The One-Loop Spectral Problem of Strongly Twisted N =4 Super Yang-Mills Theory
HHU-Mathematik-2018-11HU-EP-18/39
The One-Loop Spectral Problem of StronglyTwisted N =4 Super Yang-Mills Theory Asger C. Ipsen, Matthias Staudacher, and Leonard Zippelius
Institut für Mathematik und Institut für Physik, Humboldt-Universität zu Berlin,IRIS-Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany {acipsen,staudacher,lzippelius}@physik.hu-berlin.de
Abstract
We investigate the one-loop spectral problem of γ -twisted, planar N =4 Super Yang-Millstheory in the double-scaling limit of infinite, imaginary twist angle and vanishing Yang-Mills coupling constant. This non-unitary model has recently been argued to be a simplerversion of full-fledged planar N =4 SYM, while preserving the latter model’s conformalityand integrability. We are able to derive for a number of sectors one-loop Bethe equationsthat allow finding anomalous dimensions for various subsets of diagonalizable operators.However, the non-unitarity of these deformed models results in a large number of non-diagonalizable operators, whose mixing is described by a very complicated structure of non-diagonalizable Jordan blocks of arbitrarily large size and with a priori unknown generalizedeigenvalues. The description of these blocks by methods of integrability remains unknown. a r X i v : . [ h e p - t h ] D ec ontents N =4 Super Yang-Mills 12 Spin Chain of Strongly Twisted N =4 Super Yang-Mills 33 Fishnet Theory 74 Scaling Limit of the Twisted Bethe Equations 135 Strongly β -Twisted Theory 156 Remarks on Higher Loop Corrections 187 Conclusions and Outlook 20A Dilatation Operator of Strongly γ i -Twisted Models 21B Twisted One-Loop Bethe Equations 22C Derivatives in the Strongly β -Twisted Model 25D Nilpotency Proof 27E Walls in Fishnet Theory and Nilpotency 29 N =4 Super Yang-Mills The discovery and exploitation of the integrability of planar N =4 Super Yang-Mills theory(SYM) has been a huge success story. This was already the case when the overviewcollection [1] appeared eight years ago. Since then the scope of planar integrability asconcerns an ever increasing number of exactly computable quantities in N =4 SYM and asmall number of further, related integrable field theory models has been steadily increasing.An updated overview would certainly be warranted. On the other hand, vexingly, aconvincing explanation on why these models are integrable at finite values of their couplingconstants has so far not been discovered. Given this situation, a very interesting suggestionwas made in [2–10]. A certain non-unitary deformation of planar N =4 SYM leads in adouble-scaling limit to a decoupling of all gauge fields, a destruction of supersymmetry,and a vast simplification of the Feynman diagrammatics, while retaining integrability. Itwas then argued that, due to their perceived simplicity, a complete understanding of theintegrability of these deformed models could be reached, with the final goal of feeding theseinsights back into a possible explanation of the integrability of the mother theory, N =4SYM. And indeed, the deformed models allow for a large number of exact computationsthat do appear simpler as compared to the undeformed case. On the other hand, oddly,1he two showcase instances where integrability can be rigorously proved in N =4 SYM areobscured in the double-scaled deformed models. At strong coupling, the interpretationin terms of an integrable string sigma model is missing, see, however, [11]. And at weakcoupling, the a priori much simplified situation is also somewhat fuzzy. On the one hand,interesting connections between the Feynman diagrams of the double-scaled models andYangian invariance have been made [5, 7], integrability allowed to formulate a conjecturefor an exact series of generalized ladder graphs [12], and a connection to an integrable su (2 ,
2) Heisenberg chain was made [6]. On the other hand, we noticed that the integrableone-loop spin chain interpretation of the spectral problem of these models has not yetbeen properly exhibited. The purpose of this article is to begin a serious investigation ofthis issue. To anticipate: We find the one-loop spectral problem to be highly intricate,quite different from N =4 SYM, and certainly unsolved.For the rest of this introductory chapter, we collect a few pertinent facts on and no-tations for the strongly twisted cousins of N =4 SYM that we investigate in this paper,focusing on the points important for the investigation of the one-loop spectral problem.We refer the reader to the original papers [2–10] for motivations, derivations, and, aboveall, many more explanations and details. For a generalization to dimensions other thanfour, albeit with non-local propagators, see [13]. The prime example of a four-dimensionalintegrable interacting quantum field theory, planar N =4 SYM (see [1] for some introduc-tory material), allows for a deformation by three parameters γ i that appears to retain theintegrability of the theory [14]. The resulting theory is called planar γ -twisted N = 4SYM. However, it had been demonstrated earlier in [15] that, in contrast to undeformed N = 4 SYM, the γ -twisted theory is no longer conformally invariant, not even in the pla-nar limit, due to running double-trace operator couplings. It was suggested by the authorsof [2] that this problem can nevertheless be circumvented in the planar model by restrictingthe attention to composite operators containing at least three fields. This allowed themto define an interesting double-scaling limit of the twisted models. Defining the squaredplanar gauge theory coupling constant as g = λ π , where λ is the ‘t Hooft coupling, theysuggested to take g →
0, while some or all of the twisting parameters q j = e − iγ j / → ∞ ,such that the products g q j = ξ j are held fixed. In this paper, we call these double-scaled,twisted models simply “strongly twisted models”. Furthermore, we will mostly focus ontwo special cases. In the first one, all three ξ i are equal, i.e. ξ j = ξ , and the model willbe termed for “historic reasons” ( γ i = β ) strongly β -twisted (s β t). In the second case ξ = ξ while ξ , = 0. Here the model is often referred to as the fishnet theory (FN), asthe resulting Feynman diagrams are reminiscent of a fishing net. In fact, the name wascoined in an early paper by A. Zamolodchikov [16], where the “integrability”, in the senseof the validity of the star-triangle relation, for this class of diagrams was noticed. The twospecial cases result in the following two Lagrangian densities, respectively: L s β tint = N ξ tr( φ † φ † φ φ + φ † φ † φ φ + φ † φ † φ φ ) + iN ξ tr( ψ φ ψ + ¯ ψ φ † ¯ ψ + cyclic) , (1.1) L FNint = N ξ tr( φ † φ † φ φ ) , (1.2)where by “cyclic” we mean cyclic permutations of the three indices, and we do not show thestandard kinetic terms of complex bosonic and fermionic fields, but only the interactingpart of the Lagrangian. It was shown in [3] that the theories defined above, just liketheir unscaled counterparts [15] (see above), are not perturbatively complete, and specificdouble-trace counter terms have to be added to the Lagrangian. It was then argued in [3]2nd [8] that the couplings of these interaction terms may be fixed, such that the resultingtheory becomes indeed conformally invariant in the planar limit, see also [17]. Thus thelocal composite operators of these models should transform covariantly under the actionof the dilatation operator of the conformal algebra. It is this dilatation operator of thestrongly β -twisted model and the fishnet model in the planar limit that we investigatein detail at the leading one-loop order of perturbation theory in this paper. There, thedouble-trace interaction terms can safely be ignored for operators containing three or morefields, so we have omitted them from (1.1), (1.2). They do not contribute in the planarlimit unless they cut the color structure of a Feynman graph into two disjoint pieces.A quick look at the Lagrangians (1.1) and (1.2) suffices to realize that the hermitianconjugates of all the terms are missing. This renders these models non-unitary, with anumber of problematic technical and conceptual consequences. In particular, it leads to anon-hermitian dilatation operator that is potentially non-diagonalizable. On the positiveside, the absence of hermitian conjugates implies a dramatic reduction in the number ofFeynman diagrams for a given quantity at each order of perturbation theory. After all, thiswas the rationale behind the hope that these theories might provide interesting toy modelsfor understanding the origins and consequences of integrability in diagrammatically morecomplex planar four-dimensional quantum field theories such as N =4 SYM. N =4 Super Yang-Mills In this chapter we describe the general aspects of the spin chain picture of the stronglytwisted theories at one-loop order, in close analogy with the original, undeformed case [18].We start by introducing a few technical notions in order to be able to then quickly derivethe one-loop dilatation operator. We end the chapter by introducing the novel notion ofeclectic spin chain states with zero anomalous dimension.
We are interested in the action of the one-loop dilatation operator on single-trace operatorsbuilt from the fields of N = 4 SYM A ∈ { ∂ k φ i , ∂ k φ † i , ∂ k ψ j , ∂ k ψ j , ∂ k F , ∂ k F } , (2.1)where i ∈ { , , } , j ∈ { , , , } , and we have suppressed all spacetime indices. Wewill refer to the fields in (2.1) as letters or single-site spin chain states interchangeably.The number of letters in a single-trace operator equals the number of sites of the chainand is called the length L . Two letters appearing in the trace next to each other willbe referred to as neighboring. For a more detailed review of the relation between single-trace operators and spin chains see for example [18]. In order to describe the form of thedilatation operator, it is useful to define a map F from letters to a set of ‘flavors’ F ∅ , F := { , , , ¯1 , ¯2 , ¯3 } , F ∅ := F ∪ {∅} , (2.2)via the assignments F ( ∂ k φ c ) = c , F ( ∂ k φ † c ) = ¯ c , F ( ∂ k ψ c ) = c , F ( ∂ k ψ c ) = ¯ c , (2.3) F ( ∂ k ψ ) = ∅ , F ( ∂ k ψ ) = ∅ , F ( ∂ k F ) = ∅ , F ( ∂ k F ) = ∅ , (2.4)3here c = 1 , ,
3. We further define a ± = a ± a ± = a ± a ∈{ , , } and say two letters AB are in chiral order, if F ( B ) = ∅ , and either F ( A ) = F ( B ) + or F ( A ) = F ( B ) − , where it is understood that a = a . We say two letters AB are inanti-chiral order if BA are in chiral order.Finally, we define P − and P + as chiral respectively anti-chiral projection operators acting on neighboring letters. They annihilate improperly ordered pairs of nearest neighborstates while leaving properly ordered pairs of states invariant. N =4 Equipped with the above definitions, we are ready to derive the one-loop dilatation op-erator of the strongly twisted models from the one of the unscaled twisted models. Thedilatation operator D of the conformal algebra consists of a classical part D and quantumcorrections δ D : D = D + δ D . (2.5) D is identical for N = 4 SYM [19] and all its deformations. In this paper, we areinterested in the one-loop contribution to δ D , which we identify with the Hamiltonian H of a spin chain δ D = ξ H + O ( ξ ) , (2.6)where ξ is the coupling constant of the strongly twisted theories see section 1. Furthermore, H is the sum of local Hamiltonian densities acting on neighboring spin chain sites as H = L X n =1 H n,n +1 , (2.7)where the sum is over spin chain sites n , and L is the number of letters of the spinchain. As we consider single-trace operators, we impose periodic boundary conditions: H n,n +1 = H n, . The matrix elements of the one-loop dilatation operator density H ofunscaled γ -twisted N =4 SYM acting on a pair of neighboring letters A n , A n +1 , given in(2.1), at sites n respectively n + 1 is for L ≥ ( H γ ) A n A n +1 A n A n +1 = exp (cid:18) − i (cid:16) ( q A n ) T C q A n +1 + ( q A n +1 ) T C q A n (cid:17)(cid:19) ( H N =4 ) A n A n +1 A n A n +1 . (2.8)Here A n , A n +1 are the initial letters and A n , A n +1 the final letters as regards the action ofthis operator, and H N =4 is the complete one-loop dilatation operator of the undeformedmother theory. An explicit expression for H N =4 was obtained in [19], but note that ournormalization is different: H N =4here = 2 H N =4[19] . The twist matrix is given in the conventionsof [15] as C = − γ γ γ − γ − γ γ , (2.9)and the charges q A are vectors whose three components q iA are shown in table 1.When we scale the dilatation operator we have to keep in mind that it comes with afactor of the squared coupling constant g , which has to be combined with a factor of q Note that the Hamiltonian derived from the density in (2.8) should be multiplied by a factor of g ,not ξ , to give the one-loop contribution to δ D . We flip the sign of the exponent so as to have a notation consistent with the diagrams in [4]. ψ ψ ψ ψ F , F φ φ φ q A + − − + q A − + − + q A − − + + Table 1:
The charge vectors of the different fields in γ -twisted N =4 SYM, cf. (2.1).Conjugate fields have the opposite charges. Gauge fields do not carry and deriva-tives do not add charge.from the two coefficients in (2.8) to yield something finite in the limit, see chapter 1. Wesee that in the β -twisted model each of the two exponentials in (2.8) produces a factor of q as required, if and only if the fields change from chiral into anti-chiral order under theaction of H . Thus, as a first result, we find that the complete one-loop dilatation operatordensity of the strongly β -twisted model (1.1) can be written for L ≥ H s β t n,n +1 = P + n,n +1 H N =4 n,n +1 P − n,n +1 , (2.10)where the projection operators P ± have been defined at the end of the preceding subsection2.1. The same argument carries over for the complete dilatation operator density H FN of the much simpler fishnet theory (1.2). The difference between the models lies in thecoefficient functions in (2.8). In particular, the matrix (2.9) includes the twist parameters γ i and hence depends on how we scale the different parameters. We find that H FN n,n +1 isidentical to H s β t n,n +1 in (2.10) unless any of the four letters A n , A n +1 , A n or A n +1 from(2.8) are fermions, φ , or φ † , in which case the matrix element is zero. We could writethis with the help of an additional projection operator P FN that projects out these fields.That is, H FN n,n +1 = P FN n,n +1 P + n,n +1 H N =4 n,n +1 P − n,n +1 P FN n,n +1 . (2.11)More explicitly, for fishnet states without derivatives we find that the only non-vanishingmatrix elements are (we hope that it is obvious that here the indices on the scalar fieldsare flavor indices, and that the pairs correspond to fields sitting on neighboring sites)( H F N ) φ φ φ φ = − H F N ) φ † φ † φ † φ † = − H F N ) φ φ † φ † φ = − H F N ) φ † φ φ φ † = − . (2.12)Including derivatives, some additional combinatorial factors carry over from H N =4 n,n +1 , e.g., H F Nn,n +1 Mk ! (cid:16) ∂ k φ (cid:17) n ⊗ (cid:16) ∂ M − k φ (cid:17) n +1 = − M + 1 M X l =0 Ml ! (cid:16) ∂ l φ (cid:17) n ⊗ (cid:16) ∂ M − l φ (cid:17) n +1 . (2.13)This matrix element can be extracted, for example, from equation (B.13) of [19].The above results suffice for the two special cases we consider in the paper: The fishnetmodel and the strongly β -twisted model. It is not difficult to extend our arguments tomore general strongly γ i -twisted models, whose one-loop dilatation operator we spell outin appendix A for future use. The case L = 2 requires separate attention due to the double-trace terms that needed to be added torender (1.1) and (1.2) conformal in the planar limit; cf. our brief discussion of this in chapter 1. .3 Eclectic Spin Chains and Nilpotency of the Dilatation Operator Prior to turning to specific sectors of the strongly twisted theories, we would like to explainthe nilpotency properties of their dilatation operator. To this end, we find it convenientto introduce the notion of eclectic spin chains. As we can see from (2.10), (2.11), and thediscussion around these equations, for a given choice of vacuum, the Hamiltonian forcessome of the flavors clockwise and some of the flavors anticlockwise around the spin chain.For the rest of this paper, we will take φ as the spin chain vacuum, unless stated otherwise.Furthermore, we will work in a convention where φ , ψ α , φ † and ¯ ψ α are right-movers,their conjugates are left-movers, and the remaining fields will be called non-dispersingexcitations for reasons to become clear later. The non-dispersing excitations split into twogroups: derivatives, which can only follow other excitations around the spin chain, andthe remaining non-dispersing excitations, which simply never move at all.If we add left and right-movers to the vacuum, a sufficient number of applications of theHamiltonian will cause the excitations to meet. The excitations can however not reflect offeach other since otherwise they would travel in the wrong direction after scattering. Thereare sectors in which the excitations can also not pass through each other. Instead, they actas impenetrable walls towards one another. Then the Hamiltonian density acting on thesetwo excitations in the given order is identically zero. We say operators corresponding tothese types of spin chains have “eclectic” field content. Explicitly, eclectic field content isgiven, if at least one of the following three conditions is met. The operator contains fieldsof the three flavors { , , } , or it contains conjugate fields of the three flavors { , , } , orit contains fields and conjugate fields of the same flavor { a, a } . The fermionic or bosonicnature of the excitations is irrelevant in our definition of eclectic field content, as areexcitations of derivative type. For eclectic chains, acting with the Hamiltonian a sufficientnumber of times will push all excitations against each other and then annihilate the spinchain state. Thus the Hamiltonian is nilpotent; a rigorous proof of this statement can befound in appendix D. We conclude that a large part of the complete one-loop dilatationoperator of the strongly twisted theories has generalized eigenvalue zero. Although thesizes of the Jordan blocks are not fixed by this argument , we will focus on non-eclecticsectors for the rest of the paper. The reason is that we currently do not know if and howintegrability may be used to determine the size of these blocks.As an example consider the spin chain consisting of L − φ s , one φ and one φ . Abasis for this spin chain is: | l i = ( − l | φ φ l φ φ L − − l i − δ l, | φ φ L − φ i for 0 ≤ l ≤ L − . (2.14)The Hamiltonian acts on this basis as: H | l i = | l + 1 i , when l < L − H | L − i = 0 . (2.15)Hence the Hamiltonian is just one Jordan Block of size L − H L − = 0 on this subspace.Finally, we would like to relate the above findings to the results of [4]. In this paper,the { φ , φ , φ } sector is discussed, and its asymptotic Bethe ansatz (ABA) equations are The above argument fails, if we only have fields and conjugate fields of the same flavor. However, then H = 0. We take this special case to lie within our definition of eclectic field content. The size of the Jordan blocks is however bounded above for a given L and given excitations, as can beseen directly from the proof in appendix D. of q → q − and q → q − . This changes the chirality of some of the vertices and the { φ , φ , φ } sector in this convention should be equivalent to a { φ † , φ † , φ } sector in our convention.Luckily, this sector is indeed not eclectic. We discuss the one-loop Bethe equations of anequivalent sector, namely { φ , φ , φ † } , in section 5.1. The one-loop limit of the equationsof [4] is difficult to compare directly to our results since we work in different gradings.However, as a consistency check, it is possible to write our equations presented in section5.2 in the ABA grading and restrict to allow only specific excitation numbers to matchthe equations. We found complete agreement in this case. In this chapter, we investigate the non-eclectic sectors of the fishnet theory (1.2). Likeall the strongly twisted theories this model is non-unitary and thus has a non-hermitiandilatation operator. One therefore does not expect H to be diagonalizable. And indeed,as explained in section 2.3, it turns out to be non-diagonalizable. However, there are stillproper eigenstates of H and corresponding eigenvalues. In this chapter, we show how tofind these eigenvalues. We propose four different methods for doing so: i) explicit con-struction of creation and annihilation operators, ii) an Algebraic Bethe Ansatz [21] from astrongly twisted R-matrix, iii) a Coordinate Bethe Ansatz [22] and iv) identification of thecorrect limit of the Beisert-Roiban Bethe equations for finite twists [23]. For a comparisonof ii) and iii) as well as a discussion for finite twist see e.g. [24]. The technical details ofthe last method iv) are however deferred to the next chapter. Not all these methods workin all cases, but whenever several of these methods are available we find perfect agreementbetween them. For the broken su (2) sector we can apply all four methods, while for thesector including one derivative only iii) and iv) are available and for the full fishnet modelincluding any of the two scalars and any derivatives only iv) has yielded any results, al-though it would be interesting, if iii) were applicable. Some emerging issues, features andopen questions will be discussed at the end of this chapter. su (2) Sector The simplest non-trivial sector all the strongly twisted theories is the two-chiral-scalar-sector, where the letters of the spin chain are either φ or φ , and neither the conjugatefields φ † , φ † nor any derivatives appear. In the original N =4 model it corresponds tothe su (2) sector described by an integrable Heisenberg XXX spin chain. Now the su (2)symmetry is broken and the one-loop dilatation operator simply turns into a chiral per-mutation operator [4], as seen from (2.12). It scans the spin chain until it finds two spins The replacement is indeed only a convention. The theories before and after the replacement areidentical up to a renaming of fields. Of course there are several equivalent discrete copies of this sector, e.g. chains with only φ , φ † , etc.
7n chiral order and then exchanges them. Recall that we consider φ as a local vacuumstate and φ as a local excitation. In fact, H is the Hamiltonian of a chiral XY-model,describing a free, chiral lattice fermion with a two-body S-matrix equalling -1. Being non-hermitian, we would a priori already expect the formation of non-diagonalizable Jordan blocks. Interestingly, this is not yet the case. In fact, H may be explicitly diagonalizedby a Jordan -Wigner transformation followed by a Fourier transform, just like the XY-model [25]. We refer the reader to this classic paper for the explicit construction. Theeigenvalues E of the Hamiltonian H are neatly and explicitly found to be E = K X j =1 − α j , (3.1)1 = K Y j =1 α j , (3.2) α Lk = ( − K − , (3.3)where the spin chain length L is the total number of scalars φ , φ , and K is the numberof excitations φ . The α j = e ip j encode the lattice momenta p j of the excitations. Theyare determined by the free quantization laws (3.3) =“Bethe equations with S-matrix -1”.Unlike the case of true Bethe equations with a non-trivial S-matrix, the equations (3.3)may immediately be solved in terms of L -th roots of ( − L and K there are precisely (not necessarily cyclic) (cid:0) LK (cid:1) eigenstates, as expected.The projection to cyclic states, as required by the trace of the model’s composite operators,is encoded in (3.2). Finally, the dispersion law is read off from (3.1), clearly showing thatthere are only right-movers: E ∼ P j e − ip j . These equations possess a hidden symmetry :The spectrum for E is invariant under the replacement K → L − K , interchangingvacuum states φ and excitations φ , thereby also interchanging right and left-movers.One simple example of eigenstates in the broken su (2) sector are for L = 4 and K = 2the operators O = ±√ φ φ φ φ ) + tr( φ φ φ φ ), for which we find E = ∓ √
2, inagreement with [4].
As mentioned earlier, we only consider non-eclectic sectors, since the generalized eigenval-ues corresponding to eclectic spin chain states are zero. Thus, in order to extend beyondthe two-scalar-sector within the fishnet theory, we can only add derivatives, since anyadditional scalar would produce eclectic field content and the Hamiltonian would becomenilpotent. For simplicity, we start by introducing derivatives of only one kind, say ∂ .Individual spins are then taken out of { ∂ k φ , ∂ k φ } , and we drop the spinor indices onthe derivatives in the following to avoid too many subscripts: ∂ → ∂ . In contrast toother excitations, we can have an arbitrary number of derivatives at each spin chain site.As before, our aim is to diagonalize H , which we attempt to achieve by a coordinate Betheansatz. However, as it turns out, the Hamiltonian is non-diagonalizable in this sector, and Camille Jordan, 1838-1922, Mathematician; Pascal Jordan, 1902-1980, Physicist. The XY-model, chiral or not, does not have a “beyond the equator problem” at K > L , unlike theXXX Heisenberg spin chain: Given L , for K all values with 0 ≤ K ≤ L are allowed. We use Weyl notation for the derivatives, i.e. ∂ µ → ∂ α ˙ α . H , i.e., the energy E , is given in terms of these momenta in the dispersion law. Actingon single-excitation states, we find that only the momentum of the φ excitations entersthe dispersion law because a single derivative excitation cannot move by itself. Thus, theset of the momenta distributed between the φ has to stay the same during scattering,and we conclude that the momenta cannot be exchanged between φ excitations and ∂ excitations. This property simplifies the action of the S-matrix of spin chain excitations tobe an exchange of the excitations of type f and f multiplied by a scalar function S f ,f ,explicitly S | φ ( p ) ∂ ( p ) i = S φ ,∂ ( p , p ) | ∂ ( p ) φ ( p ) i = e ip − e − ip | ∂ ( p ) φ ( p ) i , (3.4) S | ∂ ( p ) φ ( p ) i = S ∂,φ ( p , p ) | φ ( p ) ∂ ( p ) i = 2 − e − ip e ip | φ ( p ) ∂ ( p ) i , (3.5) S | φ ( p ) φ ( p ) i = S φ ,φ ( p , p ) | φ ( p ) φ ( p ) i = −| φ ( p ) φ ( p ) i , (3.6) S | ∂ ( p ) ∂ ( p ) i = S ∂,∂ ( p , p ) | ∂ ( p ) ∂ ( p ) i = − e i ( p + p ) − e ip + 1 e i ( p + p ) − e ip + 1 | ∂ ( p ) ∂ ( p ) i . (3.7)The first three equations may be derived in the usual way by considering two-excitationstates. The scattering of derivatives requires to consider a spin chain with two ∂ exci-tations. However, since the derivatives cannot move by themselves they do not scatterunless we also add an additional φ as a transporter. To obtain the fourth equation, weconsidered a state with the three excitations: { ∂, ∂, φ } . As discussed above, if in a givenordering of excitations we assign the flavors to the momentum, the S-matrix preserves thisassignment. Put differently, if in one ordering of the excitations the particle with momen-tum p i has flavor f i , it will have flavor f i in all orderings of the excitations. This propertyallows us to make the following coordinate Bethe ansatz, using only a scalar function S σ for an eigenstate with a general number of excitations M | Ψ i = X n 2, they do become arbitrarily large withincreasing L . We demonstrate this by an explicit example below in (3.15). It isunclear to us how to determine these sizes, in general.4. There appears to be an abundance of Jordan blocks once we include derivatives.Their systematics and counting is unclear to us.5. The generalized eigenvalue of all Jordan blocks we found turned out to be E = 0 inall instances. It is natural to conjecture that this is always the case. However, so farwe could not yet find a proof of the latter.Clearly it would be interesting and important to prove these observations, especiallythe last one, and to establish a complete classification into eigenstates and Jordan cellsof all states of the fishnet model. As a possible first step, we discovered certain “wall-likestructures” that can be constructed out of a φ decorated with arbitrarily many derivatives.The simplest wall is of the form | . . . (cid:16) ( ∂φ ) φ − φ ( ∂φ ) (cid:17) . . . i , (3.14)where by . . . we indicated the rest of the spin chain. The Hamiltonian annihilates thispart of the spin chain. In addition, this structure represents an impenetrable wall forany other φ traveling around the chain. This is reminiscent of the situation we have foreclectic spin chains, and indeed we conclude that we have subspaces of states containingsuch walls where the Hamiltonian is again nilpotent. As an explicit example, consider thestate | ψ i := | ( ∂φ ) φ φ φ L − i − | φ ( ∂φ ) φ φ L − i . (3.15)It is easy to check that H L − | ψ i 6 = 0, but H L − | ψ i = 0. This means that this operatoris part of a Jordan block of size (at least) L − 2. A more detailed discussion can be foundin appendix E.We know that the Bethe ansatz does not even give all the eigenstates not belonging toJordan blocks since we found the following counterexample. Let us consider a spin chainof general length L , with a single φ and a single ∂ . The Bethe equations for such a spinchain are given in (3.10) to (3.13). We can rewrite them in terms of a single polynomialequation for the energy E as2 (cid:18) E − (cid:19) L − − (cid:18) E − (cid:19) L − − . (3.16)This equation has L − L dimensional, therefore we imme-diately see that the Bethe equations are missing one eigenstate. The missing eigenstate isexactly the wall we talked about in the last paragraph. In some physical (but not mathe-matical) sense this is a kind of 1 × E = − 2. For large enough L we can see that | E/ | has to be close to onefor the terms in (3.16) to be of roughly the same size. So for larger L the derivative is lessimpactful, and the spectrum (except for the single eigenvalue E = 0) becomes approxi-mately that of the spin chain without the derivative and without the cyclicity constraint In contradistionction to the model in [30]. We find the descendant using the Bethe equations, since we work on the level of momenta and not ofthe usual Bethe roots. The descendant still corresponds to a singular Bethe root u , but a finite e ip . This means for large L the derivative acts only as a reference point to whichthe distance of the φ can be measured and not as a properly traveling excitation.Before we mention a few open questions on the completeness of the Bethe equations,let us use the above example to see how the Bethe equations behave as one approachesthe double scaling limit from the unscaled twisted theory. Proceeding numerically, it issimplest to choose L = 3. In the unscaled theory the sector with a φ and a ∂ is notclosed. There are additional states, where the excitations have combined to form a ¯ ψ and a ψ . The eigenstates are given by two states including only φ and ∂ excitationsas above, one eigenstate including only a ¯ ψ and a ψ excitation and two eigenstatesconsisting out of the wall with some additional terms including a ¯ ψ and a ψ excitation.In the strong twisting limit the ¯ ψ and ψ decouple, hence we can look at the { φ ∂ } sector by itself. The projections of the eigenstates containing the wall onto the { φ ∂ } sector are identical, but not of Bethe form. The Bethe form only comes from the inclusionof the fermionic sector. This explains both, why the Bethe ansatz does not find these, aswell as how in general Bethe states can get lost in the double scaling limit, which allowsfor Jordan block formation. The only other possibility we have found how Bethe statescan be lost in the limit is by a collision of solutions. However, we have only observed thisphenomenon in the β -twisted theory (1.1). Let us pause to discuss the observations from the last section a bit further. After all,they are the basis for our claim that the strongly twisted models are rather different fromtheir “mother theory”, i.e. untwisted N = 4 SYM, and as such somewhat unsuitable forclarifying the origins of integrability of the latter. So far we have only found Jordan blockswith generalized eigenvalue zero. Is this a general feature or are there Jordan blocks withnon-zero generalized eigenvalues? For eclectic spin chains, as well as wall-like structuresof the last section, E = 0 immediately follows. Put differently, are there other mechanismfor Jordan block formation? Or else, is there a basis in which the Hamiltonian may bemanifestly written as the direct sum of a diagonalizable part and a wall subspace?The above questions concern the spin chain and the spectrum of the dilatation operatoras such. It is also interesting to understand how the Bethe ansatz and the quantum inversescattering method relate to these open problems. Can integrability at least find all non-zero eigenvalues? And can it quantitatively describe the Jordan cell formation of theseintegrable non-unitary model(s)? Finally one might wonder whether one can combinethe Bethe ansatz with a proper investigation of the walls to find the full spectrum of thenon-eclectic part of the spin chain? These issues are also present in the case of the more complicated β -twisted theory(1.1). Before turning to the latter, we will first expand our machinery. Indeed, it is easy to see that the solutions to (3.16) for large L are − E n / ω n exp[ − ( L − − log(2 − /ω n ) + O ( L − )] with ω n = exp[2 πin/ ( L − n = 0 , , . . . , L − One might be tempted to pose this question as follows: Can the Bethe ansatz diagonalize the wall-freesubspace? However, note that the wall-free subspace is not yet rigorously defined. While the wall subspaceas such may be defined as in appendix E, to define an orthogonal complement we would have to introducea suitable inner product, which appears unnatural to us. Scaling Limit of the Twisted Bethe Equations In order to prepare for the discussion of the spectrum of the strongly β -twisted theory(1.1), we require a way to obtain one-loop Bethe equations in more complicated sectors.Therefore we discuss in this chapter the scaling procedure on the level of the generaltwisted one-loop N = 4 Bethe equations, originally worked out in [23]. We refer to thispaper and references therein for background information on these equations. In addition,more details on the notation are given in appendix B. The scaling of the momentum-carrying roots u ,j , which encode the lattice momentum ofthe excitations, can be fixed by considering the dispersion relation. To illustrate this, wetake a plane wave of a single excitation A on an infinite chain (with φ as the vacuum), | p i = X n e ipn | A ( n ) i , (4.1)where the sum is over spin chain sites n. Acting with the β - or γ -twisted Hamiltonian(2.7), (2.8), with q = e − iβ/ or q = e − iγ / respectively we find, before taking the scalinglimit, H q | p i = E q ( p ) | p i := (4 − q c e − ip − q − c e ip ) | p i , (4.2)where c is +1, − On the other hand, the dispersion law of the Beisert-Roiban equationsis [23] E ( u ) = 2 i (cid:20) u + i/ − u − i/ (cid:21) , (4.3)with no explicit dependence on q . It follows that the relation between u and p is e ip = q c u + i/ u − i/ . (4.4)When taking the strong twisting limit q → ∞ we are focusing on states where the spinchain momenta p j remain finite. This is conveniently implemented by the change ofvariables u ,j → − i/ − iq − α ,j , j = 1 , . . . , K R , + i/ iq − α ,j − K R , j = K R + 1 , . . . , K R + K L , ˜ u ,j − K R − K L , j = K R + K L + 1 , . . . , K , (4.5)where K R ( K L ) is the number of right- (left)-movers. For e.g. a right-mover we then havethe identification e ip = α in the q → ∞ limit. The total energy becomes E = lim q →∞ q − K X j =1 E q ( u ,j ) = − K R X j =1 α ,j − K L X j =1 α ,j . (4.6) For the γ -twist the fermions will have c = ± / 2, but, since these fields decouple completely in thestrong twisting limit, we will ignore them. The wave function corresponding to a given set of Bethe roots will, in general, contain terms pairingthe different types of excitations with the momenta in all possible ways. One would thus expect to findterms where, say, a right-mover is paired with a left moving momentum. For the wave function to benon-singular with the above scaling of u it is necessary that the amplitude of such terms is zero in the q → ∞ limit (or that some kind of subtle cancellation occurs). β -twisted or fishnet Hamiltonian with den-sities (2.10) and (2.11). Before tackling the actual Bethe equations, let us deal with thezero-momentum=cyclicity constraint. According to appendix B it reads q K R − K L K Y j =1 u ,j + i/ u ,j − i/ . (4.7)Substituting our change of variables and taking the q → ∞ limit we find K R Y j =1 α ,j K L Y j =1 α ,j K − K R − K L Y j =1 ˜ u ,j + i/ u ,j − i/ . (4.8)The correct scaling of the auxiliary roots is more subtle and is highly dependent on thesector under consideration. In the next sections we will discuss several cases where we, to alarge extent, are able to check our ansätze by cross-checking the resulting Bethe equationsagainst other methods. Here we treat the full chiral fishnet sector that we already briefly mentioned in section3.2. It consists of φ (vacuum), φ (excitation) and any of the four derivatives ∂ µ ∼ ∂ α ˙ α (four further excitations). The twisted Bethe equations are given in appendix B.2, andthis specific sector is obtained by restricting auxiliary roots according to K = K = K − K = 0. We denote the number of derivatives by K ∂ = K = K , and the numberof right-movers ( φ ’s) is K R = K − K ∂ .Let us first consider a state of a single ∂ (we ignore the zero-momentum constraintfor now). This excitation has K = K = K = 1, and the Bethe equations read1 = q L u − ˜ u − i/ u − ˜ u + i/ q L u − ˜ u − i/ u − ˜ u + i/ , (4.9)and (cid:18) ˜ u + i/ u − i/ (cid:19) L = q − L ˜ u − u − i/ u − u + i/ u − u − i/ u − u + i/ . (4.10)At large q we see that we must have u / − ˜ u − i/ ∼ q − L . If we repeat this exercise forthe remaining types of derivatives (i.e. by introducing an u and/or u root) we find thesame scaling for the u and u roots, while no scaling is necessary for the u and u roots.Let us assume that this is the general pattern for states with an arbitrary number ofexcitations, and set u ,j = ˜ u ,j + i/ iq − L β ,j , u ,j = ˜ u ,j + i/ iq − L β ,j , (4.11)for j = 1 , . . . , K ∂ . It is now straightforward to take the q → ∞ limit of the Betheequations. We find α L ,k = ( − K R − K ∂ Y j =1 (˜ u ,j + 3 i/ u ,j − i/ u ,j + i/ u ,j + i/ 2) (4.12)14or the main roots, and 1 = K Y j = k u ,k − u ,j − iu ,k − u ,j + i K ∂ Y j =1 u ,k − ˜ u ,j u ,k − ˜ u ,j − i , (4.13)1 = K Y j = k u ,k − u ,j − iu ,k − u ,j + i K ∂ Y j =1 u ,k − ˜ u ,j u ,k − ˜ u ,j − i , (4.14) ˜ u ,k + i/ u ,k − i/ ! L − K R = ˜ u ,k + i/ u ,k + 3 i/ ! K R K ∂ Y j = k ˜ u ,k − ˜ u ,j − i ˜ u ,k − ˜ u ,j + i , × K Y j =1 ˜ u ,k − u ,j + i ˜ u ,k − u ,j K Y j =1 ˜ u ,k − u ,j + i ˜ u ,k − u ,j (4.15)for the auxiliary roots. Since E is independent of the ˜ u ,j , it is natural to include themamong the auxiliary roots. Here we have used the equations at nodes 3 and 5 to eliminatethe β ,j and β ,j , which has the effect of changing the ˜ u self-scattering term from the su (2) form to the sl (2) form appropriate for derivatives.In order to check that we have scaled the auxiliary roots correctly we can compare ourBethe equations to those derived in section 3.2. If we set K = K = 0 and identify e ip ∂,j = ˜ u ,j + i/ u ,j − i/ , e ip φ,j = α j , (4.16)we see that the equations are identical.Equipped with a guideline on how to scale the roots, we are now able to study twointeresting sectors of the strongly β -twisted model (1.1). β -Twisted Theory su (3) Sector The second simplest sector of strongly β -twisted theory, after the broken su (2) sectordescribed in section 3.1, consists of the three scalars { φ , φ , φ † } . In memory of theoriginal N = 4 model we will call it “broken su (3) sector”. One of these scalars has to be aconjugate scalar in order to avoid eclectic field content, which would result in a nilpotentHamiltonian as described in section 2.3. We choose φ as the vacuum and φ and φ † as(right-moving) excitations. We start with the Beisert-Roiban equations in the so-called“Beauty” grading given in appendix B.1 with a total of K excitations { φ , φ † } , and K excitations of φ † type. The scaling of the momentum-carrying roots was discussed in thelast chapter and according to (4.5) for right-movers we have u ,j = − i/ − iq − α ,j , (5.1)where u are the usual momentum-carrying Bethe roots and α = e ip is our parametrizationof the lattice momentum. To cancel the remaining factors of q in the twisted Beisert-Roiban equations we see that one option is to let u ,j = − iq − α ,j . (5.2)15e then obtain the following Bethe equations α L ,k = ( − K − K Y j =1 α ,k − α ,j , (5.3)( − K − = K Y j =1 ( α ,j − α ,k ) , (5.4)1 = K Y j =1 α ,j , (5.5) E = − K X j =1 α ,j . (5.6)These equations are a generalization of the ones for the broken su (2) sector (3.1) - (3.3).In contrast, however, they are no longer obviously solvable. At the same time, theynevertheless look simpler than “usual” one-loop Bethe equations. It would be interestingto find an analytic solution procedure. To check these equations, we can use the twistedR-matrix from [23] to apply a nested algebraic Bethe ansatz as described in [31]. Thescaled version of this R-matrix is R ( u ) = u P − + i P , (5.7)where P − is the projection operator on letters in chiral order, and P is the permutationoperator. One checks that this is the correct R-matrix by computing the Hamiltonian inthe usual fashion [21] H = − i d log Tdu (cid:12)(cid:12)(cid:12)(cid:12) u =0 , (5.8)which leads to agreement with our Hamiltonian density given in equation (2.10). We usedthis R-matrix and the corresponding transfer matrix to apply an algebraic Bethe ansatzof the form described in [31]. As a successful consistency check, we managed to rederivethe Bethe equations (5.3) - (5.6).Since the sector consisting of { φ , φ , φ } is eclectic, we know that it has a nilpotentHamiltonian, see section 2.3. Thus, one might suspect the Bethe ansatz to fail. However,it is interesting to observe the problems that arise. The construction of an R-matrix anda monodromy works. However, in the algebraic Bethe ansatz one uses so called RTTrelations to determine a set of fundamental commutation relations between the matrixelements of the monodromy. In this sector, the R-matrix has some zero entries, suchthat the RTT relations do not produce a complete set of these fundamental commutationrelations. Therefore, a Bethe ansatz, at least in its standard form, is not consistent.As mentioned above, we did not manage to explicitly solve (5.3) - (5.6) for a generalnumber of excitations. However, for one excitation of each type φ and φ † at any length L ,the equations are simple enough to be solved exactly. We find that one of the momentum-carrying Bethe roots is from either of the two sets u ∈{ exp((2 k + 1) iπ/ ( L + 1)) | ≤ k ≤ ( L − / } or (5.9) u ∈{ exp(2 kiπ/ ( L − | ≤ k ≤ ( L − / } . (5.10)The other momentum-carrying Bethe root is its complex conjugate, and hence their sum,the energy, is real. In general these two sets are disjoint, however an interesting phe-16omenon emerges when L = 4 n + 1 for some integer n . In this case the Bethe root i ap-pears in both sets, and thus the energy of the corresponding state is E = − − i + i ) = 0.Through explicit calculations we find that in this case a Jordan block of size two formsin the spectrum, with generalized eigenvalue zero. One of the two states in the block isa true eigenstate, and does correspond to this particular solution of the Bethe equations.Thus, in contrast to the derivative case discussed in 3.3, here the proper eigenstate of theJordan block is found by the Bethe ansatz. However, the remaining part of the Jordanblock is still undetermined by this ansatz. su (2 | 3) Sector Within the β -twisted model, we can extend the sector from the last section by fermions { ¯ ψ , ˙1 , ¯ ψ , ˙2 } to what we call, once more keeping the connection with unscaled N =4 SYM,a broken su (2 | 3) sector. Take again φ as the vacuum, and consider the excitations{ φ , φ † , ¯ ψ , ˙1 , ¯ ψ , ˙2 } . According to appendix B this corresponds to exciting the roots from K through K in the “Beauty” grading. This is the sector that we claim to be equivalentto the one considered in [4]. However, in [4] the so-called ABA grading is used, and thenumber of fermionic excitations is fixed explicitly to zero. This implies that the sectorconsidered in [4] reduces to a sector that is equivalent to the broken su (3) case discussed inthe last section, even though this is not immediately manifest in the form of the equations.Indeed, as mentioned earlier, we were able to match the one-loop limit of their equationswith (5.3)-(5.6)The scaling of the u and u roots is identical to the one in the broken su (3) case, andall other roots do not need to be scaled. Plugging these roots into the Bethe equations inthe Beauty grading given in appendix B we find1 = K Y j =1 α ,j , (5.11)1 = K Y j = k u ,k − u ,j − iu ,k − u ,j + i K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ , (5.12) u ,k + i/ u ,k − i/ ! K = K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ , (5.13)1 = ( − K − K Y j =1 ( α ,k − α ,j ) K Y j =1 u ,j + i/ u ,j − i/ , (5.14) α L ,k = ( − K − K Y j =1 α ,k − α ,j , (5.15) E = − K X j =1 α ,j . (5.16)We observe a curious decoupling of these equations, in the sense that (5.12) and (5.13)may be solved independently of the remaining ones, and then be used as “source terms” forthe latter. The remaining Bethe equations look strikingly similar to (5.3) and (5.6) fromthe last section. In fact, they are identical except for the additional last factor in (5.14).The reason for this phenomenon becomes clearer when looking at the action of H . In fact,in sectors without derivatives, like the one we are considering here, H does not distinguish17etween φ † and ¯ ψ , ˙ α , as can be determined from the Hamiltonian density (2.10). Howeverinterchanging positions of a φ † and a ¯ ψ , ˙ α yields two distinguishable states.As an illustrative example let us compare the spin chain given by only one statetr( φ φ † φ † ) and the spin chain given by the two states | i = tr( φ φ † ¯ ψ , ˙1 ) and | i =tr( φ ¯ ψ , ˙1 φ † ). For the second spin chain we have two eigenstates | i + | i and | i − | i witheigenvalues − | i − | i would be identically0. Hence, for this chain we have only one state, which is automatically an eigenstate witheigenvalue − 2. We conclude that the similarity between the broken su (3) sector and thebroken su (2 | 3) sector is expected, due to the identical form of H , while the differences aredue to the distinguishable nature of the excitations φ † and ¯ ψ , ˙ α . In this paper, we have mostly focused on the one-loop structure of the spectrum. Letus make a few preliminary remarks on the extension to higher loops. The analysis wepresent here will also highlight a subtlety of the relationship between the Bethe equationsderived in section 4.1 of [4] and our equations. Twisted asymptotic all-loop Bethe equationwere already given in [23], so one should be able to follow the usual procedure of solvingthese equation perturbatively around a given one-loop solution. Once wrapping and pre-wrapping [20] sets in, the analysis becomes, of course, more complicated. Here we willrestrict to determining the scaling of the momentum-carrying roots following the logic ofsection 4. Let us thus consider a single right-moving excitation with momentum-carrying root u = u ,j . At generic twist q the spin chain momentum is given by [23] e ip := q x + x − , (6.1)where x ± are the weak coupling solutions to x ± + 1 x ± = u ± i/ g , (6.2)i.e. x ± = u ± i/ g − gu ± i/ − g ( u ± i/ + O ( g ) . (6.3)Our task is to find the appropriate scaling form of u , such that the coefficients of ξ in theweak coupling expansion of e ip have a finite q → ∞ limit. As in the one-loop case, we willparametrize u by α , but now α is a series in ξ : α = ∞ X n =0 α ( n ) ξ n . (6.4)18e claim that the appropriate scaling is u = − i/ − iαq − + iξ α − , (6.5)= − i/ − iα (0) q − + iα (0) − iα (1) q − ! ξ + − iα (1) α − iα (2) q − ! ξ + O ( ξ ) . (6.6)Note that the leading term agrees with (4.5). The solution is constructed exactly suchthat x + behaves nicely for large q , in fact we have x + = − iξq α , (6.7)and it is then easy to see that e ip has a well-defined strong twisting limit, order by orderin ξ : q x + x − = α − ξ α − + O ( q − ) . (6.8)It is paramount that the eigenvalue of δ D , i.e. the anomalous dimension γ , also has afinite limit. This is guaranteed, if all the contributions γ s to the anomalous dimension ofthe individual Bethe roots are finite and indeed γ s := 2 ig (cid:18) x + − x − (cid:19) = − ξ α − + O ( q − ) . (6.9)We note that (6.1), (6.8) and (6.9) together imply the following chiral dispersion law, tobe compared with equation (4.6) of [4] γ s = q − e − ip ξ − . (6.10) We take our underlying field theories to be defined by the strong twisting limit of theperturbation series of twisted SYM. We thus expand in g (equivalently ξ ) first, and thensend q → ∞ . In contrast, ξ is kept finite as q → ∞ in section 4.1 of [4], and only after thestrong twisting limit an expansion in ξ is performed. The order of limits is thus reversedin comparison to our approach. It turns out that exchanging the order of limits leads tosubtle differences.To explicate this, let us redo the scaling analysis, but now keeping ξ finite as in [4].For clarity we use hatted variables when taking the limits in ‘reverse’ order. The finitenessof (6.1) together with (this equation follows immediately from (6.2))ˆ x + + 1ˆ x + − ˆ x − − x − = ig (6.11)imply the scaling ˆ x + ∼ q − , ˆ x − ∼ q . (6.12)From (6.2) we then immediately find (compare with Eq. (4.9) of Ref. [4])ˆ x + = ξq (cid:18) u + i/ O ( q − ) (cid:19) , ˆ x − = qξ (cid:0) ˆ u − i/ O ( q − ) (cid:1) , (6.13) The scaling form of u is unique up to a redefinition of α . γ s := 2 ig (cid:18) x + − x − (cid:19) = 2 i (ˆ u + i/ 2) + O ( q − ) . (6.14)We can compare with the usual perturbative order by identifying ˆ γ s = γ s , which leads toˆ u = − i/ iξ α − . (6.15)Referring back to (6.6), we see that the relationship between u and ˆ u is non-trivial. Let usfinally mention that the Bethe equation derived in [4] agrees with (a subsector of) those ofSec. 5.2, once this relationship is taken into account. It is thus possible that exchangingthe order of limits only leads to ‘superficial’ differences, resulting in identical results forphysical quantities. We have seen that the strongly twisted models appear to be, in comparison with theoriginal N =4 model, both much simpler in some ways as well as much more complicated inothers. The former, because the number of Feynman diagrams governing their perturbativeexpansion is vastly reduced [2–10]. The latter, since their dilatation operator ceases tobe diagonalizable. It therefore seems to us that the twisted models are, all in all, neithersimpler nor more complicated, but simply rather different from their mother theory.The main reason for the different nature of the twisted models is their non-unitarity.An immediate consequence is the non-hermiticity of the dilation generator, which we haveworked out explicitly at one-loop-order for the strongly twisted models, cf. (2.10), (2.11),(A.1). Already at this leading order it may no longer be fully diagonalized. From basiclinear algebra, the best thing one can do is to bring the non-diagonalizable sectors inJordan normal form. This was in some special cases already noticed and briefly discussedin [4], and in more detail (but still for the special case) in [6], where a connection tologarithmic conformal field theory was made. In this paper we have shown many moreexamples for non-diagonalizable states of these models. Perhaps surprisingly, even inthe (chiral) φ , φ -sector of the fishnet model with derivatives a Jordan block structureappears, which seems to be a novel result, cf. section 3.3. For the states of this sector notpart of a Jordan block we proposed novel Bethe equations, see sections 3.2 and 4.2.The lack of complete diagonalizability of the strongly twisted models should not betaken lightly. After all, one way to state the meaning of quantum integrability is thesimultaneous diagonalizability of an infinite set of charges in involution. What if none ofthem may be diagonalized in the first place? This certainly obscures the very meaningof “integrability” on a theoretical level. Recall that one of the motivations to study thetwisted models has been to get a useful insight into the reasons underlying the integrabilityof certain planar four-dimensional quantum field theories, with the hope of subsequentlytransporting these insights to full-fledged N =4 SYM.The main conceptual purpose of this paper has then been to demonstrate the significantdifferences between the N =4 theory and the twisted models that appear, in the exampleof the spectral problem, already at the leading one-loop level. Let us remember that inthe case of N =4 SYM the careful analysis of the precise one-loop structure serves as thesolid basis for the higher-loop asymptotic Bethe ansatz (ABA) and provides the necessary“initial conditions” for the exact functional equations of the quantum spectral curve (QSC)20for recent reviews of the latter, including a discussion of the twisted models, see [9, 32]).It is surely fair to ask in what sense the QSC provides an “exact solution” of the spectralproblem of a model, if a large fraction of its operators cannot be diagonalized in the firstplace.On the bright side, the novel integrable models of [2] pose an interesting mathematicalchallenge for the future: How can one systematically and fully adapt the quantum inversescattering method to integrable models with a non-hermitian (and non-pseudo-hermitian)Hamiltonian? How can one completely bring this Hamiltonian in Jordan normal form,generalizing or suitably replacing the Bethe ansatz? Are there Jordan cells with generalizedeigenvalues different from zero? If so, how to find these generalized eigenvalues? Whathappens to the Jordan cells once one takes higher loop corrections into account? Notethat the example of a cell studied in chapter 7.2 of [6] was argued to stay intact at everyorder in perturbation theory. We also feel that the interesting connections to logarithmicconformal field theory pointed out in [6] should be systematically explored. Finally, itwould be very interesting to find a physical application for the strongly twisted models. Acknowledgments We are thankful to João Caetano, Vladimir Kazakov, Gregory Korchemsky, Florian Loeb-bert, Dennis Müller and Stijn van Tongeren for inspiring discussions. We thank J. Caetano,V. Kazakov and G. Korchemsky for excellent talks on the subject. The work of ACI issupported by the Villum Foundation. LZ has been supported by the DFG-funded graduateschool GK 1504 Masse, Spektrum, Symmetrie in the preparatory phase of this project. A Dilatation Operator of Strongly γ i -Twisted Models In section 2.2 we have derived the one-loop dilatation operator of the two strongly twistedtheories that we investigate in this paper, namely the strongly β -twisted model and thefishnet model. The same arguments that led to the one-loop dilatation operator in thesetheories can also be applied to the more general deformations of strongly γ i -twisted modelswhere all three double-scaled couplings ξ i are a priori distinct and not necessarily zero. Letus parameterize the different coupling constants ξ i by setting ξ i = ξa i for some referencecoupling ξ . As before we divide the quantum corrections to the dilatation operator as δ D = ξ H + O ( ξ ). The one-loop part acquires additional factors compared to the β -twisted version given in (2.10), but stays structurally the same. It is given by( H s γ t n,n +1 ) A n A n +1 A n A n +1 = c ( a , a , a ) ( H s β t n,n +1 ) A n A n +1 A n A n +1 , (A.1)where c ( a , a , a ) depends on the fermionic or bosonic nature of the exchanged flavors.We give it case by case. • Case 1: For an exchange of two scalars c = a i , where the subscript i corresponds tothe flavor not taking part in the exchange. For example, if the dilatation operatorexchanges φ and φ , then c = a . • Case 2: For an exchange of two fermions c = a i a j , where the subscripts i, j corre-spond to the flavors that are exchanged.21 Case 3: For an exchange of a fermion and a scalar c = a i a j , where the subscriptscorrespond to the flavor of the fermion and the flavor not taking part in the exchange. • Case 4: If the fermionic or bosonic nature of the excitations changes during theflavor exchange, c is the square root of the product of two of the c ’s from theprevious cases. The factors are taken such that one factor √ c corresponds to theinitial configuration and the other factor of √ c corresponds to the configurationafter the flavor exchange. To illustrate this let us consider the matrix element of thedilatation operator corresponding to ∂φ , φ † → ¯ ψ , ψ , where we suppressed spinorindices. The c in (A.1) for this example is the product of √ c = a from case 1 and √ c = √ a a from case 2.The above analysis works for those special cases where some of the a i are zero. Thisincludes the fishnet model for which a = a = 0 and a = 1. However, one can constructeven more general double scaled theories by taking the limit q i → 0, with g/q i kept fixedfor some of the twist angles. In this case, the dilatation operator is again distinct fromthe one described in this section, and one has to go through similar calculations to obtainit. B Twisted One-Loop Bethe Equations In this appendix, we write down the twisted Bethe equations from [23] that we need.The Dynkin diagram of su (2 , | 4) admits various gradings, which leads to different sets ofBethe equations. Here we will use the “Beauty” grading [33] and the ABA grading [27]. B.1 “Beauty” Grading φ φ ψ ψ φ † φ † ψ ψ ψ ¯ ψ ∂ ∂ ψ ¯ ψ ∂ ∂ Table 2: Single excitations of the full N = 4 SYM spin chain in the “Beauty”grading. The table should be read as follows: Consider an state with the non-zero K s being K j = K j +1 = · · · = K k = 1, where 1 ≤ j ≤ ≤ k ≤ 7. The correspondingexcitation, over a vacuum of φ s, is the one listed at row j and column k .For this grading we only consider the β -twist. We thus set γ = γ = γ = β and q = e − iβ/ . The twisted Bethe equations are [23]:22 = q K − K +2 K K Y j =1 u ,j + i/ u ,j − i/ , (B.1)1 = K Y j = k u ,k − u ,j − iu ,k − u ,j + i K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ , (B.2)1 = K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ K Y j =1 u ,k − u ,j − i/ u ,k − u ,j + i/ , (B.3)1 = q K − K +2 K K Y j =1 u ,k − u ,j − i/ u ,k − u ,j + i/ × K Y j = k u ,k − u ,j + iu ,k − u ,j − i K Y j =1 u ,k − u ,j − i/ u ,k − u ,j + i/ , (B.4) u ,k + i/ u ,k − i/ ! L = q − L − K +6 K − K K Y j =1 u ,k − u ,j − i/ u ,k − u ,j + i/ × K Y j = k u ,k − u ,j + iu ,k − u ,j − i K Y j =1 u ,k − u ,j − i/ u ,k − u ,j + i/ , (B.5)1 = q L +4 K − K +2 K K Y j =1 u ,k − u ,j − i/ u ,k − u ,j + i/ × K Y j = k u ,k − u ,j + iu ,k − u ,j − i K Y j =1 u ,k − u ,j − i/ u ,k − u ,j + i/ , (B.6)1 = q − L − K +4 K − K K Y j =1 u ,k − u ,j − i/ u ,k − u ,j + i/ × K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ , (B.7)1 = K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ K Y j = k u ,k − u ,j − iu ,k − u ,j + i . (B.8)The elementary excitations are listed in Table 2. The momentum constraint (B.1) agreeswith Eq. (4.7), where K R = K − K and K L = K − K .23 .2 ABA Grading φ ψ ψ φ ψ ∂ ∂ ¯ ψ ψ ∂ ∂ ¯ ψ φ † ψ ψ φ † φ † ψ ψ φ † ψ ∂ ∂ ¯ ψ ψ ∂ ∂ ¯ ψ φ ψ ψ φ Table 3: Single excitations of the full N = 4 SYM spin chain in the ABA grading.The left table is in the conventions of [23], while the right is the R-symmetry rotatedvariant we use in connection with the β -twist. The notation is the same as Table2. γ -twist β -twist t − K − K + 2 K − K − K K − K − K t L + K − K + K − K + 2 K + 2 K t L − K − K t − L + 2 K + 2 K − L + 2 K + 2 K t L − K − K L − K − K t L + K − K + K L − K − K + 2 K Table 4: Twist factors for the ABA Bethe equation.For the Bethe equations in the ABA grading we will consider two different twists. Forthe γ -twist we set γ = γ = 0 and q = e − iγ / . For the β -twist we set γ = γ = γ = β and q = e − iβ/ . The corresponding values of t , . . . , t are given in Table 4. The elementaryexcitations are listed in Table 3. Note that for the β -twist we have performed an R-symmetry rotation compared to the conventions of [23].The zero-momentum constraint for the β -twist takes the form (4.7) with K R = K − K and K L = K . For the γ twist we focus on the elementary excitations { φ , φ † , ∂ α ˙ α } , sincethese are the one that do not decouple in the fishnet limit. This leads to the restrictions K = K and K = K , cf. Table 3. We then again find the zero-momentum constraintto be of the form (4.7), with K R = K − K = K − K and K L = K = K .24n the ABA grading the Bethe equations take the following form [23]:1 = q t K Y j =1 u ,j + i/ u ,j − i/ , (B.9)1 = q t K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ , (B.10)1 = K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ × K Y j = k u ,k − u ,j − iu ,k − u ,j + i K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ , (B.11)1 = q t K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ K Y j =1 u ,k − u ,j − i/ u ,k − u ,j + i/ , (B.12) u ,k + i/ u ,k − i/ ! L = q t K Y j =1 u ,k − u ,j − i/ u ,k − u ,j + i/ × K Y j = k u ,k − u ,j + iu ,k − u ,j − i K Y j =1 u ,k − u ,j − i/ u ,k − u ,j + i/ , (B.13)1 = q t K Y j =1 u ,k − u ,j − i/ u ,k − u ,j + i/ K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ , (B.14)1 = K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ × K Y j = k u ,k − u ,j − iu ,k − u ,j + i K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ , (B.15)1 = q t K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ . (B.16)The twisted Bethe equations in the ABA grading are also given in appendix C of [4]. Tomatch the conventions of [23], as also employed in the present paper, we find it necessaryto send q → q − and q → q − in the equations of [4]. For example, the zero-momentumconstraint is given as K Y k =1 x +4 ,k x − ,k = q − J q − J , (Eq. (C.1) of [4])with x ± ,k = g − ( u ,k ± i/ 2) + O ( g ). In the strongly β -twisted limit q = q = q → ∞ this would imply that both φ and φ are right-movers. With our conventions φ is aright-mover, but φ is a left-mover. C Derivatives in the Strongly β -Twisted Model In this section, we give Bethe equations for the sector of the strongly β -twisted modelconsisting of the excitations { φ † , ψ α , ¯ ψ α , ∂ α ˙ α } . This corresponds to the upper left part of25able 3 (right). We thus set K = K = 0. The number of right-movers is K R = K − K ,and there are no left-movers.The derivation closely follows that of section 4.2, so we will be brief. By consideringsingle excitation states we conjecture the scaling u ,j = i/ u ,j + iq − L β j , (C.1)with all other auxiliary roots unscaled. Plugging this into the equations given in appendixB.2, we find, in the q → ∞ limit, α Lk = ( − K R − K Y j =1 u ,j + iu ,j K Y j =1 ˜ u ,j − i/ u ,j + i/ K Y j = k u ,k − u ,j − iu ,k − u ,j + i K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ , (C.3)1 = u ,k u ,k + i ! K R K Y j =1 u ,k − u ,j + i/ u ,k − u ,j − i/ K Y j =1 u ,k − ˜ u ,j − i/ u ,k − ˜ u ,j + i/ , (C.4) ˜ u ,k + i/ u ,k − i/ ! L − K R = K Y j =1 ˜ u ,k − u ,j − i/ u ,k − u ,j + i/ K Y j =1 ˜ u ,k − u ,j + i ˜ u ,k − u ,j , (C.5)1 = K Y j =1 u ,k − ˜ u ,j u ,k − ˜ u ,j − i K Y j = k u ,k − u ,j − iu ,k − u ,j + i (C.6)for the auxiliary roots. As in section 4.2 we were able to eliminate the β j roots.It is interesting to try to reproduce the above equations using the coordinate Betheansatz. We proceed along the same lines as in section 3.2. For simplicity we will restrictto states with only u and u roots. This corresponds to only keeping the excitations φ † and ψ . A simple calculation shows that the S -matrix is S | φ † ( p ) φ † ( p ) i = −| φ † ( p ) φ † ( p ) i , (C.7) S | φ † ( p ) ψ ( p ) i = e ip | ψ ( p ) φ † ( p ) i , (C.8) S | ψ ( p ) ψ ( p ) i = − S ψ,ψ ( p , p ) | ψ ( p ) ψ ( p ) i . (C.9)The fermion-fermion S -matrix element cannot be fixed in the usual way by imposing theeigenvalue equation on scattering states, even if one considers more than two excitations.We thus leave it as an unspecified function. It would be interesting to see whether higherloop corrections fix S ψ,ψ ( p , p ).Since the scattering is transmission diagonal, we can use (3.9) to write down the Betheequations , e ip φ,k L = ( − K R − K Y j =1 e − ip ψ,j , (C.10) e ip ψ,k L = e ip ψ,k K R K Y j = k S ψ,ψ ( p ψ,k , p ψ,j ) − . (C.11) As always for an S -matrix it should satisfy S ψ,ψ ( p , p ) = S ψ,ψ ( p , p ) − . The boundary conditions for the fermions introduce an additional sign factor in (C.11) which cancelsagainst the explicit sign in (C.9) S ψ,ψ ( p , p ) = 1. Notethat this is also the value of the fermion-fermion S -matrix that follows from twisting theBeisert S -matrix. It might seem surprising that one is free to choose S ψ,ψ . The explanation appears tobe related to the fact that the system is very degenerate. Taking the product of (C.11)over all k we get K Y j =1 e − ip ψ,j L − K R = 1 . (C.12)Since the energy is expressed in terms of the p φ,k only, it follows that the only influencethe fermion sector has on the spectrum is through the choice of which root of unity from(C.12) to insert into (C.11) and the zero-momentum constraint. D Nilpotency Proof In this appendix we prove the following Theorem: Consider an operator tr( A · · · A L ). If there does not exist a b ∈ F such that(we use the notation introduced in section 2.1) a i := F ( A i ) ∈ { b, b + , ¯ b − } , for all i = 1 , . . . , L , (D.1)then H N | A · · · A L i = 0 for some N > 0. Here H is the one-loop Hamiltonian of eitherthe strongly β -twisted or fishnet model, H = L X n =1 H s β t n,n +1 , or H = L X n =1 H FN n,n +1 . (D.2)The Hamiltonian densities H s β t n,n +1 and H FN n,n +1 are defined in (2.10) and (2.11) respectively.As discussed in section 2.3 we call the flavor sequence ( a · · · a L ) eclectic if it satisfies thehypothesis of the theorem.The one-loop Hamiltonian density acts as H n,n +1 | A n A n +1 i = H A n A n +1 A n A n +1 | A n A n +1 i . (D.3)The crucial property of the double scaled dilatation operator leading to nilpotency is that,whenever H A n A n +1 A n A n +1 = 0, we have F ( A n ) = F ( A n +1 ) , F ( A n +1 ) = F ( A n ) , (D.4)and h F ( A n ) , F ( A n +1 ) i ∈ P − (hence h F ( A n ) , F ( A n +1 ) i ∈ P + ) . (D.5)Here h a, b i denotes an ordered pair, and we define P ± := {h a, a ± i| a ∈ F } ∪ {h a, ¯ a ∓ i| a ∈ F } . (D.6)To prove (D.4) and (D.5) one uses that H preserves the R-symmetry charges defined intable 1, and the presence of the chiral projectors P ± in (2.10) and (2.11). In fact, this particular matrix element of the S -matrix is not affected by the β -twist. a · · · a L ) we aregoing to assign an integer d ( a · · · a L ) which is bounded from above and such that, for anyeclectic sequence, d ( a · · · a L ) < d ( a a a · · · a L ) , (D.7)when h a , a i ∈ P − . By the above remark it is clear that the theorem follows.An occurrence of a subsequence ( x · · · x n ) in ( a · · · a L ) is defined to be a sequence ofindices { i m } m =1 ,...,n such that( a i a i · · · a i n ) ’ ( x x · · · x n ) , ≤ i < i < · · · < i n ≤ L , (D.8)where ’ denotes equality modulo cyclic permutations. The multiplicity of ( x · · · x n ) in( a · · · a L ), denoted mul[ x · · · x n ; a · · · a L ], is the number of occurrences (i.e. sequences { i m } m =1 ,...,n such that (D.8) holds). We now define d by d ( a · · · a L ) := X ( x ··· x n ) ∈ C + mul[ x · · · x n ; a · · · a L ] − X ( x ··· x n ) ∈ C − mul[ x · · · x n ; a · · · a L ] , (D.9)where C ± := { ( x x · · · x n ) | n ≥ , x i ∈ F ∅ , ∀ i. h x i , x i +1 i / ∈ P ∓ } . (D.10)Here and in the remainder we set x n +1 = x .It remains to show that d satisfies (D.7). For the remainder of the proof we will assumethat h a , a i ∈ P − . A little reflection now shows thatmul[ x · · · x n ; a · · · a L ] > mul[ x · · · x n ; a a a · · · a L ] . (D.11)implies that there exists an i ∈ { , . . . , n } such that x i = a and x i +1 = a . But thismeans that ( x · · · x n ) / ∈ C + . Similarly, frommul[ x · · · x n ; a · · · a L ] < mul[ x · · · x n ; a a a · · · a L ] (D.12)one can conclude that ( x · · · x n ) / ∈ C − . More loosely, the action of the Hamiltonian cannever decrease the multiplicity of the sequences in C + , and never increase the multiplicityof the sequences in C − . By the definition of d we thus have d ( a · · · a L ) ≤ d ( a a a · · · a L ) . (D.13)This holds even when ( a · · · a L ) is not eclectic. We show that the inequality is strict inthe eclectic case by a case analysis. • Case 1. a = a + , a = a : Since ( a · · · a L ) is eclectic there is an i ∈ { , . . . , L } suchthat a i ∈ { ¯ a, ¯ a + , a − , ∅} . (D.14)Now clearly mul[ a a a i ; a · · · a L ] < mul[ a a a i ; a a a · · · a L ] , (D.15)and also ( a a a i ) = ( aa + a i ) ∈ C + . (D.16)It follows that d increases by at least one. The remaining cases are similar, so wewill be more brief. 28 Case 2. a = a − , a = ¯ a : Since ( a · · · a L ) is eclectic one of the following threesubcases most hold: – Subcase 1. There is an i such that a i ∈ { a, ¯ a − , ∅} . (D.17)This is sufficient since ( a a a i ) = (¯ aa − a i ) ∈ C + . (D.18) – Subcase 2. There are i, j such that a i = ¯ a + , a j = a + , i < j . (D.19)This is sufficient since ( a a a i a j ) = (¯ aa − ¯ a + a + ) ∈ C + . (D.20) – Subcase 3. There are i, j such that a i = a + , a j = ¯ a + , i < j . (D.21)This is sufficient since ( a a a i a j ) = ( a − ¯ aa + ¯ a + ) ∈ C − . (D.22)This concludes the proof. Inspection of the case analysis shows that it would also gothrough with d ( a · · · a L ) defined by the same formula as d , but with C + := { ( aa + a − ) , ( aa + ¯ a ) , ( aa + ¯ a + ) , ( aa + ∅ ) , ( a + ¯ a ∅ ) , (¯ aa − ¯ a + a + ) | a ∈ F } (D.23)and C := { ( a − ¯ aa + ¯ a + | a ∈ F } (D.24)instead of C ± . E Walls in Fishnet Theory and Nilpotency Here we will show that the one-loop Hamiltonian of the fishnet model allows for certainstrongly bound states, which we call walls . Furthermore, the Hamiltonian is nilpotent onany state built from φ , φ and derivatives containing such a wall.Let f i , i = 1 , 2, denote the set of letters consisting of φ i with an arbitrary number ofderivatives, and set f = f ∪ f . A wall is a state on a two-site chain in the subspace ω := n | w i ∈ span (cid:8) ( A ) ⊗ ( B ) (cid:12)(cid:12) A ∈ f , B ∈ f (cid:9) (cid:12)(cid:12)(cid:12) H | w i = 0 o . (E.1)It is easy, using Eq. (2.13), to show that ω contain states with an arbitrary non-zeronumber of derivatives. The simplest example is | ( ∂ φ ) φ i − | φ ( ∂ φ ) i ∈ ω .We now consider the space W of states on a length L chain with a wall on sites oneand two, W := span (cid:8) | w i ⊗ | A · · · A L i (cid:12)(cid:12) A i ∈ f , | w i ∈ ω (cid:9) . (E.2)29ote that we are not imposing the zero-momentum constraint for the moment. Let H ij = H FN ij denote the density operator acting on sites i and j such that H = L X n =1 H n,n +1 . (E.3)From the definition of ω and due to the chiral projector P − in (2.11) it is clear that H n,n +1 annihilates any state in W for n = L, , 2. It follows that H ( | w i ⊗ | v i ) = | w i ⊗ H o | v i (E.4)for all | w i ⊗ | v i ∈ W , where H o := L − X n =2 H n,n +1 (E.5)is the Hamiltonian on an open spin chain of length L − 2. By chirality, acting repeatedlywith H o will necessarily annihilate any state the latest when the φ ’s have moved to theright of the φ ’s. We conclude that H in nilpotent in W .Finally, we need to deal with the zero-momentum constraint. This is easy, we simplyproject W onto the zero-momentum subspace. The space of operators containing walls isthus W := (cid:8) P | W i (cid:12)(cid:12) | W i ∈ W (cid:9) , (E.6)with P := 1 L L − X n =0 U n , (E.7)and where U is the (one-site) translation operator. Since P commutes with the Hamilto-nian, we find that H is nilpotent in W . References [1] N. Beisert et al. , “Review of AdS/CFT Integrability: An Overview,” Lett. Math.Phys. (2012) 3–32, arXiv:1012.3982 [hep-th] .[2] Ö. Gürdoğan and V. Kazakov, “New Integrable 4D Quantum Field Theories fromStrongly Deformed Planar N = 4 Supersymmetric Yang-Mills Theory,” Phys. Rev.Lett. no. 20, (2016) 201602, arXiv:1512.06704 [hep-th] . [Addendum: Phys.Rev. Lett.117,no.25,259903(2016)].[3] C. Sieg and M. Wilhelm, “On a CFT limit of planar γ i -deformed N = 4 SYMtheory,” Phys. Lett. B756 (2016) 118–120, arXiv:1602.05817 [hep-th] .[4] J. Caetano, Ö. Gürdoğan, and V. Kazakov, “Chiral Limit of N = 4 SYM andABJM and Integrable Feynman Graphs,” JHEP (2018) 077, arXiv:1612.05895[hep-th] .[5] D. Chicherin, V. Kazakov, F. Loebbert, D. Müller, and D.-l. Zhong, “YangianSymmetry for Bi-Scalar Loop Amplitudes,” JHEP (2018) 003, arXiv:1704.01967 [hep-th] . 306] N. Gromov, V. Kazakov, G. Korchemsky, S. Negro, and G. Sizov, “Integrability ofConformal Fishnet Theory,” JHEP (2018) 095, arXiv:1706.04167 [hep-th] .[7] D. Chicherin, V. Kazakov, F. Loebbert, D. Müller, and D.-l. Zhong, “YangianSymmetry for Fishnet Feynman Graphs,” Phys. Rev. D96 no. 12, (2017) 121901, arXiv:1708.00007 [hep-th] .[8] D. Grabner, N. Gromov, V. Kazakov, and G. Korchemsky, “Strongly γ -Deformed N = 4 Supersymmetric Yang-Mills Theory as an Integrable Conformal FieldTheory,” Phys. Rev. Lett. no. 11, (2018) 111601, arXiv:1711.04786 [hep-th] .[9] V. Kazakov, “Quantum Spectral Curve of γ -Twisted N = 4 SYM Theory andFishnet CFT,” arXiv:1802.02160 [hep-th] . [Rev. Math.Phys.30,no.07,1840010(2018)].[10] N. Gromov, V. Kazakov, and G. Korchemsky, “Exact Correlation Functions inConformal Fishnet Theory,” arXiv:1808.02688 [hep-th] .[11] B. Basso and D.-l. Zhong, “Continuum Limit of Fishnet Graphs and AdS SigmaModel,” arXiv:1806.04105 [hep-th] .[12] B. Basso and L. J. Dixon, “Gluing Ladder Feynman Diagrams into Fishnets,” Phys.Rev. Lett. no. 7, (2017) 071601, arXiv:1705.03545 [hep-th] .[13] V. Kazakov and E. Olivucci, “Biscalar Integrable Conformal Field Theories in AnyDimension,” Phys. Rev. Lett. no. 13, (2018) 131601, arXiv:1801.09844[hep-th] .S. Derkachov, V. Kazakov, and E. Olivucci, “Basso-Dixon Correlators inTwo-Dimensional Fishnet CFT,” arXiv:1811.10623 [hep-th] .[14] O. Lunin and J. M. Maldacena, “Deforming Field Theories with U(1) x U(1) GlobalSymmetry and Their Gravity Duals,” JHEP (2005) 033, arXiv:hep-th/0502086[hep-th] .S. Frolov, “Lax Pair for Strings in Lunin-Maldacena Background,” JHEP (2005)069, arXiv:hep-th/0503201 [hep-th] .[15] J. Fokken, C. Sieg, and M. Wilhelm, “Non-Conformality of γ i -Deformed N = 4 SYMTheory,” J. Phys. A47 (2014) 455401, arXiv:1308.4420 [hep-th] .[16] A. B. Zamolodchikov, “’Fishnet’ Diagrams as a Completely Integrable System,” Phys. Lett. (1980) 63–66.[17] O. Mamroud and G. Torrents, “RG stability of Integrable Fishnet Models,” JHEP (2017) 012, arXiv:1703.04152 [hep-th] .[18] J. A. Minahan, “Review of AdS/CFT Integrability, Chapter I.1: Spin Chains inN=4 Super Yang-Mills,” Lett. Math. Phys. (2012) 33–58, arXiv:1012.3983[hep-th] .[19] N. Beisert, “The Complete One-Loop Dilatation Operator of N=4 Super Yang-MillsTheory,” Nucl. Phys. B676 (2004) 3–42, arXiv:hep-th/0307015 [hep-th] .3120] J. Fokken, C. Sieg, and M. Wilhelm, “The Complete One-Loop Dilatation Operatorof Planar Real β -Deformed N = 4 SYM Theory,” JHEP (2014) 150, arXiv:1312.2959 [hep-th] .[21] L. D. Faddeev, “How Algebraic Bethe Ansatz Works for Integrable Model,” in Relativistic Gravitation and Gravitational Radiation. Proceedings, School of Physics,Les Houches, France, September 26-October 6, 1995 , pp. pp. 149–219. 1996. arXiv:hep-th/9605187 [hep-th] .[22] C.-N. Yang, “Some Exact Results for the Many Body Problems in One Dimensionwith Repulsive Delta Function Interaction,” Phys. Rev. Lett. (1967) 1312–1314.[23] N. Beisert and R. Roiban, “Beauty and the Twist: The Bethe Ansatz for TwistedN=4 SYM,” JHEP (2005) 039, arXiv:hep-th/0505187 [hep-th] .[24] M. Staudacher, “Review of AdS/CFT Integrability, Chapter III.1: Bethe Ansátzeand the R-Matrix Formalism,” Lett. Math. Phys. (2012) 191–208, arXiv:1012.3990 [hep-th] .[25] E. H. Lieb, T. Schultz, and D. Mattis, “Two Soluble Models of an AntiferromagneticChain,” Annals Phys. (1961) 407–466.[26] M. Staudacher, “The Factorized S-matrix of CFT/AdS,” JHEP (2005) 054, arXiv:hep-th/0412188 [hep-th] .[27] N. Beisert and M. Staudacher, “Long-Range psu(2,2|4) Bethe Ansatze for GaugeTheory and Strings,” Nucl. Phys. B727 (2005) 1–62, arXiv:hep-th/0504190[hep-th] .[28] W. Hao, R. I. Nepomechie, and A. J. Sommese, “Singular Solutions, RepeatedRoots and Completeness for Higher-Spin Chains,” J. Stat. Mech. (2014)P03024, arXiv:1312.2982 [math-ph] .[29] W. Hao, R. I. Nepomechie, and A. J. Sommese, “Completeness of Solutions ofBethe’s Equations,” Phys. Rev. E88 no. 5, (2013) 052113, arXiv:1308.4645[math-ph] .[30] A. M. Gainutdinov and R. I. Nepomechie, “Algebraic Bethe Ansatz for theQuantum Group Invariant Open XXZ Chain at Roots of Unity,” Nucl. Phys. B909 (2016) 796–839, arXiv:1603.09249 [math-ph] .[31] F. H. L. Essler and V. E. Korepin, “Higher Conservation Laws and Algebraic BetheAnsatze for the Supersymmetric t-J Model,” Phys. Rev. B46 (1992) 9147–9162.[32] N. Gromov, “Introduction to the Spectrum of N = 4 SYM and the QuantumSpectral Curve,” arXiv:1708.03648 [hep-th] .[33] N. Beisert and M. Staudacher, “The N=4 SYM Integrable Super Spin Chain,” Nucl.Phys. B670 (2003) 439–463, arXiv:hep-th/0307042 [hep-th]arXiv:hep-th/0307042 [hep-th]