Bethe strings in the dynamical structure factor of the spin-1/2 Heisenberg XXX chain
BBethe strings in the dynamical structure factor of the spin- / Heisenberg
XXX chain
José M. P. Carmelo,
1, 2, 3
Tilen Čadež, and Pedro D. Sacramento Center of Physics of University of Minho and University of Porto, P-4169-007 Oporto, Portugal Department of Physics, University of Minho, Campus Gualtar, P-4710-057 Braga, Portugal Boston University, Department of Physics, 590 Commonwealth Ave, Boston, MA 02215, USA Center for Theoretical Physics of Complex Systems,Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, P-1049-001 Lisboa, Portugal (Dated: 11 August 2020)Recently there has been a renewed interest in the spectra and role in dynamical properties ofexcited states of the spin- / Heisenberg antiferromagnetic chain in longitudinal magnetic fieldsassociated with Bethe strings. The latter are bound states of elementary magnetic excitationsdescribed by Bethe-ansatz complex non-real rapidities. Previous studies on this problem referredto finite-size systems. Here we consider the thermodynamic limit and study it for the isotropicspin- / Heisenberg
XXX chain in a longitudinal magnetic field. We confirm that also in that limitthe most significant spectral weight contribution from Bethe strings leads to ( k, ω ) -plane gappedcontinua in the spectra of the spin dynamical structure factors S + − ( k, ω ) and S xx ( k, ω ) = S yy ( k, ω ) .The contribution of Bethe strings to S zz ( k, ω ) is found to be small at low spin densities m and tobecome negligible upon increasing that density above m ≈ . . For S − + ( k, ω ) , that contributionis found to be negligible at finite magnetic field. We derive analytical expressions for the line shapesof S + − ( k, ω ) , S xx ( k, ω ) = S yy ( k, ω ) , and S zz ( k, ω ) valid in the ( k, ω ) -plane vicinity of singularitieslocated at and just above the gapped lower thresholds of the Bethe-string states’s spectra. As aside result and in order to provide an overall physical picture that includes the relative ( k, ω ) -planelocation of all spectra with a significant amount of spectral weight, we revisit the general problem ofthe line-shape of the transverse and longitudinal spin dynamical structure factors at finite magneticfield and excitation energies in the ( k, ω ) -plane vicinity of other singularities. This includes thoselocated at and just above the lower thresholds of the spectra that stem from excited states describedby only real Bethe-ansatz rapidities. PACS numbers:
I. INTRODUCTION
Recently, there has been a renewed interest in bound states of elementary magnetic excitations named Bethestrings known long ago [1–5]. In spite of Bethe strings being a rather theoretical issue, as they were first foundand identified within the Bethe ansatz solution of spin and electronic integrable models in some classes of energyeigenstates described by complex non-real spin rapidities [1, 2], this renewed interest is actually due, in part, to theirexperimental identification and realization in spin-chain compounds whose magnetic properties are described by thespin- / Heisenberg chain in longitudinal magnetic fields [6–10]. This applies to that model isotropic point in thecase of experimental studies of some classes of such compounds [8–11].The present paper addresses only theoretical issues of that interesting physical problem. Most previous studies onthe spin dynamical properties of the spin- / XXX chain in a longitudinal magnetic field focused on the contributionfrom energy eigenstates described by real Bethe-ansatz rapidities, which are associated with most spectral weight of thespin dynamical structure factors. Several such studies considered finite-size systems and relied on different methods.This includes for instance numerical diagonalizations [12] and evaluation of matrix elements between Bethe-ansatzstates [13–15]. Previous studies that considered the thermodynamic limit [16], were also limited to the contributionto the spin dynamical structure factors from energy eigenstates described by real Bethe-ansatz rapidities. Concerningthe specific issue of the contribution of Bethe strings to the spin dynamical properties of spin- / XXX chain in alongitudinal magnetic field, the few previous studies considered finite-size systems [8, 9].In the case of that spin- / chain, Bethe strings [1], which here we call n -strings, have for n > and in thethermodynamic limit [2] been shown to be bound states of n = 2 , ..., ∞ singlet pairs of the model physical spins / [17–19]. (The physical meaning of the form of the spin- / XXX chain’s n -strings in that limit is an issue shortlyfurther discussed below in Sec. III A.) Energy eigenstates described by only real Bethe-ansatz rapidities lack suchbound pairs and are populated by unbound singlet pairs of such physical spins [17–19]. On the other hand, thereare predictions according to which for the large spin- S Heisenberg
XXX chain in longitudinal magnetic fields, Bethestrings could rather be bound states of spin- magnons [20, 21].In this paper we address the problem of the contribution of n -strings to the spin dynamical properties of spin- / XXX chain in a longitudinal magnetic field, in the thermodynamic limit. Based on a relation between the level a r X i v : . [ c ond - m a t . o t h e r] S e p of negativity of the momentum dependent exponents that control the ( k, ω ) -plane line shape of the spin dynamicalstructure factors near singularities and the amount of spectral weight existing in their vicinity, respectively, we confirmthat in the thermodynamic limit, as in the case of finite-size systems [8, 9], the only contribution from excited energyeigenstates populated by n -strings that leads to a ( k, ω ) -plane gapped continuum in the spectrum of the spin dynamicalstructure factors refers to S + − ( k, ω ) and thus also to S xx ( k, ω ) = S yy ( k, ω ) . On the other hand, the contributionfrom n -strings states to S zz ( k, ω ) is found to be small at low spin densities and to become negligible upon increasingit beyond a spin density, ˜ m ≈ . . For the spin dynamical structure factor S − + ( k, ω ) , that contribution is found tobe negligible at any finite magnetic field.The main goal of this paper is thus the study of the line shape of the spin dynamical structure factors S + − ( k, ω ) , S xx ( k, ω ) , and S zz ( k, ω ) at and just above singularities located at the ( k, ω ) -plane gapped lower thresholds of thespectra associated with n -string states. To reach that goal, we extend the dynamical theory of Ref. [16] to a largersubspace, which allows to account for the contribution from the latter states to the spin dynamical structure factors.We then derive analytical expressions valid in the thermodynamic limit for line shape of these factors in the ( k, ω ) -planevicinity of the singularities under consideration.Complementarily and as a side result, in order to to provide an overall physical picture that includes the relative ( k, ω ) -plane location of all features with a significant amount of spectral weight, we account for the contributionsfrom all types of states that lead to gapped and gapless lower threshold singularities in the spin dynamical structurefactors. This includes both excited states with and without n -strings. (As mentioned above, the contribution fromthe latter states, shortly revisited in this paper, is known to lead to the largest amount of spin dynamical structurefactors’s spectral weight [12–16].)The paper is organized as follows. The model and the spin dynamical structure factors are the subjects of Sec. II.In Sec. III the spectral functionals that control the extended dynamical theory’s general expressions of the dynamicalstructure factors are introduced. Such factors’s spectra are studied in Sec. IV. The line shape near their singularitiesis the issue addressed in Sec. V. The subject of Sec. VI is the limiting behaviors of the spin dynamical structurefactors. Finally, the discussion and concluding remarks are presented in Sec. VII. Two Appendices provide useful sideinformation needed for the studies of this paper. II. THE MODEL AND THE SPIN DYNAMICAL STRUCTURE FACTORS
The spin- / Heisenberg
XXX chain with exchange integral J and length L → ∞ in a longitudinal magnetic field h for spin densities m ∈ ]0 , , which describes N = (cid:80) σ = ↑ , ↓ N σ physical spins / of projection σ , is a paradigmaticexample of an integrable strongly correlated system [1, 2]. Its Hamiltonian is given by, ˆ H = J L (cid:88) j =1 (cid:88) a = x,y,z ˆ S aj ˆ S aj +1 + 2 µ B h ˆ S z . (1)For simplicity, we have taken here g = 2 , ˆ (cid:126)S j is the spin- / operator at site j = 1 , ..., N with components ˆ S x,y,zj , µ B is the Bohr magneton, and ˆ S z = (cid:80) Nj =1 ˆ S zj is the diagonal generator of the global spin SU (2) symmetry algebra. Wedenote the energy eigenstate’s spin projection by S z = − ( N ↑ − N ↓ ) / ∈ [ − S, S ] where S ∈ [0 , N/ is their spin. Unitsof lattice spacing and Planck constant one are used in this paper.Due to the rotational symmetry in spin space, off-diagonal components of the spin dynamical structure factorsvanish, S aa (cid:48) ( k, ω ) = 0 for a (cid:54) = a (cid:48) where a and a (cid:48) are given by x, y, z . In addition, the two transverse compo-nents are identical, S xx ( k, ω ) = S yy ( k, ω ) . In the present case of finite magnetic fields < h < h c , one has that S zz ( k, ω ) (cid:54) = S xx ( k, ω ) . Here h c = J/µ B is the magnetic field above which there is fully polarized ferromagnetism. Thecorresponding magnetic energy scale, µ B h c = 2 J , is associated with the quantum phase transition to fully polarizedferromagnetism. In the opposite limit of zero magnetic field, one has that S zz ( k, ω ) = S xx ( k, ω ) .The dynamical structure factors S aa ( k, ω ) are given by, S aa ( k, ω ) = N (cid:88) j =1 e − ikj (cid:90) ∞−∞ dt e − iωt (cid:104) GS | ˆ S aaj ( t ) ˆ S aj (0) | GS (cid:105) = (cid:88) ν |(cid:104) ν | ˆ S ak | GS (cid:105)| δ ( ω − ω τν ( k )) for a = x, y, z . (2)Here the spectra read ω aaf ( k ) = ( E aaν − E GS ) , E aaν refers to the energies of the excited energy eigenstates thatcontribute to the aa = xx, yy, zz dynamical structure factors, (cid:80) ν is the sum over such states, E GS is the initialground state energy, and ˆ S ak are for a = x, y, z the Fourier transforms of the usual local a = x, y, z spin operators ˆ S aj ,respectively.The spin dynamical structure factor S xx ( k, ω ) can be expressed as, S xx ( k, ω ) = 14 (cid:0) S + − ( k, ω ) + S − + ( k, ω ) (cid:1) . (3)One can then address the dynamical properties of S xx ( k, ω ) in terms of those of S + − ( k, ω ) and S − + ( k, ω ) .Since S aa ( k, ω ) = S aa ( − k, ω ) for a = x, y, z and thus also S + − ( k, ω ) = S + − ( − k, ω ) and S − + ( k, ω ) = S − + ( − k, ω ) ,in this paper we consider excitation momentum values k > in the first Brillouin zone, k ∈ [0 , π ] . Another usefulsymmetry relating the spin density intervals m ∈ ] − , and m ∈ ]0 , is such that, S − + ( k, ω ) | m = S + − ( k, ω ) | − m and S + − ( k, ω ) | m = S − + ( k, ω ) | − m for m ∈ ]0 , . (4)Hence, as mentioned above, we only consider explicitly the spin density interval m = 2 S z /N ∈ ]0 , . The subspacedefined below in Sec. III A of the quantum problem studied in this paper is spanned by some classes of energyeigenstates with spin S ∈ ]0 , N/ and magnetic fields < h < h c for which the spin density belongs to the interval m ∈ ]0 , . ( N is even and odd when the states spin S is an integer and half-odd integer number, respectively. In thelatter case, the minimal spin value is / , rather than .)Some useful selection rules tell us which classes of energy eigenstates have nonzero matrix elements with the groundstate. Let | S, α (cid:105) , | S z , β (cid:105) , and | S, S z , γ (cid:105) denote energy eigenstates where S ∈ [0 , N/ is their spin, S z their spinprojection, and α , β and γ represent all other quantum numbers needed to uniquely specify these states, respectively.The selection rules given in the following are derived from the properties of the operators ˆ S zk and ˆ S ± k by straightforwardmanipulations involving their operator algebra [15].At vanishing magnetic field, h = 0 , the following selection rules hold in the thermodynamic limit, (cid:104) S, α | ˆ S ak | S (cid:48) α (cid:48) (cid:105) = 0 for S = S (cid:48) = 0 and a = z, ±(cid:104) S, α | ˆ S ak | S (cid:48) α (cid:48) (cid:105) = 0 for | S − S (cid:48) | (cid:54) = 0 , a = z, ±(cid:104) S z , β | ˆ S ± k | S z (cid:48) , β (cid:48) (cid:105) = 0 for S z (cid:48) (cid:54) = S z ± (cid:104) S z , β | ˆ S zk | S z (cid:48) , β (cid:48) (cid:105) = 0 for S z (cid:48) (cid:54) = S z . (5)On the other hand, for finite magnetic fields < h < h c of most interest for our study, the following selection rulesare valid in that limit, (cid:104) S, S, γ | ˆ S ± k | S (cid:48) , S z (cid:48) , γ (cid:48) (cid:105) = 0 for S (cid:48) (cid:54) = S ± S z (cid:48) (cid:54) = S ± (cid:104) S, S, γ | ˆ S zk | S (cid:48) , S z (cid:48) , γ (cid:48) (cid:105) = 0 for S (cid:48) (cid:54) = S and S z (cid:48) (cid:54) = S . (6)The spin dynamical structure factors satisfy the following sum rules, π (cid:90) π − π dk (cid:90) ∞ dω S + − ( k, ω ) = (1 + m )12 π (cid:90) π − π dk (cid:90) ∞ dω S − + ( k, ω ) = (1 − m )12 π (cid:90) π − π dk (cid:90) ∞ dω S zz ( k, ω ) = 12 (1 − m ) . (7)The selection rules in Eq. (5) reveal that at h = 0 and thus m = 0 when S xx ( k, ω ) = S yy ( k, ω ) = S zz ( k, ω ) , thelongitudinal dynamical structure factor S zz ( k, ω ) is fully controlled by transitions from the ground state for which S z = S = 0 to excited states with spin numbers S z = 0 and S = 1 . That according to such rules the transversedynamical structure factors are at h = 0 controlled by transitions from that ground state to excited states withspin numbers S z = ± and S = 1 , does not prevent the equality S zz ( k, ω ) = S xx ( k, ω ) imposed by the spin SU (2) symmetry.This is different from the case for magnetic fields < h < h c considered in this paper. According to the selectionrules, Eq. (6), the factor S zz ( k, ω ) (cid:54) = S xx ( k, ω ) is then controlled by transitions from the ground state with spinnumbers S z = − S to excited states with the same spin numbers S z = − S . According to the same selection rules, thedynamical structure factors S + − ( k, ω ) and S − + ( k, ω ) are controlled by transitions from the ground state with spinnumbers S z = − S to excited states with spin numbers S z ± − S ± . III. THE SPECTRAL FUNCTIONALS OF THE EXTENDED DYNAMICAL THEORY
As reported in Sec. I, the general goal of this paper is the study of the contribution from n -string states to thespin dynamical structure factors given in Eq. (2) within the spin- / XXX chain in a longitudinal magnetic field,Eq. (1). The dynamical theory used in our studies refers to an extension of that introduced for the present modelin Ref. [16]. In that reference, only the contribution to the spin dynamical structure factors from energy eigenstatesdescribed by real Bethe-ansatz rapidities was considered.The theory of that reference is directly related to that introduced for the one-dimensional Hubbard model in Ref.[22]. The related dynamical theories of Refs. [16, 22, 23] are equivalent to and account for the same microscopicprocesses [19] as the mobile quantum impurity model scheme of Refs. [24, 25] in the case of integrable models.The main difference of such an extended theory to that considered in Ref. [16], refers to the Hamiltonian, Eq.(1), acting onto an extended subspace, including n -string states. This involves different new forms for the spectralfunctionals that control the momentum dependent exponents in the spin dynamical structure factors’s expressionsobtained in this paper for ( k, ω ) -plane regions near specific types of spectral features. For simplicity, we do not providehere the details of the extended dynamical theory that are common to those already given in Ref. [16], and ratherfocus on the differences associated with its extension to the contribution from n -string states.The use of the extended dynamical theory provides useful information on the ( k, ω ) -plane distribution of the excitedenergy eigenstates’s spectra that contain in the thermodynamic limit most spectral weight of S + − ( k, ω ) , S xx ( k, ω ) ,and S zz ( k, ω ) . Such spectra are schematically represented in Figs. 1, 2, and 3, respectively.After introducing the quantum problem’s extended subspace, the general expressions of the spectral functionalsunder consideration are introduced in the following. Specific expressions of the needed spectral functionals suitableto the line-shape near the four types of spectral features considered in our study are obtained. Finally, the issueconcerning the k intervals where the corresponding momentum dependent exponents are valid is also addressed. A. The present quantum problem extended subspace
The quantum problem considered in this paper refers to the Hamiltonian, Eq. (1), in a subspace spanned by twoclasses of energy eigenstates, populated and not populated by n -strings, respectively. Our corresponding study ofthe spin dynamical structure factors relies on the representation of such energy eigenstates in terms of n -particleoccupancy configurations, which is that suitable to the dynamical theory used in this paper. Here n = 1 , ..., ∞ is thenumber of singlet pairs of physical spins / that refer to their internal degrees of freedom. The studies of Ref. [16]only involved n = 1 particles that in such a reference were named “pseudoparticles”.In the thermodynamic limit, the Bethe-ansatz rapidities have the general form given in Eq. (B3) of Appendix B[2]. For n = 1 such Bethe-ansatz rapidities are real and otherwise their imaginary part is finite. In that equation,the Bethe-ansatz rapidities are partitioned in a configuration of strings, where a n -string is a group of l = 1 , ..., n rapidities with the same real part Λ n ( q j ) . The number n is called in the literature the string length and the real partof the number n of rapidities, Λ n ( q j ) , is called the string center.For n > the n -particle internal degrees of freedom refer to a n -string, whereas the n -band momentum q j inthe argument of the real part of the set of l = 1 , ..., n rapidities, Λ n ( q j ) in Eq. (B3) of Appendix A, describes itstranslational degrees of freedom. Each n -string contains a number n = 2 , ..., ∞ of bound singlet pairs of physical spins / , whose number thus equals the length of the n -string [17–19]. The l = 1 , ..., n imaginary parts, i ( n + 1 − l ) , of theset of n rapidities of a n -string describe the binding of the l = 1 , ..., n pairs of physical spins / . Consistently, thatimaginary part vanishes at n = 1 , the internal degrees of freedom of the -particles corresponding to a single unboundsinglet pair of physical spins [17–19]. Their translational degrees of freedom refer again the -band momentum q j .Energy eigenstates that are not populated and are populated by n -particles with n > pairs, are described by onlyreal Bethe-ansatz rapidities and both real and complex non-real such rapidities, respectively.The ground states with spin densities < m < and corresponding longitudinal magnetic fields < h < h c are notpopulated by n -strings. Concerning the amount of spectral weight of the spin dynamical structure factors originatedfrom transitions from such ground states to n -string states, transitions to excited energy eigenstates populated by asingle -particle are found to be dominant by far, as justified below. This is consistent with results for large finite-sizesystems [8, 9].The - and -particles carry -band and -band discrete momentum values q j , respectively, Eq. (B5) of AppendixB, whose spacing is q j +1 − q j = 2 π/L . Accounting for /L contributions, the ground state at a given spin density < m < and corresponding longitudinal magnetic field < h < h c is populated by a number N = N ↓ of -particles . . . . . ω + − / J m = 0 . ( a ) ¯1 2 . . . . . m = 0 . ( c ) . . . . . k/π ω + − / J m = 0 . ( b ) . . . . . k/π m = 0 . ( d ) Figure 1: The two ( k, ω ) -plane lower and upper continuum regions where for spin densities (a) m = 0 . , (b) m = 0 . ,(c) m = 0 . , and (d) m = 0 . there is in the thermodynamic limit more spectral weight in S + − ( k, ω ) . The sketch ofthe ( k, ω ) -plane distributions represented here and in Figs. 2 and 3 does not provide information on the relative amount ofspectral weight contained within each spectrum’s grey continuum. The three reference vertical lines mark the momenta (a)[ k = k F ↑ − k F ↓ = 3 π/ , k = k F ↓ = 17 π/ , k = 2 k F ↓ = 17 π/ ]; (b) [ k = k F ↑ − k F ↓ = π/ , k = k F ↓ = 3 π/ , k = 2 k F ↓ = 3 π/ ];(c) [ k = k F ↓ = π/ , k = k F ↑ − k F ↓ = 2 k F ↓ = π/ ]; (d) [ k = k F ↓ = π/ , k = 2 k F ↓ = π/ , k = k F ↑ − k F ↓ = 3 π/ ]. Thelower and upper continuum spectra are associated with excited energy eigenstates without and with n -strings, respectively.In the thermodynamic limit, the ( k, ω ) -plane region between the upper threshold of the lower continuum and the gappedlower threshold of the upper n -string continuum has nearly no spectral weight. In the case of the gapped lower thresholdof the n -string continuum, the analytical expressions given in this paper refer to near and just above that threshold whosesubintervals refer to branch lines parts represented in the figure by solid and dashed lines. The latter refer to k intervals wherethe momentum dependent exponents plotted in Fig. 4 are negative and positive, respectively. In the former intervals, S + − ( k, ω ) displays singularity peaks. that fill a -band Fermi sea as follows, q j ∈ [ q − F , q +1 F ] where q j +1 − q j = 2 π/L and q ιF = ιk F ↓ − ι πL for N even and ι = ± q ιF = ιk F ↓ − ( ι ± πL for N odd and ι = ± . (8)Here ι = +1 and ι = − refer to the -band right and left Fermi points, respectively, k F ↓ = π (1 − m ) (as given inEq. (B7) of Appendix B), and in the case of N odd, ± refers in ( ι ± πL to two alternative Fermi point’s values.In the present thermodynamic limit, we often use continuous momentum variables q that replace the discrete - and -bands momentum values q j such that q j +1 − q j = 2 π/L . We can then consider for the studies of some propertiesthat q ιF = ιk F ↓ and thus a ground-state -band occupied Fermi sea, q ∈ [ − k F ↓ , k F ↓ ] .As reported in Sec. I, there is a direct relation between the values of the momentum dependent exponents thatwithin the dynamical theory used here control the line shape in the ( k, ω ) -plane vicinity of the spin dynamical . . . . . ω xx / J m = 0 . ( a ) . . . . . m = 0 . ( c ) . . . . . k/π ω xx / J m = 0 . ( b ) ¯1 2 . . . . . k/π m = 0 . ( d ) Figure 2: The two ( k, ω ) -plane lower and upper continuum regions where for the same spin densities as in Fig. 1 there is inthe thermodynamic limit more spectral weight in S xx ( k, ω ) . The notations and the momenta associated with the referencevertical lines are the same as in Fig. 1. The additional part of the lower continuum relative to that of S + − ( k, ω ) in Fig. 1stems from the contributions from S − + ( k, ω ) . As a result, for some k intervals the upper n -string continuum overlaps with thelower continuum. structure factors spectral features and the amount of spectral weight located near them: Negative exponents impliesthe occurrence of singularities associated with a significant amount of spectral weight in their ( k, ω ) -plane vicinity.The use of this criterion, reveals that in the present thermodynamic limit and for magnetic fields < h < h c , theonly significant contribution to S + − ( k, ω ) from excited energy eigenstates populated by n -particles refers to thosepopulated by a number N ↓ − of -particles and a single -particle. There is as well a much weaker contribution atsmall spin densities from states populated by a number N ↓ − of -particles and a single -particle.The only significant yet weak contribution to S zz ( k, ω ) from n -string states, refers to energy eigenstates populatedby a number N ↓ − of -particles and a single -particle. On the other hand, the contribution from such excitedenergy eigenstates to S − + ( k, ω ) is found to be negligible, since all relevant exponents are both positive and large.The contribution to S + − ( k, ω ) from energy eigenstates populated by a number N ↓ − of -particles and a single particle that occurs for small values of the spin density is very weak. It is actually inexistent in the vicinity ofthe ( k, ω ) -plane singularities to which the analytical expressions obtained in our study refer to. Indeed, except forvery small spin densities, m → , the latter very weak contributions occur in ( k, ω ) -plane regions of higher excitationenergy ω , above the gapped lower threshold of the spectrum continuum associated with energy eigenstates populatedby a number N ↓ − of -particles and a single -particle whose expression is given below in Sec. III B. That spectrumrefers to the upper continuum shown in Fig. 1.The above spectral-weight analysis refers to the thermodynamic limit. Its results are fully consistent with corre-sponding results reached by a completely different method in the case of large finite-size systems [8].The subspace of the quantum problem studied in this paper is thus spanned by an initial ground state for a givenspin density < m < and a corresponding longitudinal magnetic field < h < h c and its following excited energyeigenstates: States described by both real and complex non-real Bethe-ansatz rapidities populated by a number . . . . . ω zz / J m = 0 . ( a ) ¯12 . . . . . m = 0 . ( c ) . . . . . k/π ω zz / J m = 0 . ( b ) . . . . . k/π m = 0 . ( d ) Figure 3: The ( k, ω ) -plane continuum region where for the same spin densities as in Figs. 1 and 2 there is in the thermodynamiclimit more spectral weight in S zz ( k, ω ) . The reference vertical lines mark the same momenta as in these figures. Contributionsfrom excited states containing n -strings are much smaller than for S + − ( k, ω ) and S xx ( k, ω ) and do not lead to an uppercontinuum. The gapped lower threshold of such states is though shown in the figures. Only when that threshold coincides withthe ¯1 (cid:48) -branch line, which only occurs for spin densities < m < ˜ m where ˜ m ≈ . , singularities occur near and just abovethe ¯1 (cid:48) -branch line, which is represented by a solid (green) line. In the remaining parts of the gapped lower threshold, whichfor spin densities ˜ m < m < means all of it, the momentum dependent exponents are positive and there are no singularities.This is equivalent to a negligible amount of spectral weight near such lines. N ↓ − of -particles and a single -particle whose internal degrees of freedom refer to a n -string of length n = 2 ;States populated by a number N ↓ of -particles that are described only by real Bethe-ansatz rapidities. B. General expressions of the extended dynamical theory’s spectral functionals
The following number and current number deviations under transitions from a ground state with -band momentumdistribution given in Eq. (8) to the excited energy eigenstates that span the present subspace play an important rolein the extended dynamical theory’s expressions, δN F ,ι for ι = 1 , − , left) 1 − particles δN F = (cid:88) ι = ± δN F ,ι and δJ F = 12 (cid:88) ι = ± ι δN F ,ι δJ = ι δN ( q ) | q = ι ( k F ↑ − k F ↓ ) where ι = ± . (9)Under some of the transitions from the ground state to the excited energy eigenstates of the present subspace,the number of -particles and/or that of -particles changes. This leads to number deviations δN and/or δN ,respectively. The specific number deviations δN F ,ι in Eq. (9) refer only to changes of the -particles numbers at theleft ( ι = − or right ( ι = 1) 1 -band Fermi points, Eq. (8). Exactly the same information is contained in the twoFermi points number deviations δN F ,ι , on the one hand, and in the corresponding Fermi points number deviations δN F = (cid:80) ι = ± δN F ,ι and current number deviations δJ F = (cid:80) ι = ± ι δN F ,ι , on the other hand. The overall -particlesnumber deviation δN can be expressed as, δN = δN F + δN NF . (10)Here δN NF refers to changes in the number of -particles at -band momenta other than those at the Fermi points,Eq. (8).For the current subspace, the -band number deviations may read δN = 0 or δN = 1 . The -band is emptyin the ground state. For that state its unoccupied momentum values q j such that q j +1 − q j = 2 π/L refer to therange q j ∈ [ − ( k F ↑ − k F ↓ − /L ) , ( k F ↑ − k F ↓ − /L )] where k F ↓ = π (1 − m ) and k F ↑ = π (1 + m ) (as given in Eq.(B7) of Appendix B.) The -band momentum range that is of interest for our studies rather refers to excited energyeigenstates populated by a single -particle. For these states, the available discrete momentum values belong to theinterval q j ∈ [ − ( k F ↑ − k F ↓ ) , ( k F ↑ − k F ↓ )] where q j +1 − q j = 2 π/L . The -particle can occupy any of such -banddiscrete momentum values whose number is N ↑ − N ↓ + 1 . Only when the -particle is created at one of the two -band’s limiting values, q = − ( k F ↑ − k F ↓ ) or q = ( k F ↑ − k F ↓ ) , that process leads to a current number deviation δJ = − / and δJ = 1 / , respectively, Eq. (9).Within the extended dynamical theory, the line shape at and just above the gapped lower thresholds of the n -stringstates of S + − ( k, ω ) , S xx ( k, ω ) , and S zz ( k, ω ) is for spin densities < m < and momenta in the range k ∈ ]0 , π [ ofthe following general form, S ab ( k, ω ) = C ∆ ab (cid:16) ω − ∆ ab ¯ n ( k )) (cid:17) ζ ab ¯ n ( k ) for ( ω − ∆ ab ¯ n ( k )) ≥ n = 2 , ¯1 , ¯1 (cid:48) , (cid:48) and ab = + − , xx, zz (valid when ∆ ab gap > . (11)Here C ∆ ab is a constant that has a fixed value for the k and ω ranges associated with small values of the energy deviation ( ω − ∆ ab ¯ n ( k )) ≥ and ∆ ab ¯ n ( k ) are the spectra that define the ( k, ω ) -plane shape of the gapped lower thresholds of the n -string states’s in Figs. 1, 2, and 3. The analytical expressions of such spectra are given below in Sec. III B and thegeneral expression of the exponents ζ ab ¯ n ( k ) also appearing in Eq. (11) is provided below. The indices ¯ n = 2 , ¯1 , ¯1 (cid:48) , (cid:48) insuch spectra and exponents label the branch lines or branch line sections that are part of the corresponding gappedlower thresholds in some specific k intervals defined below in Sec. IV A. Branch lines are types of spectral featuresthat are defined as within the dynamical theory of Ref. [16].The quantity ∆ ab gap in Eq. (11) is the gap between the upper thresholds of the lower continua associated with excitedstates described only by real Bethe-ansatz rapidities and the gapped lower thresholds of the n -string states’s spectradisplayed in Figs. 1, 2, and 3. Only for ab = xx there is overlap for small spin densities and some k intervals betweenthe lower continuum and the n -string states’s upper continuum, as shown in Fig. 2 for spin densities m = 0 . and m = 0 . . In the corresponding k intervals, one has that ∆ xx gap < and the general line-shape expression given in Eq.(11) does not apply.Indeed, that expression is valid provided there is no spectral weight or nearly no spectral weight below the gappedlower thresholds of the n -string states. In the present thermodynamic limit, the amount of spectral weight just belowsuch thresholds either vanishes or is extremely small. In the latter case, the very weak coupling to it leads to a higherorder contribution to the line shape expressions given in that equation that can be neglected in such a limit. Hence,the general expression of the spin dynamical structure factors given in Eq. (11) is an excellent approximation forsmall values the energy deviation ( ω − ∆ ab ¯ n ( k )) ≥ .On the other hand, the line shape of the spin dynamical structure factors S ab ( k, ω ) where ab = + − , − + , xx, zz atand just above their lower thresholds of the lower spectra that for ab = + − , xx, zz are shown in Figs. 1, 2, and 3 andare associated with excited energy eigenstates described only by real Bethe-ansatz rapidities has been derived withinthe dynamical theory of Ref. [16]. It has the following general form, similar to that given in Eq. (11), S ab ( k, ω ) = C ab (cid:16) ω − ω ablt ( k ) (cid:17) ζ ab ( k ) for ( ω − ω ablt ( k )) ≥ ab = + − , − + , xx, zz . (12)Again, here C ab are constants that have a fixed value for the k and ω intervals for which the energy deviation ( ω − ω ablt ( k )) ≥ is small. The lower thresholds under consideration refer to a single -branch line that except for S − + ( k, ω ) has two k interval sections. The ab = + − , − + , zz lower threshold’s spectra ω + − ( k ) , ω − + ( k ) , and ω zz ( k ) in that deviation are given in Eqs. (A10), (A11), and (A12) of Appendix A, respectively.There is no spectral weight below the lower thresholds associated with the line-shape expression, Eq. (12). Thegeneral expression of the spin dynamical structure factors given in that expression is thus exact for small values ofthe energy deviation ( ω − ω ablt ( k )) ≥ .The branch-line exponents that appear in both Eqs. (11) and (12) have the same following general form, ζ ab ¯ n ( k ) = − (cid:88) ι = ± (Φ ι ( q )) for ¯ n = 2 , ¯1 , ¯1 (cid:48) , (cid:48) , . (13)The differences relative to the dynamical theory of Ref. [16], refer to the form of the spectral functionals Φ ι ( q ) inthis general exponent’s expression that is suitable to each type of branch line. In the following, the forms of suchfunctionals specific to the four types of branch line involved in our study are introduced.Consistent with the occurrence of an infinite number of conservation laws associated with the present quantumproblem integrability, there is a representation of the n -particles for which they only undergo zero-momentum forward-scattering events. The corresponding phase shifts fully control the spectral and dynamical properties. The extendeddynamical theory uses such a n -particle representation. Within it, -particles carry discrete canonical momentum values ¯ q j = ¯ q ( q j ) defined below such that ¯ q j +1 − ¯ q j = 2 π/L + h . o . , rather than -band momentum values q j directlyrelated to Bethe-ansatz quantum numbers, Eq. (B5) of Appendix B. The higher order (1 /L ) terms in the spacing ¯ q j +1 − ¯ q j have no physical meaning in the present thermodynamic limit. The key property of the -particles canonicalmomentum representation is the lack of energy interaction terms. This is what ensures the occurrence of only zero-momentum forward-scattering events.The initial ground state is populated by a macroscopic number N = N ↓ of -particles and contains no n -particleswith n > . Therefore, -particles contribute to the dynamical properties both as scatterers and scattering centers,whereas the -particle contributes to them as a scattering center only. As justified below, in the case of scatteringcenters the same results are obtained in the thermodynamic limit within the representations for which such centerscreated under transitions to excited states carry momentum q j and canonical momentum ¯ q j = ¯ q ( q j ) , respectively.While canonical momentum can also be introduced for the -particle, for simplicity we thus remain using -bandmomentum q j for it.From straightforward yet lengthly manipulations of the Bethe ansatz equations, Eqs. (B1)-(B2) of Appendix B,one finds that for the excited energy eigenstates that span the present subspace, the -band rapidity functional Λ ( q j ) can be written in terms of the corresponding ground-state rapidity function Λ ( q j ) as follows, Λ ( q j ) = Λ (¯ q j ) for j = 1 , ..., N ↑ . (14)Here ¯ q j = ¯ q ( q j ) where j = 1 , ..., N ↑ are the following discrete canonical momentum values, ¯ q j = ¯ q ( q j ) = q j + 2 πL Φ ( q j ) = 2 πL (cid:0) I j + Φ ( q j ) (cid:1) , (15)and I j are the Bethe-ansatz -band quantum numbers given in Eq. (B6) of Appendix B. The lack of energy interactionsfollows from in terms of canonical momentum values the -band rapidity function having for the excited energyeigenstates the same form, Λ ( q j ) = Λ (¯ q j ) , as for the corresponding initial ground state. (For the ground state, -band momentum values and canonical momentum values are actually the same.)The general expression of the -band functional Φ ( q j ) in Eq. (15) is in the case of the present subspace given by, Φ ( q j ) = N ↑ (cid:88) j (cid:48) =1 Φ , ( q j , q j (cid:48) ) δN ( q j (cid:48) ) + N ↑ − N ↓ + N (cid:88) j (cid:48) =1 Φ , ( q j , q j (cid:48) ) δN ( q j (cid:48) ) , (16)where Φ , ( q j , q j (cid:48) ) and Φ , ( q j , q j (cid:48) ) are as further discussed below phase shifts in units of π . They are defined byEqs. (B29)-(B31) of Appendix B. The deviations δN ( q j (cid:48) ) and δN ( q j (cid:48) ) also appearing in Eq. (16) read, δN n ( q j ) = N n ( q j ) − N n ( q j ) for n = 1 , . (17)Here N n ( q j ) and N n ( q j ) are the -band and -band momentum distributions of the excited energy eigenstate andground state, respectively. Such momentum distributions appear in the functional representation of the Bethe-ansatzequations, Eqs. (B1)-(B2) of Appendix B. The ground-state distribution N ( q j ) is associated with the -band compactoccupancy, Eq. (8), whereas N ( q j ) = 0 for q j ∈ [ − ( k F ↑ − k F ↓ − /L ) , ( k F ↑ − k F ↓ − /L )] .In the Φ ( q j ) ’s general expression, Eq. (16), the momentum values q j and q j (cid:48) are associated with scatterers andscattering centers, respectively. As mentioned above, one could associate canonical momentum values both with the0scatterars and with the scattering centers created under the transitions from the ground state to the excited states.However, in the case of the scattering centers, the form of the expression on the right-hand side of Eq. (16) revealsthat this leads to contributions of order (1 /L ) that have no physical meaning in the thermodynamic limit. (Thevalidity of the corresponding dynamical theory refers to that limit.)For δN ( q j (cid:48) ) = ± , the quantity ± Φ , ( q j , q j (cid:48) ) = Φ , ( q j , q j (cid:48) ) δN ( q j (cid:48) ) in Eq. (16) is the scattering phase shiftin units of π acquired by a -particle of canonical momentum ¯ q j = ¯ q ( q j ) (scatterer) upon creation of one -hole( − Φ , ( q j , q j (cid:48) ) ) and one -particle ( +Φ , ( q j , q j (cid:48) ) ) at a momentum q j (cid:48) in the -band unoccupied and occupied Fermisea (scattering centers), respectively. (Given the one-to-one relation between the canonical momentum ¯ q j and themomentum q j , Eq. (15), scatterers of canonical momentum ¯ q j can also be labelled by momentum q j , which refers totwo representations of the same -particle.)On the other hand, the quantity Φ , ( q j , q j (cid:48) ) = Φ , ( q j , q j (cid:48) ) δN ( q j (cid:48) ) in Eq. (16) is for δN ( q j (cid:48) ) = 1 the scatteringphase shift in units of π acquired by a -particle of canonical momentum ¯ q j = ¯ q ( q j ) (scatterer) under creation of one -particle at a momentum q j (cid:48) in the -band interval q j (cid:48) ∈ [ − ( k F ↑ − k F ↓ ) , ( k F ↑ − k F ↓ )] (scattering center).Hence Φ ( q j ) is in Eq. (16) the overall phase shift in units of π acquired by a -particle of canonical momentum ¯ q j = ¯ q ( q j ) under a transition from the ground state to an excited energy eigenstate belonging to the present subspace.Important quantities for the dynamical properties, are the following ι = ± deviations δq ιF from the values of the -band ground-state Fermi momenta, Eq. (8), under transitions to excited energy eigenstates, δq ιF = ι πL δN F ,ι = 2 πL (cid:16) ι δN ,F ,ι + Φ (cid:17) . (18)Here δN ,F ,ι are the deviations in the number of -particles at the Fermi points which either vanish or are positive ornegative integer numbers. On the other hand, the actual number deviations, δN F ,ι = δN ,F ,ι + ι Φ , can as well behalf-odd integers. Their extra term, ι Φ where ι = ± , stems from contributions from a non-scattering phase shift Φ regulated by the boundary conditions in Eq. (B6) of Appendix B. It shifts all -band’s discrete momentum valuesas, q j → q j + (2 π/L ) Φ , and is given by, Φ = 0 for for δN even= ±
12 for δN odd . (19)It follows from the form of Eqs. (15) and (16) that for the initial ground state the equality ¯ q j = q j holds.Canonical momentum values ¯ q j are different from the corresponding -band momentum values q j only for excitedenergy eigenstates. Hence the initial ι = ± ground-state -band Fermi momenta q ιF have exactly the same values,Eq. (8), for the -particle momentum q j and canonical momentum ¯ q j representations.The corresponding ι = ± Fermi canonical momentum values deviations δ ¯ q ιF fully control the momentum and spindensity dependence of the exponents ζ ab ¯ n ( k ) , Eq. (13), in the spin dynamical structure factors’s general expressions,Eqs. (11) and (12). The deviations δ ¯ q ιF are obtained from the excited state’s -band Fermi momentum values q ιF + δq ιF under the momentum - canonical-momentum transformation, Eq. (15), as follows, q ιF + δq ιF → ¯ q ( q ιF + δq ιF ) = q ιF + δq ιF + 2 πL Φ ( q ιF + δq ιF ) = q ιF + δq ιF + 2 πL Φ ( ιk F ↓ ) + h . o . . (20)Neglecting contributions of order (1 /L ) and accounting for the values of -band Fermi momenta being in the case ofthe initial ground state the same for the momentum and canonical-momentum representations, this gives, δ ¯ q ιF = δq ιF + 2 πL Φ ( ιk F ↓ ) = 2 πL (cid:0) ι δN F ,ι + Φ ( ιk F ↓ ) (cid:1) = 2 πL (cid:16) ι δN ,F ,ι + Φ + Φ ( ιk F ↓ ) (cid:17) . (21)Indeed, expanding πL Φ ( q ιF + δq ιF ) in both the O (1 /L ) corrections in the ground-state expression, q ιF = ιk F ↓ + O (1 /L ) ,Eq. (8), and in the deviation δq ιF , leads to πL Φ ( ιk F ↓ ) plus contributions of order (1 /L ) . Those have no physicalmeaning in the thermodynamic limit to which the validity of the dynamical theory refers. Here k F ↓ = π (1 − m ) , Eq.(B7) of Appendix B.That there are no n -particles with n > in the ground state dictates why only the ι = ± Fermi canonicalmomentum values fluctuations play an active role in the dynamical properties. Within the present extended dynamicaltheory, such fluctuations though account for the creation of the -particle through the phase-shift Φ , ( q j , q j (cid:48) ) in unitsof π appearing on the right-hand side of Eq. (16).That the ι = ± Fermi canonical momentum values fluctuations associated with the deviations δ ¯ q ιF fully controlthe excitation momentum k and spin density dependence of the exponents, Eq. (13), follows from the ι = ± spectral1functionals Φ ι in that equation being such deviations δ ¯ q ιF , Eq. (21), in units of the quantum momentum spacing π/L , Φ ι = δ ¯ q ιF (2 π/L ) = ι δN F ,ι + Φ( ιk F ↓ ) = ι δN ,F ,ι + Φ + Φ( ιk F ↓ )= ι δN F ,ι + N ↑ (cid:88) j (cid:48) =1 Φ , ( ιk F ↓ , q j (cid:48) ) δN ( q j (cid:48) ) + N ↑ − N ↓ + N (cid:88) j (cid:48) =1 Φ , ( ιk F ↓ , q j (cid:48) ) δN ( q j (cid:48) ) . (22)The specific phase shifts in units of π , Φ , ( ιk F ↓ , q ) and Φ , ( ιk F ↓ , q ) where ι = ± , of the -particles with momentumand canonical momentum values at the -band Fermi points that appear in the last expression of this equation, aredefined by Eq. (B32) of Appendix B. Limiting behaviors of such phase shifts are provided in Eqs. (B33)-(B36) ofthat Appendix.In the case of the four types of branch lines considered in our study, some of the deviations δN ( q j (cid:48) ) and δN ( q j (cid:48) ) inEq. (22) refer to -band momenta q j (cid:48) = ιk F ↓ and -band momenta q j (cid:48) = ι ( k F ↑ − k F ↓ ) and q j (cid:48) = 0 where ι = ± . As aresult, the expressions of the functionals Φ ι , Eq. (22), specific to the corresponding branch-line exponents, Eq. (13),involve the phase-shifts related parameters ξ and ξ
01 2 . Those are defined by Eqs. (B37)-(B41) and (B42)-(B43),respectively, of Appendix B. Corresponding number and current number deviations δN F , δJ F , δN , and δJ , Eq.(9), then emerge in such expressions of the spectral functionals Φ ι ( q ) that control the momentum and spin densitydependences of the branch-line exponents, Eq. (13).We start by providing the three specific forms of the ι = ± general spectral functionals Φ ι , Eq. (22), suitable to the - and (cid:48) -branch lines, ¯1 -branch lines, and ¯1 (cid:48) -branch lines, respectively, that are part of the gapped lower thresholdsof the upper n -string spectra in Figs. 1-3. (The spectra of such branch lines are given below in Sec. V A.)In the case of the - and (cid:48) -branch lines, the form of the ι = ± spectral functionals Φ ι ( q ) is, Φ ι ( q ) = ι δN F ξ
11 1 + ξ
11 1 δJ F + Φ , ( ιk F ↓ , q ) for s − and s (cid:48) − branch lines , (23)where ξ = 1 + (cid:80) ι = ± ( ι ) Φ , ( k F ↓ , ιk F ↓ ) (see Eq. (B37) of Appendix B.) For the excited energy eigenstates thatcontribute to the singularities at and above the - and (cid:48) -branch lines, the maximum interval of the -band momentum q in Eq. (23) is q ∈ [0 , ( k F ↑ − k F ↓ )[ or q ∈ ] − ( k F ↑ − k F ↓ ) , .For the ¯1 (cid:48) -branch lines, the form of the spectral functionals is, Φ ι ( q ) = ι δN F ξ
11 1 + ξ
11 1 ( δJ F − δJ ) − Φ , ( ιk F ↓ , q ) for ¯1 (cid:48) − branch lines , (24)where it was accounted for that the phase shift Φ , ( ιk F ↓ , ± ( k F ↑ − k F ↓ )) can be written as ∓ ξ
11 1 (see Eq. (B38) ofAppendix B.)In the case of the ¯1 -branch lines, the spectral functionals Φ ι , Eq. (22), have the following form, Φ ι ( q ) = ι δN F ξ
11 1 + ι ξ
01 2 ξ
11 1 δJ F − Φ , ( ιk F ↓ , q ) for ¯1 − branch lines , (25)where ξ
01 2 = Φ , ( k F ↓ , and it was accounted for that Φ , ( ιk F ↓ ,
0) = ι Φ , ( k F ↓ , (see Eq. (B42) of Appendix Band text below it.) The maximum interval of the -band momentum is q ∈ ] − k F ↓ , k F ↓ [ in both Eqs. (24) and (25).Finally, the expressions of the spectral functionals already considered in Ref. [16] that control the momentum andspin density dependence of the exponents associated with the -branch lines are provided. Those refer to parts of thelower thresholds of the lower continua in Figs. 1-3. The spectra of such thresholds are given in Eqs. (A10)-(A13) ofAppendix A. The corresponding spectral functionals are of general form, Φ ι ( q ) = ι δN F ξ
11 1 + ξ
11 1 δJ F ∓ Φ , ( ιk F ↓ , q ) where for 1 − branch lines − → maximum interval q ∈ ] − k F ↓ , k F ↓ [+ → maximum intervals q ∈ [ − k F ↑ , − k F ↓ [ and q ∈ ] k F ↓ , k F ↑ ] . (26)Here − and + is the phase-shift sign in ∓ Φ , ( ιk F ↓ , q ) suitable to -branch lines involving -hole and -particlecreation, respectively, at a -band momentum q belonging to the maximum intervals given in the equation.The values of the - and -bands number and current number deviations that in the case of the transverse andlongitudinal excited states are for excitation momentum k > used in Eqs. (23)-(25) to reach the specific expressionsof the branch-line exponents given in Sec. V A are provided below in Tables I and II, respectively. Those of the -band number and current number deviations that are used in Eq. (26) to derive the expressions of the branch-lineexponents provided in Sec. V B are given below in Table III.2 C. Constrains to the momentum dependent exponents’s k intervals The exponents of general form, Eq. (13), that control the spin dynamical structure factors’s expressions, Eqs. (11)and (12), at and above the -, ¯1 -, ¯1 (cid:48) -, (cid:48) -branch lines and the -branch lines, respectively, depend on the excitationmomentum k ∈ ]0 , π [ through the - or -band momentum values in the arguments of the spectral functionals, Eqs.(23)-(26).The ¯1 - and ¯1 (cid:48) -branch lines’s exponents and those of the -branch lines involving -hole creation are valid in k rangescorresponding to a maximum -band momentum interval q ∈ [ − ( k F ↓ − δq ) , ( k F ↓ − δq )] . On the other hand, theexponents of the -branch lines involving -particle creation are valid in k ranges corresponding to maximum -bandmomentum intervals such that | q | ∈ [( k F ↓ + δq ) , k F ↑ ] . In both cases, δq , such that δq /k F ↑ (cid:28) for < m < , isfor the different branch lines very small or vanishes in the thermodynamic limit.In the very small k intervals corresponding to the -band intervals q ∈ [ − ( k F ↓ + δq ) , − ( k F ↓ − δq )] and q ∈ [( k F ↓ − δq ) , ( k F ↓ + δq )] , the exponents that control the line shape of the spin dynamical structure factors have adifferent form, as given in Ref. [16]. (See Eqs. (77)-(82) of that reference.)Similarly, the - and (cid:48) -branch lines’s exponents that control the line shape at and above some parts of the gappedlower thresholds refer to k ranges corresponding to -band momentum maximum intervals q ∈ [ − ( k F ↑ − k F ↓ − δq ) , or q ∈ [0 , ( k F ↑ − k F ↓ − δq )] . Here δq , such that δq / ( k F ↑ − k F ↓ ) (cid:28) for < m < , is for the different branch linesagain very small or vanishes in the thermodynamic limit. (And again, the spin dynamical structure factors expressionsare different and known for q ∈ [ − ( k F ↑ − k F ↓ ) , − ( k F ↑ − k F ↓ − δq )] and q ∈ [( k F ↑ − k F ↓ − δq ) , ( k F ↑ − k F ↓ )] yet arenot of interest for our study.)In the present thermodynamic limit, the maximum -band q intervals corresponding to k intervals for which theexponents, Eq. (13), are valid, are represented in the following as q ∈ ] − k F ↓ , k F ↓ [ and | q | ∈ ] k F ↓ , k F ↑ ] . Similarly, the -band q intervals corresponding to such k intervals are represented as q ∈ ] − ( k F ↑ − k F ↓ ) , and q ∈ [0 , ( k F ↑ − k F ↓ )[ .Around the specific excitation momentum k values where along a gapped lower threshold or a lower threshold twoneighboring branch lines or branch line sections cross, there are small momentum widths in which the correspondinglower threshold refers to a boundary line that connects the two branch lines or branch line sections under consideration.In the thermodynamic limit, such momentum intervals are in general negligible and the corresponding small spectra’sdeviations are not visible in the spectra plotted in Figs. 1-3. In the cases they are small yet more extended, the twobranch lines or branch line sections run very near and just above the (gapped or gapless) lower threshold and there isvery little spectral weight between it and such lines. In this case, the singularities on the two branch lines or branchline sections remain the dominant spectral feature.We again account for such negligible effects by replacing parenthesis [ or ] by ] or [ , respectively, at the k ’s limitingvalues that separate (gapped or gapless) lower thresholds’s k intervals associated with two neighboring branch linesor branch line sections. IV. DYNAMICAL STRUCTURE FACTORS’S SPECTRA
Here the spectra associated with the ( k, ω ) -plane regions that contain most spectral weight of the spin dynamicalstructure factors are introduced, with emphasis in those associated with n -string states. The ( k, ω ) -plane distributionof such spectra is represented for S + − ( k, ω ) , S xx ( k, ω ) , and S zz ( k, ω ) in Figs. 1, 2, and 3, respectively, for spindensities (a) m = 0 . , (b) m = 0 . , (c) m = 0 . , and (d) m = 0 . . Note that the sketch of the ( k, ω ) -planedistributions represented in these figures does not provide information on the relative amount of spectral weightcontained within each spectrum’s grey continuum.In the case of the spin dynamical structure factors S + − ( k, ω ) and S xx ( k, ω ) , the figures show both a lower continuum ( k, ω ) -plane region, whose spectral weight is associated with excited states without n -strings, and an upper continuumwhose spectral weight stems from excited states populated by such n -strings. In the case of S zz ( k, ω ) , the contributionto spectral weight from excited states containing n -strings is much weaker than for S + − ( k, ω ) and S xx ( k, ω ) and doesnot lead to an upper continuum. The gapped lower threshold of such states’s spectrum is represented in Fig. 3 by awell defined ( k, ω ) -plane line.At finite magnetic fields < h < h c the contribution to the spectral weight from excited states containing n -stringsis negligible in the case of S − + ( k, ω ) . Therefore, its lower continuum spectrum is not plotted here. (It was previouslystudied in Ref. [16].) The additional part of the lower continuum in Fig. 2 for S xx ( k, ω ) = ( S + − ( k, ω ) + S − + ( k, ω )) relative to that of S + − ( k, ω ) represented in Fig. 1 stems actually from contributions from S − + ( k, ω ) . As a result ofsuch contributions, for small spin densities and some k intervals the upper n -string continuum of S xx ( k, ω ) overlapswith its lower continuum.In the case of both S + − ( k, ω ) and S zz ( k, ω ) , there is in the present thermodynamic limit for spin densities The information on the constrains to the k intervals provided in Sec. III C applies to the branch-line spectrastudied below in Sec. V. On the other hand, in the particular case of the small k intervals corresponding to thesmall -band intervals q ∈ [ − ( k F ↓ + δq ) , − ( k F ↓ − δq )] and q ∈ [( k F ↓ − δq ) , ( k F ↓ + δq )] and -band intervals q ∈ [ − ( k F ↑ − k F ↓ ) , − ( k F ↑ − k F ↓ − δq )] and q ∈ [( k F ↑ − k F ↓ − δq ) , ( k F ↑ − k F ↓ )] , such constrains do not apply to thetwo-parametric spectra of the ( k, ω ) -plane regions where there is more spectral weight. They also do not apply tothe corresponding one-parametric spectra of the gapped lower thresholds and lower thresholds themselves. Only thegeneral line-shape expressions, Eqs. (11) and (12), are not valid in such small k intervals.The remaining constrains mentioned in Sec. III C associated with small k intervals of the (gapped and gapless)lower thresholds in the vicinity of crossing points of two branch lines or branch line sections though apply to theexpressions of the one-parametric spectra of the gapped lower thresholds and lower thresholds given in the followingand in Appendix A. This applies, for instance, to the small excitation momentum intervals in the vicinity of k = ˜ k (inthe transverse case) and other k values whose expression involves the momentum ˜ k (in the logitudinal case) definedby the relations given in Eq. (28)The ( k, ω ) -plane upper continuum shown in Fig. 1 is associated with the gapped upper spectrum of S + − ( k, ω ) thatstems from transitions from the ground state to excited energy eigenstates populated by a number N s = N ↓ − of -particles and a single -particle. It is given by, ω + − ∆ ( k ) = − ε ( q ) + ε ( q (cid:48) ) and k = ιk F ↓ − q + q (cid:48) for ι = ± q ∈ [ − k F ↓ , k F ↓ ] , q (cid:48) ∈ [0 , ( k F ↑ − k F ↓ )] for ι = 1 and q ∈ [ − k F ↓ , k F ↓ ] , q (cid:48) ∈ [ − ( k F ↑ − k F ↓ ) , 0] for ι = − . (30)This spectrum has two branches corresponding to ι = ± , such that, k = k F ↓ − q + q (cid:48) ∈ [0 , π ] and k = − k F ↓ − q + q (cid:48) ∈ [ − π, . (31)As mentioned above, ε ( q ) and ε ( q ) are here the - and -band energy dispersions, respectively, defined by Eqs.(B8), (B9), and (B11)-(B17) of Appendix B.We denote by ∆ ab ( k ) where ab = + − , xx, zz the spectra of the n -string excited states’s gapped lower thresholds ofthe spin dynamical structure factors S ab ( k, ω ) . These gapped lower thresholds play an important role in our study.For some of their k intervals there are singularities of power-law form, Eq. (11), in the spin dynamical structurefactors at and just above them. (In that equation, the extra lower index ¯ n in ∆ ab ( k ) refers to the different branchlines or branch line sections that coincide with the gapped lower threshold for well-defined k intervals given below inSec. V A.)The spectra of the transverse gapped lower thresholds are such that, ∆ xx ( k ) = ∆ + − ( k ) for k ∈ [0 , π ] . (32)(The equality ∆ − + ( k ) = ∆ + − ( k ) also holds, yet, as reported previously, the amount of S − + ( k, ω ) ’s spectral weightproduced by n -string states is negligible in the thermodynamic limit at finite magnetic fields.) The spectrum of thelongitudinal gapped lower threshold is also related to ∆ + − ( k ) as follows, ∆ zz ( k ) = ∆ + − ( π − k ) for k ∈ [0 , π ] . (33)For smaller spin densities in the range m ∈ ]0 , ˜ m ] , the spectrum of the lower threshold is of the form, ∆ + − ( k ) = ε ( k ) for k ∈ [0 , ( k F ↑ − k F ↓ )]= 4 µ B h − ε ( k F ↑ − k ) for k ∈ [( k F ↑ − k F ↓ ) , ˜ k [= 4 µ B h − W − ε ( k F ↓ − k ) for k ∈ ]˜ k, k F ↓ ]= ε ( k − k F ↓ ) for k ∈ [2 k F ↓ , π ] . (34)For larger spin densities m ∈ [ ˜ m, , the spectrum of the lower threshold is slightly different and reads, ∆ + − ( k ) = ε ( k ) for k ∈ [0 , ˜ k [= 4 µ B h − W − ε ( k F ↓ − k ) for k ∈ ]˜ k, k F ↓ ]= ε ( k − k F ↓ ) for k ∈ [2 k F ↓ , π ] . (35)5The expressions of two-parametric transverse gapless spectra ω − + ( k ) and ω + − ( k ) previously studied in Ref. [16],whose superposition gives ω xx ( k ) , and that of the longitudinal gapless spectrum ω zz ( k ) that (except for ω − + ( k ) ) referto the lower continua in Figs. 1-3, are given in Eqs. (A1)-(A3) of Appendix A. The corresponding excited energyeigenstates are described by only real Bethe-ansatz rapidities. The related expressions of the one-parametric spectraof their upper thresholds ω − + ut ( k ) , ω + − ut ( k ) , ω xxut ( k ) , and ω zzut ( k ) and lower thresholds ω − + lt ( k ) , ω + − lt ( k ) , ω xxlt ( k ) , and ω zzlt ( k ) are also provided in that Appendix.We consider the following energy gaps that refer to ( k, ω ) -plane regions with a negligible amount of spectral weightin the thermodynamic limit, ∆ + − gap ( k ) = ∆ + − ( k ) − ω + − ut ( k ) ≥ xx gap ( k ) = ∆ xx ( k ) − ω xxut ( k )∆ zz gap ( k ) = ∆ zz ( k ) − ω zzut ( k ) ≥ , (36)where, ∆ xx gap ( k ) = ∆ + − ( k ) − ω + − ut ( k ) for k ∈ [0 , k xxut ]∆ xx gap ( k ) = ∆ + − ( k ) − ω − + ut ( k ) for k ∈ [ k xxut , π ] , (37)and ∆ zz gap ( k ) = ∆ + − gap ( π − k ) for k ∈ [0 , π ] . (38)The momentum k xxut > k F ↑ − k F ↓ in Eq. (37) is that at which the equality, ω − + ut ( k xxut ) = ω + − ut ( k xxut ) , holds. As confirmedfrom analysis of Figs. 1-3, one has that ∆ + − gap ( k ) ≥ and ∆ zz gap ( k ) ≥ for k ∈ ]0 , π [ , whereas ∆ xx gap ( k ) is negative forsmall spin densities m ∈ ]0 , ¯ m ] , where ¯ m ≈ . , in the k intervals given in Eq. (29).It follows from the k intervals given in that equation that at k = π the energy gap ∆ xx gap ( k ) has negative and positivevalues for spin densities m < ¯ m and m > ¯ m , respectively, where ¯ m ≈ . . Its corresponding negative, vanishing,and positive limiting values are provided in Eq. (A24) of Appendix A.The upper threshold spectra ω − + ut ( k ) , ω + − ut ( k ) , ω xxut ( k ) , ω zzut ( k ) in Eqs. (36)-(38) are given in Eqs. (A4)-(A7) ofAppendix A. The upper thresholds of the lower continua in Figs. 1, 2, and 3 refer to the spectra ω + − ut ( k ) , ω xxut ( k ) , and ω zzut ( k ) , respectively.In Appendix A, the k dependent expressions and the limiting values of the energy gaps considered here are provided. V. THE LINE SHAPE NEAR THE SINGULARITIES OF THE DYNAMICAL STRUCTURE FACTORS The power-law singularities, Eq. (11), in the spin dynamical structure factors S + − ( k, ω ) , S xx ( k, ω ) , and S zz ( k, ω ) ’sexpressions occur in the k intervals for which the corresponding momentum dependent exponents are negative. Thisapplies to ( k, ω ) -plane regions at and just above the ¯ n -branch lines that are part of the corresponding gapped lowerthresholds of the n -string states’s spectra. As discussed in Sec. III A, the main contributions to the line shape inthese ( k, ω ) -plane regions stems from transitions to excited states populated by a number N ↓ − of -particles anda single -particle. The momentum interval, k ∈ ]0 , π [ , of the gapped lower thresholds under consideration is dividedinto several subintervals, which correspond to a set of -, ¯1 (cid:48) -, ¯1 -, and (cid:48) -branch lines, respectively. (In the case of S zz ( k, ω ) , the order of the ¯1 (cid:48) - and ¯1 -branch lines’s k subintervals is the reverse, ¯1 - and ¯1 (cid:48) -.)On the other hand, the lower thresholds of the gapless spectra associated with the lower continua in Figs. 1-3 thatwere already studied in Ref. [16], which stem from transitions to excited states that are populated by a number N ↓ of -particles, either correspond to a single -branch line or to two sections of such a line.Following the information provided in Sec. III C, the constrains to the k intervals where the expressions of theexponents in the general line-shape expressions, Eqs. (11) and (12), are valid imply replacements of parenthesis [ and/or ] by ] and/or [ , respectively, in the limits of such intervals. A. The line shape at and just above the gapped lower threshold of S + − ( k, ω ) , S xx ( k, ω ) , and S zz ( k, ω ) In the case of S xx ( k, ω ) , the present study of the line shape at and just above the gapped lower thresholds of thespectra plotted in Fig. 2 refers to k intervals for which the corresponding energy gap ∆ xx gap ( k ) obeys the inequality ∆ xx gap ( k ) > . At small spin densities, this excludes the k intervals given in Eq. (29).6 branch line k in terms of q δN F δJ F δN NF δJ δN k = q (for -band q ) − (cid:48) k = k F ↑ − q (for -band q ) / − / k = k F ↓ − q (for -band q ) / − (cid:48) k = 2 k F ↓ + q (for -band q ) − Table I: The excitation momenta k > of S + − ( k, ω ) and S xx ( k, ω ) expressed in terms of - and -bands momenta q andcorresponding number and current number deviations for excited states populated by a number N ↓ − of -particles and asingle -particle and thus described by both real and complex non-real rapidities in the case of the -branch line, ¯1 (cid:48) -branchline, ¯1 -branch line, and (cid:48) -branch line. For the momentum intervals given in Eqs. (39)-(42), such branch lines are part of thecorresponding gapped lower thresholds. (In the case of S xx ( k, ω ) , the data given here do not apply to branch-line intervalsoverlapping with those given in Eq. (29.)branch line k in terms of q δN F δJ F δN NF δJ δN k = ( k F ↑ − k F ↓ ) + q (for -band q ) − − k = k F ↑ − q (for -band q ) − − / − (cid:48) k = k F ↑ − q (for -band q ) − / − − / (cid:48) k = π + q (for -band q ) − Table II: The same information as in Table I for the -branch line, ¯1 -branch line, ¯1 (cid:48) -branch line, and (cid:48) -branch line that for themomentum intervals given in Eqs. (44)-(47) are part of the gapped lower threshold of S zz ( k, ω ) . Otherwise, the line shape near the gapped lower thresholds of S + − ( k, ω ) , S xx ( k, ω ) , and S zz ( k, ω ) is given by Eq.(11). The exponents in the expression given in that equation have a general form provided in Eq. (13). The ι = ± spectral functionals Φ ι ( q ) in that exponent’s expression are in the case of the (i) - and (cid:48) -branch lines, (ii) ¯1 (cid:48) -branchlines, and (iii) ¯1 -branch lines given in Eqs. (23), (24), and (25), respectively.The relation of the excitation momentum k to the -band momentum q or -band momentum q in the Φ ι ( q ) ’sargument is branch-line dependent. Hence it is useful to revisit the expressions of the spectra of the gapped lowerthresholds, Eqs. (32)-(35) and (33), for each of their branch lines or branch line sections, including information onthe relation between k and the - or -band momenta q . Additional -band and -band constrains reported in Sec.III C associated with the validity of the corresponding expressions of the momentum dependent exponents are alsoaccounted for. This ensures that the branch-line spectra’s expressions are given in the following for the k intervals forwhich the dynamical structure factor’s expression is of the form, Eq. (11).In the case of S + − ( k, ω ) , the gapped lower threshold spectrum ∆ + − ( k ) is divided into the following branch-lineintervals, ∆ + − ( k ) = ε ( k ) and k = q where k ∈ [0 , ( k F ↑ − k F ↓ )[ and q ∈ [0 , ( k F ↑ − k F ↓ )[ for m ∈ ]0 , ˜ m ] k ∈ [0 , ˜ k [ and q ∈ [0 , ˜ k [ for m ∈ [ ˜ m, , (39) ∆ + − ¯1 (cid:48) ( k ) = 4 µ B h − ε ( k F ↑ − k ) and k = k F ↑ − q where k ∈ ]( k F ↑ − k F ↓ ) , ˜ k [ and q ∈ ]( k F ↑ − ˜ k ) , k F ↓ [ for m ∈ ]0 , ˜ m ] , (40) ∆ + − ¯1 ( k ) = 4 µ B h − W − ε ( k F ↓ − k ) and k = k F ↓ − q where k ∈ ∈ ]˜ k, k F ↓ [ and q ∈ ] − k F ↓ , ( k F ↓ − ˜ k )[ for m ∈ ]0 , ˜ m ] k ∈ ]˜ k, k F ↓ [ and q ∈ ] − k F ↓ , ( k F ↓ − ˜ k )[ for m ∈ [ ˜ m, , (41)and ∆ + − (cid:48) ( k ) = ε ( k − k F ↓ ) and k = 2 k F ↓ + q where k ∈ ]2 k F ↓ , π [ and q ∈ ]0 , ( k F ↑ − k F ↓ )[ for m ∈ ]0 , . (42)7 . . . . . − . − . . . ζ ¯ n ( k ) ( a )¯ n = 2 . . . . . − . − . − . − . ( b )¯ n = ¯1 . . . . . k/π − . − . − . − . ζ ¯ n ( k ) ( c )¯ n = ¯1 m = 0 . m = 0 . m = 0 . m = 0 . m = 0 . m = 0 . m = 0 . . . . . . k/π − . − . . . ( d )¯ n = 2 Figure 4: The momentum dependence of the exponents that in the k intervals for which they are negative control the S + − ( k, ω ) line shape near and just above the (a) ¯ n = 2 , (b) ¯ n = ¯1 (cid:48) , (c) ¯ n = ¯1 , and (d) ¯ n = 2 (cid:48) branch lines for several spin densities. Thepresent branch lines are part of the gapped lower threshold of the n -strings continuum displayed in Fig. 1. The same exponents,in the k intervals for which they are negative, also control the S xx ( k, ω ) line shape near and just above the corresponding branchlines in the n -strings continuum displayed in Fig. 2. In the case of the ¯ n = ¯1 and ¯ n = 2 (cid:48) branch lines, this only applies to the k intervals for which there is a gap between them and the upper threshold of the lower continuum (See Fig. 2). The corresponding k dependent exponents of general form, Eq. (13), that appear in the expression, S + − ( k, ω ) = C ∆+ − ( ω − ∆ + − ¯ n ( k )) ζ + − ¯ n ( k ) , Eq. (11) for ab = + − and β = 2 , ¯1 (cid:48) , ¯1 , (cid:48) , are given by, ζ + − ( k ) = − (cid:88) ι = ± (cid:18) − ι ξ + Φ , ( ιk F ↓ , q ) (cid:19) for q = k where k ∈ ]0 , ( k F ↑ − k F ↓ )[ for m ∈ ]0 , ˜ m ] and k ∈ ]0 , ˜ k [ for m ∈ [ ˜ m, ζ + − ¯1 (cid:48) ( k ) = − (cid:88) ι = ± (cid:18) − ξ − Φ , ( ιk F ↓ , q ) (cid:19) for q = k F ↑ − k where k ∈ ]( k F ↑ − k F ↓ ) , ˜ k [ for m ∈ ]0 , ˜ m ] ζ + − ¯1 ( k ) = − (cid:88) ι = ± (cid:18) ι ξ 01 2 ξ − Φ , ( ιk F ↓ , q ) (cid:19) for q = k F ↓ − k where k ∈ ]˜ k, k F ↓ [ for m ∈ ]0 , ˜ m ] and k ∈ ]˜ k, k F ↓ [ for m ∈ ]0 , ˜ m ] ζ + − (cid:48) ( k ) = − (cid:88) ι = ± (cid:18) − ι ξ + ξ + Φ , ( ιk F ↓ , q ) (cid:19) for q = k − k F ↓ where k ∈ ]2 k F ↓ , π [ . (43)The three ι = ± spectral functionals Φ ι ( q ) in the general expression, Eq. (13), specific to the exponents given inEq. (43) for the S + − ( k, ω ) ’s - and (cid:48) -branch lines, ¯1 (cid:48) -branch line, and ¯1 -branch line are provided in Eqs. (23), (24),8and (25), respectively. The corresponding suitable specific values of the number and current number deviations, Eq.(9), used in such functionals are for the present branch lines given in Table I.The S + − ( k, ω ) ’s -, ¯1 (cid:48) -, ¯1 -, and (cid:48) -branch line exponents whose expressions are given in Eq. (43) are plotted as afunction of k in Fig. 4. In the k intervals of the gapped lower threshold of the n -string continuum in Fig. 1 for whichthey are negative, which are represented by solid lines, there are singularities at and just above the corresponding ¯ n = 2 , ¯1 (cid:48) , ¯1 , (cid:48) branch lines in the expression S + − ( k, ω ) = C ∆+ − ( ω − ∆ + − ¯ n ( k )) ζ + − ¯ n ( k ) , Eq. (11) for ab = + − .The related S xx ( k, ω ) ’s expression, Eq. (11) for ab = xx , also valid at and just above the gapped lower thresholdof the n -string continuum in Fig. 2, is similar to that of S + − ( k, ω ) and involves exactly the same exponents. Thisthough applies provided that in the corresponding k intervals there is no overlap between that continuum and thelower continuum associated with excited states described by only real Bethe-ansatz rapidities. For small spin densities,this thus excludes the k intervals given in Eq. (29).In the case of S zz ( k, ω ) , the expressions of the gapped lower threshold spectrum ∆ zz ( k ) , Eqs. (36) and (38), arefor the k intervals of each corresponding branch line given by, ∆ zz ( k ) = ε ( k − ( k F ↑ − k F ↓ )) and k = ( k F ↑ − k F ↓ ) + q where k ∈ ]0 , ( k F ↑ − k F ↓ )[ and q ∈ ] − ( k F ↑ − k F ↓ ) , 0[ for m ∈ ]0 , , (44) ∆ zz ¯1 ( k ) = 4 µ B h − W − ε ( k F ↑ − k ) and k = k F ↑ − q where k ∈ ]( k F ↑ − k F ↓ ) , ( π − ˜ k )[ and q ∈ ] − ( k F ↓ − ˜ k ) , k F ↓ [ for m ∈ ]0 , ˜ m ] k ∈ ]( k F ↑ − k F ↓ ) , ( π − ˜ k )[ and q ∈ ] − ( k F ↓ − ˜ k ) , k F ↓ [ and for m ∈ [ ˜ m, , (45) ∆ zz ¯1 (cid:48) ( k ) = 4 µ B h − ε ( k F ↓ − k ) and k = k F ↓ − q where k ∈ ]( π − ˜ k ) , k F ↓ [ and q ∈ ] − k F ↓ , − ( k F ↑ − ˜ k )[ for m ∈ ]0 , ˜ m ] , (46)and ∆ zz (cid:48) ( k ) = ε ( k − π ) and k = π + q where k ∈ ]2 k F ↓ , π [ and q ∈ ] − ( k F ↑ − k F ↓ ) , 0[ for m ∈ ]0 , ˜ m ] k ∈ ]( π − ˜ k ) , π [ and q ∈ ] − ˜ k, 0[ for m ∈ [ ˜ m, . (47)The corresponding k dependent exponents of general form, Eq. (13), that appear in the expression, S zz ( k, ω ) = C ∆ zz ( ω − ∆ zz ¯ n ( k )) ζ zz ¯ n ( k ) , Eq. (11) for ab = zz and ¯ n = 2 , ¯1 , ¯1 (cid:48) , (cid:48) , read, ζ zz ( k ) = − (cid:88) ι = ± (cid:18) − ιξ − ξ + Φ , ( ιk F ↓ , q ) (cid:19) for q = k − k F ↑ + k F ↓ where k ∈ ]0 , ( k F ↑ − k F ↓ )[ ζ zz ¯1 ( k ) = − (cid:88) ι = ± (cid:18) − ιξ + ι ξ 01 2 − ξ − Φ , ( ιk F ↓ , q ) (cid:19) for q = k F ↑ − k where k ∈ ]( k F ↑ − k F ↓ ) , ( π − ˜ k )[ for m ∈ ]0 , ˜ m ] and k ∈ ]( k F ↑ − k F ↓ ) , ( π − ˜ k )[ for m ∈ ]0 , ˜ m ] ζ zz ¯1 (cid:48) ( k ) = − (cid:88) ι = ± (cid:18) ι ξ + ξ − Φ , ( ιk F ↓ , q ) (cid:19) for q = k F ↓ − k where k ∈ ]( π − ˜ k ) , k F ↓ [ for m ∈ ]0 , ˜ m ] ζ zz (cid:48) ( k ) = − (cid:88) ι = ± (cid:18) − ιξ + Φ , ( ιk F ↓ , q ) (cid:19) for q = k − π where k ∈ ]2 k F ↓ , π [ for m ∈ ]0 , ˜ m ] and k ∈ ]( π − ˜ k ) , π [ for m ∈ [ ˜ m, . (48)Also in the present case of S zz ( k, ω ) , the three ι = ± spectral functionals Φ ι ( q ) in the general expression, Eq.(13), specific to the - and (cid:48) -branch lines, ¯1 -branch line, and ¯1 (cid:48) -branch line are provided in Eqs. (23)-(25). Thecorresponding suitable values of the number and current number deviations, Eq. (9), given in Table II and used insuch functionals to reach the expressions in Eq. (48) are though different relative to those used for S + − ( k, ω ) .9 . . . . . k/π − . . ζ zz ¯1 m = 0 . m = 0 . m = 0 . Figure 5: The same as in Fig. 4 for the ¯1 (cid:48) branch line of S zz ( k, ω ) . For that dynamical structure factor, this exponent isthe only one that is negative and refers to singularities near the corresponding small momentum intervals of the gapped lowerthreshold of the n -string continuum in Fig. 3. Such singularities only emerge in S zz ( k, ω ) for spin densities < m < ˜ m where ˜ m ≈ . . -branch line k in terms of q k interval δN F δJ F δN NF − + k = k F ↑ − q k ∈ ]( k F ↑ − k F ↓ ) , π [ 0 − / − − k = k F ↑ + q k ∈ [0 , ( k F ↑ − k F ↓ )[ 0 − / − k = k F ↑ − q k ∈ ]( k F ↑ − k F ↓ ) , π [ 2 − / − zz k = k F ↓ − q k ∈ ]0 , k F ↓ [ 1 1 / − zz k = k F ↓ + q k ∈ ]2 k F ↓ , π ] − / Table III: The excitation momenta k > of (i) S − + ( k, ω ) and (ii) S + − ( k, ω ) and S zz ( k, ω ) expressed in terms of -bandmomenta q and corresponding number and current number deviations for the (i) -branch line and (ii) two -branch linesections that coincide with the corresponding lower thresholds of the lower continua shown in Figs. 1-3. The corresponding behaviors of the spin dynamical structure factor S zz ( k, ω ) are also qualitatively different fromthose of S + − ( k, ω ) . Except for ζ zz ¯1 (cid:48) ( k ) , the exponents in Eq. (48) are positive for all their k intervals. The ¯1 (cid:48) -branchline’s exponent is plotted as a function of k in Fig. 5. It is negative for its whole k subinterval, which is part of the k interval of the gapped lower threshold in Figs. 3 (a) and (b). The ¯1 (cid:48) -branch line’s m -dependent subinterval is thougheither small or that line is not part of the S zz ( k, ω ) ’s gapped lower threshold at all. Its momentum width decreasesupon increasing m up to a spin density ˜ m ≈ . . For ˜ m < m < , the ¯1 (cid:48) -branch line is not part of the S zz ( k, ω ) ’sgapped lower threshold spectrum. This is why for m = 0 . > ˜ m and m = 0 . > ˜ m that line does not appear in thegapped lower threshold in Figs. 3 (c) and (d), respectively.One then concludes that gapped lower threshold’s singularities only emerge in S zz ( k, ω ) for spin densities < m < ˜ m at and just above the ¯1 (cid:48) -branch line, the corresponding line shape reading, S zz ( k, ω ) = C ∆ zz ( ω − ∆ zz ¯1 (cid:48) ( k )) ζ + − ¯1 (cid:48) ( k ) . Thatbranch line k subinterval width though strongly decreases upon increasing m up to ˜ m .These behaviors are consistent with the S zz ( k, ω ) ’s spectral weight stemming from n -string states being suppressedupon increasing the spin density m within the interval m ∈ ]0 , ˜ m ] . Within it, that weight decreases upon increasingthe spin density, becoming negligible for ˜ m < m < . B. The line shape near the lower thresholds of the spin dynamical structure factors In order to provide an overall physical picture that accounts for all gapped lower threshold’s singularities and lowerthreshold’s singularities in the spin dynamical structure factors, here we shortly revisit their line shape behavior atand just above the lower thresholds of the lower continua in Figs. 1-3. The corresponding dominant contributions0 . . . . . k/π − . − . − . − . − . − . ζ xx m = 0 . m = 0 . m = 0 . m = 0 . m = 0 . m = 0 . m = 0 . Figure 6: The momentum dependence of the exponent that controls the S xx ( k, ω ) ’s line shape near and just above the lowerthreshold of the lower continuum in Fig. 2 for several spin densities. For k ∈ ]0 , ( k F ↑ − k F ↓ )[ and k ∈ ]( k F ↑ − k F ↓ ) , π [ thatexponent corresponds to that of S + − ( k, ω ) and S − + ( k, ω ) , respectively. are from excited states described by only real Bethe-ansatz rapidities [16]. Such lower continua contain most spectralweight of the corresponding spin dynamical structure factors.In the case of the transverse dynamical structure factor, S xx ( k, ω ) = ( S + − ( k, ω ) + S − + ( k, ω )) , we consider thetransitions to excited states that determine the line shape in the vicinity of the lower thresholds of both S + − ( k, ω ) and S − + ( k, ω ) , respectively. The lower threshold’s spectrum ω xxlt ( k ) of S xx ( k, ω ) is given in Eq. (A13) of Appendix A. Thecorresponding two-parametric spectrum ω xx ( k ) results from the superposition of the corresponding two-parametricspectra, ω + − ( k ) and ω − + ( k ) , Eqs. (A1) and (A2) of that Appendix, respectively. It refers to the ( k, ω ) -plane lowercontinuum in Fig. 2.For spin densities < m < and momenta k ∈ ]0 , π [ , the line shape of the spin dynamical structure factors S ab ( k, ω ) where ab = + − , − + , xx, zz at and just above their lower thresholds has the general form given in Eq. (12). In thecase of S xx ( k, ω ) , this expression can be expressed as, S xx ( k, ω ) = S + − ( k, ω ) for k ∈ [0 , ( k F ↑ − k F ↓ )[= S − + ( k, ω ) for k ∈ ]( k F ↑ − k F ↓ ) , π [ . (49)The k dependent exponents appearing in the spin dynamical factors’s expression, Eq. (12), are of general form, Eq.1 . . . . . k/π − . − . − . − . − . ζ zz m = 0 . m = 0 . m = 0 . m = 0 . m = 0 . m = 0 . m = 0 . Figure 7: The momentum dependence of the exponent that controls the S zz ( k, ω ) line shape near and just above the lowerthreshold of the lower continuum in Fig. 3 for several spin densities. (13). Their specific expressions for the different -branch lines and -branch line sections under consideration read, ζ − +1 ( k ) = − (cid:88) ι = ± (cid:18) − ξ − Φ , ( ιk F ↓ , q ) (cid:19) for q = k F ↑ − k and k ∈ ]( k F ↑ − k F ↓ ) , π [ ζ + − ( k ) = − (cid:88) ι = ± (cid:18) − ξ , ( ιk F ↓ , q ) (cid:19) for q = k − k F ↑ and k ∈ ]0 , ( k F ↑ − k F ↓ )[ ζ + − ( k ) = − (cid:88) ι = ± (cid:18) ιξ − ξ − Φ , ( ιk F ↓ , q ) (cid:19) for q = k F ↑ − k and k ∈ ]( k F ↑ − k F ↓ ) , π [ ζ zz ( k ) = − (cid:88) ι = ± (cid:18) ι ξ + ξ − Φ , ( ιk F ↓ , q ) (cid:19) for q = k F ↓ − k and k ∈ ]0 , k F ↓ [ ζ zz ( k ) = − (cid:88) ι = ± (cid:18) − ι ξ + ξ , ( ιk F ↓ , q ) (cid:19) for q = k − k F ↓ and k ∈ ]2 k F ↓ , π [ . (50)The corresponding ι = ± spectral functionals Φ ι ( q ) in the exponent expression, Eq. (13), have for the present -branch lines and -branch line sections the general form given in Eq. (26). The values of the -band number andcurrent number deviations, Eq. (9), that are used in Eq. (26) to reach the above specific exponents’s expressions areprovided in Table III.As confirmed by the form of the expressions given in Eqs. (A10) and (A11) of Appendix A, the lower threshold’spectra equality ω + − lt ( k ) = ω − + lt ( k ) holds for k ∈ [( k F ↑ − k F ↓ ) , π ] . On the other hand, ω + − lt ( k ) = ω xxlt ( k ) for k ∈ [0 , π ] ,as confirmed by inspection of Eqs. (A11) and (A13) of that Appendix. In the k ∈ ]( k F ↑ − k F ↓ ) , π [ interval, the lineshape of S xx ( k, ω ) = ( S + − ( k, ω ) + S − + ( k, ω )) is controlled by the smallest of the exponents ζ − +1 ( k ) and ζ + − ( k ) inEq. (50), which turns out to be ζ − +1 ( k ) . In the k ∈ ]0 , ( k F ↑ − k F ↓ )[ interval it is rather controlled by the exponent2 ζ + − ( k ) . The resulting exponent ζ xx ( k ) is thus given by, ζ xx ( k ) = − (cid:88) ι = ± (cid:18) − ξ , ( ιk F ↓ , q ) (cid:19) for q = k − k F ↑ and k ∈ ]0 , ( k F ↑ − k F ↓ )[= − (cid:88) ι = ± (cid:18) − ξ − Φ , ( ιk F ↓ , q ) (cid:19) for q = k F ↑ − k and k ∈ ]( k F ↑ − k F ↓ ) , π [ . (51)This exponent is plotted as a function of k in Fig. 6. On the other hand, the -branch line exponent ζ zz ( k ) whoseexpression is given in Eq. (50) is plotted as a function of that excitation momentum in Fig. 7.Both such exponents are for spin densities < m < negative in the whole momentum interval k ∈ ]0 , π [ . It followsthat there are singularities at and just above the corresponding lower thresholds. (Due to a sign error, the minus signin the quantity − ξ / appearing in Eq. (51) was missed in Ref. [16] where the exponent ζ xx ( k ) was named ξ t . Itsmomentum dependence plotted in Fig. 6 corrects that plotted in Figs. 2-4 of that reference.) VI. LIMITING BEHAVIORS OF THE SPIN DYNAMICAL STRUCTURE FACTORS At zero magnetic field, h = 0 , and thus spin density m = 0 , the spin dynamical structure factor S − + ( k, ω ) equalsthat obtained in the m → limit from m > values. On the other hand, at h = 0 and m = 0 the spin dynamicalstructure factor S + − ( k, ω ) equals that obtained in the m → limit from m < values. This is consistent with therelation, Eq. (4). As required by the SU (2) symmetry, this confirms as well that S − + ( k, ω ) = S + − ( k, ω ) at h = 0 and m = 0 . That symmetry also requires that at h = 0 and m = 0 the overall equality S xx ( k, ω ) = S yy ( k, ω ) = S zz ( k, ω ) holds. (For the ranges < h < h c and < m < considered in this paper, the equality S xx ( k, ω ) = S yy ( k, ω ) remainsvalid, whereas one has that S zz ( k, ω ) (cid:54) = S xx ( k, ω ) .)The corresponding two parametric spectrum at h = 0 and m = 0 of the spin dynamical structure factors equalsthat obtained in the m → limit from the spectrum ω − + ( k ) whose expression is given in Eq. (A1) of Appendix A for < h < h c and < m < . On the other hand, in that limit both the spectra ω + − ( k ) and ω zz ( k ) in Eqs. (A2) and(A3) of that Appendix become one-parametric and coincide with the lower threshold of the two-parametric spectrumof S xx ( k, ω ) = S yy ( k, ω ) = S zz ( k, ω ) and thus also of S − + ( k, ω ) = S + − ( k, ω ) at h = 0 and m = 0 .What is then the origin of the two-parametric spectrum S zz ( k, ω ) at h = 0 and m = 0 ? At vanishing magnetic field, h = 0 , the selection rules, Eq. (5), reveal that S zz ( k, ω ) is fully controlled by transitions from the S = S z = 0 groundstate to triplet excited states with spin numbers S = 1 and S z = 0 . This is different from the case considered in thispaper, for which the initial ground state refers to < h < h c and < m < . According to the corresponding selectionrules, Eq. (6), S zz ( k, ω ) (cid:54) = S xx ( k, ω ) is then controlled by transitions from the ground state with spin numbers S z = S or S z = − S to excited energy eigenstates with the same spin numbers, S z = S or S z = − S , respectively.For the initial ground state at h = 0 and m = 0 , one has more generally that (i) S zz ( k, ω ) and (ii) S + − ( k, ω ) and S − + ( k, ω ) are fully controlled by transitions to spin triplet S = 1 excited states with (i) S z = 0 and (ii) S z = ± ,respectively. Their -band two-hole spectrum is obtained in the m → limit from that of S + − ( k, ω ) for m < andfrom that of S − + ( k, ω ) for m > . One then finds that, ω xx ( k ) = ω zz ( k ) = − ε ( q ) − ε ( q (cid:48) ) = J π q ) + cos( q (cid:48) )) where k = ιπ − q − q (cid:48) and ι = ± q ∈ [ − π/ , π/ 2] and q (cid:48) ∈ [ − π/ , π/ . (52)These results are consistent with spin SU (2) symmetry implying that the triplet S = 1 and S z = 0 excited energyeigenstates that control S zz ( k, ω ) have exactly the same spectrum, Eq. (52), as the triplet S = 1 and S z = ± excitedstates that control S + − ( k, ω ) and S − + ( k, ω ) .On the other hand, the two parametric spectrum given in Eq. (30) refers to the S + − ( k, ω ) ’s gapped continuumof the n -string states shown in Fig. 1. It was obtained in this paper for for < h < h c and < m < . In the m → limit from such m > values, it becomes again a one-parametric line that coincides both with its ¯1 - and ¯1 (cid:48) -branch lines, which extend to k ∈ ]0 , π [ , and with the lower threshold of S xx ( k, ω ) = S yy ( k, ω ) = S zz ( k, ω ) and S − + ( k, ω ) = S + − ( k, ω ) at h = 0 and m = 0 .It is confirmed in the following that, in spite of the singular behavior concerning the class of excited states thatcontrol the spin dynamical structure factors for magnetic fields h = 0 and h (cid:54) = 0 , respectively, the same line shape atand above the lower thresholds of such factors is obtained at m = 0 and in the m → limit, respectively.3 A. Behaviors of the spin dynamical structure factors in the m → limit Taking the m → limit from m > values, one confirms that the lower threshold’s spectrum, Eq. (A10) ofAppendix A, of S − + ( k, ω ) expands to k ∈ [0 , π ] . One then finds that within the present dynamical theory thecorresponding line shape at and just above the -branch line becomes valid for k ∈ ]0 , π [ . An exactly equal spectrumis obtained for the lower threshold of S + − ( k, ω ) in the m → limit from m < values. In the m → limit, the lowerthreshold’s spectrum of S xx ( k, ω ) = ( S + − ( k, ω ) + S − + ( k, ω )) is then found to read, ω xxlt ( k ) = − ε ( π/ − k ) = J π k for k ∈ ]0 , π [ . (53)On the other hand and as reported above, the gapped continuum, Eq. (30), of the n -string states becomes inthe m → limit a ( k, ω ) -plane line whose spectrum is as well given by ∆ + − ¯1 (cid:48) ( k ) = ∆ + − ¯1 ( k ) = J π sin k for k ∈ ]0 , π [ .By use of the limiting behaviors lim m → Φ , ( ± k F ↓ , q ) = ± / (2 √ for q (cid:54) = ± k F ↓ , lim m → Φ , ( ± k F , 0) = ± / √ ,and lim m → ξ = 1 / √ provided in Eqs. (B34), (B35), and (B41) of Appendix B, one finds that the correspondingexponents ζ + − ¯1 (cid:48) ( k ) and ζ + − ¯1 ( k ) , Eq. (43), as well as the exponent ζ xx ( k ) , Eq. (51), become in the m → limit equaland read, ζ xx ( k ) = ζ + − ¯1 (cid:48) ( k ) = − (cid:88) ι = ± (cid:18) − ξ − Φ , ( ιπ/ , q ) (cid:19) = − ζ + − ¯1 ( k ) = − (cid:88) ι = ± (cid:18) ι ξ 01 2 ξ − Φ , ( ιk F ↓ , q ) (cid:19) = − . (54)The exponent ζ xx ( k ) , Eq. (51), has for < m < two expressions associated with two k intervals, respectively. Themomentum width of one of such intervals vanishes in the m → limit. This is why only one of its two expressionscontributes to the equality in this equation, which only occurs in that limit.Consistent with spin SU (2) symmetry requirements, we confirm that in the m → limit in which the two parametricspectrum ω zz ( k ) in Eq. (A3) of Appendix A becomes a one-parametric line that coincides with the lower threshold’sspectrum of S zz ( k, ω ) , Eq. (A12) of that Appendix, and in addition becomes equal to the lower threshold’s spectrum ω xxlt ( k ) , Eq. (53), and thus reads ω zzlt ( k ) = ω zz ( k ) = J π sin k for k ∈ ]0 , π [ .Similarly, the S zz ( k, ω ) ’s gapped ¯1 - and ¯1 (cid:48) -branch line spectra in Eqs. (45) and (46), respectively, are in the m → limit found to become equal and gapless and to expand to the whole k ∈ ]0 , π [ interval. In that limit, they coincidewith the S zz ( k, ω ) ’s lower threshold, their spectrum thus again reading, ∆ zz ¯1 (cid:48) ( k ) = J π sin k for k ∈ ]0 , π [ .One then finds that in the m → limit the corresponding exponents ζ zz ¯1 ( k ) and ζ zz ¯1 (cid:48) ( k ) , Eq. (48), obey theinequality, ζ zz ¯1 (cid:48) ( k ) < ζ zz ¯1 ( k ) . The smaller exponent ζ zz ¯1 (cid:48) ( k ) is associated with the spectrum in Eq. (46). That inequalityhas a physical meaning, at it reveals that the line shape is controlled by the exponents ζ zz ¯1 (cid:48) ( k ) and ζ zz ( k ) such that ζ zz ¯1 (cid:48) ( k ) = ζ zz ( k ) in that limit. By the use of the behaviors provided in Eqs. (B34) and (B41) of Appendix B, one thenfinds that in the same limit the exponents ζ zz ( k ) , Eq. (50), and ζ zz ¯1 (cid:48) ( k ) , Eq. (48), equal as well those given in Eq.(54) and read, ζ zz ( k ) = ζ zz ¯1 (cid:48) ( k ) = − (cid:88) ι = ± (cid:18) ι ξ + ξ − Φ , ( ιπ/ , q ) (cid:19) = − . (55)Also in this case, the exponent ζ zz ( k ) given in Eq. (50) has for < m < two expressions associated with two k intervals, respectively. Only one of such expressions contributes to the equality in this equation, which only holds inthe m → limit. This follows from the momentum width of one of the two < m < excitation momentum k ’sintervals considered in Eq. (50) vanishing in that limit.The above results then confirm that in the m → limit the line shape at and just above the lower threshold of thespin dynamical structure factors is controlled by the exponent − / , which is a result known to hold at zero magneticfield [24, 28, 29]. The corresponding line-shape expression reads, S aa ( k, ω ) = C ( ω − ω ( k )) − / where ω ( k ) = J π k for ]0 , π [ , (56)where aa = xx, yy, zz and C is a constant that has a fixed value for the k and ω ranges corresponding to small valuesof the energy deviation ( ω − ω ( k )) .4 B. Behaviors of the spin dynamical structure factors in the m → limit The sum rules, Eq. (7), reveal that both the spin dynamical structure factrs S − + ( k, ω ) and S zz ( k, ω ) vanish inthe m → limit. This implies that in that limit and thus in the h → h c limit, only S xx ( k, ω ) dominates the spindynamical structure factor. Here h c = J/µ B is the critical field at which fully polarized ferromagnetism is achieved.At h = h c the line-shape expressions of general form given in Eqs. (11) and (12) of the extended dynamical theoryare not valid. Indeed, at that magnetic field the problem refers to a different quantum phase associated with fullypolarized ferromagnetism and the line shape rather becomes of δ -function like type, given by, S xx ( k, ω ) = π δ ( ω − ω xxlt ( k )) where ω xxlt ( k ) = J (1 + cos k ) for [0 , π ] . (57) VII. DISCUSSION AND CONCLUDING REMARKS In this paper, the contribution to the spin dynamical structure factors S + − ( k, ω ) , S xx ( k, ω ) , and S zz ( k, ω ) fromexcited energy eigenstates populated by n -strings has been studied for magnetic fields < h < h c , in the thermo-dynamic limit. (The contribution to S − + ( k, ω ) from such states was found to be negligible.) In that limit, there isnearly no spectral weight in the ( k, ω ) -plane gap region between the upper threshold of the lower continuum shownin Figs. 1 and 3 for S + − ( k, ω ) and S zz ( k, ω ) , respectively, and the gapped lower threshold of the n -string states’sspectrum. The same applies to S xx ( k, ω ) in Fig. 2 for the spin densities and k intervals for which there is no overlapbetween the n -string states’s upper continuum and the lower continuum.Concerning the negligible amount of spectral weight in the ( k, ω ) -plane gap regions, let us consider for instancethe more involved case of S + − ( k, ω ) . Similar conclusions apply to the simpler problems of the other spin dynamicalstructure factors. The behavior of spin operators matrix elements between energy eigenstates in the selection rulesvalid for magnetic fields < h < h c , Eq. (6), has important physical consequences. It implies that the spectral weightstemming from excited energy eigenstates described by only real Bethe-ansatz rapidities existing in finite systems ina ( k, ω ) -plane region corresponding to the momentum interval k ∈ [2 k F ↓ , π ] and excitation energy values ω above theupper threshold of the lower continuum in Fig. 1, whose spectrum’s expression is given in Eq. (A2) of Appendix A,becomes negligible in the present thermodynamic limit for a macroscopic system.Our thermodynamic limit’s study is complementary to results obtained by completely different methods for finite-size systems [8, 9, 15]. The spectral weight located in that ( k, ω ) -plane region is found to decrease upon increasing thesystem size [8]. This is confirmed by comparing the spectra represented in the first row frames of Figs. 3 (a) and (b)of Ref. [8] for two finite-size systems with N = 320 and N = 2240 spins, respectively, in the case under considerationof the spin dynamical structure factor S + − ( k, ω ) .More generally, the selection rules in Eqs. (5) and (6) are behind in the thermodynamic limit nearly all spectralweight generated by transitions to excited energy eigenstates described only by real Bethe-ansatz rapidities beingcontained in the ( k, ω ) -plane lower continuum shown in Fig. 1, whose spectrum is given in Eq. (A2) of Appendix A.Let us consider for instance the ( k, ω ) -plane spectral weight distributions shown in Fig. 18 of Ref. [15] for S + − ( k, ω ) .As reported in that reference, due to the interplay of the selections rules given in Eqs. (5) and (6) for h = 0 and < h < h c , respectively, the spectral weight existing between the continuous lower boundary (cid:15) L and the upperboundary (cid:15) U at h = 0 becomes negligible for finite magnetic fields < h < h c . In addition, the spectral weightexisting between the continuous lower boundary (cid:15) L and the upper boundary (cid:15) U for small finite-size systems, becomesnegligible in the thermodynamic limit for a macroscopic system. This is indeed due to the selection rules, Eq. (6),as discussed in that reference. As also reported in it, only the spectral weight below the continuous lower boundary (cid:15) L ( q ) , located in the ( k, ω ) -plane between the lower boundary (cid:15) L and the upper boundary (cid:15) U has a significantamount of spectral weight.This refers to the ( k, ω ) -plane region where, according to the analysis of Ref. [15], for magnetic fields < h < h c a macroscopic system has nearly the whole spectral weight stemming from transitions to excited energy eigenstatesdescribed by only Bethe-ansatz rapidities. Consistent with the spectral weight in the present gap region beingnegligible, the ( k, ω ) -plane between the continuous lower boundary (cid:15) L and the upper boundary (cid:15) U in Fig. 18 of thatreference corresponds precisely to the lower continuum shown in Fig. 1, whose spectrum is provided in Eq. (A2) ofAppendix A.Our results have focused on the contribution from n -string states. This refers to the line shape at and just abovethe ( k, ω ) -plane gapped lower threshold’s spectra ∆ ab ¯ n ( k ) where ab = + − , xx, zz and ¯ n refers to different branch lines.In well-defined m -dependent k subintervals, Eqs. (39)-(42) and (44)-(47), such branch lines coincide with the gappedlower thresholds under consideration. In these physically important ( k, ω ) -plane regions, the spin dynamical structurefactors S ab ( k, ω ) have the general analytical expression provided in Eq. (11). In the case of S + − ( k, ω ) and S xx ( k, ω ) ,5such gapped lower thresholds refer to the n -string states’s upper continua shown in the ( k, ω ) -plane in Figs. 1 and 2,respectively.The above results concerning the spectral weight in the gap regions being negligible in the present thermodynamiclimit, are consistent with the amount of that weight existing just below the ( k, ω ) -plane gapped lower thresholds ofthe n -string states’s spectra shown in Figs. 1-3 being vanishingly small or negligible. This is actually behind thevalidity at finite longitudinal magnetic fields < h < h c and in the thermodynamic limit of the analytical expressionsof the spin dynamical structure factors of general form, Eq. (11), obtained in this paper.The momentum dependent exponents that control the spin dynamical structure factors’s line-shape in such ex-pressions are given in Eq. (43) for S + − ( k, ω ) and S xx ( k, ω ) and in Eq. (48) for S zz ( k, ω ) . In the former case, theexponents associated with the ( k, ω ) -plane vicinity of the − , ¯1 (cid:48) − , ¯1 − , and (cid:48) -branch lines are plotted in Fig. 4.Such lines refer to different k intervals of the gapped lower threshold of the n -string states’s spectra of S + − ( k, ω ) and S xx ( k, ω ) . The solid lines in Figs. 1 and 2 that belong to that gapped lower threshold correspond to k intervals forwhich the exponents are negative. In them, singularities occur in the spin dynamical structure factors’s expression,Eq. (11), at and above the gapped lower thresholds.In the case of S xx ( k, ω ) , the expression given in that equation does not apply for small spin densities in the range m ∈ [0 , ¯ m ] where ¯ m ≈ . to the k intervals given in Eq. (29). For these momentum ranges, there is overlap betweenthe lower continuum and upper n -string states’s continuum, as shown in Figs. 2 (a) and (b). The two k intervalsprovided in Eq. (29), k ∈ [¯ k , π ] for m ∈ ]0 , ¯ m ] and k ∈ [¯ k , ¯ k ] for ] ¯ m , ¯ m ] where ¯ m ≈ . , apply to the spectraplotted in these two figures, respectively. For spin density (a) m = 0 . , the momentum ¯ k is given by ¯ k ≈ . π whereas for spin density (b) m = 0 . the two limiting momenta read ¯ k ≈ . π and ¯ k ≈ . π .On the other hand, the contribution to S zz ( k, ω ) from excited states populated by n -strings is much weaker thanfor S + − ( k, ω ) and S xx ( k, ω ) . It does not lead to a ( k, ω ) -plane continuum. The gapped lower threshold of such statesis shown in Fig. 3. There the k subinterval associated with the ¯ n = ¯1 (cid:48) branch line is the only one at and above whichthere are singularities. Out of the four branch-line’s exponents whose expressions are provided in Eq. (48), only thatof the ¯ n = ¯1 (cid:48) branch line is indeed negative. That line is represented in the gapped lower threshold of S zz ( k, ω ) shown in Fig. 3 by a solid (green) line. The corresponding exponent is plotted in Fig. 5. That line’s k subinterval isthough small. Its momentum width decreases upon increasing the spin density within the range < m ≤ ˜ m where ˜ m ≈ . . For spin densities ˜ m ≤ m < , that line is not part of the gapped lower threshold, so that the contributionto S zz ( k, ω ) from excited states populated by n -strings becomes negligible. Consistent, in Fig. 3 (c) for m = 0 . and(d) for m = 0 . that line is lacking.In order to to provide an overall physical picture that includes the relative ( k, ω ) -plane location of all spectra witha significant amount of spectral weight, we also accounted for the contributions from all types of excited energyeigenstates that lead to gapped and gapless lower threshold singularities in the spin dynamical structure factors.This includes excited energy eigenstates described only by real Bethe-ansatz rapidities and thus without n -strings,which are known to lead to most spectral weight of the sum rules, Eq. (7) [12–16]. Their contribution to S + − ( k, ω ) , S xx ( k, ω ) , and S zz ( k, ω ) leads to the ( k, ω ) -plane lower continua shown in Figs. 1, 2, and 3, respectively.While the present work is purely theoretical, the singularities to which ( k, ω ) -plane vicinity the analytical line-shapeexpressions obtained and studied in this paper refer to are observed in inelastic neutron scattering experimentson spin-chain compounds [8–11]. All cusp singularities at and near both the gapped lower thresholds and lowerthresholds found in this paper to occur in the thermodynamic limit, indeed correspond to peaks shown in Fig. 4of Ref. [8] for CuCl · D ) and in Fig. 5 of that reference for Cu(C H N )(NO ) at the finite values of themagnetic field considered in these figures and suitable values of the exchange integral J . Also in that reference sucha correspondence was found within the spin- / XXX chain, for finite-size systems. CRediT authorship contribution statement The three authors contributed equally to the formulation of the research goals and aims. J. M. P. C. extended asuitable dynamical theory to a subspace that accounts for the contribution from n -string states to the spin dynamicalstructure factors and prepared the original draft. T. Č. and P. D. S. have designed the computer programs to solvethe integral equations that define the phase shifts and spectra and T. Č. has implemented them. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that couldhave appeared to influence the work reported in this paper.6 Acknowledgments J. M. P. C. would like to thank the Boston University’s Condensed Matter Theory Visitors Program for support andBoston University for hospitality during the initial period of this research. He acknowledges the support from FCTthrough the Grants PTDC/FIS-MAC/29291/2017, SFRH/BSAB/142925/2018, and POCI-01-0145-FEDER-028887.J. M. P. C. and T. Č. acknowledge the support from FCT through the Grant UID/FIS/04650/2013. T. Č. gratefullyacknowledges the support by the Institute for Basic Science in Korea (IBS-R024-D1). P. D. S. acknowledges thesupport from FCT through the Grants UID/CTM/04540/2013 and UID/CTM/04540/2019.J. M. P. C. and T. Č. contributed equally to this work. Appendix A: Gapless continuum spectra and energy gaps of the n -string states Within a k extended zone scheme, the S − + ( k, ω ) and S + − ( k, ω ) ’s spectra associated with lower ( k, ω ) -plane con-tinua, which for S + − ( k, ω ) is shown in Fig. 1, read, ω − + ( k ) = − ε ( q ) − ε ( q (cid:48) ) where k = ιπ − q − q (cid:48) and ι = ± q ∈ [ − k F ↓ , k F ↓ ] and q (cid:48) ∈ [ − k F ↓ , k F ↓ ] , (A1)and ω + − ( k ) = ε ( q ) − ε ( q (cid:48) ) where k = ιπ + q − q (cid:48) and ι = ± | q | ∈ [ k F ↓ , k F ↑ ] and q (cid:48) ∈ [ − k F ↓ , k F ↓ ] , (A2)respectively. Here ε ( q ) is the -band energy dispersion given in Eq. (B8) of Appendix B. The spectrum ω xx ( k ) of the transverse dynamical structure factor S xx ( k, ω ) associated with the lower continuum in Fig. 2 results fromcombination of the two spectra ω − + ( k ) and ω + − ( k ) in Eqs. (A1) and (A2), respectively.The spectrum ω zz ( k ) associated with the lower continuum in Fig. 3 is again within a k extended zone scheme givenby, ω zz ( k ) = ε ( q ) − ε ( q (cid:48) ) where k = q − q (cid:48) for | q | ∈ [ k F ↓ , k F ↑ ] and q (cid:48) ∈ [ − k F ↓ , k F ↓ ] . (A3)The upper thresholds of the two-parametric spectra, Eqs. (A1) and (A2), have the following one-parametric spectrafor spin densities m ∈ ]0 , , ω + − ut ( k ) = 2 µ B h − ε ( k F ↓ − k ) where k = k F ↓ − q for k ∈ [0 , k F ↓ ] and q ∈ [0 , k F ↓ ] , = ε ( q ) − ε ( q (cid:48) ) where k = π + q − q (cid:48) for k ∈ [ k F ↓ , π ] and v s ( q ) = v s ( q (cid:48) )with q ∈ [ − k F ↑ , − k F ↓ ] and q (cid:48) ∈ [ − k F ↓ , , (A4)and ω − + ut ( k ) = − ε (cid:18) π − k (cid:19) where k = π − q for k ∈ [( k F ↑ − k F ↓ ) , π ] and q ∈ [ − k F ↓ , , (A5)respectively. The function v ( q ) is in Eq. (A4) the -band group velocity defined in Eq. (B10) of Appendix B.The upper threshold spectrum ω xxut ( k ) of the combined spectra, Eqs. (A1) and (A2), is given by, ω xxut ( k ) = ω + − ut ( k ) for k ∈ [0 , k xxut ]= ω − + ut ( k ) for k ∈ [ k xxut , π ] , (A6)where the momentum k xxut is such that ω + − ut ( k xxut ) = ω − + ut ( k xxut ) .On the other hand, the one-parametric upper threshold spectrum associated with the two-parametric longitudinalspectrum, Eq. (A3), reads for m ∈ ]0 , , ω zzut ( k ) = ε ( q ) − ε ( q (cid:48) ) where k = q − q (cid:48) for v s ( q ) = v s ( q (cid:48) ) and k ∈ [0 , k F ↑ ] with q ∈ [ k F ↓ , k F ↑ ] and q (cid:48) ∈ [0 , k F ↓ ] , = 2 µ B h − ε ( k F ↑ − k ) where k = k F ↑ − q for k ∈ [ k F ↑ , π ] and q ∈ [ − k F ↓ , . (A7)7At k = 0 , k F ↓ , π and k = 0 , k F ↑ − k F ↓ , π , the upper threshold spectra, Eqs. (A4) and (A5), respectively, are givenby, ω + − ut (0) = W h = 2 µ B h ; ω + − ut ( k F ↓ ) = W = 2 µ B h + W p ; ω + − ut ( π ) = 0 ω − + ut ( k F ↑ − k F ↓ ) = 0 ; ω − + ut ( π ) = 2 W p . (A8)Here W = W p + W h , W p , and W h are the energy bandwidths of the full -band, its occupied Fermi sea, andunoccupied sea, respectively, in Eqs. (B22) and (B24) of Appendix B.At k = 0 , k F ↑ , π the upper threshold spectrum ω zzut ( k ) reads, ω zzut (0) = 0 ; ω zzut ( k F ↑ ) = W = 2 µ B h + W p ; ω zzut ( π ) = W h = 2 µ B h . (A9)The spin dynamical structure factors’s line shape near the lower thresholds of the spectra, Eqs. (A1), (A2), and(A3), has the general form provided in Eq. (12). In the case of (i) S − + ( k, ω ) and (ii) S + − ( k, ω ) and S zz ( k, ω ) suchlower thresholds refer to (i) a single -branch line and (ii) two sections of a -branch line, respectively. Their spectracan be expressed in terms of the excitation momentum k or of the -band momentum q and are given by, ω − + lt ( k ) = − ε ( k F ↑ − k ) and k = k F ↑ − q where k ∈ [( k F ↑ − k F ↓ ) , π ] and q ∈ ] − k F ↓ , k F ↓ ] , (A10) ω + − lt ( k ) = ε ( k − k F ↑ ) and k = k F ↑ + q where k ∈ [0 , ( k F ↑ − k F ↓ )] and q ∈ [ − k F ↑ , − k F ↓ ] , = − ε ( k F ↑ − k ) and k = k F ↑ − q where k ∈ [( k F ↑ − k F ↓ ) , π ] and q ∈ [ − k F ↓ , k F ↓ ] , (A11) ω zzlt ( k ) = − ε − k F ↓ [ F ↓ − k ) and k = k F ↓ − q where k ∈ [0 , k F ↓ ] and q ∈ [ − k F ↓ , k F ↓ ] , = ε ( k − k F ↓ ) and k = k F ↓ + q where k ∈ [2 k F ↓ ) , π ] and q ∈ [ k F ↓ , k F ↑ ] . (A12)The lower threshold’s spectrum ω xxlt ( k ) of S xx ( k, ω ) has the same expression as that of S + − ( k, ω ) , ω xxlt ( k ) = ε ( k − k F ↑ ) and k = k F ↑ + q where k ∈ [0 , ( k F ↑ − k F ↓ )] and q ∈ [ − k F ↑ , − k F ↓ ] , = − ε ( k F ↑ − k ) and k = k F ↑ − q where k ∈ [( k F ↑ − k F ↓ ) , π ] and q ∈ [ − k F ↓ , k F ↓ ] . (A13)Finally, the k dependent expressions and the limiting values of the energy gaps in Eqs. (36)-(38) are provided. Theenergy gap ∆ + − gap ( k ) is finite and positive for < m < . For spin densities m ∈ ]0 , ˜ m ] where ˜ m ≈ . it reads, ∆ + − gap ( k ) = − µ B h + ε ( k ) + ε ( k F ↓ − k ) for k ∈ [0 , ( k F ↑ − k F ↓ )]∆ + − gap ( k ) = 2 µ B h − ε ( k F ↑ − k ) + ε ( k F ↓ − k ) for [( k F ↑ − k F ↓ ) , ˜ k [∆ + − gap ( k ) = 2 µ B h − W for k ∈ ]˜ k, k F ↓ ]∆ + − gap ( k ) = 4 µ B h − W − ε ( k F ↓ − k ) + ε ( q ) − ε ( k + q − π ) for k ∈ [ k F ↓ , k F ↓ ] and q ∈ [ − ( k • − k F ↑ + k F ↓ ) , + − gap ( k ) = ε ( k − k F ↓ ) + ε ( q ) − ε ( k + q − π ) for k ∈ [2 k F ↓ , π ] andfor spin densities m ∈ ]0 , ˜ m ] . (A14)For spin densities m ∈ [ ˜ m, its expression is, ∆ + − gap ( k ) = − µ B h + ε ( k ) + ε ( k F ↓ − k ) for k ∈ [0 , ˜ k [∆ + − gap ( k ) = 2 µ B h − W for k ∈ ]˜ k, k F ↓ ]∆ + − gap ( k ) = 4 µ B h − W − ε ( k F ↓ − k ) + ε ( q ) − ε ( k + q − π ) for k ∈ [ k F ↓ , k F ↓ ] and q ∈ [ − ( k • − k F ↑ + k F ↓ ) , + − gap ( k ) = ε ( k − k F ↓ ) + ε ( q ) − ε ( k + q − π ) for k ∈ [2 k F ↓ , π ] and q ∈ [ − k F ↓ , − ( k • − k F ↑ + k F ↓ )]for spin densities m ∈ [ ˜ m, . (A15)8In the above equations, the momentum ˜ k is defined by the relations, Eq. (28), and the momentum k • satisfies thefollowing equation, v ( k • ) = v ( k • − k F ↑ + k F ↓ ) where k • > k F ↓ . (A16)On the other hand, for spin densities m ∈ ]0 , ˜ m ] , the energy gap ∆ − +gap ( k ) = ∆ − + ( k ) − ω − + ut ( k ) where ∆ − + ( k ) =∆ + − ( k ) reads, ∆ − +gap ( k ) = ε ( k ) for k ∈ [0 , ( k F ↑ − k F ↓ )]∆ − +gap ( k ) = 4 µ B h − ε ( k F ↑ − k ) + 2 ε (cid:18) π − k (cid:19) for k ∈ [( k F ↑ − k F ↓ ) , ˜ k [∆ − +gap ( k ) = 4 µ B h − W − ε ( k F ↓ − k ) + 2 ε (cid:18) π − k (cid:19) for k ∈ ]˜ k, k F ↓ ]∆ − +gap ( k ) = ε ( k − k F ↓ ) + 2 ε (cid:18) π − k (cid:19) for k ∈ [2 k F ↓ , π ] , (A17)whereas for m ∈ [ ˜ m, it is given by, ∆ − +gap ( k ) = ε ( k ) for k ∈ [0 , ˜ k [∆ − +gap ( k ) = 4 µ B h − W − ε ( k F ↓ − k ) for k ∈ ]˜ k, ( k F ↑ − k F ↓ )]∆ − +gap ( k ) = 4 µ B h − W − ε ( k F ↓ − k ) + 2 ε (cid:18) π − k (cid:19) for k ∈ [( k F ↑ − k F ↓ ) , k F ↓ ]∆ − +gap ( k ) = ε ( k − k F ↓ ) + 2 ε (cid:18) π − k (cid:19) for k ∈ [2 k F ↓ , π ] . (A18)The energy gap ∆ + − gap ( k ) is given by the constant energy scale µ B h − W where W is the energy bandwidth ofthe -band in Eqs. (B22) and (B24) of Appendix B for the following k and spin density m values and intervals, ∆ + − gap ( k ) = 2 µ B h − W k = 0 for m ∈ ]0 , k = k F ↑ − k F ↓ for m ∈ ]0 , / k ∈ ]˜ k, k F ↓ ] for m ∈ ]0 , . (A19)The energy scale µ B h − W ≥ has the following limiting values, µ B h − W = 0 for m → µ B h − W = J for m → . (A20)At k = π the energy gap ∆ + − gap ( k ) reads, ∆ + − gap ( π ) = 4 µ B h for m ∈ ]0 , . (A21)It has the following limiting values, ∆ + − gap ( π ) = 0 for m → + − gap ( π ) = 4 J for m → . (A22)The energy gap ∆ − +gap ( k ) has the following values at k = 0 , k F ↑ − k F ↓ , π , ∆ − +gap (0) = 4 µ B h − W for m ∈ ]0 , − +gap ( k F ↑ − k F ↓ ) = 4 µ B h for m ∈ ]0 , ˜ m ]∆ − +gap ( π ) = 4 µ B h − W p for m ∈ ]0 , . (A23)For small spin densities m ∈ ]0 , ¯ m ] where ¯ m ≈ . and the momentum intervals given in Eq. (29) the inequality ∆ − +gap ( k ) < holds. In such a k intervals, ∆ − +gap ( k ) equals the energy gap ∆ xx gap ( k ) = ∆ − +gap ( k ) < . At k = π such gapsthus have negative and positive values for spin densities m < ¯ m and m > ¯ m , respectively, where ¯ m ≈ . . Thecorresponding limiting values read, ∆ xx gap ( π ) = ∆ − +gap ( π ) = − πJ for m → 0= 0 for m = ¯ m ≈ . J for m → . (A24)9 Appendix B: Some useful quantities In this Appendix some quantities needed for our study are defined and corresponding useful limiting behaviors areprovided. The quantum problem studied in this paper is described by the spin- / XXX chain in a longitudinalmagnetic field, Eq. (1), acting in the subspace considered in Sec. III A. It involves a subset of Bethe ansatz equations.That associated with the -band is given by, q j = 2 arctan (Λ ( q j )) − L N ↑ (cid:88) j (cid:48) =1 N ( q j (cid:48) ) arctan (cid:18) Λ ( q j ) − Λ ( q j (cid:48) )2 (cid:19) − L N ↑ − N ↓ + N (cid:88) j (cid:48) =1 N ( q j (cid:48) ) (cid:26) arctan (Λ ( q j ) − Λ ( q j (cid:48) )) + arctan (cid:18) Λ ( q j ) − Λ ( q j (cid:48) )3 (cid:19)(cid:27) where j = 1 , ..., N ↑ and N = 0 , . (B1)That associated with the -band reads, q j = 2 arctan (cid:18) Λ ( q j )2 (cid:19) − L N ↑ (cid:88) j (cid:48) =1 N ( q j (cid:48) ) (cid:26) arctan (Λ ( q j ) − Λ ( q j (cid:48) )) + arctan (cid:18) Λ ( q j ) − Λ ( q j (cid:48) )3 (cid:19)(cid:27) where j = 1 , ..., N ↑ − N ↓ + N and N = 0 , . (B2)In these equations, N ( q j (cid:48) ) = 1 and N ( q j (cid:48) ) = 1 for occupied momentum values q j (cid:48) and N ( q j (cid:48) ) = 0 and N ( q j (cid:48) ) = 0 for unoccupied momentum values q j (cid:48) .The subspace considered in Sec. III A is spanned by excited energy eigenstates populated either by a number N = N ↓ of -particles or by a number N = N ↓ − of -particles and a single -particle. In the case of the latterclass of states, the Bethe-ansatz equation, Eq. (B2), does not include a third term, that in the case of the full Hilbertspace involves the spin rapidity differences Λ ( q j ) − Λ ( q j (cid:48) ) . Indeed, when δN = 1 one has that such a term onlycontributes for q j = q j (cid:48) at which values it vanishes.The -band Bethe ansatz rapidity is real and associated with the rapidity function Λ ( q j ) in the above Bethe-ansatzequations. In the case of general n -strings of length n > , the corresponding complex non-real Bethe ansatz rapiditieshave in the thermodynamic limit the following form [2], Λ n,l ( q j ) = Λ n ( q j ) + i ( n + 1 − l ) for l = 1 , ..., n and j = 1 , ..., L n where L n = N n + N hn and N hn = 2 S + ∞ (cid:88) n (cid:48) = n +1 n (cid:48) − n ) N n (cid:48) . (B3)This general expression also applies for n = 1 , Λ , ( q j ) being real and equal to Λ ( q j ) . In the general n = 1 , ..., ∞ case, Λ n ( q j ) are real rapidity functions defined by the set of coupled Bethe-ansatz equations associated with the fullHilbert space not given here, q j such that q j +1 − q j = 2 π/L are the corresponding n -band discrete momentum valueswhose number is L n , N n is both the number of n -band occupied momentum values and the number of n -particles andthus of n -strings when n > , N hn is that of n -band unoccupied momentum values, and S is the energy eigenstate’sspin such that S gives the number of unpaired physical spins / that are not paired and thus are not part of the n particles’s internal degrees of freedom.In the present case of the subspace considered in Sec. III A, the problem simplifies. The -band rapidity function Λ ( q j ) that appears in Eqs. (B1) and (B2) is the real part of the following two Bethe ansatz complex rapiditiesassociated with a n -string of length n = 2 , Λ ,l ( q j ) = Λ ( q j ) + i (3 − l ) for l = 1 , j = 1 , ..., N ↑ − N ↓ + N . (B4)This expression refers to that given in Eq. (B3) for n = 2 and l = 1 , .The momentum values q j in Eqs. (B1), (B2), and (B4) are given by, q j = 2 πL I nj for n = 1 , , (B5)where the quantum numbers I nj are either integers or half-odd integers according to the following boundary conditions[2], I j = 0 , ± , ± , ... for N ↑ odd= ± / , ± / , ± / , ... for N ↑ even I j = 0 , ± , ± , ... for N = 1 . (B6)0In the thermodynamic limit, we often use continuous momentum variables q that replace the discrete - and -bandsmomenta q j such that q j +1 − q j = 2 π/L . They read q ∈ [ − k F ↑ , k F ↑ ] and q ∈ [ − ( k F ↑ − k F ↓ ) , ( k F ↑ − k F ↓ )] , respectively.In that limit, the momenta k F ↓ and k F ↑ are given by, k F ↓ = π − m ) ; k F ↑ = π m ) ; k F = π , (B7)for the spin-density interval, m ∈ ]0 , where k F = lim m → k F ↓ = lim m → k F ↑ .The energy dispersions ε ( q ) and ε ( q ) that appear in the spectra of the excited energy eigenstates are defined as, ε ( q ) = ¯ ε (Λ ( q )) for q ∈ [ − k F ↑ , k F ↑ ] where ¯ ε (Λ) = (cid:90) Λ B d Λ (cid:48) Jη (Λ (cid:48) ) , (B8)where B is defined below and, ε ( q ) = 4 µ B h + ε ( q ) for q ∈ [ − ( k F ↑ − k F ↓ ) , ( k F ↑ − k F ↓ )] where ε ( q ) = ¯ ε (Λ ( q )) and ¯ ε (Λ) = (cid:90) Λ ∞ d Λ (cid:48) Jη (Λ (cid:48) ) , (B9)respectively.The corresponding - and -bands group velocities are given by, v ( q ) = ∂ε ( q ) ∂q and v ( q ) = ∂ε ( q ) ∂q . (B10)The distribution Jη (Λ) appearing in Eq. (B8) is the solution of the integral equation, Jη (Λ) = 4 J Λ(1 + Λ ) + (cid:90) B − B d Λ (cid:48) G (Λ , Λ (cid:48) ) 2 Jη (Λ (cid:48) ) . (B11)The kernel G (Λ , Λ (cid:48) ) appearing here is given by, G (Λ , Λ (cid:48) ) = − π (cid:18) 11 + ((Λ − Λ (cid:48) ) / (cid:19) . (B12)The values of the distribution Jη (Λ) in Eq. (B9) are determined by those of Jη (Λ) as follows, Jη (Λ) = J (cid:16) (cid:0) Λ2 (cid:1) (cid:17) − π (cid:90) B − B d Λ (cid:48) Jη (Λ (cid:48) )1 + (Λ − Λ (cid:48) ) − π (cid:90) B − B d Λ (cid:48) Jη (Λ (cid:48) )1 + (cid:0) Λ − Λ (cid:48) (cid:1) , (B13)where the distribution Jη (Λ) is the solution of Eq. (B11).The rapidity distribution function Λ ( q ) where q ∈ [ − k F ↑ , k F ↑ ] in the argument of the auxiliary dispersion ¯ ε inEq. (B8) is defined in terms of its -band inverse function q = q (Λ) where Λ ∈ [ −∞ , ∞ ] . The latter is defined bythe equation, q = q (Λ) = 2 arctan(Λ) − π (cid:90) B − B d Λ (cid:48) πσ (Λ (cid:48) ) arctan (cid:18) Λ − Λ (cid:48) (cid:19) for Λ ∈ [ −∞ , ∞ ] . (B14)The rapidity distribution function Λ ( q ) where q ∈ [ − ( k F ↑ − k F ↓ ) , ( k F ↑ − k F ↓ )] in the argument of the auxiliarydispersion ¯ ε in Eq. (B9) is also defined in terms of its -band inverse function q = q (Λ) where Λ ∈ [ −∞ , ∞ ] asfollows, q (Λ) = 2 arctan (cid:18) Λ2 (cid:19) − π (cid:90) B − B d Λ (cid:48) πσ (Λ (cid:48) ) arctan (Λ − Λ (cid:48) ) − π (cid:90) B − B d Λ (cid:48) πσ (Λ (cid:48) ) arctan (cid:18) Λ − Λ (cid:48) (cid:19) for Λ ∈ [ −∞ , ∞ ] . (B15)The distribution πσ (Λ) in Eqs. (B14) and (B15) is the solution of the integral equation, πσ (Λ) = 21 + Λ + (cid:90) B − B d Λ (cid:48) G (Λ , Λ (cid:48) ) 2 πσ (Λ (cid:48) ) , (B16)1whose kernel is given in Eq. (B12). Such a distribution obeys the sum rule, π (cid:90) B − B d Λ 2 πσ (Λ) = (1 − m ) . (B17)The parameter B = Λ ( k F ↓ ) appearing in the above equations has the limiting behaviors, B = Λ ( k F ↓ ) with lim m → B = ∞ and lim m → B = 0 . (B18)Useful reference values of the rapidity functions Λ ( q ) and Λ ( q ) are, Λ (0) = 0 and Λ ( ± k F ↑ ) = ±∞ Λ (0) = 0 and Λ ( ± ( k F ↑ − k F ↓ )) = ±∞ . (B19)The -band energy dispersion, ε ( q ) = ¯ ε (Λ ( q )) where ¯ ε (Λ) = (cid:90) Λ ∞ d Λ (cid:48) Jη (Λ (cid:48) ) , (B20)whose zero-energy level is for m > shifted realtive to that of ε ( q ) defines the spin density curve as follows, h ( m ) = − ε ( k F ↓ )2 µ B | m =1 − k F ↓ /π ∈ ]0 , h c [ . (B21)Here h c = J/µ B is the critical field for fully polarized ferromagnetism achieved when m → and thus k F ↓ → .The - and -band energy dispersions ε ( q ) and ε ( q ) , Eqs. (B8) and (B9), respectively, have limiting values, ε (0) = − W p ; ε ( ± k F ↓ ) = 0 ; ε ( ± k F ↑ ) = W h = 2 µ B hε (0) = W h − W = 4 µ B h − W ; ε ( ± ( k F ↑ − k F ↓ )) = 4 µ B h , (B22)where W = W p + W h , W p , and W h are the -band energy bandwidth, occupied Fermi sea energy bandwidth, andunoccupied sea energy bandwidth, respectively, and W is the -band energy bandwidth. Such energy scales have thelimiting behaviors, lim m → W = W p = π J and lim m → W = 0 , (B23)and lim m → W = W h = 2 µ B h c = 2 J and lim m → W = J . (B24)In the m → limit, the -band does not exist in the ground state. In that limit, it reduces to q = 0 with ε (0) = 0 for energy eigenstates for which N = 1 . In the same limit, the -band energy dispersions and group velocity can bewritten as, ε ( q ) = ε ( q ) = − J π q and v ( q ) = J π q for q ∈ [ − π/ , π/ 2] and m → . (B25)For (1 − m ) (cid:28) , the -band energy dispersions and group velocity, Eq. (B10), behave as, ε ( q ) = − J (cos q − 1) + J (1 − m ) sin q arctan (cid:18) 12 tan (cid:16) q (cid:17)(cid:19) ε ( q ) = − J (cos q + 1) + J (1 − m ) sin q arctan (cid:18) 12 tan (cid:16) q (cid:17)(cid:19) v ( q ) = J sin q + J (1 − m ) (cid:40) sin q (cid:0) q (cid:1) + cos q arctan (cid:18) 12 tan (cid:16) q (cid:17)(cid:19)(cid:41) for q ∈ (cid:104) − π m ) , π m ) (cid:105) and (1 − m ) (cid:28) . (B26)2Also for (1 − m ) (cid:28) , the behaviors of the -band energy dispersion and group velocity are, ε ( q ) = 4 J − J q ) + J − m ) sin q (cid:26) arctan (cid:16) (cid:16) q (cid:17)(cid:17) + arctan (cid:18) 23 tan (cid:16) q (cid:17)(cid:19)(cid:27) ε ( q ) = ε ( q ) − Jv ( q ) = J q + J − m ) sin q (cid:40) 11 + 3 sin (cid:0) q (cid:1) + 34 + 5 cos (cid:0) q (cid:1) (cid:41) + J − m ) cos q (cid:26) arctan (cid:16) (cid:16) q (cid:17)(cid:17) + arctan (cid:18) 23 tan (cid:16) q (cid:17)(cid:19)(cid:27) for q ∈ [ − πm, πm ] and (1 − m ) (cid:28) . (B27)From analysis of the above expressions, one finds that for (1 − m ) (cid:28) the following equality holds, v ( k F ↓ ) = v ( k F ↑ − k F ↓ ) = J π − m ) . (B28)The -particle phase shifts play an important role in the dynamical properties. They are given by, π Φ ,n ( q, q (cid:48) ) = 2 π ¯Φ ,n (Λ , Λ (cid:48) ) where Λ = Λ ( q ) and Λ (cid:48) = Λ n ( q (cid:48) ) . (B29)The rapidity phase shifts π ¯Φ ,n (Λ , Λ (cid:48) ) on the right-hand side of the above equality are functions of the rapidity-related variables Λ for the n -bands. In the case of the excited energy eigenstates that span the subspace consideredin Sec. III A, the quantum number n has the values and . In units of π , the corresponding n = 1 , rapidity phaseshifts π ¯Φ ,n (Λ , Λ (cid:48) ) are defined by the following integral equations, ¯Φ , (Λ , Λ (cid:48) ) = 1 π arctan (cid:18) Λ − Λ (cid:48) (cid:19) + (cid:90) B − B d Λ (cid:48)(cid:48) G (Λ , Λ (cid:48)(cid:48) ) ¯Φ , (Λ (cid:48)(cid:48) , Λ (cid:48) ) , (B30)and ¯Φ , (Λ , Λ (cid:48) ) = 1 π arctan(Λ − Λ (cid:48) ) + 1 π arctan (cid:18) Λ − Λ (cid:48) (cid:19) + (cid:90) B − B d Λ (cid:48)(cid:48) G (Λ , Λ (cid:48)(cid:48) ) ¯Φ , (Λ (cid:48)(cid:48) , Λ (cid:48) ) . (B31)The kernel G ( r, r (cid:48) ) in these equations is given in Eq. (B12).The phase shifts in units of π that appear in the expressions of the branch-line exponents, Eqs. (43), (48), (50),and (51), are given by, Φ , ( ιk F ↓ , q ) = ¯Φ , ( ιB, Λ ( q )) and Φ , ( ιk F ↓ , q ) = ¯Φ , ( ιB, Λ ( q )) where ι = ± . (B32)In the m → limit, the rapidity phase shift ¯Φ , (Λ , Λ (cid:48) ) (in units of π ) is given by, ¯Φ , (Λ , Λ (cid:48) ) = i π ln Γ (cid:16) + i (Λ − Λ (cid:48) )4 (cid:17) Γ (cid:16) − i (Λ − Λ (cid:48) )4 (cid:17) Γ (cid:16) − i (Λ − Λ (cid:48) )4 (cid:17) Γ (cid:16) i (Λ − Λ (cid:48) )4 (cid:17) for Λ (cid:54) = ι ∞ = ι √ ι ∞ and Λ (cid:48) (cid:54) = ι ∞ = ι (cid:18) √ − (cid:19) for Λ = Λ (cid:48) = ι ∞ where ι = ± , (B33)where Γ( x ) is the usual gamma function.From the use of Eq. (B33), one finds that in the m → limit for which k F = π/ the phase shift Φ , ( ιπ/ , q ) =lim m → Φ , ( ιk F ↓ , q ) reads (in units of π ), Φ , ( ιπ/ , q ) = ι √ q (cid:54) = ιk F = ι (cid:18) √ − (cid:19) for q = ιk F where ι = ± . (B34)3In the m → limit, the phase shift Φ , ( ιπ/ , q ) = lim m → Φ , ( ιk F ↓ , q ) has in units of π the following value, lim m → Φ , ( ιπ/ , q ) = Φ , ( ιπ/ , 0) = ι √ . (B35)In the opposite m → limit in which k F ↓ → , the phase shifts Φ , ( ιk F ↓ , q ) and Φ , ( ιk F ↓ , q ) behave as, lim m → Φ , ( ιk F ↓ , q ) = Φ , (0 , q ) = − π arctan (cid:18) 12 tan (cid:16) q (cid:17)(cid:19) lim m → Φ , ( ιk F ↓ , q ) = Φ , (0 , q ) = − π arctan (cid:16) (cid:16) q (cid:17)(cid:17) − π arctan (cid:18) 23 tan (cid:16) q (cid:17)(cid:19) . (B36)The phase-shift related parameters ξ and ξ 01 2 also appear in the expressions of the branch-line exponents, Eqs.(43), (48), (50), and (51). The -band Fermi-points phase-shift parameter ξ is such that [30], ( ξ ) ± = 1 + (cid:88) ι = ± ( ι ) ± Φ , ( k F ↓ , ιk F ↓ ) . (B37)It is also related to the phase shift Φ , ( k F ↓ , q ) in Eq. (B32) as follows, ξ = − Φ , ( ± k F ↓ , ( k F ↑ − k F ↓ ))= Φ , ( ± k F ↓ , − ( k F ↑ − k F ↓ )) . (B38)From manipulations of the phase-shift integral equation, Eq. (B30), and of Eq. (B37) one finds that the parameter ξ given by, ξ = ξ ( B ) . (B39)The function ξ ( r ) on the right-hand side of this equation at Λ = B is the solution of the integral equation, ξ (Λ) = 1 + (cid:90) B − B d Λ (cid:48) G (Λ , Λ (cid:48) ) ξ (Λ (cid:48) ) , (B40)where the kernel G (Λ , Λ (cid:48) ) is given in Eq. (B12).The parameter ξ continuously increases upon increasing the spin density from ξ = 1 / √ as m → to ξ = 1 for m → , so that its limiting values are, lim m → ξ = 1 √ m → ξ = 1 . (B41)Finally, the parameter ξ 01 2 that also appears in the momentum dependent exponents is given by, ξ 01 2 = 2Φ , ( k F ↓ , , (B42)where the phase shift Φ , ( k F ↓ , q ) is defined in Eq. (B32). At q = 0 , it is such that Φ , ( ιk F ↓ , 0) = ι Φ , ( k F ↓ , .This justifies why ι ξ 01 2 = 2Φ , ( ιk F ↓ , 0) = ι , ( k F ↓ , for ι = ± .The parameter ξ 01 2 continuously decreases upon increasing the spin density from ξ 01 2 = √ as m → to ξ 01 2 = 0 for m → . Consitent, it follows from Eqs. (B35) and (B36) that, lim m → ξ 01 2 = √ m → ξ 01 2 = 0 . (B43) [1] H. Bethe, Z. Phys. , 205 (1931).[2] M. Takahashi, Progr. Theor. Phys. 46 (1971) 401.[3] M. Gaudin, Phys. Rev. Lett. 26 (1971) 1301.[4] M. Takahashi, M. Suzuki, Progr. Theor. Phys. 48 (1972) 2187.[5] M. Gaudin, The Bethe wavefunction, Cambridge University Press, 2014.4