The Integrable (Hyper)eclectic Spin Chain

We refine the recently introduced notion of eclectic spin chains by including a maximal number of deformation parameters. These models are integrable, nearest-neighbor n-state spin chains with exceedingly simple non-hermitian Hamiltonians. They turn out to be non-diagonalizable in the multiparticle sector (n>2), where their"spectrum"consists of an intricate collection of Jordan blocks of arbitrary size and multiplicity. We show how and why the quantum inverse scattering method, sought to be universally applicable to integrable nearest-neighbor spin chains, essentially fails to reproduce the details of this spectrum. We then provide, for n=3, detailed evidence by a variety of analytical and numerical techniques that the spectrum is not"random", but instead shows surprisingly subtle and regular patterns that moreover exhibit universality for generic deformation parameters. We also introduce a new model, the hypereclectic spin chain, where all parameters are zero except for one. Despite the extreme simplicity of its Hamiltonian, it still seems to reproduce the above"generic"spectra as a subset of an even more intricate overall spectrum. Our models are inspired by parts of the one-loop dilatation operator of a strongly twisted, double-scaled deformation of N=4 Super Yang-Mills Theory.


Introduction and Overview
The phenomenon of integrability of certain one-dimensional quantum spin chains was discovered in 1931 by Hans Bethe [2]. He solved what is now known as the periodic XXX Heisenberg spin chain of length L, whose Hamiltonian reads where σ is essentially the spin operator at site , expressed in terms of the three Pauli matrices. H is a 2 L × 2 L hermitian matrix acting on the tensor product space (1.2) If we denote the canonical basis vectors of C 2 by |1 = 1 0 and |2 = 0 1 , the canonical basis vectors of (1.2) are |n 1 n 2 . . . n L with n j = 1, 2. The Hamiltonian H in (1.1) acts in a very transparent way on the canonical basis once one expresses it in terms of the nearest-neighbor permutation operator P , +1 (where P L,L+1 := P L,1 ): P , +1 with P , +1 | . . . n −1 n n +1 n +2 . . . = | . . . n −1 n +1 n n +2 . . . . (1.3) Written in exactly the same form, this Hamiltonian immediately generalizes to the one of an integrable 3-state spin chain with n j = 1, 2, 3 acting on (1.4) Actually, the general case of an n-state model, where the Hamiltonian acts on L copies of C n , is also integrable: It takes again the same form (1.3), except that now n j = 1, . . . , n.
The other six cases (+, +, −), (+, −, +), (−, +, +), (+, −, −), (−, +, −), (−, −, +) are different from the eclectic case. However, they are once again equivalent to each other: One checks that they are related by the six possible permutations of the three states 1, 2, 3 (a mere relabelling) followed by a suitable redefinition of the twist parameters ξ ± j . They all correspond to a refinement of the integrable spin chain model introduced as "broken su(3) sector" in section 5.1 of [1]. This model is quite different from the eclectic model and will be investigated separately 4 .
The eclectic three-state spin chain Hamiltonian studied in this paper will then be, after simplifying the above notation by defining ξ j := ξ + j , whereP |11 = 0P |22 = 0P |33 = 0 (1.14) This leads to a novel family of, on first sight, exceedingly simple looking spin chain models with non-hermitian Hamiltonians. We will discuss their relevance and inspiration from a certain double-scaling limit of strongly twisted planar N =4 Super-Yang-Mills theory (SYM) [3][4][5] in section 2.2. We will also verify the integrability of these models for arbitrary complex parameters ξ j in 2.1. However, except for trivial sectors, where the states are made up from only two of the three excitations, these Hamiltonians are nilpotent on any state containing all three excitations [1]. They are therefore totally non-diagonalizable. The most one can do is to bring them into Jordan normal form (JNF). However, as we will outline in section 2.3 and then demonstrate in more detail in chapter 3, the quantum inverse scattering method mostly fails 5 , at least in its traditional form, to be helpful for this enterprise. This is vexing, in particular since we shall find, through some caseby-case numerical studies in chapter 4 up to moderately large lengths L, that the sizes and multiplicities of the appearing Jordan block show intriguingly regular patterns. We shall also find evidence for a certain universality of the "spectrum" of Jordan blocks: Its dependence on complex parameters ξ j is relatively weak, in a sense to be explained, as long as these are suitably "generic". A non-generic situation arises e.g. when some of the parameters ξ j are zero (which is allowed!). The most extreme case is that two of them are zero, say ξ 1 = ξ 2 = 0. We may then set without loss of generality ξ 3 = 1 to obtain (again with P L,L+1 = P L,1 ) This novel system must surely be the simplest one among all integrable three-state spin chains. We call it the hypereclectic model. While looking trivial on first sight, we will demonstrate in chapter 4 that its intricate "spectrum" of Jordan blocks is actually even richer than the one of the more general and more complicated looking model (1.13), (1.14) for generic parameters ξ j .

Quantum Integrability of the Eclectic Spin Chain Models
Let us begin by establishing the quantum integrability of the spin chain Hamiltonians defined in chapter 1. We will assume some basic familiarity with the quantum inverse scattering method, and in particular with the Algebraic Bethe Ansatz (ABA) technique, see e.g. [6,7] for excellent introductions. Since we are discussing three-state models, an additional complication is the necessity to consider a nested ABA. For a recent pedagogical introduction see e.g. [8]. The starting point is an R-matrix of size, in the three-state case, 9 × 9 that acts on the tensor product C 3 ⊗ C 3 . In the twisted case it reads where u is termed spectral parameter, and the upper indices 1, 2 label the two copies of the space C 3 in the tensor product. One now uses this R-matrix as a so-called Lax-operator 6 and builds up a quantum monodromy matrix Here · denotes 3 × 3 matrix multiplication in an auxiliary space, which is another copy of C 3 . The entries of this 3 × 3 matrix act on (1.4). Therefore,M a,L (q 1 ,q 2 ,q 3 ) (u) acts on the tensor product of (1.4) and the auxiliary space a. It is customary to drop the index L. One then takes the trace over the space a, and obtains the transfer matrix acting on (1.4): It is well-known (e.g. [6,7]) that it yields for u = 0 the shift operator of the spin chain, i.e.
It is also known that a nearest-neighbor Hamiltonian may be extracted as the logarithmic derivative of the transfer matrix at u = 0, and for (2.1) one finds precisely (1.5): The R-matrix (2.1) of the twisted model satisfies the Yang-Baxter equatioñ ) for arbitrary complex values of q 1 , q 2 , q 3 . This implies [6] the following relation intertwining the monodromy matricesM a (q 1 ,q 2 ,q 3 ) with the R-matrixR 12 (q 1 ,q 2 ,q 3 ) : Taking the traces over the spaces labelled by 1 and 2, one derives the following commutation relations for arbitrary complex values u, u : Since the transfer matrix may be interpreted as a generating function of a sufficiently large number of independent charges in involution, and since the Hamiltonian is commuting with these, one calls, by definition, the model quantum integrable.
Let us now consider the strong twisting limits of the above R-matrices. The upshot is, that quantum integrability as defined above survives the limiting procedure. Here we will only consider the limit to the eclectic spin chain model, as discussed in chapter 1: We multiply the R-matrix (2.1) by ε, put q i := ε −1 ξ j , and replace u by ε u. We then take ε to zero. This results in Taking the logarithmic derivative at u = 0 as in (2.5) results precisely in the Hamiltonian H (ξ 1 ,ξ 2 ,ξ 3 ) in (1.13),(1.14). It is still integrable, as one easily checks that the Yang-Baxter equation (2.6) also holds withR (q 1 ,q 2 ,q 3 ) replaced by the eclectic R-matrixR (ξ 1 ,ξ 2 ,ξ 3 ) . For completeness we also state the R-matrix for the hypereclectic model defined in chapter 1: (2.10) It also satisfies the Yang-Baxter equation and generates the integrable hypereclectic Hamiltonian H in (1.15),(1.16).

Interlude: Origin and Relevance of the Models
Let us briefly pause to discuss the origin of the models introduced in chapter 1 and section 2.1. Applying twisted boundary conditions to quantum spin chains has a long history in condensed matter theory. They are of physical interest, preserve integrability as we just recalled, and yield a better understanding of the inner workings of the quantum inverse scattering method, to be discussed below. However, twisted integrable spin chains are also of relevance in certain conformal quantum field theories in four (as well as in three and two) dimensions, see [9], and in particular [10], where their relation to the so-called three-parameter γ j -deformed versions of N = 4 SYM is explained. This was the starting point for a series of papers [3][4][5] that introduced and started to analyze a novel class of integrable conformal four-dimensional quantum field theories. Let us recall the double-scaling limit used in [3][4][5]: Defining the square of the planar gauge theory coupling constant as g 2 = λ 16π 2 , where λ is the 't Hooft coupling, it was suggested to take g → 0, while some or all of the twisting parameters either turn to q j = e −iγ j /2 → ∞ or else to q j = e −iγ j /2 → 0, such that, respectively, either the products g q j or quotients g q −1 j are held fixed. As in [1], we shall call these double-scaled, twisted models simply "strongly twisted models". A neat way to systematically treat this total of 2 3 = 8 different strong twisting limits of γ j -deformed versions of N = 4 SYM is to use the parameter ε introduced in (1.9): Write q j := ε ∓1 ξ ± j , replace g → ε g, and take ε to zero. This results in, respectively, the limits g q j → g ξ + j and g q −1 j → g (ξ − j ) −1 . In all eight cases, the gauge fields decouples from the interacting part of the Lagrangian. As in [3][4][5], we will simply ignore them altogether 7 . The limit where all three couplings are scaled as g q j → g ξ + j (i.e. all q 1,2,3 → ∞) reduces the interacting part of the γ j -deformed Lagrangian of N = 4 SYM to where by "cyclic" we mean cyclic permutations of the three indices. We do not show the standard kinetic terms of the complex bosonic and fermionic fields. Note that the fourth fermion ψ 4 decouples from the interactive part. Conversely, in the limit g q −1 j → g (ξ − j ) −1 (i.e. all q 1,2,3 → 0) we obtain The other six limits are more subtle. If e.g. we take the limits g, q 3 → 0, q 1,2 → ∞, i.e.
we end up (setting for simplicity the redundant overall coupling g = 1) with If we put ξ + 1 = 0, ξ + 2 = ξ 2 and ξ − 3 = ξ −1 3 we recover as a special case the model mentioned in equations (8),(9) of [4]. Now, as argued in [3,4] and explained in more detail in [1], the one-loop dilatation operator of the above models is closely related to the integrable Hamiltonians discussed in chapter 1 and section 2.1. The precise statement is as follows: If we analyze the strongly twisted quantum field theories (QFTs) with the above Lagrangians, and choose to only study local composite operators 8 containing the three partonic chiral fields φ 1 , φ 2 , φ 3 , the QFTs dilatation operator D is given by where D 0 is the classical dilatation operator 9 , and the one-loop dilatation operator is g 2 times the spin chain HamiltonianĤ (ξ ± 1 ,ξ ± 2 ,ξ ± 3 ) in (1.10). Once again, in this paper we focus on the eclectic (+, +, +) case with ξ j := ξ + j , where the Hamiltonian isĤ (ξ 1 ,ξ 2 ,ξ 3 ) in (1.13), (1.14), with extra attention given to the special hypereclectic case, where ξ 1 = ξ 2 = 0, ξ 3 = 1 with Hamiltonian H in (1.15), (1.16). The gauge theory interpretation of this model is as follows. The one-loop dilatation operator still acts on composite operators containing the three chiral fields φ 1 , φ 2 , φ 3 , which are to be interpreted as the one-site states 1, 2, 3 in the spin chain interpretation, respectively. However, now the dilatation operator merely chirally exchanges φ 1 , φ 2 , while φ 3 becomes a non-moving "spectator field": (2.16) As we shall see below, these spectator fields act as impenetrable domain walls within the spin chain, leading to very intricate effects. 8 That is, we do not consider operators containing any derivatives, fermions, anti-chiral fields φ † 1 , φ † 2 , φ † 3 , nor the (in any case decoupled) gauge fields. 9 Here D0 only counts the total number of φ's of the composite operator = length L of the spin chain.
For the remainder of this paper we will ignore the gauge theory origin and interpretation of the integrable eclectic and hypereclectic spin chains, and simply study their intricate "spectrum" on its own right. We shall begin by explaining in the next section 2.3 why and how the spectral problem for diagonalizable spin chains is to be replaced, in the case of the non-diagonalizable eclectic and hypereclectic chains, with the problem of finding the appropriate JNF of the transfer matrix and the Hamiltonian. Correspondingly, we shall furthermore demonstrate how and why the direct application of the ABA to the integrable spin chains built from the R-matrix (2.9), (2.10) is bound to fail.

Non-Diagonalizability, Jordan blocks, and Failure of the ABA
For generic, finite, complex twist parameters q i the transfer matrixT (q 1 ,q 2 ,q 3 ) (u) in (2.3) of the three-state spin chain is diagonalizable with 3 L eigenvaluesT j (q 1 ,q 2 ,q 3 )(u) , j = 1, . . . , 3 L : Because of (2.8) the linearly independent eigenstates |ψ j are also eigenstates 10 of the shift operator U =T (q 1 ,q 2 ,q 3 ) (0) in (3.53). Defining (2.18) one has, from the condition U L = I, In light of (2.5), the |ψ j are of course also eigenstates of the HamiltonianH (q 1 ,q 2 ,q 3 ) : Integrability allows to find this spectrum, for example by means of the nested ABA (see e.g. [8]). The ansatz for an eigenstate with L − M states 1, M − K states 2, and K states 3 is where the creation operators B a (a = 1, 2) act on the so-called reference state |11 · · · 1 , which is exclusively made of fields of type 1 at all L sites. Each B 1 or B 2 creates a linear combination of new states, where one of the 1's is replaced by a 2 or 3, respectively. The B 1 (u), B 2 (u) are obtained as two of the components of the monodromy matrix (2.2), written as a 3 × 3 operator-valued matrix in auxiliary space a as In the nested Bethe ansatz, F a 1 ···a M is an eigenstate of a secondary, inhomogeneous rankone spin-chain of length M . It may be constructed by acting K times with the rank-one creation operators b(v i ) on a pseudo-vacuum with inhomogeneities u 1 , . . . , u M : The ansatz (2.21) works, i.e. |ψ j indeed becomes an eigenstate of the transfer matrix, if and only if both the level-one Bethe roots u 1 , . . . , u M as well as the level-two Bethe roots v 1 , . . . , u K are meticulously chosen as solutions of a nested Bethe ansatz, see section 3.1 below. The transfer matrix eigenvaluesT j (q 1 ,q 2 ,q 3 ) (u) are then determined explicitly through the level-one Bethe roots. This procedure is known to yield, as long as the q i are generic, the complete spectrum of 3 L eigenstates, once all solutions are of the BAE are found.
In contrast, the eclectic model is very different. Let us already summarize the gist of our findings (for more details, see the following two sections). For the Hamiltonian H (ξ 1 ,ξ 2 ,ξ 3 ) of (1.13), (1.14) one has to replace (2.20) by Here the |ψ m j j are so-called generalized eigenstates, associated to generalized eigenvalues The statement is then that, for a non-diagonalizable 3 L ×3 L matrix such as the Hamiltonian where b is the total number of Jordan blocks, and their sizes add up to The last equation shows that for each Jordan block, labeled by j, there is exactly one true (i.e. non-generalized) eigenstate |ϕ 1 j with eigenvalueÊ j (ξ 1 ,ξ 2 ,ξ 3 ) . The two-state sectors, where the matrix acts on spin chains that only contain two out of the three fields, are diagonalizable, i.e. the corresponding "Jordan blocks" are all of size l j = 1, and their exact spectrum is easily found [1]. However, in the three-state sectors there are many Jordan blocks of size l j > 1, even though some l j = 1 blocks remain. In fact, as already argued in [1], the generalized eigenvalues are all zero: The Hamiltonian becomes nilpotent of degree l j , see (2.24). Note however that the generalized eigenstates are still ordinary eigenstates of the shift operator U =T (q 1 ,q 2 ,q 3 ) (0) (compare to (2.19)): for all m j = 1, . . . , l j . Nevertheless, the transfer matrixT (q 1 ,q 2 ,q 3 ) (u) is only diagonalizable at u = 0. As we shall see below, for u = 0 the situation is analogous to the one for the Hamiltonian: (2.17) only holds in the two-particle sectors, while in general one has the same situation as in (2.24): for generalized eigenstates |ψ m j . Surprisingly, in the three-particle 11 sectors of the eclectic model the generalized eigenvalues turn out to be u-independent (incidentally explaining via (2.5) the result (2.30)), and are in fact equal to the eigenvalues of the shift operator: Just like (2.26), the transfer matrix may be brought by a suitable 3 L × 3 L similarity transform S (u) into the JNF 12 is u-dependent and as such different from S in (2.26). This is in stark contrast to ordinary integrable spin chains, where the similarity transform that diagonalizes the transfer matrix is u-independent and simultaneously diagonalizes all commuting charges including the Hamiltonian, cf. (2.8).
Given this structure, one would hope that a generalization of the ABA exists that would allow the construction of the generalized eigenstates |ψ m j , in order to extract the number and sizes of Jordan blocks of the eclectic model. Unfortunately, we have not yet been able to find such a generalization. The first and foremost problem seems to be that the ansatz (2.21) simply fails from the very start: For all values of the F a 1 ···a M and u 1 , . . . , u M the ansatz does not span the full state space needed to build the wanted generalized eigenstates. Disappointingly, for the majority of Jordan blocks j, it does not even account for the m j = 1 states of the block, which are ordinary eigenstates 13 . This vexing fact will be seen to be a consequence of our results in the next chapter. See especially section 3.2, where we will argue that the eigenvectors of the finitely twisted chain in a given cyclicity sector at fixed L, M , K all become collinear with a certain "locked" state. 11 In the diagonalizable (lj = 1) 2-particle sectors theT j (ξ 1 ,ξ 2 ,ξ 3 ) (u) are instead known polynomials in u. 12 The Jordan block structure of the transfer matrix (2.33) a priori does not have to coincide with the one of the Hamiltonian, (2.26). However, we suspect that the two Jordan decompositions coincide. We have found no counterexamples so far, but also could not yet find a proof of this conjecture.
13 Every Jordan block possesses exactly one non-generalized, ordinary eigenstate, the state with mj = 1.

Collapse of the Bethe States in the Strong Twisting Limit
In this chapter, we will study the strong twisting ε → 0 transition from the integrable three-state spin chain model twisted by three parameters q 1 , q 2 , q 3 to the general eclectic model with twisting parameters ξ 1 , ξ 2 , ξ 3 on the level of the ABA. In particular, we would like to understand the fate of the Bethe states (2.21) that diagonalize the transfer matrix (2.17) for any ε > 0, and investigate whether any useful information on the generalized eigenstates (2.32) may be gathered from the limiting procedure.
In section 3.1 we will analyze the BAE and their solutions in terms of Bethe roots in the ε → 0 limit. We uncover rather intricate and rich behavior, with, generically, fractional scaling behavior of the Bethe roots. Reassuringly, this will be shown to lead to the correct generalized eigenvalues (2.33) of the transfer matrix, thereby also explaining from the point of view of the ABA why all three-particle eigenstates of the eclectic chain are zero-energy states. However, despite the rich scaling behavior discovered in 3.1, it appears that no further non-trivial information on the structure of the Jordan blocks (i.e. their number and sizes) or the structure of the generalized eigenstates may be extracted from the ABA in the scaling limit: We find strong evidence in 3.2 that all Bethe states collapse to, in each cyclicity sector, a single "locked state" that happens to be the one true eigenstate of the largest Jordan block.
In the below, we will only study the transition of the ABA to the eclectic spin chain with "generic" scaled twist parameters ξ 1 , ξ 2 , ξ 3 . The limit to the hypereclectic chain would have a different singularity structure and will not be analyzed. Therefore we may, without loss of generality, 14 assume that the number of 1's making up the spin chain states is larger or equal than the number of 2's, while the latter is larger or equal than the number of 3's. Denoting these three numbers by L − M , M − K and K, respectively, this is represented by the following inequalities: These inequalities are saturated in the case L = 3K, M = 2K for arbitrary K. This clearly corresponds to the special case where the spin chain is made up from an equal number of 1's, 2's and 3's.

Bethe Ansatz for the Eclectic Spin Chain
In a rather routine if somewhat tedious fashion one may derive a nested BAE for the levelone Bethe roots u 1 , . . . , u M as well as the level-two Bethe roots v 1 , . . . , v K from the ABA (2.21) and (2.23) for the eigenstates. The procedure employs integrability by using the so-called fundamental commutation relations derived from the Yang-Baxter based (2.7), see e.g. [8]. The final result 15 for the system of BAE of the three-state model for general twisting parameters q 1 , q 2 , q 3 reads 16 while the eigenvalues of the transfer matrix (2.17) are then expressed as The shift operator eigenvalue (2.19) is extracted as while the energy, i.e. the eigenvalue of the Hamiltonian (2.20), is given bỹ We have omitted the index j, labelling the eigenstates, onT j (q 1 ,q 2 ,q 3 ) (u) and E j (q 1 ,q 2 ,q 3 ) in (3.4), (3.5), (3.6). Of course the Bethe roots strongly depend on j.
Let us now understand what happens to the BAE and their solutions in the strong twisting limit ε → 0, where one replaces according to (1.9) (in the case (+, +, +) with in terms of new, finite deformation parameters ξ j . Inserting this into (3.2),(3.3) one obtains where we have introduced for conciseness of notation Interestingly, for the special case the powers of ε drop out, and the BAE, and hence their solutions, are identical to the ones of the twisted model before taking the scaling limit, (3.2),(3.3), with the original q j 's simply replaced by the ξ j 's. See 3.1.2 below. In all other cases the powers of ε lead to inconsistencies, unless the Bethe roots exhibit a suitably singular behavior so as to compensate these powers. Finding the correct scaling of the roots can be quite tricky, especially for the level-two roots {v m }. We will now present a near-complete classification (see 3.1.5 for the still unclear cases) of the possible scaling behaviors, and their consequences for the "observables" (3.4), (3.5), (3.6).

K = 0 Sector
This corresponds to an n = 2 chiral XY model, where only the u m Bethe roots are kept. It is not necessary to assume the filling condition L ≥ 2M . The nested BAE (3.8) turn into the unnested set It is rather obvious, and discussed in considerable detail in [1], that the Bethe roots should scale as u m = ε u − m in the ε → 0 limit. The BAE (3.12) then reduce to The solutions {u m } are immediately seen to be given in terms of subsets of the L L-th roots of unity. It is easy to demonstrate completeness of states of these equations, in line with the fact that the K = 0 sector is completely diagonalizable, despite the strongly twisted two-state model's nonhermiticity. No non-trivial (i.e. of size larger than one) Jordan blocks appear. The transfer matrix eigenvalue (3.4) turns intô (3.14) The shift operator eigenvalue (3.5) is then given bŷ (3.15) while the energy, i.e. the eigenvalue of the Hamiltonian (3.6), is elegantly given bŷ (3.16)

L = 3K, M = 2K Sector
In the ε → 0 limit, the BAE (3.8) and (3.9) look singular except for the special case where the ε-dependence simply drops out. In this curious case, for the strongly twisted model the same BAE as for the original generic twisted model are satisfied, except that the three twist parameters q i are replaced by ξ i in (3.8), (3.9). This means that the Bethe roots of a given state do not change at all as one approaches the strong twist limit! However, the effects of the limit are still seen on the level of the expressions for the "observables".
In particular, since we also need to replace u → u along with (3.7), we find that the expression for the transfer matrix eigenvalue (3.4) becomes a u-independent constant, and thus identical to the shift operator eigenvalue (3.5): while the energy formula (3.6) formally yieldŝ This is rather remarkable: Even though the BAE keep their full complexity in the strong twisting limit, and thus their solutions their full intricacy, the effect on the energy is irrelevant: Since the roots stay finite, and the energy formula (3.6) has to be multiplied by ε, one trivially obtains zero in the limit. Likewise, even the result for the transfer matrix eigenvalue does not really depend on the details of the Bethe roots, as the last of the identities in (3.18) is a simple, direct consequence of the BAE. 17 In conclusion, for L = 3K, M = 2K at infinite twist there is no point in solving the BAE! The only two things one needs to know are that they stay non-singular (i.e. = 0, −1) and finite in the limit, and that they yield a complete set of states for this sector. On the other hand, the sector turns out to become non-diagonalizable in the scaling limit, with an abundance of Jordan blocks appearing. Unfortunately, this already gives a strong hint that the conventional Bethe ansatz is useless as regards the explanation of the model's Jordan block structure. We will soon see better why this is the case, cf. section 3.2. However, for completeness, let us first understand the scaling of the Bethe roots away from the special, symmetric filling conditions of the L = 3K, M = 2K sector.

M = 2, K = 1 sector
It is illuminating to first analyze the two-excitation case M = 2, K = 1 with general L > 3 in some detail. 18 From (3.8), (3.9) we then have a set of three BAE, which explicitly read From these, one notices that the Bethe roots should have the following specific scaling form in the ε → 0 limit: where α, γ are positive exponents and the "scaled" Bethe roots u ± andv are finite. Except for the general powers of ε, the situation is as described in [1]: One of the excitations is a "right-mover", the other a "left-mover". However, here we do not assume α = 1. Then, inserting this ansatz into the BAE, one finds that the powers are generically 19 given by It is interesting to note that the exponent α is a fractional number. This is quite unexpected, and has not been noticed before in the existing literature on the subject. The scaled Bethe roots then satisfy at generic ξ i the simplified BAE The solutions of these equations can be found explicitly in terms of the L-th roots and the (L−1)-th roots of unity. One easily shows completeness of all L(L−1) states of this sector. These solutions show a somewhat intricate dependence on the scaled twist parameters ξ i . We will not exhibit them here, as they are not really needed, in close analogy with the previous section 3.1.2. As we can see from (3.24), the scaling exponent α is bounded by 1 3 ≤ α < 1. Hence, the energy formula (3.6) yields again in the ε → 0 limit and the detailed values of the scaled Bethe roots u + , u − are wiped out. The same is again true for the transfer matrix eigenvalue (3.4), which once more becomes a u-independent constant, and thence identical to the shift operator eigenvalue (3.5): While this expression appears to depend on the detailed solution for u + ,u − , one sees from (3.25) that the constant must indeed be equal to the eigenvalue ω k L of the shift operator. Vexingly, while we are able to find the scaling limit of the sector's BAE, and could even solve them exactly, we see no hint, in full analogy with section 3.1.2, on how to derive the sector's Jordan block structure. The latter is actually quite simple here, and given by a single Jordan block of size L−1 in each of the L cyclicity sectors, cf. chapter 4.

Generic (L, M, K) Sector with L > 3(M − K)
Let us generalize the analysis of the M = 2, K = 1 case of 3.1.3, and assume that the level-one roots u m (m = 1, . . . , M ) can be split into classes (I) and (II) of right/left-movers: (II) u l+M = −1 + ε β u + l , l = 1, · · · , K . For each class (II) root, we associate a level-two Bethe root v k , which we term class (III). For its scaling behavior we make the ansatz Our main assumption is that the Bethe roots in each class become degenerate in the leading order of ε → 0, with small deviations appearing at subleading order with positive powers α, β, γ of ε. While α = β for the case of M = 2, K = 1 in (3.23), we need to assume α = β for the general case. Inserting this ansatz into the system of BAE (3.8), (3.9), the class (I) roots from (3.8) satisfy Similarly, (3.8) for the class (II) roots gives Finally, (3.9) for the class (III) roots along with (3.37) yields recalling the notation ω L = exp(2πi/L) in (2.18). For each root u + l with l = 1, · · · , K, the K integer exponents n l should be distinctly chosen among the set {1, · · · , L − M }. Once the u + l are determined, the Bethe rootsv l are uniquely fixed by (3.35). In the same way, the Bethe roots u − j can be determined from (3.31): where the M integer exponents i j should be chosen without repetition from the set {1, · · · , L}. Let us quickly check completeness of states. The number of solutions of the BAE may easily be computed using the standard combinatorial identity where we recall M = M − K. Notice that this is exactly the dimension of the transfer matrix in the (L, M, K) sector, and hence the number of its generalized eigenvalues. This shows that the above set of solutions of the BAE in the scaling limit is indeed complete. Using our scaling solution, we may now again formally compute the transfer matrix eigenvalue (3.4), which once more becomes a u-independent constant, entirely due to the first of the three terms in (3.4): Here we made use of the fact that α, β < 1. Note that the power of ε cancels with the scaling exponents α, β from (3.38),(3.39). Also, using the explicit solutions (3.43) and (3.44), we find the eigenvalues are given bŷ Since the exponent k is an integer, the eigenvalues are indeed the L-th roots of unity, as expected. Note that k explicitly depends on the quantum numbers introduced to label the solutions for the Bethe roots. In appendix A, we show that this result is consistent with the famous Pólya enumeration theorem for cyclic states. The expression (3.46) actually also comprises the results for the earlier cases (3.18) (where α = β = 0) and (3.27). Clearly, we find from (3.46) that, once again, the (generalized) energy eigenvalues, given by logarithmic derivatives of the transfer matrix eigenvalues at u = 0, are all found to beÊ (ξ 1 ,ξ 2 ,ξ 3 ) = 0 .
(3.48) Curiously, while we were able to understand the rather involved scaling limit of this large sector's BAE, and could even solve them exactly, we see no hint, in full analogy with the more special results of sections 3.1.2, 3.1.3, on how to derive the sector's Jordan block structure. As opposed to section 3.1.3, the latter is actually very complicated here, and we currently do not understand its general pattern, cf. chapter 4.

Remaining Values for (L, M, K)
The results of the previous sections do not fully cover all the regions in (3.1). They are not valid for the (relatively small) "window" where (3.1) holds but (3.41) fails, namely, The special case L = 3K, M = 2K in 3.1.2 actually happens to be the boundary case of this domain, where the above inequalities turn into equalities. As we argued in 3.1.2, there the Bethe roots do not scale at all and remain finite. We therefore suspect that the Bethe roots of the missing region exhibit some mixed scaling behavior, where some of the roots stay finite, while others turn to zero or −1. While it would be technically pleasing to untangle this, we are convinced that it will not really bring any new insights: The (generalized) eigenvalue of the transfer matrix will presumably still be given by the r.h.s. of (3.46) as a u-independent constant equal to the eigenvalue of the shift operator. The (generalized) energy eigenvalue will once more be zero. And no information on the Jordan block structure will be contained in the details of the Bethe roots, unfortunately. For these reasons we will not pursue the further study of this region.

Collapse of the Bethe States
As explained in section 2.3, in the framework of the standard quantum inverse scattering method, the eigenstates are constructed by acting with certain creation operators on the level-one and level-two reference states as in (2.21) and (2.23), respectively. Each set of Bethe roots satisfying the BAE (3.2), (3.3) defines a corresponding eigenstate. This leads to an explicit construction of these eigenstates, in principle. Now we want to study the eigenstates of the strongly twisted model by taking the ε → 0 limit of these Bethe states. We have already shown that the Bethe roots become highly degenerate in this limit, u j = 0, −1 and v l = −2, in the leading order as summarized in (3.28), (3.29), (3.30) with small corrections suppressed by various positive powers of ε. Hence, it is natural to expect that the eigenstates will also become degenerate.
Then, the action of the shift operator on a state gives In terms of these notations, we can find eigenstates 22 of the shift operator for a given configuration f as follows: where π means a shift by steps. The case of k = 0 corresponds to the cyclic states. Now the simultaneous eigenstate |ψ Λ,j in (3.53) can be expressed as a linear combination of the eigenstates |f k of the shift operator: where the sum is over all possible configurations that are not related by shifts. The Bethe states in (2.21) constructed by the creation operators and associated Bethe roots, in principle, determine the coefficients a j (f ), although actual computations for generic (L, M, K) can be very complicated. However, in the strong twisting limit ε → 0, it is possible to find a configuration f that makes the coefficient a(f ) most dominant in powers of ε by analyzing the creation operators and their actions on the reference state. The creation operators B 1 =M 12 and B 2 =M 13 in the Bethe state (2.21) are composed of theR matrices as defined in (2.2). Our strategy is to express the Bethe states graphically by two-dimensional square lattices, where each vertex corresponds to Boltzmann weights defined by theR matrix. In the ε → 0 limit, we can determine how these Boltzmann weights scale with ε, thereby yielding the leading configurations for each eigenstate.
TheR a,n (u) in (2.1) may be written as (3.56) where "even/odd" in the sum mean even/odd permutations of (ijk). Here we use the concise notation e a ij , e ij for 3 × 3 matrices acting on auxiliary space (a) or the -th quantum space, respectively, whose elements are given by (e ij ) ab = δ ai δ bj . We should evalu-ateR a, (u j ) with a Bethe root u j since the arguments of the creation operators in the For the Bethe roots u I of class (I),R a, in (3.56) can be expanded as If we interpretR a, as the Boltzmann weights on a vertex with a horizontal line for the auxiliary space and a vertical line for the quantum space, the first and the fourth terms of (3.58) represent crossings of two different states; the second term represents reflection while the third one arises when both lines carry identical states. These Boltzmann weights are given graphically in Figures 3.1 and 3 |0 may then be represented by a two-dimensional square lattice as shown in Figure 3.3. Here the indices a i and b j are either 2 or 3, with the condition that the total numbers of 2's and 3's should be M and K, respectively. The Boltzmann weights imposed on the vertices belonging to the top M horizontal lines are given by (3.57) (Fig. 3.1); those belonging to the bottom K lines by (3.58) (Fig. 3.2). The configuration |11 · · · 11 at the bottom of the graph defines the reference state |0 ; that of the top |n 1 n 2 · · · n L defines the Bethe state. Among L numbers n 1 , · · · , n L , the number of 2's should be M , that of 3's be K, and that of 1's be L − M , since the indices a i and b j on the left side, which are either 2 or 3, should appear on the top side since the numbers of 1, 2, 3 states are individually conserved by the Boltzmann weights. (Notice that states on the bottom and the right sides are all 1's.) Therefore, the resulting state should contain exactly the same number of configurations as (3.45).
Since the creation operators do not commute, the orderings of the operators in the definition the Bethe state (2.21) matter. Among possible orderings, we start with a state M 12 (u I 1 ) · · · M 12 (u I M )M 13 (u II 1 ) · · · M 13 (u II K )|0 . In fact, as we will show later, this is only ordering which can contribute in the ε → 0 limit. A generic configuration (and its shifts) n 1 n 2 n 3 n 4 n L−1 n L · · · · · · · · · · · · · · · b K . . . . . .
The state | π (f ) is represented graphically in Fig. 3.4. The blue and red horizontal lines entering from the left side can move only in the right or the upward directions, as one can see from the Boltzmann weights listed above. Therefore, all blue and red lines should be contained in the large box at the center. We define S(f ) for sum of all the weights for many different paths of blue, red, and black lines in this box. Although this factor turns out to be quite complicated, see below, it depends only on f and is obviously independent of .
The vertices in other parts of the lattice are "frozen" in the sense that the states (colors) are all fixed. While the right part of the box contains only black lines, the left part contains horizontal color lines, such that only vertical black lines are allowed. Since states are arranged uniquely, it is straightforward to compute the products of the vertex weights. Using the Boltzmann weights in Fig. 3.1 and Fig. 3.2, we find that the products of the vertex weights in the upper-right, upper-left, lower-right, and lower-left dash boxes in Fig. 3.4 are, respectively, given by 1, (3.60) Combining, and factoring out constant terms, we obtain 23 (3.61) · · · · · · · · · · · · · · · · · · · · · · · · . . . . . .
The power of ε inside the sum, , vanishes from (3.38) and (3.39) so that all terms in the sum are of the same order in ε as required. The factor inside the square bracket in (3.61) has been identified already as a root of unity ω −k L in (3.46) and (3.47). This proves that the vertices outside the box give the eigenstates of the shift operator |f k . It is important to notice that only this special combination of ξ i 's and the Bethe roots can yield this root of unity, and that the other orderings such as M 13 (u I 1 )M 12 (u I 2 )M 13 (u II 1 )M 12 (u II 2 )|0 cannot yield the required states. 24 In summary, the Bethe state in the strong twist limit can be written, up to an f -independent constant, as This determines the coefficients a(f ) in (3.55). Now we need to compute S(f ) from the vertices inside the box. Since these are not frozen, there are too many possibilities to compute all their weights in general. Therefore, we will content ourselves with finding the most dominant configuration for a rather simple yet non-trivial case: M = 3, K = 1 with arbitrary L. Based on this result, we will formulate a general conjecture for generic (L, M, K) sectors.
For b = 0, the allowed sub-configurations for M = 3, K = 1 are (223), (232), (322). The vertices that contribute most to these are illustrated in Fig. 3.5 (a), (b), and (c), along with the powers of ε computed from the Boltzmann weights. From this, we can find that the most dominant f is (223).
For b = 1, we need to consider sub-configurations with 1 state inserted into (223); (2123) and (2213). The vertices for these are illustrated in Fig. 3.5 (d) and (e) along with the powers of ε computed from the Boltzmann weights, where we include an ε −bβ factor in (3.62). Again, we can find that these are sub-leading compared with (223) since 1 > α > β. If b ≥ 2, more vertices inside the box with two different color lines crossing appear and add more positive powers of ε. Therefore, these are more suppressed.
Although we have no rigorous proof for the generic cases, we conjecture from this analysis in conjunction with some further specific tests that the most dominant state for This implies that the Bethe eigenstates in (3.62) all collapse in the strong twisting limit to |f k , up to a constant of proportionality. We call (3.63) and in fact all associated cyclicity eigenstates |f k locked states. This is a disappointing result, considering the model's rich structure of Jordan blocks, see next chapter 4. For each Jordan block, there is exactly one true eigenstate. However, the limiting process of the standard quantum inverse scattering method yields only one of them per cyclicity sector: A locked state. This shows that the understanding of the Jordan block structure from integrability needs a new approach; the traditional methodology of the ABA does not suffice.

Jordan Normal Forms for (Hyper)eclectic Spin Chains
As we argued in the last chapter 3, the ABA does not seem to yield any useful information on the eclectic spin chain models: In the strong twisting limit, all Bethe states collapse to a family of locked states with fixed shift operator eigenvalue, despite the fact that the models stay integrable in the limit. This is closely related to the fact that the Hamiltonian, and actually the entire transfer matrix at non-zero spectral parameter, fail to be diagonalizable in this limit. As already explained in section 2.3, the standard basis change to a diagonal form is to be replaced by a basis change to JNF, see (2.26), (2.34). However, it is currently unclear how to find this basis change from integrability. In this chapter, we will demonstrate, by way of examples, that the JNF, while highly intricate, shows very promising regularity properties and seemingly iterative patterns. We will only show results for the Hamiltonian. They were mostly obtained by using the computer algebra programs Wolfram Mathematica and Matlab, even though, in a few of the simplest cases, manual calculations based on basic combinatorial considerations suffice. Accordingly, the examples of this chapter should be considered experimental and explorative, to be systematically explained in future work.

Eclectic Spin Chain
Let us recall the Hamiltonian (1.13),(1.14) of the general eclectic model, acting on 3 ⊗L . In an alternative but standard notation, we may also write it aŝ ξ 3 e 12 e +1 21 + ξ 1 e 23 e +1 32 + ξ 2 e 31 e +1 13 , (4.1) where e 12 acts only on site , with turning a state 2 into a state 1 being its only non-zero action, etc. It is clear that this Hamiltonian does not change the length L of the spin chain, nor the numbers L−M of states 1, M −K of states 2, K of states 3, respectively. Thus its action is closed on all the states spanned by and all of its possible permutations. We call the associated state space the (L, M, K) sector. The dimension of this vector space is obviously which, therefore, gives the size of the Hamiltonian matrix in this sector, see also (3.45).
It is furthermore clear that  In general, the specifics of the Jordan decomposition also depend on the scaled twist parameters ξ 1 , ξ 2 , ξ 3 . However, we found that there always exists a generic JNF for generic parameters. It holds for "most" triplets of parameters. However, by finetuning the parameters different decompositions might appear. Let us give the simplest non-trivial example of this phenomenon: For L = 3, M = 2, K = 1 one has six states. For generic ξ i 's one has S ·Ĥ (3,2,1) where the three 2×2 Jordan blocks correspond to the three cyclicity sectors with k = 0, 1, 2.
On the other hand, by way of example, if the three twist parameters are all equal, i.e. ξ := ξ 1 = ξ 2 = ξ 3 , only the k = 0 sector stays non-diagonalizable, while the two non-cyclic sectors k = 1, 2 become diagonalizabe. One therefore has instead of (4.5) Before proceeding, let us introduce the concept of a multiset, which provides a useful notation for our JNF. Multisets, unlike sets, allow for multiple occurrences for each of their elements. The union ∪ of two multisets means joining them, adding the multiplicities of identical elements. The relative complement \ of two multisets is defined in an analogous fashion. In multiset notation, we would describe the Jordan decomposition (all three cyclicity sectors) of (4.5) by the multiset {2, 2, 2}, while for (4.6) we would write {2, 1, 1, 1, 1}. An even more concise notation, which we will use extensively below in Tables 2, 3 would be 2 3 for the first example, and 1 5 2 for the second example.
Let us next discuss the generalization of (4.5) to general L ≥ 3. It is possible to analytically prove (and easily checked numerically for various values of L) that in this case the generic JNF is where the L Jordan blocks of size L−1 correspond to the L cyclicity sectors (k = 0, . . . L−1). This simple pattern quickly gets significantly more involved as one increases M and K. Let us show some of the emerging structure by fixing K = 1 (if K > 1, the complexity of the decompositions further increases). This has the advantage that Polya counting trivializes, as one may consider the position of the single state 3 as a marker on the spin chain: There is an equal number of states in all L sectors labelled by (L, M, 1, k), namely We list the decomposition ofĤ (L,3,1,k) (ξ 1 ,ξ 2 ,ξ 3 ) , any cyclicity sector k, for L = 5, . . . , 10 in Table 1. Based on this, it is fairly straightforward to formulate a conjecture for the JNF; it reads, in (multi)set notation, Another point of view to see this are recursion relations. Let S (M ) L (with K = 1) be the multiset of the Jordan block sizes for a given L and M in the cyclic (k = 0) sector. Then, the following relation holds S with initial conditions S  Table 1: JNF for L, M = 3, K = 1 (numerical analysis).
As M increases, the JNF decomposition patterns get much richer. To uncover these, it is necessary to go to sufficiently high values of L. This is not straightforwardly done for the eclectic chain. However, we can get the wanted results for lengths up to L ∼ 20 for the hypereclectic model, see next section 4.2. As we shall explain, they are then also expected to hold for the generic eclectic model.

Hypereclectic Spin Chain
The Hamiltonian of the hypereclectic model (1.15), (1.16) is obtained from the one of the eclectic chain by setting ξ 1 = ξ 2 = 0, ξ 3 = 1. In the alternative notation of (4. (4.13) In the multiset notation introduced in the previous section, this JNF decomposition reads {9, 5, 1} = 1 5 9 . (4.14) For the eclectic model, this is valid for "most" values of ξ 1 , ξ 2 , ξ 3 , including the hypereclectic case ξ 1 = ξ 2 = 0, ξ 3 = 1. If we permute the three states 1, 2, 3, this decomposition will remain true for the generic eclectic model for all six permutations of these states due to symmetry. On the other hand, the hypereclectic model behaves very different under permutations of the states. The reason is, that the dynamics only involves the states 1 and 2, while 3 is just an inert "spectator", forming some kind of wall. So it makes a significant difference whether 3 is the least numerous state, second least numerous state, or else the most numerous state. In our specific L = 7 example, for the cyclic sector, we have again In conclusion, the JNF decompositions of the hypereclectic chain are definitely not invariant under a permutation of the states. This means, its "spectrum" is much richer than the one of the generic eclectic chain, despite its simpler looking Hamiltonian.
(4.18) Acting with the Hamiltonian H on this state, any 2 situated to the left of a 1 exchanges with it, thereby moving one step to the right. None of the remaining 1's, 2's and none of the 3's can move. In a hopefully intuitive notation we have Clearly, one can immediately write down an eigenstate that trivially annihilated by the action of the Hamiltonian, a "locked state", cf. (3.63): Here it is easy to write down the lowest Jordan descendent:  Knowing the a s clearly determines the structure of the Jordan normal form decomposition of H. And (4.24) allows to efficiently compute the a s by Gaussian elimination on a computer, wherewith one brings H s into echelon form. Clearly we may restrict s to be at most the size of H plus one. The result of this procedure for M = 4, K = 1 is summarized for L = 6, . . . 21 in Table 2. One noticeable feature is that some Jordan blocks are repeating several times with definite multiplicities denoted as exponents in the table. One may again conjecture some simple recursion relations, in generalization of (4.10), (4.11). Let S (4) L be the multiset of the Jordan block sizes for a given L with M = 4 and K = 1. Then, the following relation appears to hold in general: On easily verifies the result (4.23) for the size of the largest Jordan block for all entries of Table 2. By the same method, one finds the Jordan block decompositions for M = 5, K = 1. The emerging structure is even more intricate, as shown in Table 3 for L = 8, . . . , 18. If we denote the multisets of block sizes for a given L with M = 5, K = 1 as S (5) L , we find the following recursion relation:  Once again one may verify the result (4.23) for the size of the largest Jordan block for all entries of Table 3.

Conclusions and Open Questions
We have begun the systematic study of a class of non-diagonalizable, integrable, chiral spin chains which were christened eclectic spin chains in [1]. They still contain three free, complex twist parameters (ξ 1 , ξ 2 , ξ 3 ), and were originally inspired by parts of the oneloop dilatation operator of a strongly twisted, double-scaled deformation of N = 4 Super Yang-Mills Theory [3][4][5]. However, here we systematically study these models on their own right, without further exploring their relation to gauge theory. We also introduced a seemingly even simpler version of these models, which we called hypereclectic spin chain.
Being non-diagonalizable, the goal is to bring Hamiltonian and transfer matrix of these models into JNF. We found ample evidence for a highly intricate yet subtly structured "spectrum" of Jordan blocks, in dire need of a systematic understanding. Interestingly, the spectrum of the hypereclectic chain is richer than the one of the eclectic model at generic twist parameters (ξ 1 , ξ 2 , ξ 3 ), even though the first model possesses a simpler Hamiltonian.
A puzzling aspect is that, despite the easily demonstrated integrability of these models, the traditional means of the quantum inverse scattering method appear to fail to describe these models' Jordan decompositions. We demonstrated this in some detail for the ABA method. Particularly vexing is the fact that the BAE remain sensible in the limit, and exhibit a mathematically rich set of solutions that even lead to the correct counting of states. However, the associated Bethe states collapse to a small set of locked states that are, apparently, essentially useless for finding the spectrum of Jordan blocks.
To summarize, the ultimate goal for the future is then to solve the eclectic models, i.e. to find their intricate spectrum of Jordan blocks, by using integrability. Here we would like to draw attention to [11], where this was understood for a different non-diagonalizable spin chain model. However, the model is quite different; in particular, it only contains blocks of size one or two.
In our case, it is not clear which model will be "easier" to treat: The eclectic or hypereclectic one? It is also not obvious whether it is best to concentrate on their scaled R-matrix, or to better proceed from the finitely twisted R-matrix in conjunction with a suitable limiting procedure. Furthermore, is it better to concentrate on the models' Hamiltonians, or else on their commuting transfer matrices? Finally, could it be that these models may be solved by some suitable combinatorial methods, thereby bypassing the power of integrability?