arXiv:cond-mat/0309247v1 [cond-mat.str-el] 10 Sep 2003 eh nazsltosars h lnrregime, planar the across solutions ansatz Bethe erblt rsrig xenlmgei field. magnetic external preserving) tegrability p h dniyo h omri rsre yconser- by preserved is interaction. mutual former their the notwithstanding of laws vation identity The close investigated produce be up. that can excitations configurations collective particular quasiparticle the integrable completely systems, In composition. including and quasiparticle excitations thorough vacuum their collective physical relevant a the dynamically is of the nature dy- systems the low-temperature of many-body understanding the quantum for of data namics computational or tal neatn opstso emos nesadn the Understanding fermions. of composites interacting = ∆ case the the for ( eigenstates of all 0 structure of general solutions ansatz the Bethe of identification the namely egatfroant( antiferromagnet berg osdrbeiprac ntesuyo the of study the in importance considerable their quasiparticles, interactions. studying their for and framework transformations ansatz ideal Bethe the an situations, is config- both In quasiparticle the differently. and urations vacuum physical the affects nstoyi h 1D the the pre- controls in which anisotropy also ∆, anisotropy parameter The exchange integrability. uniaxial serves a instead ducing quasipar- spinons, relevant (magnons, dynamically ticles the and vacuum physical well. as change to collective subject the the in are Along quasiparticles excitations of transition. configurations phase the physical quantum way, the a changes how across occasionally, observe abruptly and, to gradually transforms possible vacuum pa- it continuous make on rameters depend that Hamiltonians Integrable H h e oamaigu nepeaino experimen- of interpretation meaningful a to key The t∆=0alsae a ecaatrzda non- as characterized be can states all 0 = ∆ At h anfcsi hsppri natcnclpitof point technical a on is paper this in focus main The eety eivsiae h eaopoi fthe of metamorphosis the investigated we Recently, XX = . n X N =1 1 oe) hswl aiiaetakn all tracking facilitate will This model). pcrmadtasto ae fthe of rates transition and Spectrum egsh nvri¨tWpetl aheec Naturwiss Fachbereich Universit¨at Wuppertal, Bergische  S | rniinrtsfrcan ihu to ap up We with expressions. chains product for to rates them transition reduce fer and lattice functions rates wave the transition for and expressions disc determinantal ansatz) We present We (Bethe momenta. quasiparticles soluti magnon magnon complex ansatz and or Bethe real the with of states structure include general the identify We n x ∆ S spr fasuyta netgtstednmc fthe of dynamics the investigates that study a of part As | n x < .INTRODUCTION I. +1 3 2 ,w ics h iglrntr fteBteast equatio ansatz Bethe the of nature singular the discuss we 1, egsh nvri¨tWpetl aheec hsk D- Physik, Fachbereich Universit¨at Wuppertal, Bergische + eateto hsc,Uiest fRoeIln,Kingsto Island, Rhode of University Physics, of Department ailBiegel, Daniel s S XXX = n y S ψ 2 1 n y +1 , XXZ ψ oe)b ayn h (in- the varying by model) ∆ + ∗ fte1D the of ) S model, 1 n z S ihe Karbach, Michael n z +1 5

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N r Now let us assume that among the set of rapidities c2yi + i c1(yi yj ) + i(1 yiyj ) = − − (2.5) y1,...,yr, there is one critical pair, c2yi i c1(yi yj ) i(1 yiyj )  −  j=6 i − − − Y 0 0 . . yj yj∗ =1 kj + kj∗ = π (mod 2π), (2.11) for i =1, . . . , r, with c1 = cot γ and c2 = cot(γ/2). Tak- ⇔ ing the logarithm yields and that it is a real pair. Substituting the ansatz

r 0 yi yj y = y + c δ , i =1, . . . , r, (2.12) Nφ(c y )=2πI + φ c − i =1, . . . , r, i i 1 i 2 i i 1 1 y y 6 i j Xj=i  −  into Eqs. (2.6) and taking the limit c 0 then yields the (2.6) 1 . non-critical and critical rapidities from→ successive orders with φ(x) = 2 arctan(x). Expressions (2.2) for energy in a c -expansion. The non-critical rapidities y , i = j, j∗ and wave number now read 1 i 6 are given by Eqs. (2.9) as in regular solutions. The c1- r −1 expansion of Eqs. (2.6) leads to the following equation 2 yi yi 0 0 E = −1 −1− , (2.7) determining the critical rapidities yj ,yj∗: − (c2y ) + c2y c2 + c2 i=1 i i X r − σj 2π − 2 σj yj k = πr Ii. (2.8) φ(y )= φ . (2.13) − N j i=1 N " 1 ξj # ! X − The trigonometric Bethe ansatz equations (2.6) have the ± 1 0 0 Here we have introduced y = (y y ∗ ) and we use advantage that each solution is characterized by a set j 2 j ± j of (integer or half-integer) Bethe quantum numbers Ii. r + . 2 yj sin ki These discriminating markers are used to count and clas- ξ = − , (2.14) j + −1 sify the solutions and are essential for numerical algo- N ∗ (yj ) sin ki i=6 j,j − rithms designed to find solutions. X y+ 2 j . Ij −Ij∗ sgn = (Ij + Ij∗ ), σj = ( 1) . (2.15) ξj ! −N − A. XX-limit The non-critical ki are from (2.9). Criticality implies − Here we describe the general structure of the solutions (y+)2 (y )2 = 1. Note that the contributions of the j − j of the Bethe ansatz equations (2.6) in the limit ∆ 0, critical rapidities to the energy (2.10) cancel out. Equa- → implying c1 0 and c2 1. There exist regular so- tion (2.15) implies that the Bethe quantum numbers of lutions and singular→ solutions.→ The latter are character- the critical pair must satisfy ′ ized by the occurrence of pairs of rapidities yi,yi with ′ the property yiyi = 1. For any such critical pair, the Ij + Ij∗ = N/2. (2.16) argument of φ on the right-hand side of Eqs. (2.6) is in- | | determinate and must, therefore, be treated as a limit The generalization of this solution to the case of s real process. We shall see that some limiting singular solu- critical pairs, tions are real while others are complex. An important 0 0 fact is that the simplified structure of the Bethe ansatz y y ∗ =1, l =1, . . . s, (2.17) jl jl 3 with Bethe quantum numbers satisfying I +I ∗ = N/2 D. Generic case jl jl is straightforward. The non-critical rapidities| are| as in Eq. (2.9) and the critical pairs are determined from Suppose we have t critical pairs of complex conjugate solutions, y = u + iv = y∗,...,y = u + iv = y∗ , − σjl 1 1 1 2 2t−1 t t 2t ∗ 2 σjl y − jl and s pairs of critical real solutions, l = jl, jl 2t+ φ(yjl )= φ , l =1,...,s (2.18) J { }⊂{ N "1 ξjl # ! 1,...,r , l =1,...,s. Then the real solutions in the limit − c 0} are of the form 1 → with ξjl , σjl as defined in (2.14) and (2.15). s π y0 = tan I , i/ 1,..., 2t , (2.26) i N i ∈{ } ∪ Jl l=1 C. Complex critical pairs [

2 σ Next we consider the case of a single complex conjugate − − jl φ(yj )= φ σjl yj /(1 ξjl ) , l =1, . . . , s, critical pair l N l −  
(2.27) ∗ with y1 = y2 = u + iv (2.19) among the set of rapidities y ,...,y . Equations (2.6), r + 0 t + 0 −1 1 r . 2 yjl sin ki yjl (ui ) ξjl = − +2 − . rewritten as real equations, read  +−1 0 +−1 0 −1  N i=2t+1 y sin k y (u ) ∗ jl i i=1 jl i r  i6=Xjl,jl − X −  c1(yi yj)  (2.28) Nφ(c2yi) = 2πIi + φ − 1 y y For the complex solutions we obtain j=3  i j  Xj6=i − 2 0 N 0 2 2c [(y u)(1 y u)+y v ] 1+ v 1 ξi + v 1 i i i i = i , (2.29) +φ − − 2 (2.20) 0 − 0 (1 y u)2 +(y v)2 c [(y u)2+v2] 1 v 1 ξi v  − i i − 1 i −   − i   − − i  (u0)2 + (v0)2 = 1, i =1,...,t (2.30) for i =3,...,r and i i

r with 2c2u Nφ 2 2 2 =2π(I1+I2)+ [2πIi Nφ(c2yi)], 1 c2(u +v ) − r 0 0 t 0 0 −1  −  i=3 . 2 ui sin kj ui (uj ) X ξi = − +2 − . (2.21) N  0−1 0 (u0)−1 (u0)−1  j=2t+1 ui sin kj j=1 i j − j6=i − 2 2  X X  2c2v 4vc1(1 u v )  (2.31) Nϕ 2 2 2 = ϕ 2 −2 2 − 2 1+c (u +v ) (1 u v ) +(2vc1) Energy and momentum of the state are determined by  2   − −  r 2c v(1 y2) the non-critical roots alone: 1 − j + ϕ 2 2 2 2 2 (2.22) r (1 yju) +(yjv) +c [(yj u) +v ] 2π j=3 − 1 − ! E = cos I , (2.32) X − N i . i=2t+1 i6=j ,...,j∗   with ϕ(x) = 2atanh (x) for the critical pair. The XX X1 l 0 1 limit is performed along the path u = u c1u , v = r 0 1 − 2π v c1v . The only possible complex solutions that can k = π(r s t) Ii(mod 2π). (2.33) − − − − N i=2t+1 survive the limit c1 0 are critical pairs. The non- ∗ i6=jX1,...,j critical roots are again→ as in (2.9). The critical pair of l rapidities is now determined by the equation This prescription for handling Bethe ansatz solutions in the XX limit will guarantee that all XXZ Bethe eigen- 0 N 0 2 1+ v 1 ξ1 + v 2 2 states can be traced continuously across the point ∆ = 0. 0 = − 0 , u0 + v0 =1, (2.23) 1 v 1 ξ1 v Because of the higher symmetry at ∆ = 0 and the asso-  −   − −  ciated level degeneracies,12,13,14 the relationship between where the (real and complex) Bethe eigenstates and the (always r 0 real) Jordan-Wigner eigenstates of the XX chain is non- 2 u sin ki ξ − , (2.24) trivial. The two representations will now be compared 1 ≡ N (u0)−1 sin k i=3 i for all energy levels. X − u 2 sgn 0 = (I + I ). (2.25) ξ −N 1 2  1  III. SPINONS VERSUS FERMIONS 1 As is custom, we shall replace I1 + I2 by a single Bethe quantum number I(∗) N/2. We then have I(∗) = 0 for The full spectrum of the XXZ model can be accounted all complex critical pairs.± for as composites of interacting spinons with spin 1/2 and 4

16 semionic exclusion statistics. For the XX case an alter- where the allowed values of the fermion momenta pi de- native and equivalent interpretation of the complete spec- pend on whether the number Nf of fermions in the sys- trum can be established on the basis of noninteracting tem is even or odd: spinless lattice fermions. How are spinon configurations p (2π/N)(n +1/2) , (N even) (3.6a) related to fermion configurations in single non-degenerate i ∈ { } f eigenstates and in groups of degenerate eigenstates? pi (2π/N)n , (Nf odd), (3.6b) N ∈ { } Consider the 2 -dimensional Hilbert space for even for n = 0, 1,...,N 1. The number of fermions (0 N divided into subspaces characterized by n+ spinons − z ≤ Nf N) is related to the quantum number ST in the spin with spin up and n− spinons with spin down. Equivalent ≤ z representation: ST = N/2 Nf . No matter whether Nf quantum numbers are the total number of spinons and is even or odd, there are N− distinct one-particle states. the z-component of the total spin: Every one-particle state can be either empty or singly N z occupied, yielding a total of 2 distinct many-particle 2n = n + n−, 2S = n n−. (3.1) + T + − states with energy and wave number The dimensionality of each such subspace is16 Nf Nf E E = cos p , k = p . (3.7) − F i i i=1 i=1 dσ + nσ 1 W (n+,n−) = − , (3.2) X X nσ z σ The lowest-energy state in each ST -subspace is unique. Y   1 1 The fermion configuration in reciprocal space of each dσ = (N + 1) (nσ′ δσσ′ ), (3.3) lowest-energy state for N = 8 is shown in Fig. 1. 2 − 2 − σ′ X p 0 1 2 4 6 i 3 5 7 8 where σ = denotes the spinon polarization. Summing ± Sz = 4 W (n+, 2n n+) over n or n+ yields T − 3

N/2 2n 2 N N +1 W = , W = , (3.4) 1 n+ 2n 0 n=1   n+=0   X X −1 respectively. The double sum yields 2N . In Table I we −2 list the subspace dimensionalities for the case N = 8. −3 −4

TABLE I: Dimensionalities W (n+, n−) of the invariant sub- FIG. 1: Configuration in reciprocal space of the Nf = N/2 − z − z z spaces with 2n spinons and spin ST = n+ − n for a chain ST fermions in the lowest-energy eigenstate for given ST of of length N = 8. The equivalent quantum numbers in the a chain with N = 8 spins. The positions of the fermions are z z fermion representation are Nf = N/2 − ST , nc = n − |ST |. denoted by full circles, those of vacancies by open circles. The fermion momentap ¯i are in units of 2π/N. The vertical bars − z − ¯ n+ − n n = 0 n = 1 n = 2 n = 3 n = 4 n W are at wave numbers (also in units of 2π/N) kc = N/4−ST /2 ¯+ z and kc = 3N/4+ ST /2. 8 – – – – 1 P1 6 – – – 7 1 8 z 4 – – 15 12 1 28 For the further subdivision of each ST -subspace, we introduce the wave numbers 2 – 10 30 15 1 56 k± 1 Sz 0 1 16 36 16 1 70 c =1 + T (3.8) π ± 2 N −2 – 10 30 15 1 56   −4 – – 15 12 1 28 shown as vertical bars in Fig. 1, dividing the band into −6 – – – 7 1 8 two regions, one in the center, one in the wings. Given −8 – – – – 1 1 that the lowest-energy state at Sz 0 has all particles T ≥ W 1 36 126 84 9 256 in the center region, we can characterize all excitations n+ as nc-particle states. Likewise, all excitations from the P z lowest-energy state at ST 0 can be characterized as nc- hole states. The range of this≤ second quantum number is The XX Hamiltonian in the fermion representation, n =0, 1,...,N/2 Sz . The number of fermion states transformed from (1.1) at ∆ = 0 by the Jordan-Wigner c T characterized by n− |particles| or n holes in the above mapping to a system of free spinless fermions, reads17 c c sense is then found to be expression (3.2) previously used

† for spinons, now with Hf = cos pc cp, (3.5) p z z z p n = nc + S , n± = nc + S S . (3.9) X | T | | T |± T 5

z The next goal is to bring the mapping down to The two-spinon triplets with ST = 1 are easy to handle energy levels (at fixed wave numbers) within each numerically. All rapidities yi are finite and real. Two- z z (n+,n−)-subspace or, in the fermion language, (ST ,nc)- spinon triplets with ST = 0, by contrast, tend to pose subspace.18 We note that for given Sz 0 the num- some computational challenges. At ∆ = 1 all such states T ≥ ber of fermions is equal to the number of magnons in start out with one ki = 0 magnon, implying yi = . Bethe ansatz states. Moreover, the fermion energy- While this singular behavior is benign,20 numerical prob-±∞ momentum relation (3.7) is equivalent to the magnon lems are caused by the fact that in some states yi stays energy-momentum relation (2.10). The exception are the infinite at ∆ < 1, whereas in other states it becomes fi- critical magnon pairs, whose momenta are different (real nite. The majority of states from this set have one real or complex) from the corresponding fermion momenta critical pair at ∆ = 0. To ensure continuity of the Bethe (always real). Critical pairs only occur in degenerate ansatz solutions when ∆ is varied, it is sometimes neces- levels and do not contribute to the energy of the state. sary to change the values of two Bethe quantum numbers This mapping between energy levels provides a use- leaving their sum invariant. ful tool for determining the Bethe quantum numbers of The two-spinon singlets are far more difficult to handle eigenstates directly from the momenta of the fermion numerically. Each such state has one complex-conjugate configurations. For all non-critical rapidities, we have pair of rapidities. A numerical analysis of two-spinon sin- glets at ∆ = 1 was reported in Ref. 20. Again, continuity 2π of the solutions may make it necessary to change some Ii = π pi. (3.10) Bethe quantum numbers along the way. At ∆ = 0 each N − state from this set with one complex-conjugate critical For real critical pairs, the Bethe quantum numbers are, pair becomes degenerate with a two-spinon triplet state in general, shifted relative to the positions predicted by that has one real critical pair. (3.10), but in such a way that Ij + Ij∗ = N/2 is main- tained. Complex critical pairs| are specified| by a single Bethe quantum number I(∗) = 0 as discussed in Sec. IIC. In the following, we look more closely at the two-spinon B. XX limit excitations in the two representations. According to the scheme developed in Sec. III, the two- z spinon states with ST = 1 comprise all configurations in IV. TWO-SPINON SPECTRUM which no state with positive energy in the fermion band is occupied. These are the fermion configurations with A. Planar XXZ model nc = 0. The N/2 + 1 single-particle states which are accessible under this restriction and among which the N/2 1 fermions can be distributed does indeed produce Returning to the XXZ model (1.1), we note that there − z N(N + 2)/8 distinct eigenstates. All possible configu- exist two-spinon states with ST = 0, 1. At∆ = 1 ± rations for N = 8, generated systematically across the these states are either triplets (with total spin ST = 1) 1 allowed range in reciprocal space, are depicted in Fig. 2. or singlets (ST = 0). There are 8 N(N + 2) triplet 1 levels and 8 N(N 2) singlet levels, distinguishable by − 19,20 their Bethe quantum numbers. Integrability guaran- p 0 1 2 3 4 5 6 7 8 tees that each eigenstate anywhere in the planar regime i can be traced back to the point of higher symmetry with- k = 1 ε = 0.906 ε out ambiguity. This justifies that we use the ST multiplet 2 = 1.199 labels for states at ∆ < 1. 3 ε = 1.906 At finite N and ∆ < 1, the two-spinon triplet compo- 3 ε = 0.906 nents with Sz = 1 remain degenerate (because of reflec- 4 ε = 1.613 T ε tion symmetry in± spin space) but are no longer degener- 5 = 1.906 z 4 ε = 1.919 ate with the two-spinon triplet components with ST = 0. ε We shall see that at ∆ = 0, a subset of 1 N(N 2) 5 = 0.906 8 6 ε = 1.199 states from the latter set become degenerate with− the 7 ε = 0.906 two-spinon singlets. We have tracked the Bethe ansatz solutions of every Ii 43 2 1 0 −1 −2 −3 −4 two-spinon state for N = 8 from∆ =1 to ∆= 0, in- z deed across this highly singular point of the Bethe ansatz FIG. 2: Set of N(N + 2)/8 two-spinon states with ST =1 at equations all the way to ∆ = 1. Here we briefly report ∆ = 0 for N = 8 as specified by the fermion momenta (top on what we found along the− stretch 0 < ∆ < 1.21 For scale) or Bethe quantum numbers (bottom scale). The wave numbers k¯ and fermion momentap ¯i are in units of 2π/N. all states, solutions of the Bethe ansatz equations were − The dashed lines represent k¯ and k¯+. The energies are ǫ = found for which all magnon momenta ki (or all rapidities c c E − EG relative to the ground state with EG = −2.613. yi) vary continuously across the parameter regime. 6

The magnon momenta, rapidities, Bethe quantum nature of the two-spinon singlets and 1-string nature of numbers, and energies of these states, now sorted accord- the two-spinon triplets at ∆ = 1. ing to wave numbers, are listed in Table II. No critical In the states with real critical pairs, a solution may pairs can be formed in any of these states, which makes only exist for Bethe quantum numbers that are shifted it straightforward to calculate transition rates for fairly relative to those predicted by (3.10) but still satisfy long chains (see Sec. VI). (2.16). The latter are indicated in Fig. 3 by open cir- cles. In states with complex conjugate critical pairs, two of the Bethe quantum numbers (3.10) are replaced by a TABLE II: Wave number, energy, magnon momenta, Bethe single number I(∗) = 0 representing the critical pair. quantum numbers, and rapidities of the N(N + 2)/8 two- z spinon triplet components with ST = 1 at ∆ = 0 for N = 8. The actual Bethe ansatz solutions for N = 8 pertain- The quantities k,¯ k¯i are in units of 2π/N. The ground-state 2 z ing to all N /4 two-spinon states with ST = 0 are listed energy is EG = −2.613. in Table III (triplets) and Table IV (singlets). In some in- stances, different configurations of Bethe quantum num- ¯ ¯ ¯ ¯ k E − EG k1 k2 k3 I1 I2 I3 y1 y2 y3 bers lead to equivalent Bethe wave functions. Only one 22 1 0.906 2 3 4 2 1 0 1 0.414 0 configuration is represented in Fig. 3. 2 1.199 2 3 5 2 1 −1 1 0.414 −0.414 3 0.906 2 4 5 2 0 −1 1 0 −0.414 p 0 2 3 4 6 3 1.906 2 3 6 2 1 −2 1 0.414 −1 i 1 5 7 8 4 0.199 3 4 5 1 0 −1 0.414 0 −0.414 k = 0 ε = 0 4 1.613 2 4 6 2 0 −2 1 0 −1 5 0.906 3 4 6 1 0 −2 0.414 0 −1 3 ε = 1.307 5 1.906 2 5 6 2 −1 −2 1 −0.414 −1 4 ε = 1.848 6 1.199 3 5 6 1 −1 −2 0.414 −0.414 −1 5 ε = 1.848 7 0.906 4 5 6 0 −1 −2 0 −0.414 −1 6 ε = 1.307

4 ε = 0.765 z The two-spinon states with ST = 0 comprise all (nc = ε 1)-states relative to the ground state configuration in 5 = 1.307 the fermion band. Removing one of N/2 fermions from 6 ε = 1.307 the ground-state configuration and placing it into one of 7 ε = 0.765 the N/2 empty single-particle states yields N 2/4 distinct eigenstates but only N(N + 2)/8 distinct levels in the 1 ε = 0.765 (k, E)-plane because of degeneracies. All possible con- 2 ε = 1.307 figurations for N = 8 including the ground state, gener- ε ated systematically across the allowed range in reciprocal 3 = 1.307 space, are depicted as rows of full circles in Fig. 3. 4 ε = 0.765 There are N(N 2)/8 twofold degenerate levels in the 2 ε = 1.307 (k, E)-plane with the− two states distinguished by one pair of fermions. The two fermions in question (full circles 3 ε = 1.848 connected by a line in Fig. 3) add up to π mod(2π), 4 ε = 1.848 thus contributing nothing to the energy (3.7). Addition- 5 ε = 1.307 ally, there exist N/2 non-degenerate (nc = 1)-states at wave-numbers Nk/2π =1, 3,...,N 1 representing the highest-energy two-spinon state for those− wave numbers. I i 43 2 1 0 −1 −2 −3 −4 2 In summary, the N /4 (nc = 1)-states in the fermion FIG. 3: Ground state and set of N 2/4 = N(N + 2)/8 + representation at ∆ = 0 originate from the set of z z N(N − 2)/8 two-spinon states with ST = 0 for N = 8. The N(N + 2)/8 two-spinon triplets with ST = 0 and the set positions of the full circles in relation to the top scale denote of N(N 2)/8 two-spinon singlets. Each singlet becomes − the fermion momenta and those in relation to the bottom scale degenerate with a triplet at ∆ = 0, while N/2 triplets re- the Bethe quantum numbers predicted via Eq. (3.10). The main non-degenerate. When analyzed via Bethe ansatz, critical pairs among them are connected by a solid line. The the two states of each degenerate two-spinon level with open circles and squares denote the actual Bethe quantum z ST = 0 have one critical pair of magnon momenta as numbers associated with the critical pairs for real and complex described in Sec. II. Our analysis shows that the criti- rapidities, respectively. The wave numbers k¯ and fermion cal pair is real for the triplet state and complex for the momentap ¯i are in units of 2π/N. The dashed lines represent ¯− ¯+ singlet state. This is consistent with the known 2-string kc and kc . 7

TABLE III: Wave number, energy, magnon momenta, Bethe quantum numbers, and rapidities of the N(N + 2)/8 two-spinon z ¯ ¯ triplet components with ST = 0 at ∆ = 0 for N = 8. The quantities k, ki are in units of 2π/N.

k¯ E − EG k¯1 k¯2 k¯3 k¯4 I1 I2 I3 I4 y1 y2 y3 y4 0 0.000 2.5 3.5 4.5 5.5 1.5 0.5 -0.5 -1.5 0.668 0.199 -0.199 -0.668 1 0.765 2.5 3.5 4.5 6.5 1.5 0.5 -0.5 -2.5 0.668 0.199 -0.199 -1.497 2 1.307 2.5 3.5 5.110 6.890 1.5 0.5 -1.5 -2.5 0.668 0.199 -0.466 -2.147 3 1.307 2.5 4.5 4.728 7.272 1.5 -0.5 -0.5 -3.5 0.668 -0.199 -0.294 -3.402 1.848 2.5 3.5 5.5 7.5 1.5 0.5 -1.5 -3.5 0.668 0.199 -0.668 -5.027 4 0.765 3.5 4 4.5 0 0.5 -0.5 -0.5 -3.5 0.199 0 -0.199 -∞ 1.848 2.5 4 5.5 0 1.5 -0.5 -1.5 -3.5 0.668 0 -0.668 -∞ 5 1.307 0.728 3.272 3.5 5.5 3.5 0.5 0.5 -1.5 3.402 0.294 0.199 -0.668 1.848 0.5 2.5 4.5 5.5 3.5 1.5 -0.5 -1.5 5.027 0.668 -0.199 -0.668 6 1.307 1.110 2.890 4.5 5.5 2.5 1.5 -0.5 -1.5 2.147 0.466 -0.199 -0.668 7 0.765 1.5 3.5 4.5 5.5 2.5 0.5 -0.5 -1.5 1.497 0.199 -0.199 -0.668

TABLE IV: Wave number, energy, magnon momenta, Bethe quantum numbers, and rapidities of the N(N − 2)/8 two-spinon singlets at ∆ = 0 for N = 8. The quantities k,¯ k¯i are in units of 2π/N.

∗ (∗) k¯ E − EG k¯ k¯3 k¯4 I I3 I4 u v y3 y4 2 1.307 4 3.5 2.5 0 0.5 1.5 -0.765 -0.644 0.199 0.668 3 1.307 4 4.5 2.5 0 -0.5 1.5 -0.541 -0.841 -0.199 0.668 4 0.765 4 4.5 3.5 0 -0.5 0.5 0 1 -0.199 0.199 1.848 4 5.5 2.5 0 -1.5 1.5 0 1 -0.668 0.668 5 1.307 4 5.5 3.5 0 -1.5 0.5 0.541 0.841 -0.668 0.199 6 1.307 4 5.5 4.5 0 -1.5 -0.5 0.765 0.644 -0.668 -0.199

V. MATRIX ELEMENTS VIA BETHE ANSATZ FOR ∆ = 0 ±1 ± 2 ± ( y ) det Γ M ±(q)= Kr 1 { i} | | , We start from the determinantal expressions for the λ ( y0 ) det K( y ) det K( y0 )  r i  i i transition rates K { } { } { }(5.4) where µ 2 . ψ0 Sq ψλ M µ(q) = |h | | i| , µ = z, +, (5.1) λ ψ 2 ψ 2 − cos γ K(y ,y ) k 0k k λk a b : a = b N κ(ya) 6 K  r between XXZ eigenstates characterized by real ki for ab = cos γ K(y ,y ) (5.5) 1 a j : a = b the spin operators  − N κ(y ) 6 a Xj=a  1  Sµ = eiqnSµ, µ = z, +, . (5.2)  q √ n − N n X r 0 . 0 1 G(yj ,yb) These expressions, which were derived in Ref. 10 based Γab = FN (y ,yb) a G(y0,y ) G(y ,y ) on previous work reported in Refs. 6,7,8 have the form a b j=1 j b Y  r ∗ 0 0 Γ 2 1 2 1 G (yj ,yb) N r( y ) det + , (5.6) M z(q)= K { i } | − N | , (5.3) G∗(y0,y ) G∗(y ,y )  λ 4 ( y ) det K( y0 ) det K( y ) a b j=1 j b Kr { i} { i }
{ i} Y  8

0 r 0 Switching back from the non-critical rapidities via (2.9), + . 0 G(yr+1,yb ) G(yj ,yb ) Γ = FN (ya,y ) y = cot(k /2), to the non-critical magnon momenta k , ab b  G(y ,y0) G(y0,y0) i i i a b j=1 j b we can bring expression (5.4) into the form Y  ∗ 0 r ∗ 0 G (yr+1,y ) G (yj ,y ) r r+1 0 + b b , (5.7) cos2 ki +kj G∗(y ,y0) G∗(y0,y0) 2 a b j=1 j b + i=1 j=1 + 2 Y Mλ (q)= det , (5.19) r+1 Q Q r k0+k0  2 ki+kj 2 i j | S | cos 2 cos 2 i

r 2/N . − . : b =1 ...r 1 r( yi ) = K(yi,yj ) , (5.9) = sin k0 sin k − , a =1, . . . , r, K { } | | ab a b i

transitions the rate is 2 λ ω

) 8 π ( z 1 - λ Mλ (q)= (5.23) M 3/2 N ) 4 N π ( - λ 200 in agreement with a well-known result derived in the M 23 3/2 0 fermion representation. N 0 1 2 In the following application of the transition rate ex- ωλ pressions to a T = 0 spin dynamic structure factor of the XX model we have chosen a situation where excited states without critical rapidities are important. The cal- 0 culation of transition rates with critical pairs requires 0 1 2 further developmental work. ωλ

¯ (2) FIG. 4: Scaled two-spinon transition rates M−+(π,ω) = VI. TWO-SPINON TRANSITION RATES 3/2 − N Mλ (π) at ∆ = 0 (data for chains of size N = 512, 1024, 2048, 4096). The inset shows the same quantity 2 From recent studies in the framework of the algebraic multiplied by ωλ. analysis for the infinite chain,24,25 we know that the rel- ative integrated intensity of the two-spinon contribution to the dynamic structure factor

− (2) S−+(q,ω)=2π M (q)δ (ω ωλ) (6.1) λ − the two-spinon dynamic structure factor S−+(q,ω) van- Xλ ishes in the limit N . Hence the singularity struc- →¯ (2) ∞ probing the spin fluctuations perpendicular to the sym- ture of the function M−+(q,ω) has no direct bearing on metry axis of the XXZ model is 73% for the Heisenberg the singularity structure of S−+(q,ω). A distinct sin- case (∆ = 1) and steadily growing toward 100% on ap- gularity structure and spectral-weight distribution which proach of the Ising case (∆ = ). is a property of all 2m-spinon excitations combined will ∞ emerge in the limit N . The nonzero two-spinon intensity is reflected in the → ∞ reciprocal finite-N scaling behavior of the transition rates (2) − Consequently, the exactly known leading singularities M−+(q,ωλ)= NM (q) and the scaled density of states λ at ω = 0, 1, 2,... in the frequency-dependent spin auto- +π (2) −+ D (q,ωλ)=2π/[N(ωλ+1 ωλ)], (6.2) correlation function Φ0 (ω)= −π (dq/2π)S−+(q,ω), as − worked out in Ref. 27, for example, are not attributable R which makes the product to specific 2m-fermion excitations, because the integrated intensity of each 2m-fermion contribution taken in isola- S(2) (q,ω)= M (2) (q,ω)D(2)(q,ω) (6.3) tion is likely to vanish in the limit N . −+ −+ → ∞ converge toward a piecewise smooth function in the limit To reconstruct the spectral-weight distribution of N , representing the result of the infinite chain. S (q,ω) at ∆ = 0 and to determine its singularity → ∞ −+ A qualitatively different finite-N scaling behavior is structure in the limit N from finite-N data for found in the XX case (∆ = 0) for the two-spinon tran- excitation energies and transition→ ∞ rates we need to be sition rates contributing to S−+(q,ω). For the tran- able to properly handle Bethe wave functions with criti- sition rates to converge toward a non-vanishing piece- cal pairs of rapidities. We already know (Sec. II) how to wise smooth function they must be scaled differently: solve the Bethe ansatz equations for all eigenstates in the ¯ (2) 3/2 − M−+(q,ω) = N Mλ (q). This is illustrated in Fig. 4 limit ∆ . One challenging problem for the calcula- for q = π. In the main plot we show data for N = tion of transition→ ∞ rates is that Bethe wave functions with 3/2 − 512, 1024, 2048, 4096 of N Mλ (π) versus ωλ. The scal- critical pairs thus obtained vanish identically as pointed ing is near perfect across the band. The divergence build- out in Ref. 28. However, our numerical analysis strongly −2 ing up as N in this quantity is stronger, ω , suggests that the vanishing norm ψλ in the denomina- than the known→ infrared ∞ singularity in the dynamic∼ struc- tor of (5.1) is compensated by thek vanishingk transition 26 −3/2 ture factor, S−+(q,ω) ω , as documented by the rate in the numerator to produce a unique finite ratio in inset to Fig. 4. ∼ the limit ∆ 0.29 → 10

1 M. Takahashi, Thermodynamics of one-dimensional Solv- facilitates the tasks of solving the Bethe ansatz equations able Models (Cambridge University Press, Cambridge, and calculating transition rates. United Kingdom, 1999). 16 F. D. M. Haldane, Phys. Rev. Lett. 67, 937 (1991). 2 V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quan- 17 E. Lieb, T. Schulz, and D. Mattis, Ann. Phys. 16, 407 tum Inverse Scattering Method and Correlation Functions (1961). (Cambridge University Press, Cambridge, 1993). 18 A one-to-one mapping between individual states is not pos- 3 M. Karbach and G. M¨uller, Phys. Rev. B 62, 14871 (2000). sible for degenerate levels because of the different natural 4 M. Karbach, D. Biegel, and G. M¨uller, Phys. Rev. B 66, eigenbases in the two representations. 054405 (2002). 19 L. D. Faddeev and L. A. Takhtajan, Phys. Lett. A85, 375 5 Throughout this paper, the exchange constant is used as (1981). energy unit: J = 1. 20 M. Karbach, K. Hu, and G. M¨uller, Comp. in Phys. 12, 6 V. E. Korepin, Commun. Math. Phys. 86, 391 (1982). 565 (1998). 7 A. G. Izergin and V. E. Korepin, Commun. Math. Phys. 21 D. Biegel, Diplomarbeit, Das eindimensionale Spin-1/2 99, 271 (1985). XX-Modell als Grenzfall des XXZ-Modells, Bergische 8 N. Kitanine, J. M. Maillet, and V. Terras, Nucl. Phys. B Universit¨at Gesamthochschule Wuppertal (2000). 554, 647 (1999). 22 Alternative sets of Bethe quantum numbers for the two 9 D. Biegel, M. Karbach, and G. M¨uller, Europhys. Lett. 59, states with k¯ = 4 in Table IV are (3.5, 0.5, 0.5, −0.5) and 882 (2002). (3.5, 1.5, 0.5, −1.5), respectively. 10 D. Biegel, M. Karbach, and G. M¨uller, J. Phys. A: Math. 23 S. Katsura, T. Horiguchi, and M. Suzuki, Physica 46, 67 Gen. 36, 5361 (2003). (1970). 11 J. des Cloizeaux and M. Gaudin, J. Math. Phys. 7, 1384 24 M. Karbach, G. M¨uller, A. H. Bougourzi, A. Fledderjo- (1966). hann, and K.-H. M¨utter, Phys. Rev. B p. 12510 (1997). 12 T. Deguchi, K. Fabricius, and B. M. McCoy, J. Stat. Phys. 25 A. H. Bougourzi, M. Karbach, and G. M¨uller, Phys. Rev. 102, 701 (2001). B 57, 11429 (1998). 13 K. Fabricius and B. M. McCoy, J. Stat. Phys. 103, 647 26 A. Luther and I. Peschel, Phys. Rev. B 12, 3908 (1975). (2001). 27 G. M¨uller and R. E. Shrock, Phys. Rev. B 29, 288 (1984). 14 K. Fabricius and B. M. McCoy, J. Stat. Phys. 104, 573 28 K. Fabricius and B. M. McCoy, J. Stat. Phys. 111, 323 (2001). (2003). 15 29 The commonly used rapidities are zi = 2atanh yi. There D. Biegel, M. Karbach, G. M¨uller, and K. Wiele (unpub- exist Bethe ansatz solutions with real magnon momenta ki lished). for which | tanh(zi/2)| > 1. Whereas some zi thus become imaginary, all yi stay real. Using the latter significantly