Observations on the Bethe ansatz solutions of the spin-1/2 isotropic anti-ferromagnetic Heisenberg chain: the chiral string conjecture

Calculations of low lying excited states in even length L → ∞ chains begun in 2019 J. Phys. Commun. 3 025007 in the total spin S = 0 sector are here extended to S > 0. Bethe string configurations that were observed to transition from their definition at small L to different ‘apparent’ configurations at large L are now understood to be a manifestation of the approximate chiral symmetry that develops with increasing chain length. It is conjectured that when chiral symmetry is built into the configuration definitions a complete explicit labelling of all low lying excited states by strings is obtained. The structure is of the form of a product of Bethe strings defining left and right chiral states of spin sL and sR and a central complex ‘string’ necessary to bind s ⃗ L and s ⃗ R into total S ⃗ . Rules based on the string content of a state yield its momentum and asymptotic energy, including the amplitude of the leading ln(L) correction.


Introduction and main results
Bethe [1] (for an English translation see [2]) already showed in his now classic paper that the approximate string configurations defining the eigenstates of modest length L, spin s=½, Heisenberg chains with periodic boundary conditions did not necessarily apply for large L and by inference in the limit L→∞. This result has been rediscovered many times since (cf. [3,4]) but has never been resolved in terms of a proposal for the configurations valid for L→∞. Indeed, resolution by purely analytical methods is probably impossible; what is needed is guidance from a well chosen set of numerical solutions on very long chains. The impediment to the latter is that the set must contain complex solutions and, because of the very large parameter space involved, has always been understood to be a hard problem. The impasse of finding a significant number of such eigenstates was partially broken in [5] but because it was mostly restricted to the total spin S=0 sector it was too limited to make many credible inferences for general S. The present paper, while mostly restricted to S=1, when combined with [5] contains sufficient information to lead to a conjecture for the configurations for all low lying exited states in the large L limit. The result has a simple intuitive structure. Excitations at the left and right Fermi 'surface' points bind separately into chiral spin states s L and s R . These excitations are in general 1-strings from the filled Dirac sea that have moved into available left and right holes together with what are clearly recognizable Bethe (n>1)-strings. If no other excitation is present, the chiral states combine into a stretched state S=s L +s R . Otherwise a central n-complex, n>1, provides the coupling necessary to bind the chiral spins into S=s L +s R −n+1. While an n-complex might be similar to a Bethe n-string, it is more commonly a very distorted n-string and may not even be recognizable as such 1 . The chiral state picture is not exclusively limited to large L. As a simple example, the first excited singlet state made from L/2 overturned spins from the ferromagnetic ground state can be considered as s L =s R =½ bound into S=0 by a central 2-complex that is exactly the Bethe 2-string λ=±i. The first excited triplet state made from L/2−1 overturned spins requires no further excitation to form s L =s R =½ into the stretched configuration S=s L +s R =1.
To describe the conjectured solution I begin by establishing the notation in this paper which follows closely that in [5]. The states of (even) length L periodic chains of s=½ spins anti-ferromagnetically coupled are defined by the Hamiltonian å s s where i labels both the sites and distance along the chain and the components of s i   are the Pauli spin matrices.
Symmetry dictates that eigenstates of (1) can be labelled by total S and S z and (quasi) momentum K. Each stretched state (S z =S) is constructed from N=L/2−S overturned spins from the totally aligned spin configuration. Any eigenstate of (1) is defined by (quasi) momentum eigenvalues k n , −π<k n π, each associated with an overturned spin. These eigenvalues satisfy the Bethe  and is clearly unaffected by k n sign reversal. For L not too large, the {λ} in a BAE solution group into approximate strings; an ideal n-string being the list of n numbers λ=λ r +i(n−1), λ r +i(n−3), K, λ r +i(1−n) with common real part λ r . If we denote the number of n-strings or 'particles' in a particular solution with given L and S=S z by p n , then the set {p n } is a partition of N=L/2−S with the total number of particles P=∑p n and N=∑np n . Bethe [1]  with each binomial the number of ways p n particles and h n holes can be arranged in p n +h n slots. He also confirmed that (5) summed over p(N), the unrestricted partitions of N, gives the correct total number of states. The product (5) and its sum over p(N) is easily generalized (cf. [5] leading to equation (45) ) in (6) accounts for the empirical observation that the Gaussian binomial product is centered at K=0(L/2) if the number of particles P is even(odd). It is to be understood that periodicity will be used to shift terms following the first equality in (6) into the interval (first Brillouin zone) defined following the second equality. That is, appropriate integer multiples of L will be added to n in any term ∝ q n . An explicit result for the sum over p(N) in (6) It is known [8] that with increasing L a chiral symmetry develops in which the total excitation energy, momentum and spin of the chain reduces to the sum of left and right energy, momentum and spin of (approximately) independent excited states at the two Fermi 'surface' points above a filled Dirac sea. The main conclusion of the present paper is that the left and right chiral towers of excitation have a natural Bethe string labelling parameterized by spin s and available 1-string holes h 1 . An additional important point is that the Dirac sea is a packed configuration of 1-strings only in the case that the total spin S is combined from s L and s R in the stretched arrangement, i.e. S=s L +s R . Otherwise the Dirac sea contains an n>1 excitation complex with S=s L +s R −n+1. These excitations do not change the total momentum from its left plus right chiral sum; nor do they change the energy in the L=∞ limit. Rather they are responsible for the coupling of the (approximately) independent s L  and s R  into total S  and result in the leading correction to scaling by an energy shift ∝ −   L s s ln .
The concept of cusp states introduced in [5] for the S=0, L→∞ state counting can be applied to the description of the chiral tower states and remains useful as a way of separating 1-string excitations from those of (n>1)-strings.
for non-negative integral 2s and ν=s, s+1, s+2, K . The [:] q factors in the product are symmetrized Gaussian binomials defined in (7). The sum in (9) is over modified partitions p Ñ( )obtained by adding one to all non-zero elements of the parts of every partition in p N . ( ) A useful notation for any specific partition is the product n p n over distinct n; examples of partition lists and associated p Ñ( )are Clearly the modified partitions p N , ( ) N>0, satisfy the modified sum rule ∑ n>0 (n−1)p n =N. When applied to the Bethe rule h 1 =2s+2∑ m>1 (m−1)p m we obtain h 1 =2ν as written explicitly in (9) to emphasize the special connection of h 1 to the (chosen) independent parameter ν. We identify each incremented n with an n-string and p n its multiplicity, i.e. the number of n-string 'particles', in analogy to Bethe's counting rules. The n=1 factor is excluded from the product in (9) because it does not lead to any new cusp states; instead it is the contribution to every cusp state in (9) from the excitation of the background 1-strings in the Dirac sea. Since the 'particles' p 1 →∞ as chain length L→∞ this factor can be evaluated as the limit for all s=s z states, while a further sum over s greater than some minimum s z yields where the last equality is the known result that 1/(q) ∞ is the generator for the partitions p(n). We can identify as the generator for the multiplicity m(s z , n) of tower states at s z and level n with the convention that minimum energy states at s z0 =0 or s z0 =±½ are level n=0. Comparing this with (13) yields m(s z , n)=p(n−s z 2 +s z0 2 ) in agreement with the recursion relations given in [8] based on the Kac-Moody algebra for the Wess-Zumino-Witten (WZW) model. Since (13) is based on (11), this agreement is a step towards confirmation that conjecture (9) provides a consistent labelling of all states in terms of Bethe's strings.
As noted in [5] the low lying cusp states in the L→∞ chain are characterized by left and right 1-string holes h L and h R in addition to total spin S and (scaled) momentum K (3). The empirical evidence from many examples is that the generator for the multiplicity m κ of cusp states at κ=K−K c , momentum relative to a central K c , is a sum of products of the cusp tower generators (9) given by å å å = = = P = + + -+ < + P The replacement q→1/q in the left tower accounts for the negative momenta of left moving excitations while the sum ∑ Δ restricts s L and s R to the triangle condition Δ(S, s L , s R ), i.e. |s L −s R |Ss L +s R . Since S is integral, 2s L and 2s R together with h L and h R are either all odd or all even. If h L , say, and L/2 are both odd or both even the central momentum K c =0; otherwise K c =L/2. This rule replaces the particle P based rule in (6) for determining K c . The last equality in (14) is a schematic specifying the labelling convention adopted in this paper for the chiral cusp states. It displays the string content of C h h S , L R ( ) which is that of the contributing towers T h s 2 ( ) supplemented by a central n-complex, n=s L +s R −S+1 if S<s L +s R . This n-complex acts as a single 'particle' like an n-string but does not contribute to the momentum κ and therefore is not explicit in the m κ in (14). The existence of an n-complex separate from the tower strings making up the products in (14) was not recognized in [5] and consequently no general consistent product labelling could be found. That (14) is such a consistent labelling scheme cannot of course be proved by just numerical work on finite chains but to date no counter example has been found.
An alternative to the chiral representation C h h which is just (6) restricted to cusp states and generalizes equation (86) in [5] to include all S. Explicitly, and contains no reference to chirality 2 . In particular (15) is not supplemented with any central n-complex excitation. As in (14) the momentum κ is relative to a central K c =((h L +L/2) mod 2)L/2 and the equality of the m κ determined from (14) and (15) has been confirmed numerically to high order. The labelling of a configuration is in general different in (14) and (15) and where such differences exist, the numerical evidence is that configurations that are described by (15) at small L morph into configurations described by (14) with increasing L. The chain length beyond which the chiral representation (14) is the better approximation is state dependent and varies widely. How widely was not appreciated in [5] and requires a correction; eqn. (105) in [5] is now plausibly eliminated as a counter example by additional numerical work to L=3200 and extrapolation leading to L≈10 5 as the estimate for crossover to chiral behaviour (cf. figure 6). Indeed, I conjecture that for L→∞ the chiral labelling (9) correctly describes all strings explicitly contained in C h h S , . This does not preclude the central n-complex being possibly unrecognizable as an ideal Bethe n-string.
A clear advantage of (14) is that it is easily generalized to include energy in part by introducing left and right tower arguments e q 2 / and e q 2 respectively for a generator in which tower energies and (signed) momenta are additive. Based on the work of [8], the energy for any state in a long chain is expected to take the form / / which provides the multiplicity m ω,κ at momentum κ and energy ω, including the leading logarithmic correction to scaling. The denominators in the middle term in (17) are the 1-string contributions as in (11) and are to be understood as an expansion in e . 2 As in (14), the explicit strings from the tower generators are supplemented by a central n-complex which here associates with the correction to scaling factor d e .
L ln ( ) / A final sum of (17) over both h L and h R from zero to infinity yields the generator for all low lying states in any sector S.
The remainder of the paper elaborates on the results given above. Section 2 begins with the tower conjecture (9) and provides the proof for a number of sum rules including (13) which shows that (9) is consistent with what is known from the WZW model. The conjectures such as (9) and (17) rely heavily on the S=0 sector calculations in [5] and on new S=1 calculations here. For the latter it was useful to derive a complete list of S=1, L=16 BAE solutions to serve as initial conditions for larger L and that data is available in the supplementary data file 'L16_triplet.txt' as described at the end of section 2. Section 3 describes some of the properties of the central n-complex states and in particular outlines the resolution of the apparent paradox that adding excitations to the Dirac sea still leaves the chiral states at the Fermi surfaces essentially unchanged. and (17) define the correct labelling for all states in the L→∞ or WZW model limit. Section 4 is devoted mostly to a comparison of the present 'chiral' string interpretation with the 'apparent' string results in [5]; section 5 treats mostly the new S=1 results. Conclusions appear in section 6.

Chiral tower generator properties
It is useful to rewrite the conjecture (9) to emphasize the separate contributions m nf s q 2 ; s ( ) from different numbers μ of (n>1)-strings. The structure that follows from where the sum S¢ is constrained to those modified partitions satisfying m S = > p .
The final expression in (19) is a special case of a general product to which all sums can be reduced. Its derivation below is greatly facilitated by first summing (18) over all s in the interval s z sν with s−s z (and s−ν) integral.
which will be confirmed analytically below. Assuming its validity, we get from (20) in the notation introduced in we note (q) n =(q; q) n and can express (22) as If we now set =z q s 1 2 z every term in the first sum following the equality vanishes and we are left with the identity The reversion to a sum in (29), recorded in (13), is the known result that 1/(q) ∞ is the generator for partitions p(n) while the shifted summation after the last equality is to be interpreted as ) thus defining the multiplicity of states at s z and energy 'level' n. Affleck, et al [8] provided a set of rules satisfied by the chiral towers in the s z representation based on the Kac-Moody algebra for the WZW model understood to be the chiral limit of the Heisenberg spin-½ antiferromagnetic chain. Their rules included one for the recursive generation of the multiplicity of states, m s n , , z ( ) where n=0, 1, 2,K is the 'level' with n=0 the lowest energy state at s z =s z0 =0 for the integral s z tower and at s z =s z0 =±½ for the half-(odd)integral s z tower. Here I show the multiplicity as given by (29) The substitution of (30) into (31) simplifies if we shift the dummy summation variable j by 2s z /3 rounded to the nearest integer and results in one of the three possible equations p n n j j n n p n n j j n n p n n j j where ¢ n is an integer depending on s z and s z0 but independent of j. The third equation is automatically satisfied by symmetry while the first two differ only in order of summation, i.e. j→−j. They are satisfied since these are Euler's recursion for p(n) based on his pentagonal number theorem.
The generator corresponding to (29) in the s representation is the subtraction as recorded in (12). Similarly, from (25) we get where the [:] q factor in (36) is the factor in (18) associated with n=2 in any configuration with m¢ = p , 2 the number of 2-string 'particles'. The number of 2-string 'holes' from (18) is The f factor on the right hand side of (36) is understood to have the same form as that on the left while accounting for the remaining (n>2)-strings. In particular the lower index μ=Σ n>1 p n on the left f is replaced by Σ n>2 p n =μ-p 2 =μ−μ′ on the right f. Similarly the upper index ν−s=h 1 /2−s on the left becomes h 2 /2−s=ν−μ−s from (37) on the right. This argument applies to every configuration and thus to the exhaustive sum of configurations in (36). Finally, having eliminated all 2-strings we perform a relabelling The prefactor to the sum after the equality is the k=0 value of Rhs leaving ratios in the sum that are of one of the two forms The sum term in (41) is now expressible as ) in (35) and completes the proof.
I close this section to show the close relation between the chiral cusp generator (18) for length L→∞ and the all state generator for finite (even) L and total spin S with 0SL/2. The generalization of Bethe's state count to counts at specified momenta is the q-generator where the sum S¢ is constrained to partitions satisfying S = p P.
n n The factor q , which because of periodicity could equally well be written in the form q P mod 2 L 2 ( ) appearing in [5] equation (45), accounts for the empirical observation that the (symmetric) f factors are centered at K=0(L/2) for even(odd) P. The main difference of (46) from (18) is the return from modified partitions to partitions, i.e.  p p, so that the product in (46) contains 1-string terms that were excluded in (18). The structure of the equations however is such that the f m n are the identically same polynomials with the identifications chiral spin s ↔ S, finite chain (integer) spin; 2ν=h 1 ↔ L, (even) length; chiral (n>1)-string count m S = > p n n 1 ↔ = S P p , n n total string count. We also have ν−s ↔ L/2 −S=N, the number of overturned spins from the ferromagnetic state. The generators differ by a factor, i.e, ) Additionally, because of periodicity, the Z polynomial can always be reduced to terms in the first Brillouin zone by the replacement q m →q k with k−m an integer multiple of L and −L/2<kL/2.
On applying the above correspondence to (35) we arrive at As an example of (48), the L=16 singlet generator after reduction to the first Brillouin zone is

The central n-complex
An important observation is that in the L→∞ limit cusp states of chiral spin s L and s R coupled to total spin S=s L +s R are formed with the Dirac sea a filled configuration of 1-strings. Furthermore, for the same chiral spin s L and s R configuration, each reduction of S by one unit from the stretched s L +s R value requires the Dirac sea to accommodate one extra overturned spin because of the connection N=L/2−S. For an S reduction of 2n, n complex conjugate λ pairs are added to the Dirac sea and there is a rearrangement of the real λ such that the net result can be interpreted as a (2n+1)-complex similar to a (2n+1)-string with one real λ taken from the existing Dirac sea list. The total number of real λ does not change. For an S reduction of 2n−1, again n complex conjugate pairs are added but now one real λ is lost from the existing 1-string list. This lost 1-string does not form a hole in the Dirac sea-rather the final configuration is a (2n)-complex similar to a (2n)-string superimposed on a smoothly reduced real λ density. In all cases the new configurations do not change the asymptotic L→∞ energy or momentum from their stretched spin configuration values.
To illustrate these features of the central complex in the BAE solutions I consider the symmetric case of cusp states s L =s R =2, 0S4 generated from in (17). The states in (51) are particularly simple in that the chiral spin s=2 at each end is associated with a pair of spins from the ferromagnetic state which have not been overturned and hence carry no string labels. If we adopt as our labelling convention the triad (left chiral string |central complex| right chiral string) as indicated in (14) then the label for S=4 in (51) is (0|0|0) and the labels for S<4 are (0|(5-S) 1 |0). For comparison, the Bethe cusp state labels are simply 0 (no (n>1)-string) for S=4 and (5-S) 1 for S<4 resulting in the explicit identification central n-complex ↔ n-string.
In figure 1 the calculated energies E for all 0S4 in (51) are displayed in the reduced form is confirmed as shown by the straight lines converging to ε=16 from (51). The associated BAE solutions at each L are characterized by the density ρ(λ)=−∂n/∂λ of the real λ Bethe roots for each S and by the central complex pairs λ=±Λi for S<4. A practical definition for the density based on the numerical BAE real ordered solution arrays λ n with λ n+1 <λ n is The empirical evidence confirming (55) is seen in the λ<0 curves ρ (S) (λ)−δρ (S) (λ) in figure 2 that are almost indistinguishable from ρ (4) (λ). Except for Λ=1, the numerical Λ values in (55) differ substantially from the n−1, n−3, K, 1−n for an ideal n-string. Indeed, each Λ ≠ 1 is well approximated by Λ=a+bln(L), b>0 at large L with no indication of saturation as L→∞. An additional observation based on (55) and which appears to be general is that any central complex consisting of n complex λ pairs is either a (2n)-complex or a (2n+1)-complex depending on the presence or absence of a pair λ≈a±i.
The δρ in (53) and (54) have an analytic basis, subject to certain approximations, in the BAE. To show this I use the BAE in the form of the continuous function Combining (57) and (58) to first order in the λ difference. The effects of multiple complex pairs are additive so that the above calculations contain all the necessary ingredients for S=1 and S=0 also. As already mentioned, the numerical Λ listed in (55) and used for the λ<0 curves in figure 2 differ substantially from the values for ideal n-strings. To understand the origin of this difference consider again the BAE (56) for S=2 but now for λ=Λi, Λ>2. In this case it is essential to introduce a cutoff λ c defined by and for n (2) (Λi) to be real the expression in braces {} must vanish. It is useful to write ρ (4) (λ)=ρ H (λ)+Δρ (4) where Λ<λ c has been assumed and the equality is the sum rule ò lr l The integral in (70) is estimated as ≈0.6 with almost no L dependence for large L. Thus, within the continuum approximation, we confirm the logarithmic divergence Λ>2/(π√7)ln(L)=0.2406ln(L). For comparison, an estimate by extrapolation of the exact data from L3200 to L→∞ gives the divergent part of Λ as (0.243±0.001)ln(L), in reasonable agreement.

Transition to chiral states: S=0
The many examples of 'apparent' string labelling replacing Bethe string labelling in [5] can now be understood as a necessary requirement of the chiral symmetry that develops with increasing chain length L. Consider first the symmetric examples of table 1 in [5] beginning with the transition Bethe 2 1 3 1 4 1 →apparent 3 3 shown in figure 2 in [5]. The s=s L =s R =1 inferred in [5] from the observed δ/ln(L) correction to scaling implies a central 3-complex in the chiral representation as described in section 3. The h 1 =2∑ m (m−1)p m =12 holes in [5] splits into h L =h R =6 1-string holes in the chiral representation. This cusp state and all others with the same s and h values arise from the tower generator with chiral labels (=apparent labels 3 3 and 2 4 3 1 ) below and the Bethe labels above taken from lines 11, 14, 15 and 16 in table 1 in [5]. Missing in (72) is a Bethe label 2 4 3 1 for the final cluster of 3 states but because this is a label for 11 overturned spins it cannot appear in a state list generated from L=20. One can carry out a similar analysis for all other cusp states in table 1 in [5] and find agreement between chiral and apparent string labels in every case.
Another comparison with [5] is with the states on the special energy versus momentum lines ε=4s 2 +2|κ| where s=s L =s R is the uniquely contributing chiral spin. In terms of the chiral tower products contributing to where the central complex is the S=0 specialization of that given in (14) and is understood to be (2s+1) 1 only if s>0. Because chiral s is fixed in (73) the corresponding expression for the Bethe string labelling of configurations cannot be a chiral independent form such as (15). The correct formula was conjectured to be eqn.
with again chiral labels from (73) below and Bethe labels from (75) above. Asterisks mark the Bethe states that must transition, the first being Bethe 3 1 →chiral (0|2 1 |2 1 ) at κ=3. The 3-string real λ at L=16 combines with a 1-string at L=20 to become a complex pair with small imaginary part. With increasing L the two complex pairs begin to approximate a quartet a±bi, a±(2−b)i with b increasing such that root collision occurs at L≈253. For L>253 the two approximate 2-string pairs a 1 ±i, a 2 ±i drift such that a 1 remains deep inside the Dirac sea of 1-strings whereas a 2 approaches a fixed distance from λ F , the 'Fermi' 1-string that is the boundary of the Dirac sea. This completes the transition to a clearly identifiable chiral (0|2 1 |2 1 ). The same root movement pattern occurs at κ=10 and 11 with transitions 2 1 3 1 →(0|2 1 |2 2 ) occurring at L≈975 and L≈213 respectively. The transition Bethe 4 1 →chiral (0|2 1 |3 1 ) at κ=8 has the completely different pattern shown in figure 3. No root collision occurs. Instead there is a smooth transition from approximate 4-string roots a±i, a±3i at L=12 to a central approximate a 1 ±i and an a 2 ±2i at L=16 384 8 with a 2 approaching a fixed distance from λ F resulting in a clearly identifiable chiral (0|2 1 |3 1 ). The next predicted transition from (79) is Bethe 5 1 →chiral (0|2 1 |4 1 ) at κ=15. It is plausible to suppose that of the approximate 5-string roots a, a±2i and a±4i the first three will change with L analogous to the change in the 3-string roots in the κ=3 transition. This requires the real root to combine with a Dirac sea 1-string but based on calculations to L=1024 I estimate that the real root will not overlap with the Dirac sea until L≈8000. A further (possibly large) increase in L would be required for a complex pair to develop, the two complex pairs to become a quartet and finally a root collision to result in the (approximate) 2 1 4 1 configuration a 1 ±i and a 2 ±i, a 2 ±4i. Confirmation would require such where asterisks mark Bethe labels that will transition to chiral as in (79). The first transition 4 1 →(0|3 1 |2 1 ) at κ=4 is similar to that seen in figure 3 in that only smooth drifts in the complex pair components of the Bethe 4-string occur. However the clear distinction (0|2 1 |3 1 ) ≠ (0|3 1 |2 1 ) is manifest in that here it is the ≈a 1 ±i component of the Bethe 4-string that approaches a fixed distance from λ F indicative of a 2-string. The ≈a 2 ±3i component on the other hand remains deep within the Dirac sea with the imaginary part diverging ∝ ln(L) as expected for a central 3-complex as discussed in section 3. Both asymptotic behaviours are already well established by L=256. The same drift pattern occurs in the transitions 2 1 4 1 →(0|3 1 |2 2 ) for κ=12, 13, 14 where the Bethe 2-string λ in all cases is well separated from the 4-string λ and acts only as spectator with little direct influence on the transition. The transition 5 1 →(0|3 1 |3 1 ) at κ=10 is also similar in that the complex component with largest imaginary part smoothly separates and drifts deep into the Dirac sea and can be identified as the central 3-complex. The remaining part of the 5-string remains close to λ F and so identifies as the chiral 3-string. More interesting is the transition 2 1 4 1 →(0|3 1 |3 1 ) at κ=11 illustrated in figure 4. The complex component of λ 4 with largest imaginary part also drifts deep into the Dirac sea but follows a more convoluted path. The magnitude of the imaginary part first decreases to a minimum of 2.366 at L=164 and then increases as expected for a central 3-complex. The two λ≈a 1 ±i, a 2 ±i at small L collide just beyond L=104 to form an approximate quartet a+bi, a+(2-b)i in the interval 108L168 which at L=172 has separated into a 3-string and a 1-string that becomes part of the Dirac sea. Note that this sequence is essentially just the reverse of the sequence for the transition 3 1 →(0|2 1 |2 1 ) at κ=3 described following (79).
For 2s=1 we observe from (79) that the fraction of Bethe labelled states that transition to a different chiral label is 1/2 and 3/5 for μ=1 and 2 respectively. For 2s=2 these fractions obtained from (80) are 1/3 and 5/9. If we accept the chiral conjecture leading to (73) and the independent conjecture (75) we can determine these  ( ) given by (9) and a relevant example in (71). Note that in this case direct Bethe and chiral label comparison with the elimination of common label pairs without having any BAE solutions would still correctly identify the three configurations that transition in (82). In general, without energy information, we cannot know that a common label pair refers to the same energy and eliminating it might be incorrect-implying the procedure only provides a lower bound on the transition number. If we accumulate results for all possible κ for given h L and h R we arrive at the transition fraction bounds in table 2. They suggest that the exact fractions for the limited states contributing to table 1 are not anomalously large and might indeed be typical. I conclude this section with an elaboration and correction of two results in [5]. Equation (104) in [5] is identified in the current notation as the specific product elements arising from the indicated Bethe labelled configurations in the final equality. BAE solutions to L=640 in [5] have been extended to larger L and these together with extrapolation are shown in figure 5 and confirm the assignments made in [5]. The 5-string complex pair with imaginary part of largest magnitude is a smooth function of L whereas the remainder undergoes transitions often seen for a 3-string, namely 3-string→quartet→2-string pair. In this case one of these 2-strings together with the other 5-string complex pair identifies as the chiral central 4-complex 10 labelled λ 4c in figure 5. Note that each chiral 3 1 or 2 2 appears to asymptote to the same 3 1 or 2 2 configuration as L→∞ for all states in but unlike the agreement between (104) in [5] and (84), the doubly degenerate states in (105) in [5] were labelled as 'apparent' (3 1 |2 1 |2 1 4 1 ) and not the chiral (2 1 3 1 |2 1 |2 1 3 1 ) in (85). This case can no longer be considered as a counter-example to the chiral hypothesis of this paper because the discrepancy has been resolved by extending the BAE solution from L=640 in [5] to L=3200 and extrapolating the data to L≈500 000. The results are diplayed in figure 6 where one can see that with data only at L=640 as given in (105) in [5] that 'apparent' Figure 5. BAE complex λ for states in (84) supplemented with λ F as in figure 4. Symmetry allows the data shown to be restricted to 2 1 3 1 5 1 →(2 2 |4 1 |3 1 ) (solid lines) which is one of a doubly degenerate pair and the non-degenerate, ±λ symmetric, 3 2 4 1 →(3 1 |4 1 |3 1 ) at Re(λ)<0, 2 3 5 1 →(2 2 |4 1 |2 2 ) at Re(λ)>0 (dashed lines). The solution labelled λ q is an approximate quartet that starts at L=224 and converts in a root collision at L≈6880 into two 2-strings. Crosses mark the largest L for which BAE solutions have been determined; curves at larger L are extrapolations.
(2 1 4 1 |2 1 |3 1 ) was a reasonable assignment. By continuous graphing to L=3200 it is clear this assignment is wrong but it takes the extrapolation shown in figure 6 to identify the central complex in a chiral representation as a 2-string. Further evidence for the assignment in (85) comes from a comparison of figures 3 and 6. Except for the spectator 3-string and two 2-strings in figure 6, the qualitative similarity between the behaviour in figure 3 for 12<L<2 14 and that in figure 6 for 640<L<2 19 is striking and supports the central complex as a 2-string. This qualitative similarity extends to the imaginary parts of λ 3 which at L=640 are ≈±3.25i but by 2 14 have dropped to ≈±2.9i and by 2 16 to ≈±2.5i and would not be typical of a central 3-complex. Finally, just as for the states in (84), the chiral 2 3 a 1 1 and 2 3 b 1 1 in the states in (85) plausibly asymptote to the same common values as L→∞.

Transition to chiral states: S=1
A significant feature when total S>0 is the increase in the number of chiral spin, s L and s R , combinations that are allowed in C .
subject to the triangle condition Δ, |s L −s R |Ss L +s R . In addition to the explicit n-strings provided as labels in (86) a chiral label includes a central n-complex, n=s L +s R −S+1, for states with s L +s R >S. The product terms in (86) that contribute for S=1 are listed in table 3 together with the corresponding Bethe labelled states based on the generator (15). The placement of the Bethe cusp states is fixed by calculation at different κ with L large enough to determine asymptotic ε and correction to scaling d = -  s s 2 .
parallel to λ F . Again, by L=1000 the state is clearly identifiable, in this case as chiral (0|3 1 |2 1 ). Finally, for the state at κ=6 the real parts of the complex pairs do not run parallel to λ F as seen in figure 7 for κ=11 but slowly drift towards λ F . They are still outside the Dirac sea at L=25 600 which is the largest length for which solutions have been obtained. Extrapolation shows qualitative behaviour like that for Bethe 4 1 in figure 3, κ=8 or figure 7, κ=10 but stretched to longer lengths. Specifically, the real parts of the complex pair with largest imaginary part cross λ F at L≈6.5×10 5 ; those for the other pair cross at L≈5×10 6 . The real parts of the two pairs cross each other at L the order of 10 9 . This length is also what is estimated from the imaginary parts as the crossover point where the complex roots become clearly identifiable as λ 2c and λ 3 and thus components of the chiral (0|2 1 |3 1 ) in table 4. Another illustration of a slow drift from outside to inside the Dirac sea from table 4 is that for the Bethe labelled 2 1 3 1 state that transitions to chiral (0|2 1 |2 2 ) at κ L =2¼, κ R =10¼ and δ=5/2. This state is part of a triplet at κ=8, 9 and 10 which associate with the three possible locations of two 2-strings in three possible h 2 holes and are shown in figure 8 with the hole positions labelled 1, 2 and 3. The empty h 2 hole moves from 3, maximally outside the Dirac sea at κ=8, to 1, inside the Dirac sea at κ =10. The latter identifies with the Bethe labelled 2 3 state for all L and which by the maximum L=400 calculated is clearly seen to be the chiral (0|2 1 |2 2 ). For empty h 2 hole at κ=9 the Bethe 3-string evolves by the 'standard' route in which the real root first joins with a Dirac sea root to form a complex conjugate pair with small imaginary part. With increasing L the now two complex pairs of the Bethe 3-string form an identifiable and then increasingly accurate quartet λ q . At L≈49 000 a root collision occurs and two 2-strings emerge-completing the transition 2 1 3 1 →(0|2 1 |2 2 ). For empty hole 3 at κ=8 the Bethe 3-string drifts parallel to that at κ=9 and it is plausible that the same sequence 3-string→quartet→2-string pair will occur at some large L=O(10 7 ) and confirm the predicted transition Bethe 2 1 3 1 →chiral (0|2 1 |2 2 ).
The chiral states (2 1 |2 1 |2 1 ) at 2s L =1, 2s R =3 and δ=5/2 in table 4 associate with a 2-string at two locations for 2s L =1 and a 2-string at four locations for 2s R =3. For the four cases in which κ L =3¼ the Bethe labels are 2 1 3 1 and at small L all 3-strings lie outside the Dirac sea and drift towards λ F with increasing L. The first 3-string real root to collide with and absorb λ F is that for κ=4; at L=1024 a 3-string real root is still observed but by L=1280 the 3-string is two pairs of complex roots and by L=4096 the 3-string is an identifiable quartet with the real parts of the pairs having passed the new λ F and moved into the Dirac sea. Extrapolation suggests that a quartet root collision to form a λ 2 and λ 2c pair, which combine with the remaining spectator 2-string to complete the 2 1 3 1 →(2 1 |2 1 |2 1 ) transition, will occur at L≈200 000. For κ=5 a 3-string real root is observed at L=3200 but this has become a complex conjugate pair by L=4096. This increase in L by approximately factor 3 from the preceding κ=4 case before the appearance of the quartet configuration very likely applies to its demise in the transition to (2 1 |2 1 |2 1 ) as well. Similar results are expected at even longer lengths for κ=6 and 7 but no calculations have been done at these longer lengths. In only one of the four cases of chiral (2 1 |2 1 |2 1 ) in which κ L =4¼ is the Bethe label 2 1 3 1 . Here the Bethe 3-string has a real root for L28 and is an approximate quartet for L32. This transitions to a 2-string pair λ 2 and λ 2c at L=252.8 inferred from numerical fits to Table 4. Labels for S=1 cusp states for h L =3, h R =5 in the same format as table 3. Bethe  6  7  8  9  10  11 12 (T 3 |c|T 5 ) 2s L , 2s R roots at both shorter and longer lengths. This last example together with the preceding ones illustrates the tremendous range of lengths at which Bethe to chiral labelling transitions can occur. It needs to be emphasized however that how the energy approaches its asymptotic form seems to be quite independent of the labelling transitions and that the placement of Bethe labelled states within tables such as tables 3 and 4 requires calculations for chains of moderate length only.
Calculations have been carried out also for all other combinations h L +h R =8 as well as the smaller h L +h R =6 and 4. No counter examples to the conjectured chiral string labelling transitions have been found although for confirmation, just as noted above, some resort to extrapolation was necessary. For instance for h L =h R =3 at κ L =2¼, κ R =3¼ the transition Bethe 3 1 →chiral (0|2 1 |2 1 ) is estimated from calculations to L=5120 to culminate via quartet root collision at L≈400 000. The chiral conjecture has also been confirmed for a limited number of total S>1 spin states.

Summary and conclusions
A conjecture for the string labelling of all low lying states in any sector of total spin S in the length L→∞ chiral limit of the spin-½ Heisenberg anti-ferromagnetic chain has been presented. A central role is played by cusp states consisting of a filled Dirac sea of 1-strings between the h L (left) and h R (right) 1-string holes and a product of left and right combinations of (n>1)-string excitations forming chiral spins s L and s R that couple to S. This coupling requires an additional central (n>1)-complex 'string' if S<s L +s R . The complete picture is summarized by the generator (14), C q , that are the essential ingredients of (14) and (17) are proved to be consistent with the towers for the WZW model that is understood to be the chiral limit of the spin-½ Heisenberg anti-ferromagnetic chain (cf. (29)-(34)). This is important evidence for the chiral string conjecture of the present paper.
The chiral cusp state generator (14) has been verified numerically to high order to equal the Bethe cusp state generator (15). By starting from known solutions of the BAE at small L for states in (15) 11 and extending these solutions to large L by 'continuity' one may arrive at the chiral representations in (14). It is this study of well over Figure 8. Evolution of cusp states from table 4 that at large L are the three chiral (0|2 1 |2 2 ) with 2s L =3, 2s R =1 and δ=5/2. Solutions at κ=8, 9 (Bethe 2 1 3 1 ) and 10 (Bethe 2 3 ) have been determined to maximum L=5120, 2560 and 400 respectively as indicated by crosses. For more details see text.
one hundred such numerical cases without ever finding a counter example that provides additional convincing evidence and justification for the chiral labelling conjectures (14) and (17). The equality of (15) and (14) in the q→1 limit is a total count sum rule and an interesting binomial identity, namely å m m m where the last equality is the series expansion of the first and conveniently provides -C h s s 2 2 ( ) / as the coefficient of y 2s in the polynomial multiplying x h . Table 5 provides an example list for the right end chiral counts c(R) taken directly from (A.1) for 0h R 8. An important observation is that these coefficients arrange into a (truncated) Pascal triangle. The left end chiral sum counts Σc(L) are sums of those terms in (A.1) allowed by Δ for fixed s R and S. For example, for 2s R =0, only 2s L =2S=2 is allowed so we record the coefficients of y 2 in (A.1), namely 1, 3, 9 and 28 in the rows h L =2, 4, 6 and 8 respectively. For 2s R =1 both 2s L =1 and 3 are allowed; we read off from (A.1) the sum of the coefficients of y and y 3 to obtain 1, 3, 9 and 28 for h L =1, 3, 5 and 7 respectively. The entries for all 2s R 2S=2 are the sum of three coefficients, those for y ,  Here too we make Table 5. Data for the proof of (87) for combinations h L +h R =8 at total S=1. The large type entries are the right end counts c(R)= for fixed s R . Bold type indicates the Δ allowed 2s R when 2s L =h L . A breakdown of terms for h R = 6 and 5 appears in tables 3 and 4 respectively. the important observation that these coefficients arrange in a (truncated) Pascal triangle. Because of this structure the product c(R)×Σc(L) in any element in some h L , h R row is the product of the 'walks' from the h R =0 start to that row times the 'walks' from that row to the h L =0 finish. The sum of the product of 'walks' in any row is the totality of 'walks' from start to finish and thus independent of the row. In the S=1 example above this is 28, the single element in the h L =0, h R =8 row and also the value of the left hand side of (87). The proof of (87) thus hinges on the proof of the Pascal triangle recursion for both c(R) and Σc(L) to which I now turn. The Pascal triangle generator T=y p /(1-x(y −1 +y)) satisfies the recursion y p +x(y −1 +y)T=T with the factor y p defining the position of the triangle vertex. The analog Pascal recursion for c(R) follows directly from the first equality in (A.1) which satisfies where the unphysical s<0 term following G, i.e. xg/y, is what defines G as a 'truncated' Pascal triangle. Its source is x/y times g, the y independent part of G and has no compensating xy times a term -g/y 2 from G to cancel. For the Pascal recursion for Σc(L) I start with the sums discussed above for the S=1 example. These can be expressed as the product (y −2 +1+y 2 )G with subtractions to eliminate the misrepresented terms-these are (y −2 +1) times the coefficient independent of y and y −2 times the coefficient of y. The result is the Σc(L) generator which is a difference of two Pascal binomial towers with vertices h=h L =0 at s=s R =S and -S-1. The change in dummy sum variable from n to 2s that leads to the final equality in (A.5) also imposes the restriction that h and 2s are either both even or both odd. The form of (A.5) as the difference of two towers guarantees that all 1/y terms vanish; this provides a picture of the 'truncation' process to be the result of perfect absorption of Pascal 'walks' on the line 2s=−1. A tentative physical H S is the last equality in (A.5) to which the constraint s0 is applied. With the identification h=h L , s=s R now made explicit we have