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

Evidence is presented that the solutions of the Bethe ansatz equations for spin-½ isotropic Heisenberg chains in fixed total spin and momentum sectors are the roots of single variable polynomials with integer (or integer based) coefficients. Such solutions are used as a starting point for investigation of long chain (critical region) properties. In the total spin S = 0 sector I conjecture explicit formulae for the Bethe string configuration labelling of all left and right tower excitations in the k = 1, SU(2) Wess-Zumino-Witten model.


Introduction
This paper presents empirical observations about the states of (even) length L periodic chains of s=½ spins anti-ferromagnetically coupled as defined by the Hamiltonian 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 S z <S state can be generated by angular momentum lowering operators but will not be discussed here. Bethe [1] (for an English translation see [2]) showed (1) is soluble by associating with each overturned spin a (quasi) momentum eigenvalue k n , −π<k n π. These eigenvalues satisfy the Bethe ansatz equations (BAE) The (scaled) sum of the k n is the total momentum and is clearly unaffected by k n sign reversal. While some progress has been made in the numerical solution of the BAE (2) (cf Hao et al [3]) it remains a difficult challenge. Here I present evidence that there exist important relations satisfied by BAE solutions that can be used as easily implemented checks on existing numerical solutions and/or provide alternative methods of solution. The evidence is most apparent when instead of momenta k or rapidities λ one uses x=2cos(k). Consider the case that K=L/6, L/4, or L/3 (or their negatives) and let D K =D K (L, S) be the total number of eigenstates of (1) at the given K, L and S. I find the polynomial formed from the BAE solutions has real, rational coefficients r i and can be rationalized to form an integer coefficient polynomial I x . K ( ) One can consider the process in reverse. For any given I x K ( ) existing commercial software such as Maple will efficiently find all roots x i and a finite search algorithm can find the D K combinations of N momenta k i =±arccos(x i /2) that satisfy the BAE. In principle I x K ( ) can be found from a single 1 solution x=x 1 say, obtained to some minimum accuracy, by an integer relation algorithm such as PSLQ [4] implemented on Maple. More practically, one can combine all the solutions of the BAE that are most easily found with a less accuracy demanding PSLQ to determine I x .
K ( ) Similar considerations apply for K=0 and L/2. Here symmetry allows solution of the BAE to be determined from integer coefficient polynomials I x K ( ) of reduced degree whose roots are only the non-trivial x i .
For all other K the BAE solutions are the roots of polynomials whose coefficients are 'integer based'. What this means is that the K in the interval 0<K<L/2 group into blocks K d with M d members consisting of those K whose greatest common divisor with L is d. The number of members M d =j(L/d)/2 where j(n) is Euler's function (cf Hardy and Wright [5] section 5.5). If M d =1 the situation is that described by (5); otherwise the members of K d have the same D K and the terms cos(2πmK/L), m=0, 1, K, M d −1, are integrally independent. The root polynomial analog of (5) for any member is / from (6). If p i K ( ) is known to sufficient accuracy, the PSLQ algorithm will determine the rationals r 0,i and r m,i . In other words, if all BAE solutions for one member K are known, the polynomials P x K ¢ ( ) for all other M d members follow trivially without reference to the BAE.
All of the polynomial generated BAE solutions have been plausibly identified with Bethe string configurations and, by continuity in L, define the Bethe string content at large L. This is important for discussion of the critical behaviour of (1) which Affleck [6] showed is the k=1, SU(2) Wess-Zumino-Witten (WZW) model. Subsequently Affleck et al [7] provided additional analytic and numerical confirmation. An apparent discrepancy in the asymptotic behaviour of the ground state energy has recently been resolved [8], justifying a systematic study of other states in long chains to identify the Bethe string content of the left and right tower excitations in the WZW model in the critical region.
In the S=0 sector, the asymptotic L→∞ energy eigenstates of (1) are expected [7], based on WZW and conformal field theory arguments, to have the form

/ /
This brief synopsis of the main results of the paper is expanded in the following sections together with numerous illustrative examples of BAE solutions. Section 2 is a summary of Bethe's solution for N=2 overturned spins but is here recast in a form that leads directly to (5) and (6). Since the most efficient implementation of the PSLQ algorithm requires the number of unknown constants to be available, section 3 is devoted to deriving formulas for D K (L, S). At the symmetry points K=0 and L/2, states are either non-degenerate or 2-fold degenerate coming from the inversion symmetry k n →−k n that leaves K and E unchanged. Explicit formulas for these symmetry distinct state counts are also derived. Section 4 provides example BAE solutions at K=0 and L/2; some of these are directly derivable algebraically from (2) and provide justification for (5) that extends beyond the N=2 overturned spin case. Section 5 reports some general K results for N=3. Here confirmation of (5) and (6) is based entirely on numerical inference but is important because it shows the conjectured structure is not an accidental feature that arises because the BAE have an analytic solution when N=2. Section 6 is devoted to the example L=16, S=0. Results from sections 4, 5 and 6 of the more extensive polynomials and associated state lists are provided as text files L20_nondegen.txt, 3_overturned_spins.txt and L16_singlet.txt respectively in supplementary data. Section 7 describes the basis for the multiplicity generator (8) and the general formulas for the B x .

Two overturned spins
The BAE (2) for two overturned spins are and the equality between first and last term implies the roots of unity condition with the scaling convention (3) for the total momentum. We can also use the first equation in (11) for a second relation, , t a n , 1 2 a result that applies equally to .
2 l A useful equivalent to (13) for either λ obtained by cross multiplying and rearranging is For K=0, the solutions to (14) are given by the roots of unity condition (λ+i)/(λ−i)=exp(2πin/ (L−1)) or k n =2nπ/(L−1), n=−L/2+1, −L/2+2, K, L/2−1 from which we must exclude k n =0 as the solution for the S=L/2 uniform state. The S=L/2−2 solutions are the distinct pair combinations satisfying k 1 +k 2 =0. There are L/2−1 such (non-degenerate) pairs and these exhaust the k n list for pairs. In summary, the solution lists k K Ŝ and k K S of momentum adopting the convention of using hatted variables for the non-degenerate states at K=0 and L/2. For K=L/2, there is one non-degenerate singular solution identified with λ 2 +1=0. We write the state formally as with finite quantities such as the energy contribution to (4), ΔE Sing =−2, understood to be the result of a careful limiting procedure. The solutions arising as roots of unity are k n =2nπ/(L−2), n=−L/2+2, −L/2+3, K, L/2−1 from which we exclude k n =π as the solution for the S=L/2−1 spin-wave. The S=L/2−2 solutions are the distinct k n pairs which sum to π (modulo 2π). One such set is k 1 =2nπ/(L−2), k 2 =(L−2−2n)π/(L−2), 0<n< L 4 . ⌊ ⌋ / The negatives, −k 1 , −k 2 are also solutions and exhaust the possibilities. Since reversing the signs of all k n leaves the energy unchanged as well as the sum k 1 +k 2 =π (modulo 2π), each state is doubly degenerate. In summary, for K=L/2, where it is understood that we list only the positive half of the degenerate states. For 0<K<L/2 we first express (14) in alternative forms. By dividing through by (λ 2 +1) L/2 we get the equivalent which is useful for contributing to the discussion by Bethe [1] and Essler et al [9] of a possible complex pair k=πK/L±iy K solution for K>1. On substituting either k into (18) we find after some algebra that y K must satisfy K L y L y L K odd y L y L K even cos sinh 2 2 sinh 2 , , cosh 2 2 cosh 2 , . 19 The left hand side of (19) differs from unity by O(1/L 2 ) for large L whereas the right hand side for odd K never exceeds 1−2/L. Thus we recover the known result that for fixed odd K>1 there is always some critical length L c satisfying cos(πK/L c )=1-2/L c beyond which the complex solution transforms via y K →iy K to two real solutions. The (19) for even K always has a solution but is interesting in that the associated λ pair has imaginary parts I(λ)≈±2L 1/2 /(πK) for fixed K and L→∞ that do not approach the ideal 2-string values ±1 [10].
A second alternative forms the basis for the polynomials (5) and (6). Squaring both sides of (14) yields an equation explicitly dependent on λ 2 only which we write as λ 2 =(2+x)/(2−x), x=2cos(k). After rearranging and multiplying through by the denominator factor (2−x) L−1 and a convenient normalization we arrive at an equation for k given by where the x n are the roots of C K (x). In the process of squaring (14) we have lost k n sign information but this can be recovered by a finite m, n and sign search process in which we demand the correct signs in (20) are those for which k m +k n =2πK/L. The A and B in (20) are polynomials in x=2cos(k) of degree L−3 and L−1 respectively; explicitly,  To reduce C x K ( ) to the polynomial P x K S ( ) whose only roots are those for S=L/2-2 we must divide out the factor x K L 2 cos 2p -( ) / for the S=L/2-1 spin-wave. If K is odd we must also divide out two spurious root factors x K L x K L 2 cos 2 cos ; p p -+ ( ( ))( ( )) / / the first (k=πK/L) is easily shown to be a solution of (18) but has no pair partner for a BAE solution because the second (k= π−πK/L) leads to left and right hand sides of (18) having opposite sign. In summary, where the roots x n of P K L 2 2 -/ combine into L/2−2(L/2−1) BAE solution pairs for K odd(even). For every BAE solution of (22) one automatically has also a BAE solution for −K obtained by simply reversing all k n signs.
A summary list of the number of solutions K n which is the non-degenerate part of D K (L, S) for K=0 and L/2, v K which is one-half of the remaining degenerate part and v K =D K (L, S) for 0<K<L/2 is where the ellipsis indicates a repetition of the alternating sequence L/2−i, i=2, 1, 2, K to a total of L/2−1 terms. The total number of states from (23) is For K=L/6, L/4 or L/3 all cosine terms in (22) are rational so that (22) simplifies by elementary division to a polynomial with rational coefficients. As example, for L=12, S=4 and K=2, 3 and 4 we find the rationalized polynomials P x I x   For general K in 0<K<L/2 excluding K=L/6, L/4 and L/3 treated above, elementary algebraic division in (22) will lead to products of cosines that can always be eliminated by use of 2cos(a)cos(b)=cos(a+b)+cos (a−b). The resulting polynomial P K L 2 2 -/ has coefficients that are sums of (possibly many redundant) c nK . By using various trigonometric identities it is possible to reduce the number of c nK in the coefficient of any x i to a minimum number of integrally independent terms. As a first step in this reduction, inversion and shifts allow replacement of any c nK by c mK with 0m L 4 ⌊ ⌋ / provided we treat separately even and odd K so that the replacement rule (28) with its (−1) K factors is the same for all K in either category. Such separation with distinct rules for different groups but the same rules for every K within a group dictates that the general grouping is defined by blocks K d where d is the greatest common divisor of K and L. The number of members M d in block K d is j(L/d)/2 where j(n) is Euler's function and the division by 2 arises from our restriction 0<K<L/2. Any K d with one element will be one of L/6, L/4 or L/3 which was considered in the preceding paragraph.
Before dealing with the general c mK reduction to an integrally independent set consider the L=12, S=4 example again. The distinct blocks are K 1 =1, 5 and K 2 =2, K 3 =3, K 4 =4 so that only K 1 remains to be treated 2 . The c mK left after reduction by (28) are 1, c 1K , c 2K and c 3K but for K=K 1 =1 or 5, c K 2 1 =1/2 and c K 3 1 =0 leaving only the integrally independent 1 and c K 1 1 in which to express the result of the division (22). The explicit result for the rationalized P K 4 from ( where the L=4 M case is the trivial cos(πn/2)=0 for n odd (e.g. c K 3 1 = 0 in the L=12 example above). The result in (32) for L=4 M+2 follows from the roots of unity condition n N cos 2 / together with (28). No identities beyond (28) and (32) are needed if L/2=p, p prime >2, or L/2=2 ℓ . If L has odd divisors >1 some of the odd K in the interval 0<K<L/2 will be excluded in the construction of K 1 . We will then need as many new identities as there have been exclusions. One set of identities follows trivially from (32)whenever L is a multiple of some 4M+2, M>0, then which with f>1 supplements (32). An example is f=2, M=1 giving c K 2 1 =1/2 used in the L=12 discussion leading to (29). Other identities follow from (33) which we get by first rewriting (33) as On multiplying this by c iK 1 and again using the identity 2cos(a)cos(b)=cos (a+b)+cos(a−b) we obtain An example replacement using (34) is at L=24 (f=4, M=1) where K 1 =1, 5, 7, 11 and with i=1, c c c .

= -
Together with c K 4 1 =1/2 from (33) and c K 6 1 =0 from (32) we are left with the required four integrally independent 1, c , replaces (32) and there are corresponding replacements for (33) and (34). For any given L and divisor d, at most two c mK d relations are needed to complete the division (22) provided these are used in replacements at each step of the division process so as to always limit the maximum m in c mK d to a fixed number. Furthermore, the effort to derive the required relations from formulas such as (32)-(34) can be avoided by using a PSLQ determination instead. Specifically, for any L and d we know the number of K d elements is M d = j(L/d)/2 and the empirical evidence, based on PSLQ analysis, is that c mK d =cos(2πmK d /L), m=0, 1, K, M d −1, are integrally independent and can be used as a basis in which to express any c , mK d m M d , as a sum with rational coefficients. The PSLQ algorithm, with any K d as numerical input, will provide an analytical expression for c M K d d that suffices for d even and in addition c M ) that is required for d odd. This procedure has been confirmed for all even L to 100.
This completes the N=2 overturned spin analysis that forms the basis for (5) and (6). Many examples have shown the structure of (5) and (6), as defined by the blocks K d with M d members, remains unchanged for any NL/2 overturned spins. The N dependence lies entirely in the degree of the integer polynomials which relates directly to the number of states D K (L, S) determined in the next section.

State counting
To determine the number D(N, L, K) of states of total momentum K for N overturned spins in a length L periodic chain start with the observation that the binomial L N ( ) is the total number of configurations ψ for fixed N and L and these can be separated into exclusive classes ψ d where d is a common divisor of L and N. The distinguishing feature of class ψ d is that for configurations T n ψ d (translations by n=1, 2, K from ψ d ) the first occurrence of T n ψ d =ψ d is at n=L/d. Such configurations are formed from d repetitions of N/d overturned spins on segments of length L/d. The configurations in ψ d can be grouped into D d blocks, each block containing L/d translation related configurations T n−1 ψ, n=1, K, L/d, which provide a basis for forming, by superposition, D d states for each total K which is necessarily restricted to multiples of d. Adding together the state counts (L/d) D d of every class ψ d gives the total L N ; ( ) this is the sum rule The notation used in (36) and the following is that (M, M′) is the greatest common divisor of a pair M, M′ while m|M denotes m is a divisor, including 1 and M, of M and Σ m|M means sum over m subject to the constraint m|M. More generally, the total number of periodic configurations of period L/d is and contributing to this total are the classes ψ (N,L)/d′ for d′|(N, L)/d. The corresponding sum rule is being the special case d=1. The number of equations (37) are the number σ 0 of divisors d of (N, L) and these uniquely determine the σ 0 unknown D d . The number of states D(N, L, K) then follows as å D = / and so we confirm An explicit formula for D d is obtained as follows. Define f(d′) as the expression in the sums (37) and replace that equation list by the equivalent where μ is the Möbius function. An equivalent of (40) is obtained by the substitution d→(N, L)/d again and when the resulting D d is substituted into (38) we get the explicit state count 0K , the fully aligned states. The number of states D K (L, S) at fixed S=S z is given by the well known subtraction As an example consider L=12. We get from D(N, L, K) in (41) that D(1, 12, K)=1, D(2, 12, K)= 5+Δ 2,K , D(3, 12, K)=18+Δ 3,K , D(4, 12, K)=40+2Δ 2,K +Δ 4,K , D(5, 12, K)=66, D(6, 12, K)= 75+3Δ 2,K +Δ 3,K +Δ 6,K while a subtraction (42) gives D K (12, 0)=9+3Δ 2,K +Δ 3,K +Δ 6,K . The values D 0 (12, 0)=14 and those for other L using (41) and (42) agree with the sums D(SP01)+D(SP02) given by Fabricius et al [11] in their Table II. On the other hand D(6, 12, 0)=80 calculated here differs from their D(S z =0, K=0)=44. More detailed comparison shows that D(S z =0, K=0) in [11] incorrectly includes only even S contributions. That the state counts (41) are correct has been confirmed by many additional checks including comparison to a generalization of Bethe's [1] state counting to which I now turn.
A string configuration for a state of total spin S and S z =S on a chain of even length L with periodic boundary conditions is specified by the list (p 1 , p 2 , p 3 , K) where the p n are the number of n-strings in the configuration. Each n-string is associated with n overturned spins and this yields the constraint N=∑np n =L/2−S on the total number of overturned spins N. The Bethe formula for the number of states with this configuration, denoted below as p , with each binomial factor the number of ways p n 'particles' (i.e. strings) and h n 'holes' can be arranged in p n +h n integer slots. An important observation from (43) is that h n depends only on p m , m>n and in particular h 1 =2 S+2∑(n−1)p n is fixed by the n-string content for n2. Since the constraint N=∑np n =L/2−S also fixes p 1 =N−∑ n>1 (np n ), any configuration can equally be specified by just the list (p 2 , p 3 , K). Bethe also introduced P=∑p n for the total number of 'particles' which yields the alternative expressions h 1 =2S+2∑(n−1)p n =2S+2N−2P=L−2P, results that will be of use later (cf (54)).
Bethe shows that D L S p , , n ( { })summed over all p n { } that are the unrestricted partitions of N, gives the correct total number of states for N overturned spins but does not explicitly remark on the number of states at fixed total (scaled) momentum K=(L/2π)∑k i . However, implicit in (43) is the observation that a shift of any 'particle' or 'hole' to an adjacent slot leads to the same change |ΔK|=1. Consequently it is possible to define a generator Z L S p , , n q ( { }) which is a polynomial invariant under the interchange q↔1/q with the coefficient of q κ being the number of states at K=κ relative to a central value K=K c . This generator has the form of ⎟ replaced by the Gaussian binomial modified by a prefactor q −ph/2 for q↔1/q invariance. Explicitly, The justification for this prescription relies first on Pólya's [12] observation that the coefficient of q A in the expansion of is the number of p+h step walks between (0, 0) and (h, p) that enclose area A between the walk, the x-axis and the line x=h. Second, there is a one to one correspondence between Pólya walks and configurations of p particles and h holes and, to within an additive constant, K=A. To show this adopt the reference configuration corresponding to the zero area Pólya walk to be that of all particles to the left of all holes with the holes labelled 1, 2, K, h in sequence starting with hole 1 as the rightmost hole. A general configuration will have n i particles to the right of hole i with 0n 1 n 2 Kn h p. If this configuration is represented as a histogram of n i versus i inscribed in an h×p rectangle it will be seen to be one of Pólya's walks with A=∑ i n i . Furthermore every particle to the right of a hole is the result of an adjacent particle hole interchange and a unit increase in momentum implying ∑ i n i =K and hence A=K relative to the reference configuration momentum.
It is observed empirically that the central (symmetric) K c is either 0 or L/2 (mod L), depending on whether P=∑p n , is even or odd respectively. On incorporating this result we get as our generalization of the Bethe formula (43) the q-generator with [] q for each n given by (44). The coefficient of q K in (45) is the contribution of the particle configuration {p n } to the number of states at momentum K. Shifts of K by multiples of L are understood to bring K into the first Brillouin zone −L/2<KL/2. The relation to the total number of states (42) is where the left hand side sum is understood to be over all partitions of N=L/2−S. Consider as example L=12, S=0. Separate the partitions p(6) into even and odd P; then, in a truncated notation and {p n } written as product n , 0,   1  12  13  3  24  15   1  1  1  1  1   1  1  1  1  1  1  1  1  1   1  1  4  5  8  8  12 8  8  5  4  , 12, 0, The sum mapped to the first Brillouin zone is which agrees with D K (12, 0)=9+3Δ 2,K +Δ 3,K +Δ 6,K noted in the paragraph following (42). Neither method of calculation distinguishes between degenerate and non-degenerate states at the symmetry points K=0 and K=L/2. For that I turn to another generalization of Bethe's method.
Some of the states at K=K c arise from terms in which, in every binomial factor in (43), the particles and holes are symmetrically distributed. If the number of overturned spins is odd one of the associated Bethe wavevectors will be π but except for this isolated case the Bethe wave-vectors k i will occur in symmetric pairs 3 (k i , −k i ) and describe the non-degenerate states at K=0 or K=L/2. To obtain the number of these states note that the number of holes h n is always even in each binomial distribution and exactly half of the holes, h n /2, must occupy, say the right, half of the available slots, p 2 n ⌊ ⌋ / +h n /2. The occupancy of the left half is fixed by the required symmetry so that the symmetric (non-degenerate) state count is just the new binomial product Bethe has proved the analogous formula D(L, S, P) for the constrained total number of states by induction after first showing it satisfies the recursion The corresponding recursion here follows by replacing the binomial in the second equality in (52), which is the n=1 factor in (43), by the n=1 factor in (49) thus giving To show (51) satisfies (53) the four cases in which N and P are separately even or odd must be considered. For the even-even case set N=2 R, R=1, 2, K and P=2Q, Q=1, 2, K, R; then (53) reduces to where the two terms in braces arise from the even p 1 =2q and odd p 1 =2q+1 terms in the original p 1 sum in (53). These can be combined into a single binomial and if we define R-Q=A, Q-q-1=k the right hand side of (54) can be written with the equality verified by direct comparison of terms in the sum with terms in the hypergeometric function.
The latter is Saalschützian (cf Erdélyi et al [13] which, since R−A=Q, then confirms (54) is correct. A similar analysis for the remaining N and P even/odd cases verifies that (51) satisfies the recursion (53) in general. Furthermore, the special values Z L S , , 1 1 (51), which are easily shown to agree with the definition (50), serve as the initial conditions to complete the inductive proof of (51) for P>2. Now only a sum over P in (51) remains to obtain the total number of symmetric (non-degenerate) states. In compliance with the discussion on whether the center of symmetry K c is 0 or L/2, we have Explicit calculation of Bethe states has confirmed (57) in many cases, including all SL/2 and even L12. All other states at K=0 and L/2, necessarily including those translated from outside the first Brillouin zone, are doubly-degenerate states related by the reflection symmetry k i →−k i . The state counts (57) take a particularly simple form when related directly to the wave-vector lists that occur. These are of the form [ * , (k 1 , −k 1 ), (k 2 , −k 2 ),... ,(k n , −k n )] where * are special values comprising four cases of N * =0 to 3 wave-vectorsnull; π; π/2+i∞, π/2−i∞; π/2+i∞, π/2−i∞, πwith associated K * =0, L/2, L/2, 0 respectively. The state counts (57) now take the form The results in (58) confirm those for N * =3 for L≡2 mod 4 and N * =2 for even L in [14] (their equations (29) and (30)). These authors do not give general results for the remaining case N * =3 for L≡0 mod 4 but their specific count of 4 for L=12 with 5 overturned spins is in errordisagreeing with the count of 5 from (58), the explicit (61) arrived at by an independent calculation below, and the results reported in [15].

Non-degenerate states at K=0 and L/2
The simplest extension of BAE solutions to more than 2 overturned spins is for states of symmetrically distributed particles and holes discussed in the preceding section. These are the non-degenerate states at K=0 and L/2 and I begin with a few examples of states contributing to counts (58). The result of fixing the N * special wave-vectors is a reduced set of BAE for the remaining n independent rapidities λ j =cot(k j /2) 4 . These equations are in which the L/2 roots of the middle factor are the λ 1 =λ 2 solutions of (62) and are to be discarded. Also to be discarded are the roots of the first factor in (63) which are the singular solutions already accounted for in the N * wave-vector list. The (L/2-1)(L/2-2) roots of the R polynomial in (63) are to be paired using (62) and so exhaust the state counts (58) for n=2. For L=10, S=0, K=5 one has N * =1, n=2 and the R polynomial root equation, expressed in x, is If the roots of (64) are ordered from 1 to 12 by non-decreasing real part, the Bethe solutions are the pairs [1,9], [3,10], [5,11], [2,4], [6,7], [8,12] obtained using (62). In explicit terms and including the N * =1 root k=π (λ=0), the solutions in this sequence are, respectively corresponding to the odd partitions of 5 overturned spins, namely 1 2 3 1 (3 cases) and 1 5 , 1 1 2 2 , 5 1 (one case each) again in agreement with (49). I am unaware of any simple algebraic process that will find the analogs of polynomials (61) or (64) for n>2. On the other hand, the existence of these polynomials has been confirmed in a number of cases either by direct construction from solutions of the BAE or more simply by use of the integer relation algorithm PSLQ [4]. For any given L, N * and n one need only find one BAE solution from which to pick a wave-vector k 1 and determine x 1 =2cos(k 1 ) with a certain minimum accuracy. This x 1 is used to construct the list [1, x 1 , x 1 2 , K, x 1 nD ] where D is the state counts from (58). This list serves as input to the PSLQ algorithm and provided the accuracy is adequate, the output will be the integer coefficient list [a 0 , a 1 , a 2 , K, a nD ] in the polynomial ∑a i x i . Software packages such as Maple can efficiently find polynomial roots and what remains is then just a finite search process for D groups of n roots that satisfy the BAE.
The needed accuracy in x 1 for a successful PSLQ return is roughly nD times the number of digits in the coefficient a i of largest magnitude. This can be a severe limitation but one can always reduce the PSLQ complexity by increasing the number of BAE solutions used for input. Instead of the single root power list one constructs the array, and by linear algebra, its triangular reduction * * * * * * * which leaves, in the final row, nD−m+2 non-zero elements that become the new input list into PSLQ. Back substitution of a successful PSLQ coefficient list return into the triangular reduction array yields successive polynomial coefficients. To within the floating-point accuracy used these are either integer or rational and in the latter case the entire (tentative) list can be converted to integer by an appropriate multiplication. It is advantageous to supply the PSLQ algorithm with real coefficients; complex x i need not be discarded and instead the complex power list [1, x i , x i 2 , K, x i nD ] should be input as the two lists which are its real and imaginary parts. The largest L treated by this method has been L=20 (D=126) with PSLQ input reduced to less than 50 elements. The final 631(505) polynomial coefficients for K=0(10) appear in the supplementary data file L20_nondegen.txt as lists Ih0S0L20 I 0 0 (ˆ) and Ih10S0L20 I .
and is identified as the P=4 partition 1 1 2 1 3 1 4 1 . Here it is important to recognize that symmetry dictates that the nominal Bethe strings have the same (vanishing) real part which implies 2-fold λ root degeneracy at both ±i and 0. This degeneracy must be lifted and (67) shows it is lifted by a spitting of the roots in the imaginary direction 5 . The splitting in (67) can be emphasized by writing the λ roots as Figure 1. Non-degenerate L=20 state energies at K=0 and10, separated into partitions of 10 and grouped by particle number P=∑p n . The left-most column in each P group is the partition 1 P-1 (11-P) 1 ; partitions for the remaining columns are in the dictionary order given in Table 24.2 of AS [18]. The horizontal lines are the ferromagnetic (L/2) and limiting anti-ferromagnetic (L/2-2Lln (2) that even for L=20 are in qualitative agreement with (68). One notes that e Q becomes doubly exponentially small but does not in any way prevent the ±i and 0 root splittings from becoming exponentially small. A more interesting situation arises at L>20 when the (2 1 3 1 4 1 )-string combination with L/2-9 remaining 1-strings is kept as an S=0 excitation. The number of such excitations based on (49) is m m 6 + ( ) for L=20+4 m but I consider for each L only the one state in which the 1-string λ i are in magnitude as small as possible and sandwiched between a symmetric set of 1-string large |λ| holes 6 . The 1-strings interfere significantly with the (2 1 3 1 4 1 )-string combination and lead to an increase in both e 0 and e 1 in (68) until a complex λ i collision occurs and changes the qualitative character of the solution. This is illustrated in figure 2. That the states for L 40 are indeed the continuation of those for L<40 is confirmed by noting that the squares of the splitting between the colliding roots form a smooth sequence with a sign change at L≈39. The configuration at L=48 when viewed in isolation would almost certainly be identified as an 'apparent' 1 15 3 3 partition rather than the Bethe 1 15 2 1 3 1 4 1 . While this is just a more elaborate example of a complex root collision discussed following (19) and already observed by Bethe and others, it does illustrate that quartet configurations are typically unstable intermediate forms that facilitate transitions between states of different character.
BAE solutions for L>20 such as those shown in figure 2 have been found by Newton-Raphson (NR) iteration. The L=20 results are invaluable as a template for NR initialization for L=24. For larger L, polynomial extrapolation in L (with allowance for root collision) is usually adequate for the complex root initialization. For real root initialization it is preferable to start with numerical approximations to the density ρ=dn/dλ and extrapolate these in L. One then obtains λ n by the integration n= d Let the left hand side be numerical BAE solutions and the explicit terms with integer ε and 2 s on the right hand side be possible asymptotic WZW solutions. A sample of such paired graphs, including the (2 1 3 1 4 1 )-string combination featured in figure 2, is shown in figure 3. It is apparent that in most cases a length L=1280 is more than adequate to unambiguously establish the Bethe-WZW correspondence.
The results of the correspondence from figure 3 and many similar calculations are given in and can be understood to be the rules for all states with at most two (n>1)strings. A more comprehensive set of rules and combinatorial relations will be given in section 7 after Bethe string configurations at general K have been discussed in sections 5 and 6.
I close this section with a discussion of a very different but intriguing state. It is the single particle P=1, L/2string state which appears in figure 1 as the lowest lying S=0 excitation on the ground state of a ferromagnetic chain. This is the state [323, 341, 379, 468] which in λ representation is is the analogous single particle L/2-string for L/2 odd. The explicit (72) and (73) serve as useful templates for initial guesses for larger L and can be easily improved by NR iteration. Oscillations due to odd/even L/2 rapidly decay with increasing L and I find from an analysis of states to L=60 that the energy is where the π 2 has been inferred from numerical values but is not in doubt. Corresponding inference for the other numerical values in the series (74) has not been successful. The excitation energy ∝ 1/L implies this state is not two ferromagnetic domains separated by finite width domain walls. Another guess for a classical analog of this state is one in which the chain is cut and the ferromagnetic ground state twisted by 2π before reconnection. This state is not topologically distinct from the ground state but it is a highly degenerate stationary energy state since the vector defining the 2π rotation can have any orientation. In all such states neighbouring spins deviate by ⟩where n e and a n are chosen to guarantee the orthogonality 0.
The characteristic (eigenvalue) polynomial of this matrix agrees with that found by Bethe ansatz for all cases considered. The highest energy eigenvalue is that of the single particle L/2-string state and the spin-spin correlations found from the associated eigenvectors for even L from 4 to 20 are shown in figure 4. The j=1 correlation C 1 is related to the energy (74) by C 1 =2E L /(3L) while the factor 3 enhancement of the j=0 correlation C 0 over that of the asymptotic C 1 in figure 4 is the s=½ distinction between spin length squared s(s+1)=3/4 for a single quantum spin and the maximum s s 1 2   ⟨ · ⟩=s 2 =¼ for distinct (parallel) spins. This obvious quantum effect has no classical analog. The data in figure 4 for L=20 has the Fourier decomposition Table 1. WZW asymptotic parameters ε and s together with Bethe 1-string hole count h 1 for the lowest energy cusp state picked from every column in figure 1. Each main configuration entry is the L=20 Bethe (n>1)-string list; this is followed by a label in parentheses that is the 'apparent' large L string content if there are changes as a result of root interactions. For states labelled by F nm see text; for a state designated with +n there are additional cusp states with energies ε greater by 4 m, m=1 K n.

States of 3 overturned spins for general K
This section confirms the structures (5) and (6) for the case of 3 overturned spins and Bethe string configurations 1 3 , 1 1 2 1 and 3 1 . The results are coefficient lists reported in 3_overturned_spins.txt based on the following notation and conventions. The data for given L starts with the state count list in the form (23), versus site separation j in the single particle P=1, L/2string state for chains of length 4, 6, K, 20. Lines connecting C j , 0jL/2 at the same L are a guide to the eye. The remaining lines are polynomial in 1/L fits to C L/2 for the largest L and these extrapolate to C ∞ =-0.33(1) as shown. The inset shows the convergence of C j for L=20 when only contributions from bases 1 y ñ | through n y ñ | are kept. The extremes are C j =δ j,0 -δ j,L/2 for n=1 and the exact C j for n=126. Intermediate curves are n=2, 3, 4, 5 and 7.
For K=0, L/6, L/4, L/3 or L/2 the polynomial P x is defined by the single list 'IKS$' but for K an element of block K d with M d =j(L/d)/2 2 elements as described following (28) replaces the single list IKS$   used to define P x K ( ) as in the (29) example. The 3ν K roots of P K (x) define the 3 Bethe wave-vectors for all of the ν K states. These states are represented as the list k$S$ := [[n 1 , n 2 , n 3 ], [n 4 , n 5 , n 6 ], K, [K, n 3 K n ]] where the $ are numerical K and S as before while the |n i | are position pointers to the root list. It is to be understood that the roots x i are arranged in non-decreasing R(x i ) order with the Bethe momentum k n i associated with n i then uniquely given by (26). The list k S $ $  is energy ordered with the energy of each state given by (4). A check is provided by energy polynomial coefficient lists IeKS$   that are the analog of IKS$   but define polynomials P K (E) whose ν K roots are the energies of the states in the kKS$   lists. The analogy between IeKS$   and IKS$   extends to the combining rule (80) that is applicable to both lists. One final list Ihe S $ $   provides the energies for the states generated from Ih S $ $  at K=L/2. The results presented in 3_overturned_spins.txt include all even L, 8L26. The L=8 data, part of which is ) are the roots of the associated polynomials P K (x) of either coefficient list. Only the result following the first equality conforms to that in (6) but the second form with irrational multipliers is more typically found when obtaining the polynomials by PSLQ. Since such different forms give identical roots, supplementary data that is equivalent to (6), as in L=8 above, has been left unchanged. The remaining data in (81) can be used to verify that the roots of P K (E) determined from the list sum Ie S K Ie S 1 1 2cos are the energies calculated from lists kKS1   for K=1 and 3 using (4). The maximum L=26 exceeds the L≈21.86 critical value where (19) shows the first complex root collision for 2 overturned spins and this allows us to explore more fully string interactions. As a specific example, figure 5 shows how strings in 1 1 2 1 interact and modify bare 2 1 behaviour. The main indicators of interaction are the approximately linear drifts from the marker values and a pronounced level repulsion around the line 2k 1 =k 2 mod 2π. 2-string root collisions are first observed at L=24, a marginal shift from L=22 expected based on (19), for 22 distinct k 1 values. In contrast for the 23rd k 1 , the collision seen in the L=74 inset is suppressed until L=56. For states with root collision expected at L≈61.35 based on (19), the 2-string remains complex in two states with k 1 above the line 2k 1 =k 2 mod 2π at L=74.

L=16 singlet states for general K
This section provides further confirmation of (5) and (6) but more importantly provides the BAE solutions that serve as templates for the calculations of much longer chains. The notation and conventions follow those in section 5 and start with the S=0, L=16 state count   for K a member of the block K 1 =1, 3, 5, 7 or K 2 =2, 6 the superposition rules of (80) apply. Similarly for the energy polynomial lists IeK .
  Solutions are plausibly identified with Bethe string configurations that are partitions of 8 and confirm the counts (45). Energy versus momentum of all states, separated by partition, is shown in figures 6-8. Of particular note are cusp states defined as those for which all 1-strings occupy adjacent positions with no intervening holes. In the limit of large L these are local energy minima with respect to 1-string excitation and a particularly important set of low energy states called current excitations by Bortz et al [20]. Many such large L (≈1000) solutions have been found by NR and analyzed similarly to that described in the text leading to (70). The results for all cusp states supplementing the odd ε cases from table 1 are shown in figure 9 for ε41. The state labelling conforms to that used in figures 6-8. Combinatorial rules that predict the location of states in the L→∞ limit shown in figure 9 are found to be a simple modification of the standard Bethe rules and are described in the next section. Here I only note that while the Bethe string labelling is an essential component of these rules, a different 'apparent' string labelling is often a much better indicator of the solution rapidities in the complex λ plane.
Very clear patterns are seen in figure 9 of which the most striking is that all state counts are consistent with products of the (left and right moving) ±κ excitation counts appearing in the single row diagonals ε=(n−1) 2 +2|κ| terminating at the single n-string values at κ=0. This is as expected for the WZW model and is also explored in more detail in the next section which concludes with a conjecture for the string content of all left and right tower states in the total S=0 sector. Another observation is that any cusp state associated with WZW spin s contains at least one n-string with n>2 s; this is shown to follow from the string content conjecture.

Low energy S=0 state counting for L→∞
The cusp state examples described in sections 4 and 6 lead naturally to conjectures for the multiplicity of all low lying singlet states as L→∞. An important parameter in the cusp state classification is the number of 1-string holes h 1 =2∑(n-1)p n (cf (43) and subsequent discussion) which is necessarily even and fixed by the n-string content for n>1. Thus h 1 is decoupled from L and our analysis does not require any specific value for L beyond L even and L?h 1 . The complete list of possible p n h 1 1 > { } = (p 2 , p 3 , K) for any h 1 is the list of the partitions of h 1 /2 with every integer in a partition incremented by one. For example, for h 1 =10, the partitions of 5 (1 5 , 1 3 2 1 , 1 2 3 1 , 1 1 2 2 , 1 1 4 1 , 2 1 3 1 , 5 1 ) after incrementing are the cusp state configurations {p n>1 } 10 (2 5 , 2 3 3 1 , 2 2 4 1 , 2 1 3 2 , 2 1 5 1 , 3 1 4 1 , 6 1 ). Define P =∑ n>1 p n and Ñ =∑ n>1 np n , the total number of 'particles' and overturned spins Figure 6. Energy E versus (quasi) momentum K for Bethe 1 8-n n 1 (P=9-n) configurations. Only the (n>1)-string component is used as plot label. Lines connect upper and lower boundary states for each n as a guide to the eye. For clarity the K (mod L) for each state has been chosen such that only after including reflection about K=0(L/2) for states of even(odd) particle number P will the display be in explicit agreement with the counts (45). Diamonds replace crosses for the ground state (gs) and cusp states described in the text. Configurations with no 1-strings are also potential cusp states and are marked as squares. The horizontal line of length L (one periodic cell) marks the ferromagnetic energy L/2. respectively in the (n>1)-strings. For the example list, P =5, 4, 3, 3, 2, 2, 1 and in general Ñ = P +h 1 /2 with 1P h 1 /2.
Any particular {p n>1 } appears as a distinct cusp state for each division of h 1 into exclusively left h L and right h R holes. In the symmetric case, h L =h R =h 1 /2, there is a trivial generalization of the generator (45) to the cusp state generator 7 .  Bethe configuration labelled cusp state asymptotic energy ε from (7) versus momentum κ=K−K c , K c =0(L/2) for L/2 odd(even). States are distinguished by WZW model s=s L =s R =1/2(red), 3/2(green) and 5/2(blue). States at κ>0 are obtained by reflection about κ=0. There are hidden s=3/2 states near κ=0, ε=37; for these see the lower left corner insert which hides the s=1/2 states instead. The data for the lowest excited state shown has been carried to L=16384 in [21].
To efficiently evaluate configuration sums of expressions such as (86) it is useful to first determine amplitudes A i,j , i=1, 2, K, 1ji, which are sums of the products Π [K] q appearing in (85) and (86) subject to the constraints h 1 =2i, P =j (or Ñ =i+j). Endpoint values are A i,1 =A i,i =1 arising from configurations (i+1) 1 and 2 i respectively. Intermediate cases for the partition of 5 list above are is easily verified by noting the Gaussian binomials at q=1 are ordinary binomials after which follows a one to one correspondence between the sums here and Bethe's constrained sums D L S P , , ( )in (52). What are L and P with S=0 in (52) are here h 1 and P respectively as a consequence of our transcription of partitions into the cusp state configurations {p n>1 }. The symmetry A i,j+1 =A i,i-j allows us to restrict our explicit amplitude calculation to A i,i-j with 1j(i-1)/2 in which case every Gaussian binomial product contains some f 2 and with the replacement q→1/q we see all the terms in (93) can be identified with the terms on a single (lowest) diagonal of energy versus momentum in figure 9. / , we meet only the products 2 1 from B 2 0 ( ) times (p 2 , p 3 , K) from B .

2,2 2n
The process for getting the independent left and right cusp count generators illustrated by the examples above extends to the general case. Equations (90) and (91) rewritten as } h 1 =4 s+2ν, contributing to the sum (100) are as described in the first paragraph of this section while the product is over those n for which p n is non-vanishing. The Gaussian binomial a b q ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ is to be understood to vanish when a<b which happens when n is its maximum n max (making h n =0, cf (85)) and n max 2 s; this is the basis for the rule that there is no contribution to B q s v s 2 2 2 + ( ) ( ) from a configuration that does not contain at least one string with length greater than 2 s. Conversely, every configuration with n max >2 s contributes; for a proof it suffices to show h n −2s+n−1 0 for every n2 s in (100). Now h n =2∑ m>n (m−n)p m 2(n max −n) n max −n+1>2s−n+1 which is the required result. Agreement between (98) and (100) has been confirmed numerically to high order.
Many regularities are observed in the solutions (98) and (100). Of particular note, the terms c κ q κ in B q s s 2 2 2 n + ( ) The formulas for C (1) and C (0) =1+xC (1) can be derived using the A i,j sum rule in the discussion following which corresponds to ordinary multiplication and division (p′ 2 , p′ 3 , K)(p 2 , p 3 , K)/2 1 =(p 2 +p′ 2 -1, p 3 +p′ 3 , K). Two possibilities for the 3 1 product are required to correctly generate all terms in (A.1) with case c1 arising from the replacement 3 1 →2 2 that maintains h 1 =4 as noted in the remarks following (95).
Consider first the generation of B q .