Lineshape predictions via Bethe ansatz for the one-dimensional spin-1/2 Heisenberg antiferromagnet in a magnetic field

The spin fluctuations parallel to the external magnetic field in the ground state of the one-dimensional (1D) s=1/2 Heisenberg antiferromagnet are dominated by a two-parameter set of collective excitations. In a cyclic chain of N sites and magnetization 0<M_z<N/2, the ground state, which contains 2M_z spinons, is reconfigured as the physical vacuum for a different species of quasi-particles, identifiable in the framework of the coordinate Bethe ansatz by characteristic configurations of Bethe quantum numbers. The dynamically dominant excitations are found to be scattering states of two such quasi-particles. For N ->\infty, these collective excitations form a continuum in (q,\omega)-space with an incommensurate soft mode. Their matrix elements in the dynamic spin structure factor S_{zz}(q,\omega) are calculated directly from the Bethe wave functions for finite N. The resulting lineshape predictions for N ->\infty complement the exact results previously derived via algebraic analysis for the exact 2-spinon part of S_{zz}(q,\omega) in the zero-field limit. They are directly relevant for the interpretation of neutron scattering data measured in nonzero field on quasi-1D antiferromagnetic compounds.


I. INTRODUCTION
Advances in experimental techniques combined with improvements in sample preparation make it possible to produce data of ever increasing resolution for the quantum fluctuations and the underlying collective excitations in quasi-one-dimensional (1D) magnetic compounds. Advances in the theoretical analysis of relevant model systems combined with progress in the computational treatment of aspects that remain elusive to exact analysis make it possible to gain an ever more profound understanding of the observable collective excitations in terms of a small number of constituent quasi-particles.
There is scarcely a better case for illustrating this multi-track advancement of understanding quantum fluctuations than the 1D s = 1 2 Heisenberg antiferromagnet and the growing number of materials that have been discovered to be physical realizations of this model system. The Hamiltonian for N spins 1 2 arranged in a cyclic chain with isotropic exchange coupling J between nearest neighbors and a uniform magnetic field h, is amenable to exact analysis via Bethe ansatz 1,2 and displays dynamical properties of intriguing complexity.
The field h is a controllable continuous parameter, which leaves the eigenvectors unaltered, but changes the nature of the ground state via level crossings and thus has a strong impact on the dynamical properties, in particular at low temperatures.
At h ≥ h S = 2J the ground state of H has all spins aligned in field direction: |F ≡ | ↑↑ · · · ↑ is the reference state of the coordinate Bethe ansatz, and all eigenstates are described as excitations of interacting magnons, a species of spin-1 quasi-particles. Hence |F plays the role of the magnon vacuum. The ground state |A of H at h = 0 contains N/2 magnons. The Bethe ansatz enables us to reconfigure this state as the physical vacuum for a different species of quasi-particles -the spinons, which have spin 1 2 . The entire spectrum of the Heisenberg model (1) can also be generated as composites of interacting spinons. 3 Both descriptions are valid throughout the spectrum, but the magnon interpretation is more useful near the magnon vacuum, and the spinon picture is more useful near the spinon vacuum. The interaction energy of magnon scattering states or spinon scattering states is of O(N −1 ) as long as the number of quasi-particles in the collective excitations is of O(1). 4 In a macroscopic system, the spectrum of such states is thus indistinguishable from the corresponding free quasi-particle states. Even under these simplifying circumstances, however, the interaction of the quasi-particles remains important in the make-up of collective wave functions, and is likely to strongly affect the transition rates and lineshapes.
At intermediate values 0 < h < h S of the magnetic field, the number of magnons or spinons contained in the ground state of H is of O(N ), implying that the interaction energy for either quasi-particle species remains nonzero for N → ∞ in the ground state and in all lowlying excitations. This obscures the role of individual magnons or spinons in the collective excitations and ob-structs the interpretation of spectral data obtained by experimental or computational probes.
We can circumvent this problem by configuring the ground state |G at 0 < M z /N < 1 2 as the physical vacuum for yet a different species of quasi-particles. From the new vantage point, the dynamically relevant collective excitations are then again scattering states of few quasi-particles with an interaction energy of O(N −1 ), which greatly facilitates the interpretation of the spectra probed experimentally or computationally.

II. DYNAMIC STRUCTURE FACTOR
In an inelastic neutron scattering experiment performed at low temperature, the observable scattering events predominantly involve transitions from the ground state to a subset of collective excitations filtered from the rest by selection rules and transition rates. Under idealized circumstances, the scattering cross section is proportional to the T = 0 dynamic spin structure factor where S µ q = N −1/2 n e iqn S µ n , µ = x, y, z is the spin fluctuation operator. In a macroscopic system, the aggregate of spectral lines in (2) pertaining to scattering events with energy transfer ω λ ≡ E λ − E G , momentum transfer q ≡ k λ −k G , and transition rate | G|S µ q |λ | 2 form characteristic patterns of spectral weight in (q, ω)-space. The shape of the spectral weight distribution provides key information on how the dynamically relevant collective excitations are composed of quasi-particles with specific energy-momentum relations.
Experimentally it is possible, at least in principle, to separate the information contained in the dynamic structure factors of the spin components parallel and perpendicular to the field direction, i.e. the functions S zz (q, ω) and S xx (q, ω) = 1 4 [S +− (q, ω) + S −+ (q, ω)], respectively, for the fluctuation operators S z q and S ± q = S x q ± iS y q . At h = 0 the additional symmetry of H dictates that S zz (q, ω) = 1 2 S +− (q, ω) = 1 2 S −+ (q, ω). An anchor point for the new results presented in the following is the exact 2-spinon dynamic spin structure factor at T = 0, which was determined recently via algebraic analysis and shown to contribute 73% of the total intensity in S zz (q, ω) at h = 0. 5 Given the energymomentum relation 2 of the spinon quasi-particle, the 2-spinon states with wave numbers q = p 1 + p 2 and energy ω = ǫ sp (p 1 ) + ǫ sp (p 2 ) form a continuum confined by the boundaries 6,7 as illustrated in Fig. 1 (inset). The main plot shows the exact 2-spinon lineshapes of S zz (q, ω) at q = π/2, 3π/4, π. The most detailed experimental data available for testing these results pertain to KCuF 3 . 8 We shall see that the magnetic field causes dramatic changes in both the spectrum and the lineshapes. At the root of these changes is a change in the nature of the relevant quasi-particles. Two compounds suitable for studying magnetic-field effects on spectrum and lineshapes are Cu(

III. BETHE ANSATZ EQUATIONS
The Bethe ansatz 1 is an exact method for the calculation of eigenvectors of integrable quantum many-body systems. The Bethe wave function of any eigenstate of (1) in the invariant subspace with r = N/2−M z reversed spins relative to the magnon vacuum, |ψ = 1≤n1<...<nr≤N a(n 1 , . . . , n r )S − n1 · · · S − nr |F , (5) has coefficients of the form a(n 1 , . . . , n r ) = determined by r magnon momenta k i and one phase angle θ ij = −θ ji for each magnon pair. The sum P ∈ S r is over the permutations of the labels {1, 2, . . . , r}. The consistency requirements for the coefficients a(n 1 , . . . , n r ) inferred from the eigenvalue equation H|ψ = E|ψ and the requirements imposed by translational invariance lead to a set of coupled nonlinear equations for the k i and θ ij . A computationally convenient rendition of the Bethe ansatz equations has the form where φ(z) ≡ 2 arctan z, k i = π − φ(z i ) and θ ij = π sgn[ℜ(z i − z j )] − φ (z i − z j )/2 . Every solution of (7) is specified by a set of Bethe quantum numbers I 1 < I 2 < · · · < I r , which assume integer values for odd r and half-integer values for even r. The energy and wave number of the eigenvector thus determined are where E F = JN/4 is the energy of the magnon vacuum. We consider the class K r of eigenstates whose Bethe quantum numbers comprise, for 0 ≤ r ≤ N/2 and 0 ≤ m ≤ N/2 − r, all configurations Here we employ the solutions {z i } of the Bethe ansatz equations not only to generate spectral data via (8), which is standard practice, but also to evaluate transition rates | G|S µ q |λ | 2 for the dynamic structure factor (3) directly from the normalized Bethe wave functions |λ ≡ |ψ /||ψ||. The computational aspects of this method are discussed elsewhere. 11

IV. PHYSICAL VACUUM AND QUASI-PARTICLES
The ground-state wave function |G at 0 ≤ M z ≤ N/2 is specified by the set of r = N/2 − M z Bethe quantum numbers 12 As the magnetic field increases from h = 0 to h S = 2J, the magnetization M z increases in units of one from zero to N/2. A sequence of level crossings produces a magnetization curve (M z /N versus h) in the form of a staircase with N/2 steps of height 1/N , which converges toward a smooth line as N → ∞. 13,14,15 Depending on the reference state used for the characterization of the ground state |G , it can be regarded as a scattering state of N/2 − M z magnons excited from the magnon vacuum |F or as a scattering state of 2M z spinons excited from the spinon vacuum |A . To illustrate the distinct roles played by the two species of quasiparticles in the class-K r states, we show in Fig. 2  spinons ( ) by the vacancies across the full range of the I i 's allowed by (9) for class K r states.
Henceforth we treat |G as the new physical vacuum. At h = 0 (top row in Fig. 2) it coincides with the spinon vacuum, a state with N/2 magnons. At h = h S (bottom row) it coincides with the magnon vacuum, a state containing N spinons. All states within class K r are generated from |G by rearranging the magnons or (equivalently) the spinons into all allowed configurations.
For r = N/2 (top row) and r = 0 (bottom row) the state shown is the only possible configuration within class K r . In the fourth row, the lone magnon can be moved across the array of spinons, generating a branch (oneparameter set) of 1-magnon excitations for N → ∞. In the second row, the two spinons can be moved independently across the array of magnons, generating a continuum (two-parameter set) of 2-spinon excitations for N → ∞ with boundaries (4) as shown in Fig. 1. The center row in Fig. 2 pertains to the field at half the saturation magnetization (M z = r = N/4), the case we shall investigate extensively for various system sizes. Here |G contains twice as many spinons as it contains magnons.
The integer m with range 0 < m ≤ M z used in (9) is a convenient quantum number for the subdivision of the classes K r . Every state of K r at fixed m can then be regarded as a scattering state of m pairs of spinon-like quasi-particles. To distinguish them from the spinons, we name the new quasi-particles psinons.
The ground state |G , the only state with m = 0, is the psinon vacuum. Here the magnons form a single array flanked by two arrays of spinons (see Fig. 2). Relaxing the constraint in (9) from m = 0 to m = 1 yields a twoparameter set of states -the 2-psinon excitations. Here the array of magnons breaks into three clusters separated by the two innermost spinons, which now assume the role of psinons. The remaining 2M z −2 spinons stay sidelined. In the 4-psinon states (m = 2), two additional spinons have been mobilized into psinons. By this prescription, we can systematically generate sets of 2m-psinon excita- To illustrate the quasi-particle role of the psinons in denotes the wave number of an incommensurate soft mode. The lower 4-psinon spectral boundary is the same as the 2-psinon lower boundary but extended periodically over the entire Brillouin zone. The upper 4-psinon boundary is related to the upper 2-psinon boundary by a scale transformation (q → 2q, ω → 2ω).
The relationship between the ranges in (q, ω)-space of the 2-psinon states and the 4-psinon states does indeed reflect the fact that they are scattering states of two or four quasi-particles, respectively, of the same species. Like the spinon, the psinon is not observable in isolation via neutron scattering, but its energy momentum relation ǫ ψ (p), −π/4 ≤ p ≤ π/4, can be inferred from the data of Fig. 3 (see inset).
If there were no psinon interaction, the wave number and energy of a 2m-psinon state would be q The N = 16 data make it quite clear that the finite-size energy correction caused by the psinon interaction is stronger in the 4-psinon states than in the 2-psinon states. In both sets of collective states, the interaction energy goes to zero as the scattering events become less and less frequent in a chain of increasing length. However, it takes longer chains for finite-N 4psinon data to reach comparable convergence toward the spectral boundaries predicted for N → ∞, because for fixed N , the scattering events between psinons are more numerous in a typical 4-psinon state than in a typical 2-psinon state.
If instead of the psinon vacuum we had used the spinon vacuum as the reference state at M z = N/4, then both the 2-psinon states and the 4-psinon states would have to be described as scattering states of N/2 spinons. Although we know the energy-momentum relation of a spinon, Eq. (3), it is of little use to determine the spectral threshold in Fig. 3. Since the 2-psinon and 4-psinon states maintain a finite density of spinons in the limit N → ∞, the spinon interaction energy remains significant. This problem does not arise at M z = 0. In the 2-spinon scattering states depicted in Fig. 1, the spinon interaction energy vanishes for N → ∞ just as the psinon interaction energy does in the 2-psinon and 4-psinon scattering states depicted in Fig. 3. 16

V. DYNAMICALLY RELEVANT EXCITATIONS
At M z = 0 the spectral weight in the dynamic spin structure factor S zz (q, ω) is dominated by the 2-spinon excitations. 5 Our task here is to determine how the spectral weight of S zz (q, ω) at M z = 0 is distributed among the 2m-psinon excitations. In investigating this question, we follow the strategy of an older study 17 but with vastly improved conceptual and numerical tools.
We begin by exploring, in a chain of N = 16 spins at M z /N = 1 4 , the transition rates between the ground state |G and all 2m-psinon excitations for m = 0, 1, 2, 3, 4. The Bethe quantum numbers of the states with m = 0, 1 are shown in Fig. 4. The first row represents the psinon vacuum with its four magnons sandwiched by two sets of four spinons. The two innermost spinons (marked grey) become psinons when at least one of them is moved to another position. In the rows underneath, the psinons are moved systematically across the array of magnons while the remaining spinons stay frozen in place. These eight configurations describe all 2-psinon states with q ≥ 0.
The wave numbers, energies, and transition rates of the states shown in Fig. 4 are listed in Table I. Remarkably, almost the entire 2-psinon spectral weight is concentrated in the lowest excitation for any given q. The dynamically dominant 2-psinon states are marked by solid circles in Fig. 3. In a macroscopic system, they form the lower boundary of the 2-psinon continuum.
Next we calculate the transition rates | G|S z q |λ | 2 for the complete set of 4-psinon states. Interestingly, we observe that most of the 4-psinon spectral weight is again carried by a single branch of excitations. The dynamically dominant 4-psinon states for N = 16 are shown as full squares in Fig. 3. For large N they form a branch adjacent to the 2-psinon spectral threshold.    Table II. All other 2m-psinon excitations have transition rates that are smaller by at least two orders of magnitude at q < π/2, and still by more than one order of magnitude at q ≥ π/2. Inspection of Fig. 5 reveals an interesting pattern, indicative of the composition of the dynamically relevant collective excitations. They form a two-parameter set. The two parameters are highlighted by grey circles. Hitherto we have interpreted each group of four configurations as a branch of 2m-psinon excitations, which are seemingly arbitrary one-parameter subsets taken from 2mparameter sets of states. In a macroscopic system, all but the lowest such branches contain a macroscopic number of psinons. Hence the range of the dynamically relevant excitations in (q, ω)-space cannot be inferred from the psinon energy-momentum relation alone as was possible for the 2-psinon and 4-psinon continua, because the psinon interaction energy will remain non-negligible in most of these states for N → ∞, just as the spinon interaction energy was non-negligible in the 2-psinon and 4-psinon scattering states at M z = 0. A more natural interpretation of the pattern on display in Fig. 5 identifies one of the two parameters as a psinon (large grey circle) as before and the other parameter as a new quasi-particle (small grey circle). The latter is represented by a hole in what was one of two spinon arrays in the psinon vacuum. Instead of focusing on the cascade of psinons (mobile spinons) which this hole has knocked out of the vacuum, we focus on the hole itself, which has properties commonly attributed to antiparticles. The psinon (ψ) and the antipsinon (ψ * ) exist in disjunct parts of the psinon vacuum, namely in the magnon and spinon arrays, respectively. When they meet at the border of the two arrays, they undergo a mutual annihilation, represented by the step from the second row to the top row in Fig. 5.
We could have interpreted the small grey circle as a magnon (spin-1 quasi-particle), but when we do that  0, 1, . . . , 4). The latter form the ψψ * continuum in the limit N → ∞. The wave numbers q ≡ k − k G ≥ 0 are in units of 2π/N . we must take into account that it then coexists in the magnon vacuum with a macroscopic number of fellow magnons (small black circles). From this perspective, the collective excitation must be viewed as containing a finite density of magnons (for N → ∞), in which the magnon interaction remains energetically significant for scattering states. The nonzero interaction energy obscures the role of individual magnons.
On the other hand, when the small grey circle is interpreted as an antipsinon, then it lives in the psinon vacuum, i.e. almost in isolation. The only other particle present is a psinon (large grey circle). In a macroscopic system, the interaction energy in a psinon-antipsinon (ψψ * ) scattering state becomes negligible. Therefore, the identity of both quasi-particles is easily recognizable in the spectrum.
The energies versus the wave numbers of the 16 ψψ * states listed in Table II are shown in Fig. 6(a) as large symbols. The four branches from bottom to top pertain to m = 1, . . . , 4. Also shown in the same plot are the ψψ * states for N = 256. The lower boundary of the ψψ * continuum emerging in the limit N → ∞ touches down to zero frequency at q = 0 and q = q s = π/2. Between q s and π, it rises monotonically and reaches the value E − E G = h. A direct observation of the incommensurate soft mode at q s was made in a neutron scattering experiment on Cu(C 6 D 5 COO) 2 ·3D 2 O (copper benzoate). 9 Figure 6(b) shows the relative integrated intensity of the ψψ * excitations for various N at fixed M z /N = 1 4 . At q q s = π/2, virtually all spectral weight of S zz (q, ω) originates from ψψ * fluctuations. An extrapolation of the data points at q = π/2 suggests that the relative ψψ * spectral weight is in excess of 93%. At q q s the ψψ * contribution to the integrated intensity decreases monotonically but stays dominant over more than half the distance to the zone boundary. The width of the ψψ * continuum vanishes linearly on approach of q = π, and the relative spectral weight more slowly: S zz (q) ∼ (π −q) γ , γ ≃ 0.3. This enhances the observability of the ψψ * excitations in the narrow energy range near the Brillouin zone in spite of the low absolute intensity. Finite-N data for the integrated intensity S zz (q) are shown in the inset to Fig. 6(b). This function is peaked at q = q s , where the ψψ * spectral weight is overwhelmingly predominant.
When we lower M z , the soft mode at q s moves to the right, the number of 2m-psinon branches that contribute to the ψψ * continuum shrinks but each branch gains additional states. At M z = 1 we are left with one 2-psinon branch extending over the interior of the entire Brillouin zone. This branch is equal to the lowest branch of 2spinon states with dispersion ǫ L (q), Eq. (4). However, even for this case the psinon vacuum is different from the spinon vacuum. The former is the lowest-energy 2-spinon state (with M z = 1), whereas the latter is a state with M z = 0. The wave number of the two vacua differ by π. At M z = 0 the ψψ * excitations disappear altogether. The limit h → 0 of the infinite chain is very subtle and will be discussed elsewhere. 18 When we increase M z toward the saturation value, the soft mode moves to the left, and the number of 2m-psinon branches increases, but each branch becomes shorter. At M z = N/2 − 1, the two-parameter set collapses into a one-parameter set consisting of one 2m-psinon state each for m = 1, 2, . . . , N/2 − 1. These states are more naturally interpreted as a branch of 1-magnon excitations with dispersion ǫ 1 (q) = J(1 − cos q). Their relative spectral weight in S zz (q, ω) is now 100%, but the absolute intensity for q = 0 is only of O(N −1 ). To further illustrate the roles of the psinon and the antipsinon as the relevant quasi-particles in the collective excitations dominating the spectral weight in S zz (q, ω), we compare in Fig. 7 the energies between the ψψ * scattering states for N = 64 and the corresponding (fictitious) free ψψ * superpositions. The vertical displacement of any (•) from the associated (+) reflects the interaction energy between the two quasi-particles. This energy approaches zero for all states of this class as N → ∞.
The energy-momentum relations of the two quasiparticles can be accurately inferred from N = 2048 data for the spectral thresholds of the ψψ * states as illustrated in the inset to Fig. 7. The psinon dispersion ǫ ψ (p) is confined to the interval at 0 ≤ |p| ≤ π/4 (solid line) and the antipsinon dispersion ǫ ψ * (p) to π/4 ≤ |p| ≤ 3π/4 (dashed line). The different ranges of momentum which the two quasi-particles are allowed to have correspond to the different regions in Fig. 5 across which the circles pertaining to them can be varied.
The lower boundary of the ψψ * continuum is defined by collective states in which one of the two particles has zero energy: the psinon for 0 ≤ |q| ≤ π/2 and the antipsinon for π/2 ≤ |q| ≤ π. The upper boundary consists of three distinct segments.
Here the shape of the continuum boundary is that of the psinon dispersion. Likewise, for 3π/4 ≤ q ≤ π, the states along the upper continuum boundary are made up of a maximum-energy antipsinon (with momentum p ψ * = 3π/4 and a psinon with momentum p ψ = −3π/4 + q. Here the shape of the continuum boundary is that of the psinon dispersion. When these two delimiting curves are extended into the middle segment, 0.3935 q ≤ 3π/4, they join in a cusp singularity at q = π/2. Here the highest ψψ * state does not involve any zero-energy quasi-particles. The maximum of ǫ ψ (p ψ ) + ǫ ψ * (p ψ * ) subject to the constraint p ψ + p ψ * = q does not occur at the endpoint of any quasi-particle dispersion curve. Consequently, the ψψ * continuum is partially folded about the upper continuum boundary along the middle segment.

VI. LINESHAPES
To calculate the lineshapes relevant for fixed-q scans in an inelastic neutron scattering experiment from the spectrum and matrix elements obtained via Bethe ansatz, we exploit key properties of transition rates and densities of states of sets of excitations that form twoparameter continua in (q, ω) space for N → ∞. The ψψ * transition rates (scaled by N ) form a continuous function M ψψ * zz (q, ω) for N → ∞. The ψψ * density of states (scaled by N −1 ) becomes a continuous function D ψψ * (q, ω) for N → ∞. The ψψ * spectralweight distribution is then the product S ψψ * zz (q, ω) = D ψψ * (q, ω)M ψψ * zz (q, ω). 19 In the following, we consider three wave numbers at M z = N/4.
At q = π/2, the ψψ * continuum is gapless and the relative ψψ * spectral weight in S zz (q, ω) has a maximum. The scaled density of ψψ * states is generated from N = 2048 data of the set of points where ν * = m marks the antipsinon quantum number in the ψψ * continuum and picks the dynamically relevant branch from the set of 2m-psinon states. The psinon quantum number ν is adjusted to keep the wave number q of the ψψ * state fixed. This choice of labels produces an ordered sequence of levels. Starting at ω = 0, the graph of D ψψ * (π/2, ω ν * ) rises from a nonzero value very slowly up to near the upper band edge, where it bends into a square-root divergence as shown in Fig. 8(a). The divergence is produced by a maximum of the sequence ω ν * at the fold of the ψψ * -continuum. 20 In Fig. 8(b) we show finite-N data at q = π/2 for the scaled transition rates  (13) between the psinon vacuum and the ψψ * states at q = π/2 for N = 12, 16,20,24,28,32. (c) Lineshape at q = π/2 of the ψψ * contribution to S zz (q, ω). All results are for M z = N/4.
These data compellingly suggest the existence of a smooth function M ψψ * zz (π/2, ω) for the ψψ * transition rates in the limit N → ∞, which further highlights the physical significance of the psinon and the antipsinon as relevant quasi-particles in this situation. The function M ψψ * zz (π/2, ω) is monotonically decreasing with a divergence at ω = 0 and a cusp singularity at the upper band edge ω U ≃ 1.679J.
The product of the transition rate function and the (interpolated) density of states is shown in Fig. 8(c). 21 The curve fitted through the data points represents the ψψ * lineshape at q = π/2 in S zz (q, ω). Its most distinctive feature is the double peak due to apparent divergences at both band edges.
The divergence at ω = 0, which is caused by the matrix elements, is a power law, ω −α , with an exponent that is exactly known from field theoretic studies of the Heisenberg model. 22,23,24 For the situation at hand, the value is α = 0.4688 . . . . The divergence at ω U is caused by the diverging density of states but is weakened if the cusp singularity of M ψψ * zz (π/2, ω) starts from zero at ω = ω U . The expectation is a power-law singularity, (ω U − ω) −β with an exponent 0 ≤ β ≤ 1 2 . It is interesting to compare the ψψ * transition rate function M ψψ * zz (π/2, ω) at M z = N/4 inferred from the Bethe ansatz with the 2-spinon transition rate function M (2) zz (π, ω) at M z = 0 calculated via algebraic analysis. 5 The shape of both functions is similar, but there are some differences: M (2) zz (π, ω) has a stronger power-law divergence at ω = 0 and it approaches zero more rapidly at the upper band edge. As a result it produces a monotonically decreasing spectral-weight distribution S (2) zz (π, ω) (see Fig. 1) notwithstanding the fact that the 2-spinon  (13) between the psinon vacuum and the ψψ * states at q = π/4, 3π/4 for N = 16, 24, 32. (c) Lineshape at q = π/4, 3π/4 of the ψψ * contribution to S zz (q, ω). All results are for M z = N/4. density of states is also a monotonically increasing function terminating in a square-root divergence.
At q = π/4 the integrated intensity S zz (q) is only a third of what it was at q = π/2, but spread over a narrower range of frequencies (see Fig. 6). The bandwidth has shrunk to less than a third of the value it had at q = π/2. The relative ψψ * contribution to the intensity is even larger than at q = π/2, almost 100%. In this application, the method of analysis is stretched more closely to its limits because q = π/4 exists in fewer manageable system sizes. However, the data still make reliable lineshape predictions possible.
The density of states D ψψ * zz (π/4, ω), plotted in Fig. 9(a), rises discontinuously from zero to a finite value at the spectral threshold, ∆E ≃ 0.379J. From there it increases gradually with gradually increasing slope and ends in a cusp singularity at the upper band edge. 25 The finite-N data for the scaled transition rates shown in Fig. 9(b) again suggest a smooth ω-dependence in the form of a monotonically decreasing curve with enhanced steepness near both band edges. However, the countertrend of the density of states at the upper band edge is of sufficient strength to produce a second maximum in the line shape again.
Also shown in Fig. 9 are the corresponding data for the ψψ * density of states, transition rates, and lineshape at q = 3π/4. Here the relative spectral weight carried by the ψψ * excitations is only 83% of the value at q = π/2, but that fraction is concentrated over a frequency band that has shrunk to 65% of the width at q = π/2, while the absolute intensity remains fairly high (87% of the value at q = π/2). Both quantities, which determine the ψψ * lineshape, exhibit similar frequency dependences as we have already observed for the other two fixed-q scans. The density of states is divergent again at the upper boundary. The energy gap is now much larger, ∆E ≃ 0.899J. The fact that the lower continuum boundary at q = 3π/4 coincides with the upper continuum boundary at q = π/4 is a consequence of the quasi-particle dispersions as discussed previously.

VII. CONCLUSION
The spectrum of the completely integrable 1D s = 1 2 Heisenberg antiferromagnet (1) can be generated in more than one way from multiple excitations of quasi-particles. The external magnetic field controls the nature of the ground state. In strong fields, it becomes the vacuum of magnons and in zero field the vacuum of spinons. The dynamically relevant collective excitations of specific quantum fluctuations in the two cases are then naturally described as composites of quasi-particles from the respective species and are likely to involve only a small number of quasi-particles.
In intermediate magnetic fields, neither the magnons nor the spinons provide a useful interpretation of dynamically relevant collective excitations for the same fluctuation operators. The ground state itself contains a macroscopic number of quasi-particles from one or the other of the two species. However, when it is reconfigured as the physical vacuum for psinons and antipsinons, then it turns out that the spin fluctuation operator S z q induces predominantly transitions to ψψ * states, which contain just one particle from each kind.
Like the magnon and the spinon, the psinon and the antipsinon are interacting quasi-particles in the Heisenberg model (1). In the ψψ * scattering states, the interaction energy of the psinon and the antipsinon is of order O(N −1 ) whereas the interaction energy among magnons or spinons is of order O(1). Hence, for N → ∞, the ψψ * states join up in (q, ω)-space to form a two-parameter continuum whose spectral boundaries and density of states are fully determined by the energy-momentum relations of the psinon and the antipsinon. Moreover, the scaled ψψ * transition rates converge for N → ∞ toward a smooth function of q and ω.
We have exploited these asymptotic quasi-particle properties to extract lineshape information for the dynamic structure factor S zz (q, ω), which probes the spin fluctuations parallel to the applied magnetic field. The same quasi-particles will also play a dominant role in the spin fluctuations perpendicular to the field, but here different combinations of them make up the composition of the dynamically relevant collective excitations. In the dynamic spin structure factor S −+ (q, ω), for example, the spectral weight is almost completely carried by 2-psinon excitations. 18 In all likelihood, the psinon quasi-particles will also be useful for the analysis of thermal spin fluctuations in this model system. The peculiar spectral weight distributions found in recent complete diagonalization studies 26,27 of S zz (q, ω) at h = 0 and T > 0, for example, indicate the presence of stringent selection rules between collective states coupled by the spin fluctuation operator S z q . In zero field, psinon vacua are densely spread across the entire energy range of the model. Each psinon vacuum can be used as the reference state of a 2m-psinon expansion (9). If there are general selection rules related to psinon quasi-particles among transition rates | λ ′ |S z q |λ| 2 within a given class K r of Bethe ansatz solutions, they will have a strong impact on the spectral weight distribution in S zz (q, ω) at all temperatures.

ACKNOWLEDGMENTS
We thank Andreas Klümper, Klaus Fabricius, and Alexander Meyerovich for interesting and useful discussions.
Financial support from the URI Research Office (for G.M.) and from the DFG Schwerpunkt Kollektive Quantenzustände in elektronischen 1D Ubergangsmetallverbindungen (for M.K.) is gratefully acknowledged.