated-electron lattice sites of j ′ . Here and below we use the equality j ′ = jn * αν to denote the integer number j ′ defined above. Thus, the site j ′ is such that j ′ = j for αν = c0 operators, j ′ = jn ↑ for αν = s1 operators, j ′ = j[1 − n] for cν = c0 operators when n < 1, and j ′ = j[n ↑ − n ↓ ] for sν = s1 operators when m > 0. The following local rotated-electron operator expressions in terms of pseudofermions, whose coefficient C J is different for each operator, refer to the elementary processes (A) subspace: (i) One of the simplest processes for creation of one spin-down rotated electron at the unoccupied site j involves the creation of a local c0 pseudofermion with the operator f † xj, c0 and of a local s1 pseudofermion with the operator and thusc j, ↓ (1 −ñ j, ↑ ) = 1 CJ f xj , c0 f x j ′ , s1 refers to annihilation of one spin-down rotated electron at the singlyoccupied site j. In Eq. (46),ñ j, σ =c † j, σc j, σ is the local spin-projection σ rotated-electron density operator. To create one spin-up rotated electron at the empty site j, a simple process corresponds to create a local c0 pseudofermion with the operator f † xj, c0 ,c † to annihilation of one spin-up rotated electron at the singly-occupied site j. Such processes also involve creation and annihilation, respectively, of an empty site in the effective s1 lattice. When the initial ground state belongs to a m = 0 CPHS ensemble subspace, there is for the former process a single s1 pseudofermion hole in the excited state, which corresponds to the created site.
We note that the processes of the first expressions of Eqs. (46) and (47) also involve the annihilation of a +1/2 Yang holon, whereas the processes of the second expressions of the same equations involve the creation of a +1/2 Yang holon. Similarly, the processes of the first and second expressions of Eq. (47) involve the creation and annihilation, respectively, of a +1/2 HL spinon. In the remaining cases considered below we do not specify the elementary processes of creation or annihilation of +1/2 Yang holons and +1/2 HL spinons, which are taken into account implicitly by the pseudofermion description, as discussed above.
(ii) One of the simplest processes associated with the creation of one spin-up rotated electron at a spin-down rotated-electron singly occupied site j involves the annihilation a local c0 pseudofermion with the operator f xj , c0 and of a local s1 pseudofermion with the operator f x j ′ , s1 and the creation of a local c1 pseudofermion with the operator f † x j ′′ , c1 such that j ′ = jn ↑ and j ′′ = j[1 − n], respectively, refers to annihilation of one spin-up rotated electron at a doubly occupied site j. Moreover, to create one spin-down rotated electron at a spin-up rotated-electron singly occupied site j, a simple process corresponds to annihilate a local c0 pseudofermion with the operator f xj, c0 and to create a local c1 pseudofermion with the operator f † In this casec j, ↓ñj, ↑ = 1 CJ f † xj , c0 f x j ′ , c1 corresponds to annihilation of one spin-down rotated electron at a doubly occupied site j.
(iii) A simple process involved in the creation of two rotated electrons of opposite spin projection onto the empty site j corresponds to creation of a local c1 pseudofermion with the operator f † It follows thatc j, ↑cj, ↓ = 1 CJ f x j ′ , c1 refers to annihilation of two rotated electrons of opposite spin projection onto a doubly-occupied site j. This involves annihilation of a local c1 pseudofermion with the operator f x j ′ , c1 such that j ′ = j[1 − n].
(iv) One of the simplest processes associated with the annihilation of one spin-up rotated electron and creation of one spin-down rotated electron at the singly-occupied site j, involves the creation of a local s1 pseudofermion with the operator f † Thenc † j, ↓c j, ↑ = 1 CJ f x j ′ , s1 corresponds to annihilation of one spin-down rotated electron and creation of one spin-up rotated electron at the singly-occupied site j. This involves the annihilation of a local s1 pseudofermion with the operator f x j ′ , s1 such that j ′ = jn ↑ .
For local rotated-electron operators generating more complex processes involving creation or annihilation of several rotated electrons, the creation and annihilation of local c0 pseudofermions is always associated with creation and annihilation of rotated-electron singly occupied sites, respectively. Since the local c0 pseudofermions and c0 pseudofermion holes occupy the same sites j l and j h as the rotated-electron singly occupied sites and rotated-electron doubly-occupied and unoccupied sites, respectively, there is a one-to-one correspondence between the rotated-electron and c0 pseudofermion algebras. Here we have used the site notation of Ref. [17].
However, once the composite local αν = c0 pseudofermions have internal structure that involves 2ν rotated-electron sites with different index j, the operational relation of rotated electrons to such composite quantum objects is more involved [17]. This justifies why the expressions of the local rotated-electron operators in terms of creation and annihilation pseudofermion operators involve a superposition of different pseudofermion expressions, corresponding to the set of compatible J-CPHS ensemble subspaces. Nevertheless, creation onto the ground state of a cν pseudofermion (and sν pseudofermion) always involves ν rotated-electron doubly occupied sites and ν unoccupied sites (and ν spin-down rotated-electron singly occupied sites ν spin-up rotated-electron singly occupied sites). In general, the excited-energy-eigenstate ν rotated-electron unoccupied sites (and ν spin-up rotated-electron singly occupied sites) of a created cν pseudofermion (and sν pseudofermion) are generated by annihilating an equal number of +1/2 Yang holons (and +1/2 HL spinons) of the initial ground state.
For the creation of a local cν pseudofermion, each of the ν new created rotated-electron doubly occupied sites can result from creation of a rotated-electron pair onto an unoccupied site or of a rotated electron onto a singly-occupied site. The latter case involves always one of the elementary processes associated with the pseudofermion terms given just after Eqs. (46) and (47). On the other hand, for the creation of a local sν pseudofermion such that ν > 1, the ν involved spin-down rotated-electron singly occupied sites can result from creation of spin-down rotated electrons onto unoccupied sites or from recombination of pre-existing ground-state s1 pseudofermions, as further discussed below.
For instance, let us consider two J-CPHS ensemble subspaces contained in different CPHS ensemble subspaces which except for the occupancies of the c1 and c2 branches have the same pseudofermion numbers. For such branches, one has {N c1 = 2, N c2 = 0} for the J-CPHS ensemble subspace (I) and {N c1 = 0, N c2 = 1} for the J-CPHS ensemble subspace (II). Let us consider that the local rotated-electron operator behind the transitions to both subspaces is the same and involves creation of two rotated electrons of spin projections σ =↑ and σ =↓ onto the spin-down singly occupied site j and spin-up singly occupied site j + 1, respectively. For the subspace (I), this corresponds simply to the process of Eq. (48) for the site j and the process of Eq. (49) for the site j + 1. For the subspace (II), in order to create two rotated electrons of spin projections σ =↑ and σ =↓ onto the spin-down singly occupied site j and spin-up singly occupied site j + 1, respectively, we need to annihilate two local c0 pseudofermions with the operators f xj, c0 and f xj+1, c0 and a local s1 pseudofermion with the operator f x j ′ , s1 such that j ′ = jn ↑ and to create a local c2 pseudofermion with the operator f † It should be mentioned that in spite of the annihilation of one s1 pseudofermion, this process does not involve the corresponding creation of a s1 pseudofermion hole. Indeed, it involves the annihilation of the site j ′ = jn ↑ in the effective s1 lattice. Thus, when the initial ground state belongs to a m = 0 CPHS ensemble subspace, in spite of the annihilation of the s1 pseudofermion the excited state corresponds to a fully occupied s1 band, as the initial ground state. A similar process gives rise to creation of a local s2 pseudofermion provided that creation of the two rotated-electron doubly-occupied sites is replaced by creation of two spin-down rotated electron singly occupied sites. However, in this case there is the possibility that one (or both) the spin-down spinons needed for creation of the local s2 pseudofermion is (or are) generated from annihilation of one (or two) ground-state s1 pseudofermion(s). Such processes can dress any rotated-electron process and are behind the occurrence of an infinite number of compatible J-CPHS ensemble subspaces for each local rotated-electron operator. These non-dominant pseudofermion processes do not obey the relation (45) (which is not an exact rotated-electron selection rule) and for all finite values of U/t amount to less than 6% of the rotated-electron spectral weight. For instance, in order to create one spin-down rotated electron at the empty site j, in addition to the pseudofermion process (46) there is for instance a process corresponding to the creation of a c0 pseudofermion with the operator f † xj , c0 and of a s2 pseudofermion with the operator f † x j ′′ , s2 such that j ′′ = j[n ↑ − n ↓ ] and to the annihilation of a s1 pseudofermion with the operator f † x j ′ , s1 such that j ′ = jn ↑ , We emphasize that the amount of spectral weight transferred from the ground state by the operator (46) is much larger than that transferred by the operator (53). Indeed, the J-CPHS ensemble subspace associated with the expression (46) belongs the dominant CPHS ensemble subspace of the local rotated-electron operator, whereas the J-CPHS ensemble subspace corresponding the expression (53) does not. Note that in the present case the two competing J-CPHS ensemble subspaces can have the same pseudofermion deviation numbers and values except for the occupancies of the s1 and s2 branches. Thus, the processes generated by the operators (46) and (53) correspond to J-CPHS ensemble subspaces belonging to different CPHS ensemble subspaces such that {∆N s1 = 1, N s2 = 0} and {∆N s1 = −1, N s2 = 2}, respectively. Similar dressing processes involving creation of sν pseudofermions belonging to ν > 1 branches by annihilation of one to ν ground-state s1 pseudofermions can occur for all rotated-electron processes but correspond to very fast decreasing values of the amount of spectral weight transferred from the ground state for increasing number of pseudofermion processes. Moreover, we recall that the subspace summation on the right-hand side of the spectral-function expression (36) is limited to the compatible CPHS ensemble subspaces: their pseudofermion number deviations and numbers obey the sum rules (18), (19), (39), and (43) and selection rules (21), (40), and (42). We could present here other pseudofermion terms of increasing complexity, corresponding to the local rotatedelectron operators considered above. However, the amount of spectral weight transferred from the ground state by the pseudofermion operator terms describing the above-mentioned dressing processes involving creation of sν pseudofermions such that ν > 1 by annihilation of an increasing number of ground-state s1 pseudofermions decreases very rapidly for increasing values of ν. Also the spectral weight transferred from the ground state by the operator terms with increasing value for the index i of the expression (32) of the general operatorΘ l N , j decreases very rapidly. For instance, for the one-electron spectral weight the contributions from dressing sν pseudofermion processes for ν > 2 and the terms of index i > 1 of the expression (32) for N = 1 are typically beyond numerical measurability. Therefore, as far as numerical measurability is concerned, only a few pseudofermion terms contribute to the actual electronic spectral-weight distributions [3].

V. THE SPECTRAL FUNCTION AS A CONVOLUTION OF PSEUDOFERMION SPECTRAL FUNCTIONS
In this section we express the spectral functions (37) as a convolution of pseudofermion, independent −1/2 holon, and independent −1/2 spinon spectral functions. The excited energy eigenstates appearing on the right-hand side of Eq. (37) can be written as the following pseudofermion Slater determinant, Here and in other expressions below |0 is the pseudofermion vacuum such that fq j , αν |0 = 0 for all αν branches and canonical-momentum values. In turn, according to Eqs. (C.24) and (C.25) of Ref. [8], the ground state corresponds to a canonical-momentum densely packed occupancy for the c0 and s1 pseudofermion bands and the Slater determinant has the following simpler form, The generators F † −GS, αν and F † J−GS, αν given here correspond to densely packed distributions introduced below and the discrete canonical-momentum values of the pseudofermion operators f † qj , αν of their expressions are those of the CPHS ensemble subspace which the ground state |GS and the excited state |f.L; C of Eq. (54) belong to, respectively. The Fermi points appearing in the products of their expressions of Eq. (55) read q F αν, ±1 = ±q 0 ], respectively, where the deviation numbers ∆N 0,F αν, ±1 and ∆N F αν, ±1 are those of Eq. (35). The excited-energy-eigenstate canonical-momentum distribution function N αν (q j ) on the right-hand side of Eq. (54) can be written as, Here N −0 αν (q j ) and N −0 αν (q j ), such that N −0 αν (q j ) = N −0 αν (q j ), correspond to the excited densely packed distributions αν=c0, s1 F † J−GS, αν |0 and αν=c0, s1 F † −GS, αν |0 , respectively. Furthermore, the ground-state distribution N 0 αν (q j ) is both such thatq j = q j and N 0 αν (q j ) = N 0 αν (q j ), where N 0 αν (q j ) is the ground-state bare-momentum distribution function given in Eqs. (C.1)-(C.3) of Ref. [8]. Thus, ∆N F αν (q j ) = ∆N F αν (q j ) describes αν = c0, s1 pseudofermion addition to or removal from the Fermi points. Moreover, ∆N N F αν (q j ) = ∆N N F αν (q j ) describes αν pseudofermion creation and/or annihilation away from the Fermi points for the αν = c0, s1 branches and creation of αν pseudofermions at canonical-momentum values such that |q j | < q 0 αν for the αν = c0, s1 branches, whereas ∆N F αν (q j ) = ∆N F αν (q j ) describes creation of αν = c0, s1 pseudofermions at the limiting canonical-momentum valuesq j = ±q 0 αν . Finally, the deviation ∆N phF αν (q j ) = ∆N phF αν (q j ) corresponds to low-energy and small-momentum αν = c0, s1 pseudofermion particle-hole processes. For n = 1 (and/or m = 0) the excitation subspace is such that ∆N phF c0 (q j ) = ∆N phF c0 (q j ) = 0 (and/or ∆N phF s1 (q j ) = ∆N phF s1 (q j ) = 0). The above deviations are such that, Since for the αν = c0, s1 pseudofermion branches there is no occupancy in the initial ground state, the canonicalmomentum distribution function is such that, The generator F † f, αν given in Eq. (54) can be written as, The expressions of the generators F † J−N F, αν , F † p−h, αν , and F † N F, αν in terms of pseudofermion creation and annihilation operators are given in Eqs. (B.1), (B.4), and (B.5) of Ref. [3], respectively, and that of the generator F † J−GS, αν is provided in Eq. (55), whereas the generator F † F, αν creates αν = c0, s1 pseudofermions at q = ±q 0 αν . The generators F † p−h, αν F † J−GS, αν and F † J−N F, αν are associated with the deviations N ph αν (q j ) and ∆N N F αν (q j ) of Eq. (56), respectively, F † p−h, αν corresponds to the deviation ∆N phF αν (q j ) of the same equation and thus generates the low-energy and smallmomentum αν = c0, s1 pseudofermion particle-hole processes (C). The operators F † N F, αν and F † F, αν refer to the αν = c0, s1 branches and are associated with the deviations ∆N N F αν (q j ) and ∆N F αν (q j ), respectively. The precise expression of the spectral functions of Eq. (37) depends on the specific form of the local operator Θ l Ni, k , whose expression includes contributions from all αν branches with finite pseudofermion occupancy in the corresponding J-CPHS ensemble subspace. For each such a subspace that operator expression and the corresponding coefficient C J appearing in Eqs. (46)-(53) have the following general form, where the values of the positive real coefficient G C and momentum deviation ∆P J are given below and we recall that j ′ = jn * αν denotes the integer number closest to jn * αν . In equation (57) and in the remaining of this paper θ(x) is such that θ(x) = 1 for x > 0 and θ(x) = 0 for x ≤ 0.
The value of the coefficient G C appearing in the general expression for the operatorΘ l Ni, j given in Eq. (57) is the same for all J-CPHS ensemble subspaces belonging to a given CPHS ensemble subspace, whereas that of G J and thus of |C J | is specific to each J-CPHS ensemble subspace. Furthermore, when the expression of the local operatorÔ l N , j of Eq. (7) is independent of U/t, the same occurs for the related operatorsΘ l N , j andΘ l N0, j of Eqs. (27)-(28) and (32), respectively. It follows that in the case of the i = 0 operatorΘ l N0, j , the value of the coefficient G C appearing in its expression given in Eq. (57) is also independent of U/t. Fortunately, for the dominant CPHS ensemble subspaces considered in the previous section, such a value can be found from analysis of the problem for U/t = 0 or U/t = ∞. One then finds that G C = 1 in theΘ l N0, j = 1 GC [ ανΘ l,0 j ′ , αν ] expression corresponding to the dominant CPHS ensemble subspaces.
The operatorΘ l,i j ′ , αν appearing in Eq. (57) has the following general form for the αν branches with finite pseudofermion occupancy in the J-CPHS subspace, Here the operatorsΘ l,N F,i j ′ , αν andΘ l,F,i αν are associated with the elementary processes (A) and (B), respectively, and the phase-factor momentum l∆P 0 αν is given below. Considering that [x j ′ − x j ] = 0, it is such that −i αν ∆P 0 αν j ′ a 0 αν = −ij∆P J . Here the summation αν is over the αν branches with pseudofermion occupancy in the J-CPHS ensemble subspace and ∆P J is the momentum deviation of Eq. (57), which reads, Let us proceed by studying the phase factor e −i∆P 0 αν j ′ a 0 αν and operatorsΘ l,F,i αν andΘ l,N F,i j ′ , αν whose product gives the operatorΘ l,i j ′ , αν . We start by characterizing for each αν branch the elementary processes that originate the phase factor e −i∆P 0 αν j ′ a 0 αν . The expression of the local operatorΘ l,i j ′ , αν does not involve the generator of the elementary processes (C), F † p−h, αν , but provides the momentum l∆P phF αν for such processes through a phase factor, e −i∆P phF αν j ′ a 0 αν . Here l∆P phF αν = l[2π/L] [m αν, +1 − m αν, −1 ] is the small momentum deviation generated by the αν = c0, s1 pseudofermion particle-hole elementary processes (C) and m αν, ±1 is the number of such processes of momentum ±[2π/L] in the vicinity of the Fermi points ±q 0 F αν . Furthermore, each αν = c0, s1 pseudofermion created or annihilated at the Fermi point ι q 0 F αν by the elementary processes (B) contributes with a phase factor e −iι q 0 F αν j ′ a 0 αν or e +iι q 0 F αν j ′ a 0 αν , respectively, where ι = ±1. This leads to a phase factor e −i2q 0 cν with αν = c0, s1. Adding all these contributions leads to the above net phase factor e −i∆P 0 αν j ′ a 0 αν for the αν = c0, s1 branches whose phase-factor momentum reads l∆P 0 αν = l[∆P phF αν + ∆P F αν ]. For densities in the ranges 0 < n < 1 and 0 < m < n, the momentum ∆P F αν appearing in that phase factor is given by ∆P for the αν = c0 and αν = s1 branches, respectively. It results from the current contributions associated with the c0 and s1 Fermi points ±2k F and ±k F ↓ , respectively. Moreover, each cν = c0 pseudofermion created by the elementary processes (A) contributes with a phase factor e −i(1+ν)πj ′ a 0 cν what leads to a net phase factor e −i∆P 0 cν j ′ a 0 cν for the cν = c0 branches such that ∆P 0 cν = (1 + ν)πN cν . Finally, the phase-factor momentum vanishes for the sν = s1 branches. Thus, the phase-factor momenta contributing to ∆P J read, Next we consider the operatorΘ l,F,i αν . For the cν = c0 and sν = s1 branches, that operator creates 2νN F cν independent −1/2 holons of momentum π and 2νN F sν independent −1/2 spinons of momentum zero, respectively. (The only effect of the creation of the corresponding N F cν cν = c0 FP scattering centers and N F sν sν = s1 FP scattering centers [3], respectively, is the above contribution to the phase factor e −i∆P 0 c0 j ′ a 0 c0 and e −i∆P 0 c0 j ′ a 0 c0 e −i∆P 0 s1 j ′ a 0 s1 , respectively.) For the αν = c0, s1 branches the operatorΘ l,F,i αν is such that, Thus, that operator can be written as, where we omitted corrections of order 1/L to the momentum value q 0 αν appearing in the phase factor. However, these corrections must be considered in the momentum of the pseudofermion operators. In the above two equations the pseudofermion operators correspond to spatial and canonical-momentum variables, respectively, and in the argument of the exponentials appearing in these equations and in other equations given below i is the usual imaginary number. (It is not the index i of the operatorΘ l,F,i αν , whereas i ′ is a summation index.) Moreover, in the above equations and in the remaining of this paper Θ(x) is such that Θ(x) = 1 for x ≥ 0 and Θ(x) = 0 for x < 0.
We finish the study of the operatorΘ l,i j ′ , αν by considering the operatorΘ l,N F,i j ′ , αν associated with the elementary processes (A), whose expression refers to a given J-CPHS subspace. In order to arrive to that expression in terms of the local αν pseudofermion creation and annihilation operators, we recall that our study refers to spectral functions of form (7) whose operatorÔ l N , j expression involves N elementary electronic operators which create or annihilate electrons in a compact domain of lattice sites. For such an operator the general expression of the corresponding operatorsΘ l,N F,i j ′ , αν involves N αν local αν pseudofermion creation and annihilation operators for the αν = c0, s1 branches (and creation operators for the αν = c0, s1 branches) which refer to a compact domain of N αν effective αν lattice sites. The operators (46)-(53) involve the product of operators whose expressions involve elementary operators of a single αν branch and are particular examples of such a general expression, which has the following form, Here the spatial coordinates x 0 , x 1 ,...,x Nαν −1 correspond to the compact domain of N αν effective αν lattice sites where the number N αν is given in Eq. (57). Let us consider the operatorΘ l,i k, αν , which is the Fourier transform of the above local operatorΘ l,i j ′ , αν , This operator can be expressed in terms of the following momentum convolution, where the operatorΘ l,F,i αν was given above andΘ l,N F,i k, αν is the Fourier transform of the local operatorΘ l,N F,i j ′ , αν . It reads, Here the setq 0 ,q 1 ,q 2 ,...,q Nαν −1 refers to N αν summation canonical-momentum variables associated with the αν pseudofermion bands. We note that the canonical-momentum values in the Krönecker δ's of Eq. (58) run under the summations but not under the products appearing in that equation. For the αν = c0, s1 branches, the N αν -αν pseudofermion operatorΘ l,N F,i k, αν creates and annihilates |∆N N F αν | αν pseudofermions when ∆N N F αν > 0 and ∆N N F αν < 0, respectively, and generates N phN F αν = 0, 1, ... finite-momentum and finite-energy αν pseudofermion particle-hole processes. For the αν = c0, s1 branches, it creates N N F αν αν pseudofermions of bare-momentum q such that |q| < q 0 αν . In the case of the αν = c0, s1 branches, the general expression given in Eq. (58) for the operatorΘ l,N F,i k, αν is valid for densities such that the corresponding ratios n * αν = N * αν /N a have finite values. For the cν = c0 (and sν = s1) pseudofermion branches and electronic density n = 1 (and spin density m = 0) all pseudofermions separate into 2ν independent −1/2 holons (and 2ν independent −1/2 spinons). Therefore, the operator given in Eq. (58) does not exist. Moreover, for n = 1 (and m = 0) the above generator F † p−h, c0 (and F † p−h, s1 ) of the elementary processes (C) reduces to F † p−h, c0 = 1 (and F † p−h, s1 = 1). However, our theory also applies for electronic density n = 1 (and spin density m = 0), provided that we consider the corresponding restrictions in the c0 (and s1) excitation spectrum and take into account thatΘ l,i k, αν =Θ l,N F,i k, αν , for the cν = c0 (and sν = s1) branches. For the αν = c0, s1 branches the pseudofermion weight distribution involves the following matrix element, Our study refers to very large values of L when the commutator [F † p−h, αν , F † J−N F, αν ] = 0 vanishes and the operator Θ l,F,i αν is such thatΘ l,F,i αν F † GS, αν = F † −GS, αν and thus the matrix element can be rewritten as, When applying the generators F † f, L, αν and F † GS, αν of Eqs. (54) and (55), respectively, onto the pseudofermion vacuum to construct a given energy eigenstate, the set of αν band discrete canonical-momentum values {q j } of the pseudofermion operators f † qj , αν in the expressions of these generators are those of the CPHS ensemble subspace which that state belongs to. This rule applies when one considers the generators of the full energy eigenstates. (Below we express each J-CPHS as a direct product of subspaces. To reach the correct final results, such a rule does not apply to some of the states which span such direct-product subspaces.) The same occurs with the discrete canonical-momentum values of the pseudofermion creation and annihilation operators of the expression of any operatorΘ when it acts onto a given energy eigenstate. For instance, let |β and |β ′ be energy eigenstates. Thus, the discrete canonical-momentum values of the pseudofermion creation and annihilation operators of the expression of the operatorsΘ † andΘ inΘ † |β ′ andΘ|β are those of the CPHS ensemble subspace which the states |β ′ and |β belong to, respectively. An important property for our theory is that for L large both choices lead to the same value for the matrix element β|Θ|β ′ .
The operatorΘ l,N F,i k, αν of Eq. (58) is for the αν = c0, s1 branches such that the commutator [F p−h, αν ,Θ l,N F,i k, αν ] = 0 vanishes when it acts onto the CPHS ensemble subspace which the corresponding excited state belongs to. Furthermore, there occurs a full overlap of that operator with the generator F J−N F, αν and for the αν = c0, s1 branches there occurs a full overlap of the operatorΘ l,N F,i k, αν given in the second expression of Eq. (58) with the generator F N F, αν . The latter full overlap results from the lack of αν = c0, s1 pseudofermion occupancy of the initial ground state.
A J-CPHS ensemble subspace can be expressed as the direct product of subspaces, one for each αν branch pseudofermion occupancy. In the particular case of the αν = c0, s1 branches the low-energy and high-energy physics separate provided that L is large and one can define two of such product subspaces for each branch. They are associated with the excitation occupancy configurations generated by the finite-energy elementary processes (A) and low-energy elementary processes (B,C), respectively. We call p − h, αν = c0, s1 branch subspace the latter low-energy subspace. Thus, the number of product subspaces equals the number of αν branches with finite pseudofermion occupancy in the J-CPHS ensemble subspace plus two. Finally, for some J-CPHS ensemble subspaces the direct product also includes the independent −1/2 holon subspace and independent −1/2 spinon subspace. For such subspaces the generator F † F, αν either creates N F αν αν pseudofermions with limiting bare-momentum values q = ±q 0 αν or reads F † F, αν = 1 when N F αν = 0 and thus F † f, αν = F † N F, αν (or F † f, αν does not exist if n = 1 and αν = cν or m = 0 and αν = sν). The states which span the αν branch direct-product subspaces and p−h, αν = c0, s1 branch direct-product subspaces have the following form, |f.L; αν ≡ F † J−N F, αν |GS ; αν = c0, s1 ; |f.L; αν ≡ F † N F, αν |GS ; αν = c0, s1 , |f.L; p − h, αν ≡ F † p−h, αν F † J−GS, αν |0 ; αν = c0, s1 .
Below we express the spectral functions as convolutions of p − h, αν and αν pseudofermion spectral functions. However, in order to reach the same spectral-weight distributions as by use of the above matrix elements, it turns out that: (i) When applying the generators F † J−N F, αν and F † N F, αν onto the ground state |GS to construct a given αν branch direct-product-subspace state, the set of the αν band discrete canonical-momentum values {q j } of the pseudofermion operators f † qj , αν in the expressions of these generators must be those of the ground-state CPHS ensemble subspace; (ii) When applying the generator F † p−h, αν F † J−GS, αν onto the pseudofermion vacuum |0 to construct a given p − h, αν = c0, s1 branch direct-product-subspace state, the set of the αν band discrete canonical-momentum values {q j } of the pseudofermion operators f † qj , αν in the expressions of that generator must be those of the corresponding excited energy eigenstate.
The property (i) ensures that the above full matrix-element overlaps are reproduced. Furthermore, properties (i) and (ii) also ensure that the contribution from the unconventional orthogonality catastrophe matrix-element overlap discussed below is not counted twice.
Below we introduce the αν = c0, s1 pseudofermion spectral functions and p − h, αν = c0, s1 pseudofermion spectral functions which involve the operatorsΘ l,N F,i k−l∆P F αν , αν andΘ l,F,i αν δ k ′ , l∆P phF αν , respectively. The momentum convolution of these two operators leads to the correct expression for the operatorΘ l,i k, αν such thatΘ l,i k, αν = Here the f summations run over the states which span such subspaces and the matrix elements are given by, for the αν pseudofermion spectral functions and for the p − h, αν = c0, s1 pseudofermion spectral functions. The simple form of the matrix elements (59) follows from the full overlap of the generators F J−N F, αν and F N F, αν with the operatorΘ l,N F,i k, αν for the αν = c0, s1 and αν = c0, s1 branches, respectively. Such a full overlap also justifies that the corresponding αν spectral functions whose expression is given below have a non-interacting character. In turn, the evaluation of the matrix element (60) of the spectral function associated with the αν = c0, s1 pseudofermion elementary processes (B,C) is a more involved problem. For the αν = c0, s1 branches the phase-factor momentum l∆P 0 αν = l[∆P phF αν + ∆P F αν ] involves a term, l∆P phF αν , which arises from the elementary processes (C). Interestingly, the dynamics associated with the overlap of the αν = c0, s1 state f.L; p − h, αν| = 0|F J−GS, αν F p−h, αν | with the stateΘ l,F,i αν |GS =Θ l,F,i αν F † GS, αν |0 = F † −GS, αν |0 of the matrix element (60) is not controlled by the operator Θ l,N F,i k, αν but rather results from the different discrete canonical-momentum values of the pseudofermion creation and annihilation operators involved in the generators of each of these states. (For these branches the expression of the operatorΘ l,i k, αν does not include that of the generator F † p−h, αν , as mentioned above.) Each discrete canonical momentum value of the pseudofermion operators involved in the generators of the former state includes an extra overall canonical-momentum shift Q αν (q)/L relative to those of the latter state. If a αν = c0, s1 pseudofermion or pseudofermion hole is created at the Fermi points by the elementary processes (B) and thereafter moved from there by the elementary processes (C) generated by the operator F p−h, αν , the dynamics associated with the overlap of the excited-state occupancy configurations generated by the latter processes with the ground-state generator is controlled by the orthogonality catastrophe that occurs in the matrix element (60) due to the overall phase shift Q αν (q)/2. Such a matrix element involves N 0 αν + ∆N F αν αν pseudofermions. The occupancy configuration of the state F † −GS, αν |0 corresponds to the densely packed momentum distribution N −0 αν (q j ) for N 0 αν + ∆N F αν αν pseudofermions. The corresponding discrete canonical momentum valuesq j (occupied and unoccupied) are those of the ground state, q j = q j . The occupancy configuration associated with the state |f.L; p − h, αν = F † p−h, αν F † J−GS, αν |0 also refers to N 0 αν + ∆N F αν αν pseudofermions. However, its discrete canonical momentum values are those of the excited energy eigenstate. This feature leads to an exotic overlap for the matrix element (60). Such an overlap is behind the unusual quantum-liquid spectral properties, as further discussed below and in Ref. [3]. We note that in the absence of the αν = c0, s1 pseudofermion overall phase shifts Q αν (q)/2, Eq. (11), the matrix element (60) would vanish, except for the lowest-peak weight such that F p−h, αν = 1 and thus ∆P phF αν = 0. The spectral function expressions are additive in the contributions of each ground-state -excited-energy-eigenstate transition. For each transition, the available excited-energy-eigenstate pseudofermion discrete canonical-momentum values are in general slightly different and given by the functional (8). The important point is that for each ground-state -excited-state transition one knows the precise values of such discrete pseudofermion canonical momenta. Given these values, the αν pseudofermion creation and annihilation operators of the matrix element corresponding to the specific transition act independently for each αν excitation branch. This is behind the introduction of the above subspace direct product and follows in part from the factor δ αν, α ′ ν ′ on the right-hand side of the pseudofermion anticommutation relation (14). Thus, since the pseudofermion creation and annihilation operators of each αν branch act independently for each ground-state -excited-energy-eigenstate transition, they also do it for the whole spectral function, which is additive in the contributions of each ground-state -excited-energy-eigenstate transition. Moreover, as a result of the additive character of the energy in terms of αν pseudofermion, independent −1/2 holon, and independent −1/2 spinon single energies and of the corresponding expression of each J-CPHS ensemble subspace as the direct product of the above considered subspaces, the excited-energy-eigenstate wave-functions of the ground-state normal-ordered 1D Hubbard model factorize. It follows that the spectral functions B l,i (k, ω) of Eq. (37), generated by transitions from the ground state to a given J-CPHS ensemble subspace, can be expressed as a convolution of pseudofermion spectral functions, one for each branch with finite occupancy in such a subspace and for the independent −1/2 holons and/or independent −1/2 spinons, if they have finite occupancy in the same subspace, and two functions for the particular case of the αν = c0, s1 branches, as discussed above. It follows from the form of the matrix elements given in Eq. (60) that the contribution of the corresponding p − h, αν = c0, s1 pseudofermion spectral functions to the weight overlaps is more involved than that of the remaining pseudofermion, independent −1/2 holon, and independent −1/2 spinon spectral functions.
For each J-CPHS ensemble subspace, we introduce a dimension D associated with the elementary processes (A), where the numbers N αν are defined in Eq. (57). For a general J-CPHS subspace the function B l,i (k, ω) of Eq. (37) can be written as a convolution of the p − h, c0, s1 pseudofermion spectral function, p − h, s1 pseudofermion spectral function, one N αν -αν pseudofermion spectral function for each of the D branches such that N αν > 0, independent −1/2 holon spectral function, and independent −1/2 spinon spectral function. Thus, let us provide the general expressions of the spectral functions corresponding to such a general J-CPHS ensemble subspace. The p − h, αν = c0, s1 pseudofermion spectral function associated with the elementary processes (B,C) is given by, where the energy and momentum spectra are given below and the matrix element is that of Eq. (60). The summation J−CP HS−αν−(C) runs over the J-CPHS ensemble subspace αν = c0, s1 pseudofermion occupancy configurations generated by the elementary processes (C). The indices Q c0 and Q s1 remind us that the overall phase shifts of Eq. (11) have a specific value for each ground-state -excited-energy-eigenstate transition. In turn, it follows from the form of the matrix elements of Eq. (59) that the αν pseudofermion spectral function B l,N F,i αν (k, ω) associated with the elementary processes (A) has a non-interacting character and reads, both for the αν = c0, s1 and αν = c0, s1 branches. Here the summation J−CP HS−αν−(A) runs over the J-CPHS ensemble subspace αν pseudofermion occupancy configurations generated by the elementary processes (A). For the αν = c0, s1 branches and densities in the domains 0 < n < 1 and 0 < m < n the number of the latter occupancy configurations is given by and thus can be written as follows, where the values of N , N ↑ , and N ↓ are those of the corresponding excited-state CPHS ensemble subspace. We recall that for the cν = c0 (and sν = s1) branches and electronic density n = 1 (and spin density m = 0) the spectral function B l,N F,i cν (k, ω) (and B l,N F,i sν (k, ω)) of Eq. (63) does not exist. Finally, the form of the operatorΘ l,F,i αν implies that GS|F F, ανΘ l,F,i αν |GS = 1 for the αν = c0, s1 branches with finite occupancy in the J-CPHS ensemble subspace. This together with the non-interacting character of the −1/2 Yang holons and −1/2 HL spinons is behind the form of the independent −1/2 holon (α = c) and independent −1/2 spinon (α = s) spectral function, which reads, Here the coefficient C α is given in Eq.
The momentum spectra corresponding to such energy spectra are given by, In these equations ǫ cν (q j ) = 2νµ + ǫ 0 cν (q j ) for ν > 0, ǫ sν (q j ) = 2νµ 0 H + ǫ 0 sν (q j ) for ν > 1, the bands ǫ αν (q j ) for αν = c0, s1 and ǫ 0 αν (q j ) for αν = c0, s1 are defined by Eqs. (C. 15)-(C.21) of Ref. [8], the small energy ∆E phF αν is such that m αν, ±1 is the number of elementary αν = c0, s1 pseudofermion particle-hole processes (C) considered above, v αν ≡ v αν (q 0 F αν ), and v αν (q) = ∂ǫ αν (q)/∂q. As further discussed in Ref. [3], for densities 0 < n < 1 and 0 < m < n the elementary processes (C) leading to the spectral-function singular features include contributions from small but finite values of m αν, ±1 /N a as N a → ∞. For n = 1 (and m = 0) the latter processes do not exist for the c0 (and s1) branch and thus ∆E phF c0 = 0 and ∆P phF c0 = 0 (and ∆E phF s1 = 0 and ∆P phF s1 = 0). Let us consider the general situation when the J-CPHS ensemble subspace has finite occupancy for the c0 and s1 pseudofermion branches, D − 2 > 0 αν = c0, s1 pseudofermion branches, independent −1/2 holons, and independent −1/2 spinons. In this case the functions B l,i (k, ω) of Eq. (37) can be written as, where G l,i (k, ω) = These functions are studied elsewhere. However, the second expression of Eq. (68) refers to densities in the domains 0 < n < 1 and 0 < m < n only. The probability amplitudes | 0|F −GS, αν F † p−h, αν F † J−GS, αν |0 | 2 associated with the matrix element (60) which appear in expression (62) have the following general form, where αν = c0, s1. In expression (69) we have considered that the ground state corresponds to pseudofermion annihilation operators and the pseudofermion operators left for the excited energy eigenstate are of creation character. At this stage, for the evaluation of N -electron spectral functions, the main problem remaining is the computation of the non-trivial probability amplitude (69), which can be expressed by the following determinant, for the αν = c0, s1 branches. This result is justified by the following pseudofermion properties. First, the pseudofermions have no residual-interaction energy terms, as discussed in Refs. [3,9]. Second, the canonical-momentum shift Q αν (q j )/L, which under the ground-state -excited-energy-eigenstate transition involves all αν = c0, s1 pseudofermions of the initial ground state, is a zero-energy process [9]. Third, the elementary pseudofermion processes (C) correspond to αν = c0, s1 pseudoparticle particle-hole processes whose energy spectrum is of non-interacting character for the pseudoparticles, what implies that the energy spectrum ∆E phF αν = [2π/L] v αν m αν of Eq. (66) remains linear in m αν for small finite values of m αν /N a as N a → ∞ [3].

VI. DISCUSSION AND CONCLUDING REMARKS
The main result of this paper is the general spectral function expression defined by Eqs. (36) and (68). The expression given in the latter equation involves the p − h, c0 and p − h, s1 pseudofermion spectral functions provided in Eq. (62), whose probability amplitude | 0|F J−GS, αν F p−h, αν F † −GS, αν |0 | 2 can be expressed in terms of the determinants of Eqs. (70)-(72). An important aspect of the pseudofermion dynamical theory introduced in this paper and further developed in Ref. [3] for the metallic phase, is the different origin of the dynamics associated with the matrix-element overlaps of the αν = c0, s1 pseudofermion occupancy configurations in the vicinity and away of the Fermi points.
In reference [2] the finite-energy spectral function expressions derived by use of the pseudofermion dynamical theory introduced here are applied to the study of the spectral-weight features observed in the quasi-1D organic compound TTF-TCNQ. Interestingly, one finds quantitative agreement with the observed spectral features for the whole experimental energy band width. The microscopic mechanisms found in Ref. [1] by use of our theory are also consistent with the phase diagram observed in the (TMTTF) 2 X and (TMTSF) 2 X series of organic compounds and explain the absence of superconducting phases in TTF-TCNQ. Our theory is also of interest for the understanding of the spectral properties of the new quantum systems described by ultra-cold fermionic atoms on an optical lattice [6].