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Normal-Ordered Wave-Function Factorization of the 1D Hubbard Model for Finite Values of the On-site Repulsion U

In this paper we find that in the thermodynamic limit and for the the ground-state normal-ordered 1D Hubbard model the wave function of excited states contained in few-electron excitations factorizes for all values of the on-site Coulombian repulsion. This factorization results from the non-interacting character of the pseudofermions whose occupancy configurations describe these excited states. Our study includes the introduction of the pseudoparticle - pseudofermion unitary transformation and of an operator algebra for both the pseudoparticles and the pseudofermions. The pseudofermions are non interacting and thus have no residual interactions, in contrast to the corresponding pseudoparticles, whose statistics we classify according to the generalized Pauli principle. The physics behind the invariance of the pseudoparticles under the above transformations for specific values of the bare momentum is also studied and discussed.

for the c0 pseudoparticles, all pseudoparticle branches introduced in Ref. [29] have a composite character in terms of holons or spinons [31]. The pseudofermions are related to the pseudoparticles by a second unitary transformation. The charge or spin carried by the pseudoparticle and its holon or spinon contents remains invariant under such a transformation. It follows that except for the c0 pseudofermions, all remaining pseudofermion branches are composite objects of holons or spinons. The concepts of local pseudoparticle and effective lattice widely used in this paper are introduced in Ref. [35].
The pseudofermion operational description introduced in this paper and the associated factorization of the wave function of energy eigenstates contained in few-electron excitations are used elsewhere in the study of the finite-energy spectral properties [1,10,34]. The theoretical predictions of Refs. [1,10] seem to describe both qualitatively and quantitatively the one-electron removal spectral lines observed by photoemission experiments for finite values of the energy in real quasi-1D materials. We thus expect that our pseudofermion operational description is useful for the further understanding of the exotic properties displayed by low-dimensional materials.
The paper is organized as follows: In Sec. II we introduce the 1D Hubbard model and the rotated electrons. The elementary holon, spinon, and c0 pseudoparticle description and associated αν pseudoparticle representation are summarized in Sec. III. Moreover, in that section we introduce the operator algebra for the pseudoparticles and the statistics of these quantum objects according to the generalized Pauli principle [36]. In Sec. IV we consider the ground-state normal-ordered pseudoparticle operator description and introduce useful ground-state quantities. The pseudofermion description and the relationship between pseudoparticle and pseudofermion operators are introduced and discussed in Sec. V. This includes the introduction of the pseudofermion anticommutator algebra. In Sec. VI we study the pseudofermion energy and momentum spectra and introduce and discuss the factorization of the fewelectron Hilbert subspace of the ground-state normal-ordered 1D Hubbard model. The investigation of the laws under the pseudoparticle -pseudofermion transformation of several quantum objects and quantities is the subject of Sec. VII. Finally, in Sec. VIII we present the discussion and the concluding remarks.

II. THE 1D HUBBARD MODEL AND ROTATED ELECTRONS
In a chemical potential µ and magnetic field H the 1D Hubbard Hamiltonian can be written as, where the HamiltonianĤ SO(4) =Ĥ H − (U/2)N + (U/4) N a ;Ĥ H =T + UD , has SO(4) symmetry. HereĤ H is the "simple" Hubbard model, is the kinetic-energy operator, andD is the electron double-occupation operator. On the right-hand side of Eq. (1) we have that µ c = µ, µ s = µ 0 H, µ 0 is the Bohr magneton, and the number operators, are the diagonal generators of the η-spin and spin SU (2) algebras [37,38,39], respectively. We consider that the number of lattice sites N a is large and even and that N a /2 is odd. The electronic number operators on the right-hand side of Eq. (5) readN = σ=↑, ↓N σ andN σ = Na j=1N j, σ , where the operatorN j, σ = c † j, σ c j, σ counts the number of spin σ electrons at real-space lattice site j. On the right-hand side of Eqs. (3)-(5) the operator c † j, σ (and c j, σ ) creates (and annihilates) a spin σ electron at lattice site j = 1, 2, ..., N a . We denote the lattice constant by a and the lattice length by L = N a a.
The momentum operator is given by, Here the spin σ momentum distribution operator readsN σ (k) = c † k, σ c k, σ , where the operator c † k, σ (and c k, σ ) creates (and annihilates) a spin σ electron at momentum k. The operators c † k, σ and c k, σ are related to the above operators c † j, σ and c j, σ by the following Fourier transforms, The HamiltonianĤ SO(4) given in Eq.
(2) commutes with the six generators of the η-spin and spin SU (2) algebras and has SO(4) symmetry [37,38,39]. While the expressions of the two corresponding diagonal generators are given in Eq. (5), the off-diagonal generators of these two SU (2) algebras read respectively. Throughout this paper we use units of Planck constant one and denote the electronic charge by −e. The Betheansatz solvability of the 1D Hubbard model (1) is restricted to the Hilbert subspace spanned by the lowest-weight states (LWSs) [25,26] or highest-weight states (HWSs) [27] of the η-spin and spin algebras, that is by the states whose S α and S z α numbers are such that S α = −S z α or S α = S z α , respectively, where α = c for charge and α = s for spin. In this paper we choose the η-spin and spin LWSs description of the Bethe-ansatz solution. In this case, that solution describes energy eigenstates with electronic densities n = N/L and spin densities m = [N ↑ − N ↓ ]/L in the domains 0 ≤ n ≤ 1/a and 0 ≤ m ≤ n, respectively. Some of our results correspond to the ranges 0 < n < 1/a and 0 < m < n. The description of the states corresponding to the extended domains 0 ≤ n ≤ 1/a ; 1/a ≤ n ≤ 2/a and −n ≤ m ≤ n ; −(2/a − n) ≤ m ≤ (2/a − n), respectively, is achieved by application onto the latter states of off-diagonal generators of the η-spin and spin SU (2) algebras [31,39].
Each lattice site j = 1, 2, ..., N a of the model (1) can either be doubly occupied, empty, or singly occupied by a spin-down or spin-up electron. The maximum number of electrons is 2N a and corresponds to density n = 2/a. Besides the N electrons, it is useful to consider [2N a − N ] electronic holes. (Here we use the designation electronic hole instead of hole, in order to distinguish this type of hole from the pseudoparticle hole and pseudofermion hole.) Our definition of electronic hole is such that when a lattice site is empty, we say that it is occupied by two electronic holes. If a lattice site is singly occupied, we say that it is occupied by an electron and an electronic hole. If a lattice site is doubly occupied, it is unoccupied by electronic holes. The same definition holds for the rotated-electronic holes. We note that the lattice occupied by rotated electrons is identical to the original electronic lattice.
The electron -rotated-electron unitary transformation maps the electrons onto rotated electrons such that rotatedelectron double occupation, no occupation, and spin-up and spin-down single occupation are good quantum numbers for all values of U/t. We call c † j, σ the electrons that occur in the 1D Hubbard model (1) and (2), while the operatorc † j, σ such thatc † j, σ =V † (U/t) c † j, σV (U/t) represents the rotated electrons, where the electron -rotated-electron unitary operatorV (U/t) is defined below. Similarly, c † j, σ =V (U/t)c † j, σV † (U/t). Note that c † j, σ andc † j, σ are only identical in the U/t → ∞ limit where electron double occupation becomes a good quantum number.
The operatorŜ of Eq. (10) is uniquely defined by the following two equations, and where the HamiltonianĤ H is given in Eq. (2) and the rotated-electron double occupation operatorD reads, HereD is the electron double occupation operator given in Eq. (4). The operator (13) commutes with the 1D Hubbard model. We denote the rotated-electron double occupation by D r . It is a good quantum number for all values of U/t. The transformation associated with the electron -rotated-electron unitary operatorV (U/t) was introduced in Ref. [32]. The studies of that reference referred to large values of U/t and did not clarify for arbitrary values of U/t the relation of rotated-electron double occupation to the quantum numbers provided by the Bethe-ansatz solution. However, this transformation is uniquely defined for all values of U/t by Eqs. (10)- (12). Equations (11) and (12) can be used to derive an expression for the unitary operator order by order in t/U . The authors of Ref. [33] carried out this expansion up to eighth order (see foot note [12]).
The introduction of the pseudofermion operational description studied in this paper requires the use of an operator representation for the pseudoparticles. Thus, in this section we introduce an operational description for the αν pseudoparticles and the holons and spinons which are not part of composite pseudoparticles. An operational description for the pseudoparticles of bare-momentum q was introduced in Ref. [29]. However, such a description did not take into account the holon (and spinon) composite character of the cν pseudoparticles (and sν pseudoparticles). Moreover, the spinon description used in the studies of that reference is not valid for the whole Hilbert space. This affected the values of the entries of the pseudoparticle statistical-interaction matrix [36]. In this section we provide the correct values for the entries of such a matrix. Another limitation was the lack of a representation for the pseudoparticle operators in terms of spatial coordinates. The concepts of local αν pseudoparticle and effective αν lattice are introduced in Ref. [35]. The studies of that reference include a description of the local pseudoparticles in terms of the rotated-electron site distribution configurations. The absence of such a local operational description is one of the reasons why the studies of Ref. [29] did not provide useful information about the main issue of the relation between the pseudoparticle and electronic operators. Also the concept of rotated electron is important for the study of that issue and was not considered in the studies of such a reference. All these concepts are valuable and necessary for the application of the operational pseudofermion representation introduced in this paper to the evaluation of few-electron spectral functions [34].
In Appendix A we summarize the basic properties of the αν pseudoparticles which are needed for our studies. This includes introduction to the αν pseudoparticle bare momentum, effective αν lattice, and useful ground-state quantities.

A. THE PSEUDOPARTICLE OPERATORS AND PSEUDOPARTICLE STATISTICS ACCORDING TO THE GENERALIZED PAULI PRINCIPLE
Generation and removal of pseudoparticles is in general associated with creation and/or annihilation of electrons. Yet there are also transitions which change the numbers of these quantum objects at constant spin σ electron numbers. One can introduce elementary operators for creation or annihilation of αν pseudoparticles. In this subsection, we introduce two alternative representations corresponding to pseudoparticle operators, both in terms of the bare-momentum q and spatial coordinates. These representations refer to the bare-momentum pseudoparticles and local pseudoparticles [35], respectively.
Let us introduce the bare-momentum αν pseudoparticle creation (and annihilation) operator b † q, αν (and b q, αν ) which creates (and annihilates) a αν pseudoparticle of bare momentum q. In addition, we introduce the local αν pseudoparticle creation operator b † xj, αν and annihilation operator b xj, αν . These bare-momentum and local pseudoparticle operators are related as follows, The local αν pseudoparticle creation (and annihilation) operator b † xj, αν (and b xj, αν ) creates (and annihilates) a local αν pseudoparticle at the effective αν lattice site of spatial coordinate x j = a αν j, where j = 1, 2, ..., N * αν . The effective αν lattice constant a αν is defined in Eq. (A7) of Appendix A, where the concept of an effective αν lattice, introduced in Ref. [35] is described. The conjugate variable of the bare-momentum q j of the αν pseudoparticle branch is the space coordinate x j of the corresponding effective αν lattice. This is different to the electronic operators of Eq. (7), where the conjugate variable of the momentum k j is the space variable of the original electronic lattice. In reference [35], the pseudoparticle site distribution configurations in the effective αν lattices are related to the corresponding rotated-electron site distribution configurations.
The pseudoparticles obey a Pauli principle relative to the bare-momentum occupancy configurations, i.e. a discrete bare-momentum value q j can either be unoccupied or singly occupied by a pseudoparticle. For ν > 0 composite αν pseudoparticles, the number of discrete momentum values N * αν is not the same for all energy eigenstates. Thus, these objects cannot be classified as fermions or bosons. In order to classify the αν pseudoparticle according to the generalized Pauli principle introduced in Ref. [36], we consider the αν dimensions, where according to Eqs. (A3)-(A6) of Appendix A, N αν and N h αν are the number of αν pseudoparticles and αν pseudoparticle holes, respectively, and N * αν = N αν + N h αν . A transition to an excited energy eigenstate producing deviations ∆N αν in the αν pseudoparticle numbers, leads to the following deviations in the corresponding αν dimension, According to the generalized Pauli principle [36], the parameters g αν, α ′ ν ′ on the right-hand side of this equation are the entries of the statistical-interaction matrix. From the use of Eqs. (A3)-(A6) of Appendix A, we find that for the αν pseudoparticles such a statistical-interaction matrix has infinite dimension and its entries are given by, This fully defines the statistics of the αν pseudoparticles. We emphasize that the c0 pseudoparticle entry g c0, αν given in Eq. (17) has Fermionic character. This is related to the fact that the number of sites of the effective c0 lattice is constant and given by N a , and that the corresponding effective lattice constant a c0 equals the electronic lattice constant a. Thus, the effective c0 lattice and electronic lattice are identical. Furthermore, it is found in Ref. [35] that the sites occupied by c0 pseudoparticles (and c0 pseudoparticle holes) are the same as the sites singly occupied by rotated electrons (and doubly occupied and unoccupied by rotated electrons). While the c0 pseudoparticles do not feel the statistical-interactions of the remaining pseudoparticles, the ν > 0 composite αν pseudoparticles feel the statistical interactions of the c0 pseudoparticles, as confirmed by Eq. (17). The form of the entries given in Eq. (17) confirms that the latter composite pseudoparticles are neither fermions nor bosons [36]. The statistical interactions of the composite pseudoparticles result in part from the property that the width of the bare-momentum domain 2q αν = 2π[1/a αν − 1/L] is different for different values of the pseudoparticle numbers.
The αν pseudoparticle bare-momentum distribution functions N αν (q) play an important role in the pseudoparticle description [30,31]. These functions are for all energy eigenstates the eigenvalues of the following pseudoparticle bare-momentum distribution operators,N The bare-momentum distribution functions N αν (q) read N αν (q j ) = 1 for occupied discrete bare-momentum values q j and N αν (q j ) = 0 for unoccupied discrete bare-momentum values q j . Each LWS is uniquely specified by the values of the set of distribution functions {N αν (q)} such that ν = 0, 1, 2, ... for α = c and ν = 1, 2, ... for α = s. Physical quantities such as the energy, depend on the values of these distribution functions and numbers through the rapidity momentum functional k(q) and rapidity functionals Λ cν (q) and Λ sν (q). The value of these functionals is uniquely provided by solution of the following functional integral equations [30,31], and Here is the cν rapidity-momentum functional and the limiting bare-momentum values q ± c0 and q αν where α = c, s and ν = 1, 2, ... are given in Eqs. (A9) and (A10) and in Eq. (A11), respectively, of Appendix A. The function Θ ν, ν ′ (x) appearing in Eqs. (20) and (21) is given in Eq. (B17) of Appendix B. The equations (19)- (21) correspond to a functional representation of the thermodynamic Bethe-ansatz equations introduced by Takahashi [26]. The rapiditymomentum functional is real and the rapidity functionals are the real part of Takahashi's ideal strings [26,31]. It is useful to introduce the following c0 rapidity functional, where k(q) is the rapidity-momentum functional.

B. HOLONS, SPINONS, c0 PSEUDOPARTICLES AND THE YANG HOLON AND HL SPINON ELEMENTARY OPERATORS
In this subsection we consider a holon, spinon, and c0 pseudoparticle description for the whole Hilbert space of the 1D Hubbard model whose validity is shown elsewhere [31]. Such elementary quantum objects correspond to specific rotated-electron site occupations. This description of all energy eigenstates in terms of occupancy configurations of three elementary quantum objects only, is useful for the studies of this paper.
Let us distinguish the total η spin (and spin) value, which we denote by S c (and S s ) and the corresponding η-spin (and spin) projection, which we denote by S z c (and S z s ), from the η spin (and spin) carried by the elementary quantum objects. We call s c (and s s ) the η spin (and spin) carried by the holons, spinons, and other elementary objects and σ c (and σ s ) their η-spin (and spin) projection. The operatorsM c, σc andM s, σs which count the number of the σ c = ±1/2 holons and σ s = ±1/2 spinons have the following form [31], respectively.
Here, the operator j c † j↑ c j↑ c † j↓ c j↓ counts the number of electron doubly-occupied sites, j c j↑ c † j↑ c j↓ c † j↓ counts the number of electron empty sites, j c † j↓ c j↑ c † j↑ c j↓ counts the number of spin-down electron singly-occupied sites, and j c † j↑ c j↓ c † j↓ c j↑ counts the number of spin-up electron singly-occupied sites. The operator V (U/t) on the right-hand side of Eqs. (24) and (25) is uniquely defined for all values of U/t by Eqs. (10)- (12). The new physics brought about by the relations of Eqs. (24) and (25) is that the electron -rotated-electron unitary transformation generates the holons and spinons whose occupancy configurations describe the exact energy eigenstates. Throughout this paper we denote such holons and spinons according to their value of σ c = ±1/2 and σ s = ±1/2, respectively. The unitary rotation is such that for all values of U/t the number of emerging −1/2 holons, +1/2 holons, −1/2 spinons, and +1/2 spinons equals precisely the number of rotated-electron doubly occupied sites, empty sites, spin-down singly occupied sites, and spin-up singly occupied sites, respectively.
The holons have s c = 1/2, s s = 0, and σ s = 0. Thus, there are two types of holons, which have σ c = −1/2 and σ c = +1/2 and carry charge −2e and +2e, respectively. These objects are on-site spin-singlet rotated-electron pairs and rotated-electron unoccupied sites, respectively. An important property is that rotated-electron double occupation D r equals the value M c, −1/2 for the number of −1/2 holons. The rotated-electron double occupation D r = M c, −1/2 plays an important role in the description of the few-electron spectral properties of the quantum problem. For fewelectron spectral functions the weight distribution resulting from transitions to the Hilbert subspace spanned by excited states of rotated-electron double occupation D r corresponds to the D th r upper Hubbard band, where D r = 1, 2, ... [34]. Moreover, the spin-up and spin-down rotated-electron singly occupied sites correspond to the σ s = +1/2 and σ s = −1/2 spinons, respectively. The s s = 1/2 spinons have no charge degrees of freedom and thus describe the spin degrees of freedom of rotated-electron singly occupied sites only. The charge degrees of freedom of the rotated electrons (and rotated-electronic holes) of these singly occupied sites are described by the chargeons of charge −e (and antichargeons of charge +e) [31]. These quantum objects are part of the charge degrees of freedom, but do not contribute to the η-spin and spin SU (2) algebras. In the case of the description of the transport of charge in terms of electrons (and electronic holes) the elementary carriers of charge are the chargeons of charge −e and −1/2 holons of charge −2e (and the antichargeons of charge +e and +1/2 holons of charge +2e). The c0 pseudoparticle has no η-spin and no spin degrees of freedom and is a composite quantum object which contains a chargeon and an antichargeon. However, in how transport of charge is concerned, the chargeon and antichargeon correspond to alternative descriptions of the c0 pseudoparticle. When the transport of charge is described in terms of electrons (and electronic holes) the c0 pseudoparticle couples to charge probes through the chargeon (and antichargeon) and carries elementary charge −e (and +e).
All energy eigenstates of the 1D Hubbard model can be described in terms of occupancy configurations of holons, spinons, and c0 pseudoparticles. On the other hand, in Ref. [29] it was found that all energy eigenstates associated with the 1D Hubbard model Bethe-ansatz solution [25,26], can be described in terms of occupancy configurations of αν pseudoparticles. By merging these two representations, one finds that the cν pseudoparticles such that ν = 0 and the sν pseudoparticles are 2ν-holon and 2ν-spinon composite objects, respectively [31]. The composite cν pseudoparticles (and sν pseudoparticles) are s c = σ c = 0 (and s s = σ s = 0) objects without spin (and charge) degrees of freedom. These composite quantum objects contain an equal number ν of −1/2 holons and +1/2 holons (and −1/2 spinons and +1/2 spinons). When the transport of charge is described in terms of electrons (and electronic holes), the 2ν-holon composite cν pseudoparticles couple to charge probes through their ν −1/2 holons of charge −2e (and their ν +1/2 holons of charge +2e). Thus these composite quantum objects carry elementary charge −2ν e (and +2ν e).
The ±1/2 holons (and ±1/2 spinons) which are not part of 2ν-holon composite cν pseudoparticles (and 2ν-spinon composite sν pseudoparticles) are called ±1/2 Yang holons (and ±1/2 HL spinons). In the designations HL spinon and Yang holon, HL stands for Heilmann and Lieb and Yang refers to C. N. Yang, respectively, who are the authors of Refs. [37,38]. Note that the holons (and spinons) which are part of s c = 0 (and s s = 0) composite cν pseudoparticles (and sν pseudoparticles) remain invariant under application of the off-diagonal generators of the η-spin (and spin) SU (2) algebras given in Eq. (8) (and (9)). Application of these generators produces η-spin (and spin) flips in ±1/2 Yang holons (and ±1/2 HL spinons). Thus, the −1/2 Yang holons and −1/2 HL spinons are created by application onto the LWSs of suitable off-diagonal generators of the η-spin and spin SU (2) algebras, respectively, given in Eqs. (8) and (9). The generators (8) and (9) can be written in terms of rotated-electron operators as, and respectively. In these equations q c = π/a and q s = 0 is the momentum. The operators (26) and (27) are invariant under the electron -rotated-electron unitary transformation [31]. Therefore, they have precisely the same expression in terms of electron and rotated-electron creation and annihilation operators. Note that within the operational representation of Eqs. (26) and (27) the Yang +1/2 holons and HL +1/2 spinons are not explicitly considered. Indeed, the numbers L c, +1/2 of +1/2 Yang holons and L s, +1/2 of +1/2 HL spinons are fully determined by the numbers L c, −1/2 of Yang −1/2 holons, L s, −1/2 of HL −1/2 spinons, and the set {N αν } of the different αν pseudofermion branches. This justifies why here we consider the operators of Eqs. (26) and (27) as creation and annihilation operators for −1/2 Yang holons and −1/2 HL spinons, respectively. When applied onto LWSs, the operators d † qα, −1/2 produce energy eigenstates with finite values for the numbers L c, −1/2 and/or L s, −1/2 of the following form, Here |LW S is the LWS that corresponds to the state |{L c, −1/2 , L s, −1/2 } and L α = L α, +1/2 + L α, −1/2 = 2S α where S α is the state η-spin value (α = c) and spin value (α = s). The energy eigenstates (28) are not described by the Bethe-ansatz solution [31,39]. The pseudoparticle bare momentum q is such that q ∈ (−q αν , +q αν ) where the limiting bare-momentum values ±q αν associated with the limits of the αν pseudoparticle Brillouin zone are given in Eq. (A11) of Appendix A. According to the results of Ref. [30], the ±1/2 Yang holons, ±1/2 HL spinons, and ν > 0 composite αν pseudoparticles of bare-momentum values q = ±q αν are non-interacting and localized quantum objects. Such behavior results from the invariance of these quantum objects under the electron -rotated-electron unitary transformation. This means that the ±1/2 Yang holons, ±1/2 HL spinons, and ν > 0 composite αν pseudoparticles of bare-momentum values q = ±q αν are the same quantum objects as the corresponding rotated objects. Thus, these objects are localized and do not contribute to the transport of charge or spin. However, in general a ν > 0 composite αν pseudoparticle is different from a ν > 0 rotated composite αν pseudoparticle. The only exception is precisely for bare momentum values q such that q → ±q αν . As the bare momentum approaches its limiting values, q → ±q αν , the ν > 0 composite αν pseudoparticle and the ν > 0 rotated composite αν pseudoparticle become the same quantum object. Importantly, the electrons and rotated electrons involved in processes associated with creation of ±1/2 Yang holons, ±1/2 HL spinons, and ν > 0 composite αν pseudoparticles of bare-momentum values q = ±q αν are also the same quantum object, i.e. remain also invariant under the electron -rotated-electron unitary transformation. This always refers to localized electrons. It follows that the transport of charge (and spin) is associated with the c0 pseudoparticle and q = ±q cν composite cν pseudoparticle quantum charge fluids (and q = ±q sν composite sν pseudoparticle quantum spin fluids) where ν = 1, 2, ....
An electronic ensemble space is spanned by all energy eigenstates with the same values for the numbers N ↑ and N ↓ . An important concept is that of CPHS ensemble space [30]. This is a subspace spanned by all energy eigenstates with the same values for the numbers of ±1/2 holons and ±1/2 spinons {M α, ±1/2 }, where α = c, s. In general, an electronic ensemble space contains several CPHS ensemble spaces. Moreover, usually a CPHS ensemble space includes different CPHS ensemble subspaces. A CPHS ensemble subspace is spanned by all energy eigenstates with the same values for the sets of numbers N c0 , {N αν }, and {L α, −1/2 } such that α = c, s and ν = 1, 2, .... (According to the notation of Ref. [30], CPHS stands for c pseudoparticle, holon, and spinon.)

C. THE GENERAL ENERGY SPECTRUM AND THE MOMENTUM OPERATOR
The generators of the η-spin and spin SU (2) algebras given in Eqs. (26) and (27) commute with the c0 pseudoparticle and composite αν pseudoparticle creation and annihilation operators of Eq. (14). We emphasize that this property has the following important effect: All 2S α energy eigenstates obtained from a given regular energy eigenstate have the same pseudoparticle momentum distribution functions N c0 (q) and {N αν (q)} for all branches α = c, s and ν = 1, 2, .... Thus these states are described by similar pseudoparticle occupancy configurations and only differ in the relative numbers of +1/2 Yang holons and −1/2 Yang holons (α = c), or/and in the relative numbers of +1/2 HL spinons and −1/2 HL spinons (α = s). This reveals that the coupled functional equations (19)- (21), which involve the pseudoparticle momentum distribution functions and do not depend on the L α, σα numbers, describe both LWSs and non-LWSs. The eigenvalues E of the energy eigenstates of the Hamiltonian (1) can be written in the following form, where and On the right-hand side of Eq. (31) the numbers S z are the eigenvalues of the diagonal generators given in Eq. (5) and µ c = 2µ and µ s = 2µ 0 H are the same quantities as on the right-hand side of Eq. (1).
The values of the rapidity-momentum functional k(q) and rapidity functionals Λ αν (q) are the same for all 2S α + 1 states in the same tower. Such functionals are eigenvalues of operators which commute with the off-diagonal generators of η-spin and spin algebras. This is consistent with the η-spin and spin SU (2) symmetries, which imply that the Hamiltonian (2) commutes with these generators and thus the energy (32) is the same for the set of 2S α + 1 states belonging to the same η-spin (α = c) or spin (α = s) tower. The above operators obey equations similar to Eqs. (19)- (21), with the αν bare-momentum distribution functions N αν (q) replaced by the corresponding operatorsN αν (q) given in Eq. (18). The pseudoparticle Hamiltonian expression is also obtained by replacing in the energy expressions (31)-(33) the distribution functions N αν (q) by the operatorsN αν (q), and the rapidity-momentum and rapidity functionals by the corresponding operators.
The momentum operator given in Eq. (6) can be expressed in terms of pseudoparticle and −1/2 holon operators as follows [31], The momentum operator (34) commutes with the electron -rotated-electron unitary operator V (U/t) [31]. The momentum eigenvalues can be straightforwardly written by replacing on the right-hand side of Eq. (34) the pseudoparticle and holon number operators by the corresponding eigenvalues. Note that the number L c, −1/2 and corresponding operatorL c, −1/2 appearing on the right-hand side of Eqs. (33) and (34), respectively, are beyond the Bethe-ansatz solution. Indeed, the states which span the Hilbert subspace associated with that solution have no −1/2 Yang holons and thus L c, −1/2 = 0 in Eq. (33) for these states. However, the energy spectrum defined by Eqs. (31)-(33) and the momentum-operator expression (34) refer to the whole Hilbert space of the 1D Hubbard model. On the other hand, as a result of the SO(4) symmetry of the Hamiltonian (2), the corresponding energy spectrum (32) does not depend on the value of the number L c, −1/2 . This property is confirmed by analysis of expressions (32) and (33) and by noting that

IV. THE GROUND-STATE NORMAL-ORDERED PSEUDOPARTICLE OPERATIONAL DESCRIPTION
In this section we consider the ground-state normal-ordered pseudoparticle operational description which is needed for the introduction of the pseudofermion description. We are mostly interested in excited states generated from the ground state by processes involving changes in the occupancy configurations of a finite number of αν pseudoparticles, −1/2 Yang holons, and −1/2 HL spinons. The pseudofermion description introduced in the ensuing section corresponds to the Hilbert subspace spanned by such excited states. In the thermodynamic limit the few-electron excitations are contained in that subspace.
A. NORMAL-ORDERED NUMBER OPERATORS AND USEFUL GROUND-STATE QUANTITIES Throughout this paper the symbol :Ô : refers to the ground-state normal-ordered expression of a general operator O. Such a ground-state normal-ordered expression is given by that operator minus its ground-state expectation value.
We introduce the ground-state normal-ordered αν pseudoparticle bare-momentum distribution operator, and the −1/2 Yang holon (α = c) and −1/2 HL spinon (α = s) ground-state normal-ordered number operator, Here the operatorsN αν (q) andL α, −1/2 are defined in Eqs. (18) and (30), respectively, N 0 αν (q) is the ground-state αν pseudoparticle distribution function whose expressions are given in Eqs. (A12)-(A14) of Appendix A, and L 0 α, −1/2 = 0 is the −1/2 Yang holon (α = c) and −1/2 HL spinon (α = s) ground-state number. It follows from the expressions given in Eq. (A15) of Appendix A for the ground-state values of the number N * αν , whose general expressions are given in Eqs. (A3), (A4), and (A6) of the same Appendix, that in the case of the ground state the effective αν lattice constants (A7) are given by [35], where δ = (1/a − n) is the doping concentration. We note that the meaning of the divergences in the value of the constants a 0 αν defined in Eq. (37) is that the corresponding effective αν lattice has no sites, i.e. N 0, * αν = 0 and, therefore, does not exist for the ground state. This is the case of the effective cν lattices for half filling when ν > 0 and of the effective sν lattices for zero spin density when ν > 1. It follows that such singularities just indicate the collapse of the corresponding effective αν lattice. This is one of the reasons why some of our expressions refer to electronic densities 0 < n < 1/a and spin densities 0 < m < n, such that all ground-state effective αν lattice constants (37) have finite values.
From use of expressions (37) one can write the ground-state number N 0, * αν given in Eq. (A15) of Appendix A as, Except for terms of order 1/L, the limiting bare-momentum values defined in Eqs. (A8)-(A11) of Appendix A simplify and are given by, where the ground-state effective-lattice constants a 0 αν are these given in Eq. (37). Comparison of Eqs. (37) and (39) leads to the following ground-state expressions for the αν pseudoparticle limiting bare-momentum values, In most situations, one can disregard the 1/L corrections and use the bare-momentum limiting values given in Eq. (40). For each specific energy eigenstate the rapidity-momentum functional k(q) and the rapidity functionals Λ αν (q) become mere functions of q and are called rapidity-momentum and rapidity functions, respectively. We denote the ground-state rapidity-momentum function by k 0 (q) and we call Λ 0 cν (q) and Λ 0 sν (q) the ground-state rapidity functions. These ground-state functions are computed by solution of the integral equations obtained by introducing in Eqs. (19)-(21) the ground-state distribution functions (A12)-(A14) given in Appendix A. The solution of these equations can be written in terms of the inverse functions of k 0 (q) and Λ 0 αν (q) which we callq c (k) andq αν (Λ), respectively. Here k and Λ are the rapidity-momentum coordinate and rapidity coordinate, respectively. For the pseudoparticle description these functions refer to the ground state only. However, in the case of the pseudofermion description introduced in the ensuing section these functions apply both to the ground state and excited states. It follows that these functions play an important role in the latter description. These are odd functions such that, The domains of the rapidity-momentum coordinate k and rapidity coordinate Λ are such that −π/a ≤ k ≤ +π/a and −∞ ≤ Λ ≤ ∞, respectively. If follows that, where we introduced the ground-state c0 rapidity function, The c0 rapidity function (43) is the ground-state value of the rapidity functional defined in Eq. (23). The relations (42) are equivalent to, The Fermi points ±2k F and ±k F ↓ of the c0 pseudoparticles and s1 pseudoparticles, respectively, correspond to the following rapidity Fermi points, Then,q Interestingly, one can write the above functionsq c (k),q cν (Λ), andq sν (Λ) in terms of the two-pseudofermion phase shiftsΦ αν, α ′ ν ′ expressed in terms of the rapidity-momentum coordinate k and rapidity coordinate Λ, whose physical meaning is discussed in the ensuing section. These functions read, respectively. The two-pseudofermion phase shifts play an important role in the few-electron spectral properties and appear in the expression of the momentum carried by the pseudofermions. The two-pseudofermion phase shifts Φ αν, α ′ ν ′ expressed in terms of the rapidity-momentum coordinate k and rapidity coordinate Λ are mathematically defined by the following equations, where the two-pseudofermion phase shiftsΦ αν, α ′ ν ′ (r, r ′ ) are defined by the integral equations (B2)-(B15) of Appendix B. These are expressed in terms of the variable r defined in the arguments of the functions on the right-hand side of Eqs. (50)-(51). The values of the parameters Q and B introduced in Eq. (45) are controlled by the two-pseudofermion phase shifts and are computed by self-consistent solution of the following equations, respectively. At zero spin density the parameter Q changes from Q = k F in the limit U/t → 0 to Q = 2k F as U → ∞. At half filling and zero spin density, the U/t dependence of the parameter Q is singular at U/t = 0. It reads Q = π/2a at U/t = 0 and is given by Q = π/a for all finite values of U/t. The U/t > 0 value, Q = 2k F = π/a, is associated with a full c0 pseudoparticle band both in bare-momentum q and in rapidity-momentum k spaces and thus implies insulator behavior. In contrast, the U/t = 0 value Q = π/2a corresponds to a metallic band which is half filled in the rapidity-momentum space. Such a singular behavior is associated with the Mott-Hubbard transition [10,25]. At zero spin density the parameter B is given by B = ∞ and vanishes in the limit of spin density m → n, as the fully polarized ferromagnetic state is approached. Unfortunately, it is in general difficult to obtain closed form expressions for the ground-state functions k 0 (q) and Λ 0 αν (q) by inverting the functions defined in Eqs. (47)-(49). This can be achieved for specific limits of the parameter space only. For instance, from the use of Eqs. (47)-(49) we can obtain for zero spin density m = 0, values of the electronic density 0 ≤ n ≤ 1/a, and limiting on-site repulsion values U/t → 0 and U/t >> 1 the following closed form expressions for the ground-state functions k 0 (q), Λ 0 c0 (q), Λ 0 cν (q), and Λ 0 s1 (q), for ν > 0, and respectively. We note that for zero-spin density the ν > 1 ground-state rapidity function Λ 0 sν (q) vanishes for all values of n and U/t. For m = 0, the bare-momentum domain width of the sν pseudoparticles belonging to branches such that ν > 1 vanishes and the corresponding effective sν lattice has no sites and thus collapses. This collapsing is associated with the divergence of the effective sν lattice constant given in Eq. (37) as m → 0 when ν > 1. The same occurs for the cν pseudoparticles belonging to branches such that ν > 0 when half filling is approached.
For any energy eigenstate |ψ the normal-ordered bare-momentum distribution and number operators defined in Eqs. (35) and (36), respectively, obey the following eigenvalue equations, :N αν (q) : |ψ = ∆N αν (q)|ψ ; : Here ∆N αν (q) is the αν pseudoparticle bare-momentum distribution function deviation and ∆L α, −1/2 is the deviation in the number of −1/2 Yang holons (α = c) or of −1/2 HL spinons (α = s). These deviations are given by These values describe deviations of occupancy configurations of excited states relative to the ground-state occupancy configurations described by the bare-momentum distribution functions and numbers given in Eqs. (A12)-(A14) of Appendix A. For these excited states, the αν pseudoparticle bare-momentum distribution function and the −1/2 Yang holon and −1/2 HL spinon numbers read, From use of the ground-state distribution and number values given in Eqs. (A12)-(A14) of Appendix A, we find the following operational relations, Relations (60) are justified by the absence of the corresponding quantum objects in the initial ground state.

B. THE NORMAL-ORDERED PSEUDOPARTICLE MOMENTUM AND ENERGY FUNCTIONALS
Let us introduce the momentum and energy functionals associated with the Hilbert subspace spanned by excited states described by small deviations (58). The bare-momentum distribution deviations of these states involve a small density of pseudoparticles, −1/2 Yang holons, and −1/2 HL spinons. In the case of momentum, such a functional is obtained directly from use of the operator expression (34). Introduction of the corresponding normal-ordered expression and replacement of the bare-momentum number operators and holon number operators by their eigenvalues leads to the following momentum functional, The momentum functional (61) is linear in the bare-momentum distribution function deviations for all excited states independently of whether the values of these deviations are small or large. Thus, this expression is valid even for large values of these deviations. As for the Fermi-liquid quasiparticles [42,43], while the momentum functional (61) is linear in the bare-momentum distribution function deviations, the corresponding energy functional includes linear, quadratic, and higher-order terms in these deviations. Such an energy functional is derived by solution of the integral equations (19)-(21) for bare-momentum distribution functions of the general form (59). This leads to deviation expansion expressions for the rapidity-momentum and rapidity functionals in terms of the set of bare-momentum distribution function deviations ∆N c0 (q), {∆N cν (q)}, and {∆N sν (q)} where ν = 1, 2, .... According to Eqs. (31)-(33), the energy spectrum of the excited states can be expressed in terms of these functionals. Use in the general energy expressions (31)-(33) of the systematic expansion in the pseudoparticle bare-momentum distribution deviations and holon and spinon number deviations of the above functionals leads to a general finite-energy Landau-liquid energy functional of the following form, The terms of order i larger than one describe the residual interactions of the pseudoparticles. In contrast, the energy functional (62) is linear in the −1/2 Yang holon and −1/2 HL spinon number deviations for all excited states independently of whether the values of these deviations are small or large. Such behavior follows from the noninteracting character of these quantum objects [30]. In the case of excited states whose bare-momentum distribution function deviations involve a small but finite density of pseudoparticles, the physics is described by the energy terms on the right-hand side of Eq. (62) of all scattering orders i = 1, 2, .... However, in the limit when the density of these pseudoparticles becomes vanishing, only the first two terms become relevant. According to the results obtained in Refs. [29,30], the first and second-order terms on the right-hand side of Eq. (62) are of the following general form, and where ∆N c (q), ∆N cν (q), and ∆N sν (q) are the pseudoparticle bare-momentum distribution function deviations given in Eq. (58) and ω 0 is an energy functional linear in the −1/2 holon, −1/2 spinon, and s1 pseudoparticle number deviations ∆M c, −1/2 ∆M s, −1/2 , and ∆N s1 , respectively. (The latter functional is given below in Eq. (101) of Sec. V, with the s1 pseudofermion number deviation equaling the corresponding s1 pseudoparticle number deviation.) The coefficients of the i = 1 linear terms are the αν pseudoparticle energy bands ǫ c0 (q), ǫ s1 (q), and ǫ 0 αν (q) studied in Refs. [29,30,31]. The coefficients of the i = 2 quadratic terms are the pseudoparticle f functions f αν, α ′ ν ′ (q, q ′ ) studied in Ref. [29].
In contrast to what the energy terms (63)-(64) may suggest, we emphasize that the energy (62) is not an expansion in 1/L. The small parameters in such an expansion are the pseudoparticle bare-momentum distribution function deviations, as in the case of the quasiparticle Fermi-liquid energy functional [42,43]. Indeed, when these function deviations involve a small but finite density of pseudoparticles, the energy terms of order i on the right-hand side of Eq. (62) are not of order [1/L] i . Instead, in that case all terms of the energy expansion (62) are of the same order [1/L] −1 = L. As discussed in the ensuing section, few-electron excitations are associated with excited states generated from the ground state by a finite number of pseudoparticle, holon, and spinon processes. The deviations of these states involve a vanishing density of pseudoparticles, holons, and spinons. However, in order to achieve the correct microscopic description of the few-electron spectral properties, one must consider the limit where a vanishing density of pseudoparticles is approached rather than considering that such a density is zero from the very beginning. As the density of pseudoparticles involved in the bare-momentum distribution deviations decreases and approaches zero, the energy terms on the right-hand side of Eq. (62) of large scattering order i, also vanish and do not not contribute as they become of order [ The f functions on the right-hand side of Eq. (64), are associated with the two-pseudoparticle residual interactions and have the same role as those of Fermi-liquid theory [42,43]. Indeed, for small values of the energy ω and electronic densities n and spin densities m such that 0 < n < 1/a and 0 < m < n, respectively, the few-electron spectral properties are controlled by the residual two-pseudoparticle interactions described by the i = 2 terms (64) of the energy functional (62). In the corresponding low-energy Hilbert subspace, only the c0 and s1 pseudoparticle branches have finite occupancies. In this case, as the limit of vanishing density of pseudoparticles is approached, the general energy functional (62) acquires the form of the energy spectrum of a two-component c0 and s1 conformal field theory [21]. For the relation of the c ≡ c, 0 and s ≡ s, 1 pseudoparticle description to two-component conformal-field theory see Refs. [22,23]. The weight distribution of the few-electron spectral functions can in this case be studied by conformal field theory [24].
We thus conclude that the pseudoparticles have residual interactions associated with the terms of the general energy functional (62) of order i > 1. The pseudoparticle residual interactions control the low-energy few-electron spectral properties of the quantum liquid, as confirmed by the studies of Refs. [22,23]. In this paper, we introduce an alternative representation in terms of non-interactiong pseudofermions. For such pseudofermions the bare momentum is replaced by a momentum functional whose coefficients are two-pseudofermion phase shifts. In this case, the relevant energy spectrum is non-interacting and thus has only linear terms in the momentum distribution function deviations. For the pseudofermion description the few-electron spectral properties are controlled by such two-pseudofermion phase shifts [34]. Such shifts are associated with first-order momentum distribution function deviations only, whereas the pseudoparticle residual interactions involve scattering orders i associated with terms of the energy functional (62) such that i > 1.

C. THE CONCEPT OF J-CPHS ENSEMBLE SUBSPACE
Let us introduce the quantum number ι = sgn(q)1 = ±1 which refers to the number of right (ι = +1) and left (ι = −1) pseudoparticle movers. The number N αν, ι of αν pseudoparticles of ι character is a good quantum number. We thus introduce the αν pseudoparticle current number J αν such that, Each CPHS ensemble subspace contains in general several subspaces with different values for the sets of current numbers J c0 and {J αν } such that α = c, s and ν = 1, 2, .... We call these subspaces J-CPHS ensemble subspaces. For a given J-CPHS ensemble subspace one can introduce the J-CPHS ground state. This energy eigenstate has compact bare-momentum occupancies for the αν pseudoparticle bands. Moreover, a partial J-CPHS ground state is a state with such a type of occupancy configuration for the αν pseudoparticle branches such that αν = c0 and αν = s1 only. These concepts are associated with the right (ι = +1) and left (ι = −1) Fermi points, which separate the αν pseudoparticle occupied from the unoccupied regions of bare-momentum space. Ignoring terms of order 1/L, these Fermi bare-momentum values read, The ground state is a particular case of J-CPHS ground state such that the Fermi points of Eq. (66) read, where In most situations one can disregard the 1/L corrections and use the bare-momentum Fermi values provided in Eqs.
(66)-(68). The pseudofermion description introduced in the ensuing section, corresponds to J-CPHS ensemble subspaces spanned by energy eigenstates differing from the ground-state by the occupancy configuration of a finite number of αν pseudoparticles, −1/2 Yang holons, and −1/2 HL spinons. For such subspaces the pseudoparticle Fermi band momenta given in Eq. (66) can be written as follows, where q 0 F αν is the αν ground-state Fermi bare momentum whose values are given in Eq. (68) and thus, Here

V. THE PSEUDOFERMION DESCRIPTION
In this section we introduce the pseudofermion operational description. This includes introduction of the pseudoparticle -pseudofermion unitary transformation and of the Hilbert subspace associated with that transformation. In such a subspace, the rapidity functionals have a particular form. Moreover, we obtain the anticommutation relations of the pseudofermions.

A. THE FUNCTIONAL CHARACTER OF THE PSEUDOFERMION MOMENTUM
The αν pseudofermion carries momentumq j given by, Here Q αν (q j )/L is the momentum functional, The functions Φ αν, α ′ ν ′ (q, q ′ ) on the right-hand side of this equation are the two-pseudofermion phase shifts expressed in terms of the bare-momentum q. These phase shifts are given by, whereΦ αν, α ′ ν ′ (r, r ′ ) are the two-pseudofermion phase shifts expressed in terms of the variable r. The latter phase shifts are the unique solutions of the integral equations (B2)-(B15) of Appendix B. The value of the momentum functional given in Eq. (73) is controlled by shake-up two-pseudofermion phase-shift processes resulting from the changes in the quantum-object occupancy configurations described by the deviations ∆N α ′ ν ′ (q j ′ ) on the right-hand side of that equation.
The αν pseudofermion is related to the corresponding αν pseudoparticle by a mere unitary transformation involving the discrete bare-momentum values q j and such that, whereq j is the discrete momentum defined in Eq. (72). The number of αν pseudoparticles, N αν , equals that of αν pseudofermions, which we denote by N αν . Moreover, we introduce the αν pseudofermion momentum distribution is the inverse function of (72). Note that the momentum functional (73), is such that Q α, ν (q)/L = 0 for the initial ground state. Indeed, the pseudofermion description refers to the ground-state normal ordered 1D Hubbard model. Thus, there is a specific αν pseudofermion description for each initial ground state. For the latter state the bare momentum q j = [2π/L]I αν j equals the pseudofermion momentumq j = q j + Q αν (q j )/L. This justifies the designation bare momentum for q j . Thus, in the case of the ground state the αν pseudoparticles are identical to the αν pseudofermions. It follows that the ground state is invariant under the pseudoparticle -pseudofermion unitary transformation and plays the role of the vacuum of the pseudofermion theory.
As for the case of the pseudoparticle representation, the pseudofermion description corresponds to large values of the Hubbard chain length L and is thus compatible with Takahashi's string hypothesis [26,31]. However, while the pseudoparticle representation corresponds to the whole Hilbert space, the pseudofermion description refers to a Hilbert subspace spanned by the initial ground state and the following types of states: (A) -Excited states generated from the initial ground state by a finite number of pseudofermion processes involving creation or annihilation of c0 and s1 pseudofermions, creation of pseudofermions belonging to other αν = c0 and αν = s1 branches, and creation of −1/2 Yang holons and −1/2 HL spinons. In the thermodynamic limit, the momentum and bare-momentum distribution function deviations associated with such excited states obey the following relations, (B) -Excited states generated from the initial ground state by a finite number of c0 and s1 pseudofermion particlehole processes.
Throughout this paper, we call such states, excited states A and excited states B, respectively. We emphasize that in general the excited states involve both. However, for few-electron excitations which do not change the electronic numbers the low-energy states are of type B only. Moreover, for simplicity we are considering densities in the ranges 0 < n < 1/a and 0 < m < n. Our analysis also holds for other values of the densities, yet for the half-filling n = 1/a or zero-magnetization m = 0 phases the excitation subspace is more reduced.
The unitary transformation associated with Eq. (75), is well defined for the Hilbert subspace spanned by the initial ground state and the excited states A and B. Fortunately, in the thermodynamic limit application onto the ground state of a few-electron operator generates excited states A and B only. By few-electron operators, we mean here operators which can be written as a product of N electron creation and/or annihilation operators and N /N a → 0 as N a → ∞. The pseudofermion functional theory introduced in this paper, can be applied to the study of the few-electron spectral weight distributions for all values of energy [1,10,34]. We find below that the pseudofermions have a non-interacting character. This allows the evaluation of few-electron spectral functions for all values of energy [10,34]. The αν pseudoparticle residual interactions are cancelled by the momentum transfer term Q α, ν (q)/L of Eq. (73). The information recorded in the pseudoparticle interactions is transferred over to the momentum twopseudofermion phase shifts of the momentum term Q αν (q)/L. The pseudofermion theory is of first order in 1/L both for the energy and momentum spectra. In contrast, the methods that use the pseudoparticle residual interactions lead to expressions for these spectral functions which are valid for small values of the energy ω only [22,23,29]. Therefore, the pseudofermion description introduced below corresponds to a breakthrough in what the study of the finite-energy spectral properties of the 1D Hubbard model is concerned.
That only excited states A and B have finite overlap with few-electron excitations, is confirmed by direct evaluation of matrix elements between few-electron excitations and excited states [34]. For instance, if Q αν (q)/L = 0 for the excited states B, the contribution to the one-electron spectral function would come from one particle-hole pseudofermion processes only, and the spectral functions would have the familiar δ-function structure. This is not true for our case where Q αν (q)/L = 0 for these states. In this case, there are contributions from many particle-hole processes in the c0 and s1 pseudofermion bands as well. The largest weight comes from the one particle-hole pseudofermion processes, and increasing the number of particle-hole processes, the additional weight decreases rapidly. Most of the weight associated with the excited states B is generated by one, two, and three particle-hole pseudofemion processes [34]. In the general case of few-electron spectral functions, the contribution from excited states B involving an infinite number of particle-hole processes in the c0 and s1 pseudofermion bands vanishes in the thermodynamic limit. The same holds for the excited states A. The excited states generated from the ground state by an infinite number of processes of the types reported in (A) and (B) have vanishing overlap with few-electron excitations as L → ∞.
The cancellation of the αν pseudoparticle residual interactions by the momentum transfer term Q α, ν (q)/L of Eq. (73) is related to the form of the rapidity functionals Λ αν (q) and rapidity-momentum functional k(q) in the Hilbert subspace spanned by the excited states A and B. Introduction of the pseudoparticle bare-momentum distribution functions of general form given in Eq. (59) in the rapidity functional integral equations (19)- (21) and their expansion in the small deviations (58), permits explicit solution of these equations. This procedure leads to expressions for the rapidity functionals Λ αν (q) and rapidity-momentum functional k(q) in terms of the bare-momentum distribution function deviations introduced in Eq. (58). Solution of the integral equations (19)- (21) for distributions of the general form (59) leads to first-order in the deviations to expressions for the rapidity-momentum functional and rapidity functionals of the following form, Here Λ 0 αν (q) and k 0 (q) are the corresponding ground state functions. On the right-hand side of Eq. (77),q(q) is the αν momentum functional given in Eq. (72) with q j replaced by the continuum momentum q.
It is remarkable that in the Hilbert subspace spanned by the excited states A and B the functionals Λ αν (q) and k(q) equal the corresponding ground-state functions Λ 0 αν (q) and k 0 (q), respectively, with the bare momentum q replaced by the momentum functional (72). This property is behind the non-interacting character of the pseudofermions. The twopseudofermion phase shiftsΦ αν, α ′ ν ′ (r, r ′ ) defined by Eqs. (B2)-(B15) of Appendix B which appear in the expression of the momentum functional (73) through the relation (74) play a central role in the pseudofermion description of the few-electron spectral properties. Within that description the expression of many physical quantities involve such two-pseudofermion phase shifts. For instance, the functions defined by Eqs. (47)-(49) and the expressions of the pseudofermion energy bands given below involve the two-pseudofermion phase shifts. Furthermore, these phase shifts control the matrix elements between few-electron excitations and the excited states and thus also the few-electron spectral weight distributions [34].
According to Eq. (33) the energy spectrum depends on the quantum object occupancy configurations through the rapidity and rapidity-momentum functionals. It is this functional character that is behind the pseudoparticle residual interactions and leads to the energy expansion given in Eqs. (62)-(64). However, by re-expression of these functionals in terms of the pseudoferminon momentumq, the general energy spectrum (31)- (33) can be written in terms of pseudofermion momentum distribution functions N c0 (q j ) and N cν (q j ) as, This expression is valid for the Hilbert subspace spanned by the initial ground state and the excited states A and B. The term α µ α S α z is the same as on the right-hand side of Eq. (31) and N cν = N cν is the number of cν pseudofermions. The form of the general energy spectrum (78) justifies why the shake-up effects associated with the phase shifts of the functional (73) occur in the case of the pseudofermions in the momentum instead of in the energy. The dependence of the general energy spectrum (78) on the momentum occupancy configuration values occurs through the arguments of the ground-state rapidity and rapidity-momentum functions. Thus these functions play the role of non-interacting spectra, since they have the same form both for the initial ground state and excited states A and B. The shake-up effects associated with the two-pseudofermion phase shifts are thus felt by the non-interactiong pseudofermions as mere changes in the momentum occupancies through the twisted boundary conditions imposed by the elementary processes which generate the excited state from the ground state. This property is behind the non-interacting character of the pseudofermions. It is also behind the fact that within the pseudofermion description the functionsq c (k) andq αν (Λ) defined in Eqs. (47)-(49) refer to both the ground state and the excited states. In contrast, within the pseudoparticle representation these functions refer to the ground state only.
The pseudoparticle bare-momentum q j description is naturally provided by the Bethe-ansatz equations within Takahashi's string hypothesis [26,29,31]. We recall that the pseudoparticle discrete bare-momentum values q j are of form given in Eq. (A2) of Appendix A and according to Eq. (A1) are such that q j+1 − q j = 2π/L. The single discrete bare-momentum values q j are integer multiples of 2π/L or of π/L [31] and bare-momentum contributions of order [1/L] j such that j > 1 are outside the validity of the pseudoparticle description. These bare-momentum contributions have no physical meaning and must be considered as zero. Importantly, the same is required for the pseudofermion momentum discrete valuesq j given in Eq. (72). These discrete values are also at least of order of 1/L and contributions of order [1/L] j such that j > 1 must be considered as equaling zero. For instance, as confirmed in Appendix C, the discrete momentum level separation, is such that the second term on the right-hand side of Eq. (79) is of order [1/L] 2 , where Q αν (q)/L is the momentum functional given in Eq. (73). Thus, to first order in 1/L one finds thatq j+1 −q j = 2π/L, as for the corresponding discrete bare-momentum level separation given in Eq. (A1) of Appendix A. However, this does not imply that to first order in 1/L the pseudofermion momentum equals the bare-momentum. Indeed, we emphasize that the momenta Q αν (q j )/L on the right-hand side of Eq. (75) are of order 1/L and play a central role in the control of the fewelectron spectral weight distribution by the non-perturbative many-electron shake-up effects [34]. We note that the level separationq j+1 −q j = 2π/L is valid locally in the discrete momentum space. By that we mean the following: If in the present thermodynamic limit two momentum valuesq j andq j ′ differ by a small yet finite momentum difference ∆q =q j −q j ′ , then in general ∆q = 2π L [j − j ′ ]. In contrast, for the corresponding bare-momentum values it holds that ∆q = 2π L [j − j ′ ]. Therefore, for small but non-vanishing momentum separation the difference [Q αν (q j ) − Q αν (q j ′ )]/L is not anymore of order [1/L] 2 and has physical significance.

B. PSEUDOFERMION OPERATOR ALGEBRA
The elementary creation and annihilation operators of the αν pseudofermions can be expressed in terms of the corresponding operators of the αν pseudoparticles as follows, HereV αν is a unitary operator that we call the αν pseudoparticle -αν pseudofermion unitary operator. This operator acts onto the Hilbert subspace spanned by the initial ground state and excited states A and B. It shifts the baremomentum value q j by Q αν (q j )/L and thus is given by, The momentum distribution function N αν (q j ) is the eigenvalue of the operator, Here N αν (q j ) stands for the αν pseudoparticle bare-momentum distribution function and q j (q j ) is the inverse of the function (75). The function N αν (q j ) is the eigenvalue of the corresponding bare-momentum number operator (18). Keeping only the physical contributions that correspond to terms up to first order in 1/L, the function q j (q j ) is given by, Here, ∆N α ′ ν ′ (q j ′ ) is the deviation of the function (82) relative to its ground-state value. Since the functional (73) vanishes for the ground state, note that q j =q j for that state. Moreover, on the right-hand side of Eq. (83) Φ f αν, α ′ ν ′ (q,q ′ ) is the two-pseudfermion phase shift. It is defined as, where the function q = q(q) is the continuum version of the function (83), Φ αν, α ′ ν ′ (q, q ′ ) is given in Eq. (74), and Φ αν, α ′ ν ′ (r, r ′ ) is the two-pseudofermion phase shift expressed in the variable r defined by the integral equations (B2)-(B15) of Appendix B. Often in the expressions given in previous sections we replaced the pseudoparticle bare-momentum summations by integrals and the corresponding discrete bare-momentum values q j by a continuum bare-momentum variable q. Since according to Eq. (A1) of Appendix A, the difference q j+1 − q j = 2π/L is constant for all values of j, the use of such a continuum representation involves the replacement, In the Hilbert subspace spanned by the excited states A and B, the rapidity functional Λ αν (q) and rapiditymomentum functional k(q) equal the corresponding ground-state rapidity function Λ 0 αν (q) and rapidity-momentum function k 0 (q), respectively, with the bare-momentum q replaced by the momentumq. It follows that in such a subspace the limiting values of the continuum momentumq are given by the ground-state limiting values ±q 0 αν given in Eqs. (39) and (40). (This is confirmed in Sec. VII.) Thus, to replace the discrete momentum values by a continuum momentum variableq, one must replace the summations by the following integrals, We then introduce the momentum distribution function, The function q = q(q) on the right-hand side of Eq. (86) is the inverse of the function given in Eq. (72) and is given in Eq. (83) withq j replaced byq. Its derivative is given by, The second term on the right-hand side of this equation is of first order in the momentum distribution function deviations. For the non-interacting pseudofermion description only momentum and energy contributions up to first order in these deviations are physical. As a result, in the case of momentum distribution function deviations ∆N αν (q) one can consider that, where in contrast to the case of Eq. (87) we used dq(q)/dq = 1.
The αν pseudoparticle number operator, is invariant under the pseudoparticle -pseudofermion transformation. It equals the number of αν pseudofermions operator,N where the pseudfermion momentum distribution operatorsN αν (q) are given in Eq. (82). Like in Eq. (29), the operator L α, ±1/2 on the right-hand side of Eq. (92) is the ±1/2 Yang holon (α = c) and ±1/2 HL spinon (α = s) number operator given in Eq. (30). All results reported in Sec. III concerning pseudoparticle charge and spin transport are also valid for the corresponding pseudofermions. For instance, for finite values of U/t the transport of charge (and spin) is associated with the c0 pseudofermion and cν pseudofermion quantum charge fluids (and sν pseudofermion quantum spin fluids). It is found in Ref. [35] and discussed in Appendix A that the bare-momentum q is the conjugate of the spatial coordinate x j = a αν j associated with the effective αν lattice, where j = 1, 2, ..., N * αν . As in the case of the charge (or spin) carried by the pseudoparticles and of their composite character in terms of chargeaons and antichargeons, ±1/2 holons, or ±1/2 spinons, also the effective αν lattice remains invariant under the αν pseudoparticle -αν pseudofermion unitary transformation. Indeed, the momentum functional Q αν (q)/L which controls the pseudoparticle -pseudofermion transformation (75) does not affect the underlying effective αν lattice. That momentum functional just imposes a twisted boundary condition such that each αν pseudofermion hopping from site N * αν − 1 to site 0 of such an effective lattice will acquire a phase e iQαν (q) . From combination of that analysis with the expression for the general momentum functional Q αν (q)/L given in Eq. (73), we find that the phase shift Φ f αν, α ′ ν ′ (q,q ′ ) is such that creation into the system of a α ′ ν ′ pseudofermion of momentumq ′ imposes a twisted boundary condition on the wave function of the αν pseudofermion branch. This condition is such that when the αν pseudofermion of momentumq hops from site N * αν − 1 to site 0 of the effective αν lattice, it will acquire a phase e i 2π Φ f αν, α ′ ν ′ (q,q ′ ) . The studies of Ref. [34] reveal that within the pseudofermion description the few-electron spectral properties are fully controlled by such two-pseudofermion twisted boundary conditions associated with the phase shifts (84).
As for the case of the pseudoparticles, it is useful to introduce the local αν pseudofermion creation operator f †

xj , αν
and annihilation operator f xj, αν . These operators are related to the operators f † q, αν and fq , αν , respectively, obtained from the corresponding pseudoparticle operators through the relations given in Eq. (80), as follows, where the summations refer to the sites of the effective αν lattice. 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 = a 0 αν j, where j = 1, 2, ..., N * αν and a 0 αν is the effective αν lattice constant given in Eq. (37). (Since the pseudofermion representation refers to the Hilbert subspace spanned by the initial ground state and the excited states A and B, except for 1/L corrections we can consider that the corresponding effective αν lattice constants are the ground-state constants a 0 αν .) Thus, the conjugate variable of the momentumq j of the αν pseudofermion branch is the space coordinate x j of the corresponding effective αν lattice. The local αν pseudoparticles and corresponding local αν pseudofermions have the same effective αν lattice. It follows that the local pseudoparticle and local pseudofermion site distribution configurations which describe the ground state and the excited states A and B are the same. These configurations are expressed in terms of rotated-electron site distribution configurations in Ref. [35].
While the local αν pseudoparticles and corresponding local αν pseudofermions live in the same effective αν lattice, the values of the set of discrete bare-momentum values {q j } and momentum values {q j } such that j = 1, 2, ..., N * αν are different and related by Eq. (83). There is an one-to-one relation between these two sets of discrete values, which keep the same order because there is no level crossing. This property follows from the discrete bare-momentum and momentum separation given in Eq. (A1) of Appendix A and Eq. (79), respectively.
Finally, we consider the anticommutation relations of the pseudofermion operators. It is confirmed in Ref. [34] that such relations play a major role in the evaluation of matrix elements between energy eigenstates. Let us consider the general situation when the momentaq andq ′ of the operators f † q, αν and fq′ , αν , respectively, correspond to different J-CPHS ensemble subspaces. The anticommutator {f † q, αν , fq′ , α ′ ν ′ } can be expressed in terms of the localpseudofermion anticommutators {f † xj, αν , f x j ′ , α ′ ν ′ } associated with spatial coordinates x j and x j ′ of the effective αν and α ′ ν ′ lattices, respectively, as follows, After performing the j and j ′ summations we find that the αν pseudofermion operators obey the following algebra, and the anticommutators between two αν pseudofermion creation or annihilation operators vanish. On the right-hand side of Eq. (95) the functionals Q αν (q) and Q ′ αν (q ′ ) whose general expression is given in Eq. (73) refer to the J-CPHS ensemble subspaces where the momentaq andq ′ refer to, respectively. Note that when sin([Q αν (q) − Q ′ αν (q)]/2) vanishes the anticommutation relation (95) is the usual one, {f † q, αν , fq′ , α ′ ν ′ } = δ α, α ′ δ ν, ν ′ δq ,q ′ . In contrast, for finite values of that quantity there is an overall two-pseudofermion phase shift which arises from a non-perturbative shake-up effect associated with the functional character of the momentum Q αν (q)/L given in Eq. (73).
A case of particular importance is when one of the two J-CPHS ensemble subspaces associated with the momentaq andq ′ is that of the initial ground state. In such a case Q ′ αν (q ′ ) = 0 for the ground-state J-CPHS ensemble subspace and thus the anticommutation relation (95) simplifies to,

VI. THE PSEUDOFERMION ENERGY AND MOMENTUM SPECTRA AND THE WAVE-FUNCTION FACTORIZATION OF THE NORMAL-ORDERED 1D HUBBARD MODEL
In this section, we find that the description of the quantum problem in terms of the non-interacting pseudofermions leads in the thermodynamic limit to a wave-function factorization for the excited states A and B contained in the excitations generated by application onto the ground state of few-electron operators. Such a factorization refers to the ground-state normal-ordered 1D Hubbard model.

A. THE PSEUDOFERMION ENERGY AND MOMENTUM SPECTRA
Provided that all scattering orders are considered, the pseudoparticle energy functional (62) describes excitations which involve a small, but finite density of pseudoparticles. However, in the thermodynamic limit, the few-electron excitations are contained in a smaller Hilbert subspace. This subspace is spanned by the excited states A and B. These states are generated from the ground state by processes which involve a finite number of pseudofermions (and pseudoparticles). In the thermodynamic limit, this corresponds to a vanishing density of these objects rather than to a small but finite density. In the Hilbert subspace spanned by the ground state and the excited states A and B the energy spectrum is of the form given in Eq. (78). This spectrum can be expressed in terms of continuum momentum integrals with the result, whereN c0 (q) andN cν (q) are the pseudofermion momentum distribution functions of Eq. (87).
In contrast to the pseudoparticle energy functional (62)-(64) in terms of the pseudoparticle bare-momentum distribution function deviations, the energy spectrum derived from expressions (78) and (97) only includes first-order terms in the pseudofermion momentum distribution function deviations. In the Hilbert subspace that the energy spectrum (78)-(97) refers to, only such first-order pseudofermion momentum distribution function deviation contributions have physical meaning. The information recorded in the pseudoparticle interactions is transferred over to the momentum two-pseudofermion phase shifts of the momentum term Q αν (q)/L. The pseudofermion discrete momentum values are of 1/L order and store this information.
According to Eq. (73), the processes which generate the excited states A and B from the initial ground state lead to a collective momentum shift of all the c0 pseudofermions and s1 pseudofermions of the initial-state Fermi sea. That momentum shift reads, where the bare-momentum shift π 0 αν /L is given by, The excited states A and B are generated from the ground state by two virtual excitations: (1) a collective momentum shift of the Fermi sea c0 pseudofermions and s1 pseudofermions; (2) for the states A, a finite number of c0 pseudofermion and s1 pseudofermion creation or annihilation processes and of αν = c0 and αν = s1 pseudofermion, −1/2 Yang holon, and −1/2 HL spinon creation processes; for the states B, a finite number of c0 pseudofermion and s1 pseudofermion particle-hole processes. Consideration of both the energy associated with the virtual excitations (1) and (2), gives the energy spectrum of few-electron excitations associated with the energy (78)-(97). Such an energy spectrum corresponds to the ground-state normal-ordered 1D Hubbard model and is additive in the −1/2 holon, −1/2 spinons, and αν pseudofermion energies. It reads, where the energy parameter ω 0 is given by, Here ∆M α, −1/2 are the deviations in the numbers of −1/2 holons (α = c) and of −1/2 spinons (α = s) and ∆N s1 = ∆N s1 is the deviation in the number of s1 pseudofermions. The limiting-momentum values q 0 c0 , q 0 s1 , and q 0 αν appearing in the integrations of Eqs. (97) and (100) are given in Eqs. (39) and (40). Moreover, on the right-hand side of Eq. (100) the functions ǫ c0 (q), ǫ s, 1 (q), and ǫ 0 αν (q) are the pseudofermion energy bands. These energy bands equal the corresponding pseudoparticle energy bands appearing on the right-hand side of Eq. (63), provided that the bare momentum q is replaced by the momentumq. The latter bands are plotted in Figs. 6 to 9 of Ref. [30] as a function of the bare momentum for zero spin density and several values of electronic density and on-site Coulombian repulsion.
The shape of the pseudofermion energy bands is controlled by the two-pseudofermion phase shifts and reads, and ǫ s1 (q) = ǫ 0 s1 (q) + 2µ 0 H .
The two-pseudofermion phase shifts on the right-hand side of these equations are defined in Eqs. (50)-(51). The parameter Q is given in Eq. (45) and the functions k 0 (q) and Λ 0 cν (q) are the inverse of the functions defined in Eqs. (47) and (48), respectively. The zero-energy levels of the energy dispersions (102)-(105) are such that, The dependence on the electronic density n, spin density m, and U/t of the chemical potential µ and magnetic field H appearing in Eqs. (102) and (105) can be expressed in terms of the two-pseudofermion phase shifts as, and respectively. We emphasize that the virtual-excitation (1) energy spectrum vanishes and thus the general deviation-linear energy spectrum (100) amounts to the contributions from the excitation (2). By construction of the pseudofermion description subspace, the virtual excitation (2) of the states A and B, involves changes in the occupancy configurations of a finite number of quantum objects. However, the excitation (1) is a collective momentum shift of all the ground-state c0 pseudofermions and s1 pseudofermions. In the thermodynamic limit, the value of the corresponding ground-state numbers N c0 = N and N s1 = N ↓ approaches infinity. Thus, the self-consistency of the non-interacting pseudofermion theory implies that the energy of the excitation (1) vanishes in that limit, so that the total energy is additive in the corresponding pseudofermion energies. Let us confirm that as L → ∞ this holds true. The c0 and c1 pseudofermion momentum distribution function deviations of Eq. (100) can be written as, where ∆N (1) αν (q j ) and ∆N (2) αν (q j ) are associated with the collective pseudofermion Fermi sea excitation (1) and excitation (2), respectively. Within the first order 1/L pseudofermion description the former deviation reads, Here, the momentum Fermi value q 0 F αν of the c0 and s1 branches is given in Eq. (68) and π 0 αν /L is the corresponding bare-momentum shift of Eq. (99). To first order in 1/L, use of Eq. (110) in the energy spectrum (100) leads to the following excitation (1) energy spectrum, In order to obtain the result (111) we have used the symmetry ǫ αν (q) = ǫ αν (−q) and Eq. (106) such that ǫ αν (q 0 F αν ) = 0 for αν = c0, s1. We recall that the occupancies of other αν = c0, s1 pseudofermion branches vanish for the ground state. Since the excitation (1) involves all ground-state c0 and s1 pseudofermions, the evaluation of the 1/L higherorder contributions to the corresponding energy spectrum involves non-linear deviation terms. We find that such 1/L higher-order energy contributions also vanish as L → ∞. Thus, the self-consistency condition that the excitation (1) energy spectrum vanishes as the system length L approaches infinity is fulfilled.
The energy (101) on the right-hand side of Eq. (100) controls the finite-energy physics. The remaining energy terms correspond to gapless contributions provided that the involved pseudofermions correspond to the momentum values of the energy-band arguments of Eq. (106). For most excited states, the latter terms also lead to finite-energy contributions. The typical value of such energy contributions is of the order of the pseudofermion energy dispersion band-width per pseudofermion involved in the excited states. We note that the energy spectrum (100) of the excited states A and B, can have any finite value associated with the regions of the (k, ω) plane where the few-electron spectral functions have finite spectral weight [10,34].
Also, the excitation momentum spectrum can be written in terms of the pseudofermion momentum distribution function deviations. It is given by, The important point is that the large-L pseudofermion operational description introduced in this paper, associated with the momentum distribution function deviation first-order energy spectrum (100) and momentum spectrum (112), contains full information about the few-electron spectral properties. Besides the expression in terms of pseudofermion operators of the diagonal (in the basis of the energy eigenstates) operators associated with these spectra, such a description is suitable for the evaluation of few-electron spectral functions for finite values of energy [34].

B. WAVE-FUNCTION FACTORIZATION OF THE NORMAL-ORDERED 1D HUBBARD MODEL
It is well known that both the ground state wave function and the wave function of the excited states of the 1D Hubbard model can in the U/t → ∞ limit be constructed as a product of a spin-less fermion wave function and a squeezed spin wave function [16,18,19]. In our pseudofermion, Yang holon, and HL spinon language this factorization means that in such a limit the expression of the momentum and energy of these states is linear in the αν pseudofermion momentum distribution functions and in the −1/2 Yang holon and −1/2 HL spinon numbers.
Let us show that for finite values of U/t the energy (31) is not linear in the αν pseudofermion momentum distribution functionsN αν (q). Our analysis refers to the few-electron excitation subspace. This confirms that the above type of factorization does not occur for finite values of the on-site repulsion. For the pseudofermion Hilbert subspace spanned by the ground state and excited states A and B the form of the general energy spectrum (31) simplifies to, where ∆E is the general energy excitation spectrum given in Eq. (100), E 0 stands for the ground-state energy, N c0 is the number of c0 pseudofermions, and the term α µ α S α z is the same as on the right-hand side of Eq. (31). From combination of the ground-state occupancy configurations provided in Appendix A with the general energy spectrum given in Eq. (33) we find that the energy E 0 H on the right-hand side of Eq. (117) reads, whereN 0 c0 (q) is the ground-state c0 pseudofermion momentum distribution function. (We recall that q =q for the ground state.) From use of Eqs. (47), (50), and (84) we arrive to the following expressions, By iterative solution of the equations given in (119) one can derive a functional representation for the rapiditymomentum function k 0 (q) in terms of the ground-state distribution functionN 0 c0 (q). Use of such a functional on the right-hand side of Eq. (118) leads to the following non-linear energy functional in the ground-state momentum distribution functionN 0 c0 (q), Here the indices i = 1, 2, ... of the momentumq i label independent integration continuum variables rather than discrete momentum values. The form of expression (120) confirms that for finite values of U/t the energy (118) is highly non-linear in the ground-state pseudofermion momentum distribution functionN 0 c0 (q). Therefore, the above type of factorization does not occur in general for the 1D Hubbard model. However, one can confirm that in the limit U/t → ∞ the energy functional (120) becomes linear inN 0 c0 (q). Indeed, one finds from analysis of the integral equations provided in Appendix B that for spin density m = 0 the two-pseudofermion phase shift on the right-hand side of Eq. (120) is such that Φ f c0, c0 (q,q ′ ) → 0 as U/t → ∞. Thus, in that limit the ground-state expression (120) simplifies to, This property also holds for the excited states and is behind the full factorization of the wave functions used in the studies of Refs. [16,18,19].
Fortunately, the evaluation of few-electron spectral functions can be achieved without the full factorization of the wave functions. Such a problem can be solved by use of the ground-state normal-ordered 1D Hubbard model. When expressed in terms of pseudofermion operators, that normal-ordered Hamiltonian and associated momentum operator are given in Eqs. (113) and (114), respectively. These quantum problems correspond to the energy and momentum spectra given in Eqs. (100) and (112), respectively. The ground-state normal-ordered description is equivalent to specifying the energy eigenstates by means of the momentum distribution function deviations ∆N αν (q) = ∆N αν (q) instead of in terms of the corresponding full momentum distribution functionsN αν (q).
In the thermodynamic limit, there is a wave-function factorization for the excited states belonging to the Hilbert subspace of the pseudofermion normal-ordered Hamiltonian (113). This factorization follows from the form of the expressions (100) and (112) for the energy and momentum, respectively. The energy spectrum defined in Eqs. (100) and (101) can be written as, where the contributions from each branch are linear in the αν pseudofermion momentum distribution function deviations and read, Here the pseudofermion energy bands are given in Eqs. (102), (105), and (115). The −1/2 Yang holons, −1/2 HL spinons, and each pseudofermion branch leads to a different energy term linear in the corresponding number deviation or momentum distribution function deviation. We emphasize that the virtual-excitation (1) energy ∆E (1) c0, s1 of Eq. (111) associated with the momentum shift of Eq. (98) which involves all c0 pseudofermions and s1 pseudofermions of the ground-state Fermi sea, is also additive in the energy contributions from each of these quantum objects. Indeed, according to Eqs. (110) and (111), the energy contribution from each c0 pseudofermion and s1 pseudofermion involved in such a collective excitation vanishes. As the total energy ∆E (1) c0, s1 of the same excitation also vanishes, it is indeed additive in the energies of all involved quantum objects. Therefore, the wave function of the energy eigenstates of the normal-ordered Hamiltonian can be expressed in the pseudofermion subspace as a product of wave functions. Each wave function corresponds to a different pseudofermion branch. In excited states with finite −1/2 Yang holon and −1/2 HL spinon occupancy there is also a wave function for these objects. This factorization is associated with the deep physical meaning of the pseudoparticle -pseudofermion transformation (85): it transfers the information recorded in the pseudoparticle interactions over to the pseudofermion momentum, providing a non-interacting character to the latter objects.
In contrast, for the pseudoparticle representation the energy functional (62)-(64) includes bare-momentum distribution function deviation non-linear terms associated with the pseudoparticle residual interactions. For instance, the quadratic energy term (64) contains bare-momentum summations over products of deviations of the form f αν;α ′ ν ′ (q, q ′ ) ∆N αν (q) ∆N α ′ ν ′ (q ′ ). Such f function terms are associated with the residual two-pseudoparticle interactions. The occurrence of these energy terms mixes contributions from different branches. It follows that the pseudoparticle energy spectrum is not additive in the αν pseudoparticle branch contributions, in contrast to the pseudofermion energy spectrum given in Eqs. (100) and (122). Thus, in this case the wave function of the excited states does not factorize in the form of a product of pseudoparticle wave functions.
The number of wave functions contributing to the factorized wave function of a given energy eigenstate depends on the occupancy configurations of that state. Only the αν pseudofermion branches with finite occupancy in the state contribute the wave function. This contribution is in the form of a αν wave function factor. The same applies to the occupancy of −1/2 Yang holons and −1/2 HL spinons.

VII. TRANSFORMATION LAWS UNDER THE PSEUDOPARTICLE -PSEUDOFERMION SUBSPACE UNITARY ROTATION
In this section, we find the transformation laws under the pseudoparticle -pseudofermion Hilbert subspace unitary rotation of several quantum objects and quantities and discuss the physical meaning of these laws. For instance, the αν pseudoparticle number operator (90), is invariant under such a transformation and thus equals the corresponding αν pseudoparticle number operator (91). Also the Yang holons and HL spinons, the charge and spin carried by the pseudoparticles, and the effective pseudoparticle lattices remain invariant under such a transformation.
The pseudoparticle -pseudofermion unitary transformation is generated by the operator (81), which only shifts the pseudoparticle bare-momentum values. Thus, given a pseudoparticle of bare-momentum q, the transformation law q →q of its bare momentum q also defines the transformation law of such a quantum object under the pseudoparticle -pseudofermion unitary transformation. For instance, if a specific value of the bare momentum q remains invariant under that transformation, then the pseudoparticle carrying bare momentum q is the same quantum object as the corresponding pseudofermion.

A. TRANSFORMATION LAWS OF PSEUDOPARTICLES AT THE BARE-MOMENTUM LIMITING VALUES
The transformation laws of the limiting pseudoparticle bare-momentum values ±q α,ν under the pseudoparticlepseudofermion transformation defined in Eq. (85) provide interesting information about the physics described by the αν pseudofermions. For the excited states A and B, the limiting αν pseudoparticle bare-momentum values ι q αν , where ι = ±1, can be written as, Here q 0 αν is the ground-state limiting bare-momentum value given in Eqs. (39) and (40). We start by considering the case of ν > 0 pseudoparticle branches. In that case the deviations ∆q αν have the following form, In Appendix C we find that at q = ±q αν and for ν > 0 the functional defined in Eq. (73) equals the following value for all excited states A and B, where the quantity ∆q αν is given in Eq. (125). Here it is assumed that the bare momentum q ′ associated with the pseudoparticle bare-momentum distribution function deviations ∆N α ′ ν ′ (q ′ ) on the right-hand side of Eq. (126) belongs to the domain q ′ ∈ (−q α ′ ν ′ , +q α ′ ν ′ ) and can be such that q ′ → ±q α ′ ν ′ but q ′ = q α ′ ν ′ . Interestingly, the value of the momentum functional Q αν (q)/L at q = ι q αν is such that it precisely cancels the term ι ∆q αν appearing in the limiting bare-momentum expression on the right-hand side of Eq. (124). Thus at the limiting bare-momentum values q = ±q αν given in Eq. (124) the value of the momentum functionalq(q) defined in Eq. (72) is independent of the value of the deviation ∆q αν on the right-hand side of Eq. (124) which defines the final excited state. By use of this result in Eq. (72) we find that, ι q αν = ι q 0 αν + ι ∆q αν → ιq αν = ι q 0 αν ; α = c, s ; ν = 1, 2, ... ; ι = ±1 .
Thus, we conclude that for ν > 0 pseudoparticle branches the limiting momentum values of the corresponding αν pseudofermion bands equal for all excited states A and B the ground-state limiting bare-momentum values given in Eqs. (39) and (40), i.e. ±q αν = ±q 0 αν . While the width of the bare-momentum Brillouin zone of the αν pseudoparticles, 2q αν , has according to Eqs. (124) and (125) an exotic dependence on the state occupancy configurations, the value of the corresponding momentum width of the αν pseudofermion Brillouin zone is constant and equals 2q αν = 2q 0 αν , where q 0 αν is the ground-state limiting bare momentum given in Eqs. (39) and (40).
This confirms that the width of the αν pseudofermion momentum domain remains unchanged, as in the case of the momentum bands of non-interacting particles. This result is consistent with the expressions given in Eq. (77) for the rapidity and rapidity-momentum functions, respectively, of the excited states. When expressed in terms of the pseudofermion momentumq such functions have the same form both for the ground state and excited states. This introduces the requirement that the width of the momentum domain where these functions are defined must be the same for the ground state and the excited states. Thus we have just confirmed that this requirement is fulfilled through the cancelling associated with the value of the momentum functional (126).
On the other hand, the bare-momentum width of the c0 pseudoparticles is independent of the pseudoparticle occupancy configurations and reads 2q 0 c0 = 2π/a. In Appendix C we find that the transformation (85) leaves this width invariant and shifts the bare-momentum values q = 0 and q = ±q 0 c0 by the same amount as follows, where Q c0 (0)/L is the q = 0 value of the momentum functional Q c0 (q)/L given in Eq. (73). Thus, for the c0 band the shift is the same for these three bare-momentum values. As a result the width of the corresponding pseudofermion momentum domain remains unchanged and is given by 2q 0 c0 = 2π/a. Note that the value of such a shift is a functional which depends on the actual occupancy configurations of the excited states pseudoparticle bare-momentum distribution function deviations.

B. TRANSFORMATION LAWS OF FERMI-POINT PSEUDOPARTICLES
Let us next consider excitations corresponding to J-CPHS ground states whose Fermi points are defined in Eqs. (69)-(71). Our goal is to find out how the Fermi bare momentum q F cν, ι given in these equations transforms under the pseudoparticle -pseudofermion transformation (85). In this case, such a transformation can formally be written as, where according to Eq. (71), ∆q F αν, ι = ι [2π/L] ∆N αν, ι . In Appendix C, we find that the deviation ∆q F αν, ι on the right-hand side of Eq. (129) is given by, Here and where Φ f αν, α ′ ν ′ (q,q ′ ) is the two-pseudofermion phase shift given in Eq. (84). The pseudoparticle -pseudofermion transformation (85), refers to a subspace spanned by states of finite energy ω. Let us show that in the specific limit of small energy ω → 0 that unitary transformation is directly related to the primary-field conformal dimensions of the 1D Hubbard model two-component conformal field theory [21,22,23]. We consider the particular case of excited states with finite occupancies in the c0 and s1 pseudoparticle bands only. For electronic densities and spin densities such that 0 < n < 1/a and 0 < m < n, respectively, this corresponds to the low-energy Hilbert subspace considered in the studies of Refs. [22,23]. For simplicity, in this specific case we use the notation of these references such that c ≡ c, 0 and s ≡ s, 1. The indices αν of all quantities are replaced by the index α such that α = c, s. In the present low-energy Hilbert subspace the above general expressions simplify to, where and Here Let us combine Eqs. (134) and (135) and rewrite the number ∆N f α, ι as, Based on the form (136) of the parameters ξ j α, α ′ one can by manipulation of the integral equations given in Appendix B, define these parameters in terms of related integral equations. Importantly, comparison of these latter equations and associated expressions (136) and (137) with the results of Refs. [22,23] reveals that the number ∆N f α, ι given in Eq. (137) is such that, where the quantity ∆ ι α is the α conformal dimension of the primary fields [20,21] associated with the α = c and α = s excitation branches. Moreover, the parameters ξ 1 α, α ′ and ξ 0 α, α ′ are entries of the transpose of the dressed charge matrix and of the inverse of the transpose of the dressed charge matrix, respectively, of the low-energy two-component 1D Hubbard model conformal field theory [21,22,23].
These results confirm that for the low-energy subspace defined above and for electronic densities and spin densities such that 0 < n < 1/a and 0 < m < n, respectively, the conformal invariance of the 1D Hubbard model is directly related to the pseudoparticle -pseudofermion unitary transformation (85). For these values of the electronic and spin densities the low-energy excited states are described by pseudoparticle bare-momentum occupancy configurations such that there are both pseudoparticle and pseudoparticle holes in the c ≡ c, 0 and s ≡ s, 1 pseudoparticle bands. However, the pseudoparticle -pseudofermion unitary transformation is more general and refers to all values of energy associated with few-electron excitations.
The above results provide interesting information about the transformation laws of c0 pseudoparticles (and s1 pseudoparticles) of bare momentum values at the Fermi points, where ι = ±1. Under the pseudoparticle -pseudofermion transformation (85), these pseudoparticles are mapped onto c0 pseudofermions (and s1 pseudofermions) with momentum values at the Fermi pointsq = ι 2k F + ι 2π L 2 ∆ ι c (and q = ι k F ↓ + ι 2π L 2 ∆ ι s ) where ∆ ι c (and ∆ ι s ) is the conformal dimension of the c0 (and s) excitation branch primary field [21,22,23]. We then conclude that these conformal dimensions are such that the quantities 2 ∆ ι α are nothing but deviations in the values of c0 (α = c) and s1 (α = s) pseudofermion momentum Fermi points resulting from lowenergy excitations. (We note that the positive and negative value of the root 2 ∆ ι α refers to pseudofermion creation and annihilation, respectively.) Conformal-field theory can be used to evaluate expressions for low-energy few-electron spectral functions [21,24]. We recall that our general pseudofermion description provides the momentum deviation values resulting from finite-energy excitations for all values of pseudofermion momentum and for all pseudofermion branches. Then it is not unexpected that the general pseudofermion description introduced in this paper can be used in the evaluation of few-electron spectral function expressions for all values of energy [1,10,34].

C. INVARIANCE UNDER THE PSEUDOPARTICLE -PSEUDOFERMION UNITARY TRANSFORMATION
According to the results of Appendix A, there are no cν pseudoparticles and sν pseudoparticles belonging to ν > 0 and ν > 1 branches, respectively, in the ground state. This is consistent with the Fermi values of Eq. (68), such that q 0 F αν = 0 for these αν pseudoparticle branches. In Appendix C, it is shown that the momentum functional Q αν (0)/L given in Eq. (73) vanishes at q 0 F αν = 0 for these αν pseudoparticle branches, provided that the excited states are partial J-CPHS ground states and the bare-momentum distribution function deviations on the right-hand side of Eq. (139) are for the c0 and s1 pseudoparticle branches such that, Such excited states have vanishing momentum. We recall that partial J-CPHS ground states are states that have J-CPHS ground-state occupancy configurations for all the αν pseudoparticle branches other than the c0 and s1 pseudoparticle branches. It follows that for the former branches the vanishing pseudoparticle Fermi points, remain invariant under the pseudoparticle -pseudofermion transformation (85), or This means that a αν pseudoparticle belonging to a branch other than the c0 and s1 branches, carrying bare momentum q → 0, and involved in excitation processes associated with zero-momentum excited states whose c0 and s1 bare-momentum distribution function deviations obey Eq. (140) remains invariant under the pseudoparticlepseudofermion transformation (85). It follows that in this limiting case the q → 0 αν pseudoparticle is the same quantum object as the correspondingq → 0 αν pseudofermion and then, This invariance occurring as q → 0 for αν pseudoparticles belonging to branches such that αν = c0 and αν = s1, is associated with the free, non-interacting, and delocalized character of these quantum objects. Indeed, in this limit the residual interactions vanish and these pseudoparticles are free, non interacting, and delocalized quantum objects provided that the excitation processes they are involved in correspond to excited states of zero momentum. On the other hand, according to the discussions presented in Sec. III, as q approaches the limiting bare-momentum values (124), q → ±q αν , the αν pseudoparticles belonging to branches other than the c0 pseudoparticle branch become non-interacting and localized. Such a q → ±q αν non-interacting and localized behavior is associated with another invariance: As q → ±q αν these αν pseudoparticles become invariant under the electron -rotated-electron unitary transformation [30,31]. Thus, as the bare momentum q approaches the limiting values ±q αν the αν pseudoparticle and the rotated αν pseudoparticle become the same quantum object. In contrast, according to Eq. (127) in that limit, these αν pseudoparticles are not invariant under the pseudoparticle -pseudofermion unitary transformation. Their non-interacting and localized behavior is achieved by a cancellation of the term ι ∆q αν appearing in the limiting bare-momentum expression on the right-hand side of Eq. (124) by the momentum functional Q αν (ι q αν )/L of Eq. (126). Moreover, for other bare momentum values q such that q = 0 and q = ±q αν the αν pseudoparticle is different both from the αν pseudofermion and rotated αν pseudoparticle.
We thus conclude that as q → 0 a αν pseudoparticle involved in deviations associated with zero-momentum excited states becomes free, non-interacting, delocalized, and the same quantum object as the corresponding αν pseudofermion. As the limiting values are approached and thus q → ±q αν , the αν pseudoparticle becomes non-interacting, localized, and the same quantum object as the corresponding rotated αν pseudoparticle. In this latter case, the energy of this composite object becomes additive in the energy of its ν −1/2 holons (α = c) or ν −1/2 spinons (α = s). Note that one of the effects of increasing the bare-momentum absolute value from | q| = 0 to | q| = q αν is to enhance the localization character of the αν pseudoparticles. If a pseudoparticle has bare-momentum q and q + ∆q for an initial ground state and an excited state, respectively, then the limiting free, non-interacting, and delocalized character and non-interacting and localized character are achieved when Q αν (q)/L = 0 and Q αν (q)/L = −∆q, respectively, where Q αν (q)/L is the momentum functional (73). For other values of that momentum functional the pseudoparticle is interacting. The pseudoparticles are never invariant under both the pseudoparticle -pseudofermion and pseudoparticle -rotated-pseudoparticle unitary transformations. This is because invariance under the pseudoparticlepseudofermion unitary transformation is associated with a free, non-interacting, and delocalized character, whereas invariance under the pseudoparticle -rotated-pseudoparticle unitary transformation corresponds to a non-interacting and localized character. Therefore the impossibility of simultaneous invariance under these two transformations is related to Heisenberg's uncertainty relation.
Another interesting example of invariance under the pseudoparticle -pseudofermion transformation (85) concerns the bare-momentum difference (q − q ′ ) when q and q ′ differ by a finite multiple of 2π/L. This case is of physical importance because (q − q ′ ) corresponds to the momentum of a pseudoparticle -pseudoparticle hole excitation in the c0 or s1 band. This type of process is associated with the excited states B and plays a key role in the few-electron spectral-weight distribution [34]. In this case, we find in Appendix C that in the thermodynamic limit, (q − q ′ ) is indeed invariant under the pseudoparticle -pseudofermion transformation and thus,

VIII. CONCLUDING REMARKS
In this paper we introduced a pseudofermion operational description for the 1D Hubbard model. We found that in the thermodynamic limit the wave function of excited states belonging to the Hilbert subspace of the groundstate normal-ordered 1D Hubbard model associated with few-electron excitations factorizes for all values of U/t. This factorization results from the non-interacting character of the pseudofermions whose occupancy configurations describe these excited states. While the pseudoparticle description studied in Refs. [30,31,35] corresponds to the whole Hilbert space, the pseudofermions are defined in the Hilbert subspace associated with few-electron excitations. In such a subspace, the pseudofermions are related to the pseudoparticles by a unitary transformation. We classified the statistics of the latter quantum objects according to a generalized Pauli principle [36].
Our study included the introduction of the pseudoparticle -pseudofermion unitary transformation and of an operator algebra for both the pseudoparticles and pseudofermions. Such a transformation removes the residual interactions of the αν pseudoparticles, which are mapped onto the non-interacting αν pseudofermions. The αν pseudoparticle residual interactions are cancelled by the momentum functional Q α, ν (q)/L of Eq. (73). The information recorded in the pseudoparticle interactions is transferred over to the momentum two-pseudofermion phase shifts of that functional. These phase shifts control the few-electron spectral properties through the same unconventional momentum functional.
We introduced creation and annihilation operators for both the pseudoparticles and pseudofermions and derived the anticommutation relations of these pseudofermion operators. We also studied and discussed the transformation laws of the pseudoparticles under the pseudoparticle -pseudofermion transformation. This study included the discussion of the physics behind both these transformation laws and the invariance under the pseudoparticle -pseudofermion and pseudoparticle -rotated-pseudoparticle unitary transformations of the pseudoparticles for some specific baremomentum values. Invariance under the former (and the latter) transformation is associated with a free, noninteracting, and delocalized character (and non-interacting and localized character) for these quantum objects. Thus as a result of Heisenberg's uncertainty relation, these two invariances never occur simultaneously. We also find that under the pseudoparticle -pseudofermion unitary transformation, the c0 and s1 pseudoparticles of bare momentum at the Fermi points are mapped onto corresponding pseudofermions whose momentum Fermi points expressions are related to the conformal dimensions of the two-component conformal-field theory primary fields [20,21]. It follows that in the limit of low energy the conformal invariance of the 1D Hubbard model is related to the general pseudoparticle -pseudofermion unitary transformation.
The pseudofermion algebra introduced in this paper is used elsewhere in the evaluation of finite-energy few-electron spectral-function expressions [10,34]. Fortunately, as a consequence of the above wave function factorization, the few-electron spectral functions can be written as a convolution of pseudofermion, −1/2 Yang holon, and/or −1/2 HL spinon spectral functions for all values of energy and on-site repulsion U [34].

APPENDIX A: PSEUDOPARTICLE REPRESENTATION, EFFECTIVE PSEUDOPARTICLE LATTICE, AND GROUND-STATE QUANTITIES
In this Appendix, we summarize some aspects of the pseudoparticle, description which are needed for the studies of this paper. This includes a brief summary of the basic pseudoparticle properties, introduction to the pseudoparticle bare momentum [31], local pseudoparticle concept, and effective lattices [35]. In addition, we provide the ground-state pseudoparticle bare-momentum distribution functions, as well as other ground-state quantities.
According to the results of Ref. [35], the αν pseudoparticle bare-momentum description obtained naturally from the Bethe-ansatz solution in Refs. [22,23,29,31] is related by Fourier transform to a local αν pseudoparticle description in terms of spatial coordinates of an effective αν lattice. These concepts are needed and useful for both the operational pseudoparticle and pseudofermion representations introduced in this paper.
The αν pseudoparticles carry bare momentum q. This is the continuum bare momentum associated with the discrete bare-momentum values q j such that, These discrete values read [31], where the I αν j numbers are integers or half-odd integers [30,31]. The index number j can have the values j = 1, 2, ..., N * αν . Here the number N * αν equals the number of discrete bare-momentum values in the αν pseudoparticle band and is such that, where N αν is the number of αν pseudoparticles and N h αν is the number of αν pseudoparticle holes. The latter number is given by, where S c and S s are the values of η-spin and spin, respectively, which read Combination of Eqs. (A4) and (A5) leads to, For αν = c0 (and αν = c0) the numbers I αν j on the right-hand side of Eq. (A2) are integers (half-odd integers), if N * αν (and N a /2 − α=c,s ∞ ν=1 N αν ) is odd (even). This non-perturbative shake-up effect is related to the so called orthogonality catastrophe [41]. It reveals that the values of the available bare-momentum values q j of each αν pseudoparticle band might be different for different CPHS ensemble subspaces.
The holons, spinons, and pseudoparticles are related to the electrons through the rotated electrons, as discussed in Sec. III and in Ref. [31,35]. The unitary transformation which maps electrons onto rotated electrons is defined in Sec. II. Since the momentum operator is invariant under such a transformation, also the lattice occupied by rotated electrons has the same lattice constant a and length L as the original electronic lattice. In Ref. [35] it was shown that there is an effective αν pseudoparticle lattice for each αν pseudoparticle branch.
In contrast to the case of a non-interacting band, the value of the number of available discrete bare-momentum values N * αν defined in Eqs. (A3), (A4), and (A6) might be different for different CPHS ensemble subspaces. This value was identified in Ref. [35] with the number of sites of the corresponding effective αν pseudoparticle lattice. This lattice has the same length L = N * αν a αν as the original real-space lattice, where is the effective αν lattice constant. Since N * αν is different for different CPHS ensemble subspaces, it follows from Eq. (A7) that the value of the corresponding effective αν lattice constant a αν also changes. The spatial coordinates of the effective αν pseudoparticle lattice are x j = a αν j where j = 1, 2, 3, ..., N * αν . The value of N * αν defined in Eqs. (A3)-(A6) determines the corresponding limiting bare-momentum values of the αν pseudoparticle Brillouin zone.
In the case of the c0 pseudoparticle band, the discrete bare-momentum values q j belong to the following range, Here, forÑ c0 even and forÑ c0 odd. Note that since N * c0 = N a , the lattice constant of the effective c0 lattice equals the lattice constant a of the original electronic lattice, that is a c0 = a. Thus, for the c0 band the lattice constant a c0 has the same value for all CPHS ensemble subspaces.
For the αν pseudoparticle bands such that ν > 0, the discrete bare-momentum values q j are distributed symmetrically relative to zero, and are such that |q j | ≤ q αν . The two band momenta ±q αν are the limiting bare-momentum values associated with the limits of the αν pseudoparticle Brillouin zone. The limiting bare-momentum q αν reads, where the lattice constant a αν ≥ a is given in Eq. (A7). We close this Appendix by providing some useful ground-state quantities. Following the results of Refs. [29,30], the ground-state pseudoparticle bare-momentum distribution functions and −1/2 Yang holon and −1/2 HL spinon numbers read, Except for 1/L corrections, the Fermi values on the right-hand side of Eq. (A12) are given by q ± F c0 = ±q 0 F c0 , where the ground-state values q 0 F s1 and q 0 F c0 are given in Eq. (68). Here we have ignored terms of order of 1/L and used the expressions k F σ = πn σ and 2k F = πn. The expressions of the Fermi values including 1/L contributions are provided in Appendix C of Ref. [31].
For the ground-state CPHS ensemble subspace the expression of the number N * αν given in Eqs. (A3), (A4), and (A6) simplifies. Let us denote by N 0, * αν the ground-state value of the numbers N * αν . These numbers read, whereas N 0, * c0 = N * c0 is given by N * c0 = N a for the whole Hilbert space. In addition to the −1/2 Yang holon and −1/2 HL spinon numbers already given in Eq. (A14), the ground state belongs to a CPHS ensemble subspace with the following values for the pseudoparticle, ±1/2 holon, and ±1/2 spinon numbers, in the charge sector and in the spin sector. Finally, note that for ν > 0 the number (A3) can be expressed in terms of the ground-state values (A15) as follows, Moreover, for αν = s1 the pseudofermion hole number defined in Eqs. (A4) and (A6) can be written as, Here we provide the set of integral equations which define the two-pseudofermion phase shiftsΦ αν, α ′ ν ′ (r, r ′ ) on the right-hand side of Eqs. (50)-(51), (74), and (84).

APPENDIX C: TRANSFORMATION LAWS UNDER THE PSEUDOPARTICLE -PSEUDOFERMION SUBSPACE UNITARY ROTATION
Here we study the momentum functional Q αν (q)/L given in Eq. (73) for specific values of q. In some cases we consider general excited states obeying relations (76), whereas in other cases we consider particular cases of such general excited states.
We start by confirming that the second term on the right-hand side of Eq. (79) is of [1/L] 2 order. Let us consider a more general situation and also confirm the validity of Eq. (144) and show that (q − q ′ ) is invariant under the pseudoparticle -pseudofermion transformation when q = q ′ + N ph [2π/L] and N ph = ±1, ±2, ... is a finite integer number. Equation (79) corresponds to the particular case when N ph = 1. Thus we want to show that the quantity [Q αν (q + N ph 2π L ) − Q αν (q)]/L involving the functional (73) is of (1/L) 2 order. By expressing this quantity in terms of the derivative ∂Q αν (q)/∂q one finds, Analysis of the form of the derivative ∂Q αν (q)/∂q reveals that it is of [1/L] 0 order. This is the confirmation that the momentum contribution (C1) is of second order in 1/L. Thus, since within the present large-L pseudofermion description [1/L] 2 bare-momentum and momentum contributions vanish, (q − q ′ ) is indeed invariant under the pseudoparticle -pseudofermion transformation. The remaining of this Appendix is complementary to the studies of Sec. VI about the transformation laws of the pseudoparticle -pseudofermion subspace unitary rotation. Next we consider that q = ±q αν for ν > 0 pseudoparticle branches. Our goal is the study of the momentum functional, and confirm that it is given by expression (126) where the deviation ∆q cν is provided in Eq. (125). It is assumed that the bare momentum q ′ belongs to the domain q ′ ∈ (−q α ′ ν ′ , +q α ′ ν ′ ) and can be such that q ′ → ±q α ′ ν ′ but q ′ = q α ′ ν ′ . First we note that according to Eq. (42), the two-pseudofermion phase shift Φ αν, α ′ ν ′ (±q αν , q ′ ) expressed in terms of the pseudoparticle bare momentum on the right-hand side of Eq. (C2) is such that, Φ αν, α ′ ν ′ (±q αν , q ′ ) =Φ αν, α ′ ν ′ ±∞, where the two-pseudofermion phase shiftΦ αν, α ′ ν ′ (r, r ′ ) is defined in Appendix B. In order to achieve this result we used that Λ 0 αν (±q αν ) = ±∞, as given in Eq. (42). By manipulation of the integral equations given in Appendix B, we find that all the two-pseudofermion phase shifts of form (C3) vanish except the following ones, Use of Eq. (C4) on the right-hand side of Eq. (C2) leads to, where we also used expression (125). Finally, note that expression (C5) is equivalent to Eq. (126). Let us now consider the quantity, and confirm that it obeys relation (128). This is easily confirmed by noting that according to Eq. (42) the twopseudofermion phase shifts Φ c0, α ′ ν ′ (q ± c0 , q ′ ) on the right-hand side of Eq. (C2) are such that, where the two-pseudofermion phase shiftsΦ c0, α ′ ν ′ (r, r ′ ) are defined in Appendix B. The result (C7) implies the validity of Eq. (128). Our next task involves evaluation of the momentum functional Q αν (q)/L given in Eq. (73) for q = ι q F αν in the particular case when the excited states associated with the deviations ∆N α ′ ν ′ (q ′ ) on the right-hand side of that equation are J-CPHS ground states. The Fermi bare momentum values that limit the compact pseudoparticle baremomentum occupancy configurations of these states are given in Eqs. (70) and (71). While the above expressions derived in this Appendix refer to general excited states whose deviations obey relations (76), the following results are valid only for the particular case when these excited states are J-CPHS ground-states. Our goal is to arrive to expressions (130)-(132). This implies evaluation of the following quantity, where and the bare-momentum distribution function deviations ∆N α ′ ν ′ (q ′ ) correspond to final J-CPHS ground states. The phase shifts Φ αν, α ′ ν ′ (q, q ′ ) andΦ αν, α ′ ν ′ (r, r ′ ) have the following property, Φ αν, α ′ ν ′ (q, q ′ ) = −Φ αν, α ′ ν ′ (−q, −q ′ )Φ αν, α ′ ν ′ (r, r ′ ) = −Φ αν, α ′ ν ′ (−r, −r ′ ) .
This symmetry is found by analysis of the integral equations given in Appendix B, which define the two-pseudofermion phase shiftsΦ αν, α ′ ν ′ (r, r ′ ) associated with the phase shifts Φ αν, α ′ ν ′ (q, q ′ ) through Eq. (74). Once the ground-state rapidity functions that appear on the right-hand side of Eq. (74) in the argument of the two-pseudofermion phase shiftsΦ αν, α ′ ν ′ (r, r ′ ) are odd functions of the bare momentum, the second relation given in Eq. (C10) implies the validity of the first relation of the same equation.
Since the pseudoparticle bare-momentum distribution function deviations of the J-CPHS ground-states include creation or annihilation of pseudoparticles in the vicinity of the Fermi points only, we can replace q ′ by sgn(q ′ ) q 0 F α ′ ν ′ in the argument of the phase shift Φ αν, α ′ ν ′ (ι q 0 F αν , q ′ ) on the right-hand side of Eq. (C9). This leads to, By performing the q ′ integrations and using Eq. (C10) we arrive to, After performing the ι ′′ summation and using again Eq. (C10) we find, Note that since ∆q F αν, ι = ι 2π L ∆N f αν, ι this result is equivalent to Eqs. (130)-(132). Finally, let us confirm the validity of Eq. (139) and show that the momentum functional Q αν (0)/L given in Eq. (73) vanishes for αν pseudoparticle branches such that α, ν = c0 and α, ν = s1 provided that the deviations ∆N c0 (q) and ∆N s1 (q) on the right-hand side of that equation obey Eq. (140). In this case, the excited states are partial J-CPHS ground states. These are states that have J-CPHS ground-state occupancy configurations for αν pseudoparticles such that α, ν = c0 and α, ν = s1.
There are no αν pseudoparticles belonging to branches other than c0 and s1 in the initial ground state. It follows that excited states obeying relations (76) and having J-CPHS ground-state occupancy configurations for αν pseudoparticles such that α, ν = c0 and α, ν = s1 have only a vanishing density of these quantum objects. Such J-CPHS ground state occupancy configurations correspond to compact bare-momentum domains of vanishing width centered at q = q 0 F αν = 0. The generation from the ground state of these excited states includes creation of αν pseudoparticles such that α, ν = c0 and α, ν = s1 for bare momentum q → 0. A simple and useful example, is the creation of a single αν pseudoparticle at bare momentum q = q 0 F αν = 0. To start with, we assume that the deviations ∆N c0 (q) and ∆N s1 (q) are small but have arbitrary values, whereas the deviations ∆N αν (q) of the remaining excited-state occupied αν pseudoparticle branches correspond to J-CPHS ground state occupancy configurations associated with Fermi points of the form given in Eq. (141).
Let us next consider the same type of excited states but with small deviations ∆N c0 (q) and ∆N s1 (q) obeying Eq. (140). These excited states have vanishing momentum and their c0 and s1 current number deviations ∆J c0 and ∆J s1 , respectively, vanish, i.e. ∆J c0 = ∆J s1 = 0. In this case it follows from Eqs. (C14)-(C15) that, This result confirms the validity of Eq. (139).