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Free Fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT

In this paper we first demonstrate explicitly that the new models of integrable nearest-neighbour Hamiltonians recently introduced in PRL 125 (2020) 031604 satisfy the so-called free fermion condition. This both implies that all these models are amenable to reformulations as free fermion theories, and establishes the universality of this condition. We explicitly recast the transfer matrix in free fermion form for arbitrary number of sites in the 6-vertex sector, and on two site in the 8-vertex sector, using a Bogoliubov transformation. We then put this observation to use in lower-dimensional instances of AdS/CFT integrable R-matrices, specifically pure Ramond-Ramond massless and massive AdS_3, mixed-flux relativistic AdS_3 and massless AdS_2. We also attack the class of models akin to AdS_5 with our free fermion machinery. In all cases we use the free fermion realisation to greatly simplify and reinterpret a wealth of known results, and to provide a very suggestive reformulation of the spectral problem in all these situations.


Integrability in AdS 3 and AdS 2 backgrounds
Integrability of the AdS 3 × S 3 × S 3 × S 1 and AdS 3 × S 3 × T 4 string-theory backgrounds [1] (see also [2]), has proceeded taking the moves from the infinite-length treatment of the AdS 5 /CF T 4 spin-chain.
The massless sector displays in a very clear fashion the nature of the string integrable model as a quantum-group deformation of two-dimensional Poincaré supersymmetry. The idea goes back to [22] in the five-dimensional case, revisited by the more recent [23]. In the massless AdS 3 case one can go further [24,25,26] to the point where one can show [27,28] that there exists a change of variables which puts the massless non-relativistic S-matrix and its dressing factor in difference-form, in fact the exact same difference-form as the (non-trivial) BMN limit [11]. The massless TBA of [11] is then almost straightforwardly adapted to the full (massless) non-relativistic case [28].
The AdS 3 background allows for a mixed-flux extension [29] which modifies the traditional magnon dispersion relation [30]: where k (which we will restrict to the natural numbers) stands for the WZW level, m for the mass and h for the coupling constant -see [31] for a full study of the moduli space of this model. A particular relativistic limit has been studied in [28], where another instance of nontrivial scattering for right-right and left-left moving modes was found, leading to a family of CF T s with exact TBA description. An important wealth of work on this and similar deformations has now been done [32].
The theory admits a Z 4 automorphism [36] which is key to its classical integrability [37]. In [38] an S-matrix was derived utilising a centrally-extended psu(1|1) 2 algebra built upon the BMN ground state [39,40]. A similar series of steps were taken following the higher-dimensional recipes and comparing with the available perturbative results [40]. The AdS 2 massive modes sit in a long representation [41], and only the massless modes are in short ones. The Yangian symmetry was studied in [38,42]. It is known that comparison with perturbation theory is problematic for massless modes [43,13], which is a signal that massless S-matrices are fundamentally distinct from massive ones [7] -they rather describe certain massless renormalisation group flows between conformal field theories [9].
The BMN limit for right-right and left-left movers is once again non-trivial [44] and reminiscent of, although rather distinct from, N = 1 supersymmetric models [45]. The S-matrix has an XYZ/8-vertex structure [46,47]. The absence of a reference state prevents the familiar algebraic Bethe ansatz approach [48], see also [49]. The strategy of [44] relies on the free fermion condition [47,50] which will be crucial to this paper as well, and on inversion relations [51]. The transfer matrix was explicitly calculated up to 5 particles in [54], where a conjecture for the massless Bethe ansatz was given and for part of the massive Bethe equations. The free fermion condition holds in fact for the massive AdS 2 S-matrix as well, which we will show can be understood in the light of the results of this paper. This will provide a framework where to revisit the observations of [40,43,55] as well.

This paper
In this paper we will embed the above-mentioned integrable structures within the larger context of the classification performed in [16] -built upon the work in [56] -where a general study and a classification was made of integrable R-matrices and nearest-neighbour interaction spin-chain Hamiltonians of the 8-(or less) vertex type. Such classification includes the cases described in the previous section (with the exception of the massless AdS 2 situation which we will discuss in a separate section 5.3). Here, we will show that the remarkable property which was observed for these AdS models, namely the free fermion condition, holds in fact for the two new models introduced in [16]. We will be giving plenty of details of this extraordinary unifying feature in the following sections, especially at the beginning of section 3, where we will also provide a context for the associated literature. At this stage we remark that this confirms the universality of this condition and its far-reaching physical significance. It also furnishes an explanation of the diverse observations related to it which have been encountered so far within the lower-dimensional AdS/CF T examples.
We will give the mathematical proof of the fact that the two new models introduced in [16] satisfy the free fermion condition, both by explicit verification, and by use of the Sutherland equation. This will allow us to use a diagonalisation procedure inspired by the coordinate Bethe ansatz, which sets the transfer matrix in manifest free fermion form for an arbitrary number of sites and inhomogeneities for the 6-vertex type R-matrices. The final results are remarkably compact and suggestive, and we consider them as a powerful alternative to the Algebraic Bethe Ansatz. We will then apply this to the case of the pure Ramond-Ramond AdS 3 massless R-matrix and use the free fermion condition to revisit and drastically simplify the associated algebraic structure. We will then follow a similar approach for the mixed-flux AdS 3 relativistic case and for the massive pure Ramond-Ramond AdS 3 case.
We will then analyse the 8-vertex type models, which include the AdS 2 scattering problem. We will recast the two-site transfer matrix explicitly in a free fermion form, and in the process discover that there exists a state playing a role similar to that of the pseudo vacuum, which such models do not possess. This is because the particle-hole transformation we perform allows us to populate the spectrum in the usual fashion by adding excitations onto this state, which we therefore dub the pseudo pseudo vacuum. The construction is then paraphrased for massless AdS 2 excitations and the pseudo pseudo vacuum is shown. We then apply the formalism to the case of 16 × 16 R-matrices and Hamiltonians exhibit su(2) ⊕ su (2) symmetry and manage to recast the latter in a form which is as close as possible to free fermions (achieving free fermions in special cases). This enters the territory of the Hubbard model and of the AdS 5 integrable scattering theory.
In the Appendices, after providing a bare recursive formula for the pure Ramond-Ramond massless AdS 3 transfer matrix on an arbitrary number of sites with inhomogeneities, we will restrict to the homogeneous case and give explicit examples of the general formalism, namely the effective spin-chain Hamiltonian for pure Ramond-Ramond massless and mixed flux relativistic AdS 3 , for open as well as closed chains. We will show there how the formulas simplify for the free fermion realisation to rather minimalistic expressions.
Besides the general proof of the validity of this condition and its potential physical implications for these models, we hope that this paper will show the power of the free fermion realisation in organising the exact results for these systems in an astonishingly transparent and suggestive form.

Free fermion condition for general 8-vertex models
In [16,17], building upon the earlier [56], a classification was made of all the possible regular (i.e. nearest-neighbour) integrable spin-chains whose R-matrix admits the general 8-vertex form where r i = r i (u, v). It is the purpose of this section to verify that all integrable models of this type fall into two classes that are characterized by a general property of the R-matrix (r 1 r 4 + r 2 r 3 − r 5 r 6 − r 7 r 8 ) 2 r 1 r 2 r 3 r 4 = const. (2.2) It is then natural to consider two classes of models A Baxter condition const = 0 (2.3) B The free fermion condition const = 0 (2.4) which implies r 1 r 4 + r 2 r 3 = r 5 r 6 + r 7 r 8 . (2.5) This classification was already known for models of difference form and we find that (2.2) is the generalization to models of non-difference form, which, in particular, include the AdS/CFT integrable models.
Class A corresponds to the models 6-vertex A and 8-vertex A from [16], while class B corresponds to the models 6-vertex and 8-vertex B from [16], as we check below. All holographic integrable models fall into class B. Class A basically contains the usual XXZ and XYZ spin-chains. On the level of the Hamiltonian, the difference between both models is the presence of the S z ⊗ S z interaction term in the Hamiltonian. Class A models contain this interaction, while Class B models do not. Interesting properties can be found in [57].
In the next section we shall comment on the historical origin of free fermion condition and its use, and afterwards we shall use it to simplify specific models. In this section we shall prove that indeed the class B solutions singled out in [16] (modulo what we could call gauge transformations 1 ) exactly satisfy the free fermion condition. We will then prove the above classification directly using the Sutherland equation, which allows for possible generalizations of the free fermion conditions to other types of models.
1 See section Identifications in [16] for details.

Class A
Up to basic identifications, models from class A correspond to the usual 6-and 8-vertex model. It is not hard to see that the condition (2.3) is compatible with these identifications. Hence it remains to show that these models satisfy condition A. We will spell this out for the 8-vertex model and leave the 6-vertex model for the reader. The 8-vertex R-matrix is given in terms of Jacobi elliptic functions as where η and k are arbitrary constants. Thus, we find and indeed the Baxter condition holds.
One can then compute each term in (2.5) and obtain r 1 r 4 + r 2 r 3 = 1 = r 5 r 6 + r 7 r 8 (2.9) what proves that the free fermion condition (2.5) is satisfied for 6-vertex B.
For 8-vertex B the entries of (2.1) are , k an arbitrary constant and η(u) an arbitrary function.
Let us start by the rhs of equation (2.5).
r 5 r 6 + r 7 r 8 = 1 + k 2 sn 2 cn 2 dn 2 . (2.15) Now for the left hand side of equation (2.5), let us consider first r 1 r 4 (2.16) where the two crossing terms cancelled. Now let us compute r 2 r 3 (2.17) which brings us to Putting (2.15) and (2.18) together we have

Sutherland equation
One can in fact prove that any R-matrix of the form (2.1) satisfies the generalized condition (2.2) by using the Sutherland equations where the dot and prime denote derivative with respect to u and v, respectively.
Let us assume a Hamiltonian density of the form 2.25) and an R-matrix of the form (2.1), with r i being any function of u and v and h i being any function of u. In order to prove the relation (2.2), we start by doing the following procedure 1. Substitute (2.1) and (2.25) into the Sutherland equations (2.24).
2. Solve for the derivativesṙ i and r ′ i in terms of h i and r i (without actually solving the differential equations).
3. Then solve for some of the h i 's in terms of r i (u, v).
Remarkably, by doing this, one obtains the following set of conditions on the coefficients of the R-matrix where f (u) and g(u) are functions of the matrix elements h i (u) of the density Hamiltonian H (2.25).
By multiplying (2.26) and (2.27) and then also (2.28) and (2.29) we can see that , (2.30) and hence proving that equation (2.2) holds. Since the Sutherland equations (2.24) are obtained as a derivative of the Yang-Baxter equation, we proved that all integrable models of the form (2.1) belong to one of the two classes A and B.
Notice that this recipe can be applied to other models as well. We will work this out in Section 6 for a special case of a 16 × 16 R-matrix and Hamiltonian with su(2) ⊕ su (2) symmetry. There we will use this method to demonstrate that the AdS 5 R-matrix also satisfies similar free fermion conditions (cf. [59]).
Since all the AdS 3 and AdS 2 integrable models fall into class B, and AdS 4 relies on the AdS 5 R-matrix, we obtain the remarkable fact that all the R-matrices arising in the AdS/CF T correspondence are of free fermion type.

An alternative to the Algebraic Bethe Ansatz
As we have demonstrated, all the integrable cases we shall discuss have a common feature: the associated R-matrices satisfy the so-called free fermion condition, whose role in the context of AdS/CF T was highlighted in [59]. Therefore, according to the standard lore, they are amenable to a description in terms of free fermions. Further analysis has been performed in [60] also in the context of inversion relations, tetrahedral algebras and higher-dimensional vertex models. An extensive investigation of the free fermion realisation for a variety of models with different lattice geometries and symmetries can be found for instance in [61]. Further work can be found in [62].
This idea and its realisation goes back to old literature [63,64,50,65,66] (more recent work has appeared in [67]). The papers [63,64] were able to re-write the Hamiltonians and transfer matrices of the XY chain and of Ising-type chain and lattice models in a form which manifestly displays their free fermion nature. In order to achieve this, a particular transformation of the canonical spin-chain operators is performed. Inspired by those ideas, in this paper we will find the appropriate canonical transformations for the various situations we shall study. In doing so we will of course crucially rely on the fact which we have proven in the previous section, that all the R-matrices we are going to analyse satisfy the free fermion condition.
We will show explicitly in each case how to map the problem to one of free fermions, by displaying the transformation at the level of the transfer matrix, where everything can be made completely manifest.
We will mainly focus on the cases in which the spaces we deal with are two-dimensional, and are spanned by one boson |φ and one fermion |ψ . We introduce the following notation, suggestive of the treatment to ensue: where we have introduced canonical fermionic creation and annihilation operators c † and c, respectively, such that Later on, whenever we will need to work with multiple spaces, we shall introduce operators For example, i, j may be ranging from 0 to 2 for the transfer matrices with one auxiliary {0} and two physical spaces {1, 2}. In general, they will be ranging from 0 to N .
We notice that the graded tensor product of spaces is nicely encoded in these operators, and provides a more compact way of performing many of the Algebraic Bethe Ansatz manipulations. For instance, suppose we were to calculate the action of a matrix on the pseudovacuum such as 4) where E ab are the matrices with all zeroes but 1 in row a column b, and states are numbered as |1 = |φ and |2 = |ψ as usual, so that E ab |c = δ bc |a . In the language of creation and annihilation operators we have where |0 is used to denote now |φ ⊗ |φ by abuse of notation. The fermionic nature of the operators implies that which is the equivalent of 2 In terms of a single space, the association is therefore and clearly Let us now work out some explicit examples on how the free fermion condition can be exploited to diagonalize spin-chains. 2 We will sometimes write 1 and sometimes simply 1 to denote the identity operator. The context will always make it clear.

Set-up
Let us now demonstrate how we can use the free fermion condition to diagonalize the transfer matrix by considering 6-vertex model whose R-matrix satisfies the free fermion condition.
Thus, we consider an R-matrix of the form (2.1) with r 7 = r 8 = 0. It can be written in terms of oscillators in the following form where we suppressed the explicit dependence of r i on (u, v). Suppose the R-matrix is regular, i.e.
R(u, u) = P , with P being the graded permutation operator, then the Hamiltonian density is given by the logarithmic derivative of the R-matrix If we denote the corresponding derivative coefficients as h i = ∂ u r i , then we find in particular On the level of the Hamiltonian, the free fermion condition (2.5) which eliminates the n 1 n 2 term.
More generally, the conserved charges are generated by taking logarithmic derivatives of the transfer matrix The parameters θ i are local inhomogeneities, which for holographic models corresponds to momenta of world-sheet excitations. In case all the rapidities coincide θ i = θ, then the first logarithmic derivative corresponds to the nearest-neighbour Hamiltonian (3.12). For generic inhomogeneities, all the conserved charges have interaction range N .

Solving homogeneous spin-chains with Free Fermions
Let us now consider the Hamiltonian (3.12) for the homogeneous spin-chain. To this Hamiltonian we then apply our non-local free fermion transformation to diagonalize it. We note that it is enough to consider the one-particle sector since this will induce a canonical map between the oscillators c, c † and η, η † . Restricted to this subsector, our Hamiltonian takes the simple form (3.14) The eigenvalues and eigenvectors of H (1pt) can now be easily computed. Let x N = 1, then the eigenvectors v and eigenvalues λ are of the form . (3.15) This means that there are exactly N eigenvectors parameterised by the N th roots of unity and we can write the canonical transformation (3.16) This results in the Hamiltonian 17) which is now manifestly diagonal. Moreover, we can show that this canonical transformation also diagonalizes the full transfer matrix. Remarkably, only from N = 4, do we need the free fermion condition for this. We find 18) where The transfer matrix takes a factorized form, which reminds of separation of variables. Because of this factorized form, we can exponentiate it and read off the conserved charges where (

3.21)
Since all the number operators commute, the exponent is well-defined.

Inhomogenous spin-chains
Next we consider the case in which each spin-chain site has a corresponding inhomogeneity parameter θ i . In this case the one-magnon states are described by the inhomogeneous version of the Bethe Ansatz described in Appendix A.
Similar to the homogeneous case, we construct the free fermion map by using the one-magnon eigenstates. We only need to invert the relation between the Bethe states and the basis vectors. The onemagnon states correspond to the N solutions of the inhomogeneous Bethe equations . (3.22) Let us label the Bethe roots as v n , then we obtain the following map It is then straightforward to check, at least for small lengths, that under this map all terms in the transfer matrix are again given by simple products of the number operators. In fact we again find the following factorized form of the transfer matrix . (3.25) Concluding, by using our free fermion transformation, we were able to derive this very compact and elegant form of the transfer matrix.

Applications to AdS 3 /CF T 2
Let us now apply the formalism of free fermions to the various integrable models that arise in the AdS/CF T correspondence. We will focus in this section on AdS 3 models. For the kind of models relevant to the AdS 3 /CF T 2 correspondence, which are all of the 6-vertex type, we have constructed in the previous section a completely general formula which includes the inhomogeneities for an arbitrary number of frame particles (sites of the level-one transfer matrix in the nesting procedure). The scope of this subsection is to display explicitly, for the particular AdS 3 models considered in the literature, how the general formalism unfolds.

Transfer matrix analysis
We start by writing the R-matrix for the massless sector of the pure Ramond-Ramond AdS 3 integrable system in the oscillator formalism we have equipped ourselves with. We will use the relativistic variable θ everywhere, although one may simply replace θ → γ following [27,28] and all the formulas will hold true for the complete non-relativistic massless theory. We will also traditionally denote One has We have chosen not to write the overall normalisation factor since it will not play a role for the momentwe will reinstate it whenever necessary at a later stage. With this normalisation, we can directly compare with the 6-vertex R-matrix in [59]: where a, b, c, d are the parameters used in [59]. We can immediately see that the free fermion condition is satisfied: The R-matrix degenerates to the graded permutation operator for θ = 0.
This R-matrix is actually describing a nested Bethe ansatz, where the pseudovacuum made of all |φ is truly a level-one pseudovacuum, as opposed to the true BMN vacuum made of all |Z as described for instance in [6]. It does not make sense to calculate the Hamiltonian from it for AdS purposes, while it does of course in the approach of [16] which is using these R-matrices as generating functions themselves of integrable systems. For AdS scopes one should compute the transfer matrix with full non-equal inhomogeneities θ 1 , .., θ N , as done in [11]. Moreover, the inhomogeneities need to be interpreted as momenta and they satisfy some additional Bethe equations. We have denoted the supertrace over the auxiliary space by str 0 .
Let us now explicitly work out the free fermion map for this holographic model. For clarity, we will just focus on two physical spaces N = 2, for which the transfer matrix reads We will now perform the following canonical transformation [63,64]: having restricted ourselves to real inhomogeneities corresponding to physical momenta of the frame particles 4 . It is not difficult to see that the new creation and annihilation operators still satisfy which results in the transformation has the following effect on the transfer matrix: We could have added a piece +0 η † 2 η 2 inside the square bracket just to prove the point that the transformation we have introduced completely diagonalises the transfer matrix on two physical spaces, and explicitly shows the free fermion property.
Let us verify that we obtained the expected eigenvectors and eigenvalues as in [11]. The eigenvalues are now clearly extremely easy to compute, as it is just a matter of adding free fermion energies. This reveals an additive structure to the eigenvalues of the transfer matrix which was otherwise hidden in the old formalism. Moreover, the eigenvectors are naturally normalised if the pseudovacuum is, thanks to the canonical nature of the oscillators.
Since both η 1 and η 2 still annihilate the pseudovacuum, it is clear that |0 = |φ ⊗ |φ is an eigenstate, with eigenvalue cosh θ12 2 . Then, we have with eigenvalue − cosh θ12 2 . Afterwards, we have The inverse transformation is given by We make the choice of branch cot α = −e −θ 12 2 , so that the above exactly reproduces the result of [11] once a series of simplifications are worked out.
It is instructive to compare explicitly with the Algebraic Bethe Ansatz. The creation operator of the exact transfer matrix eigenstates is B(β; θ 1 , θ 2 ), evaluated on the solutions of the auxiliary Bethe In the case of two physical spaces there are only two solutions: β = ±∞, corresponding to two creation operators which are easily translated in our new formalism: It is immediate to see that 15) establishing the equivalence of the Algebraic Bethe Ansatz to our formalism in generating eigenstates of the transfer matrix. We can also see how the B operators do not straightforwardly create normalised eigenstates, but it is simple now to just use the η † i which do. We can notice a certain similarity of our approach with the B good strategy of [68], in the spirit of trying to find a B-type operator which automatically implements the nesting procedure of the Algebraic Bethe Ansatz. In our case we have operators η † i which do not require the intermediate step of solving the auxiliary Bethe equations, while they immediately already create (normalised) eigenstates of the transfer matrix.
We can proceed to complete the map with the remaining operators of the Algebraic Bethe Ansatz.
This allows establishing a correspondence between our free fermion realisation and the RTT relations -somewhat in the spirit of the Holstein-Primakoff oscillator realisations of Lie algebras. To be more precise, we have access from our formalism only to the RTT relations for the B (and, shortly, C) operators evaluated on the solutions to the auxiliary Bethe equations (we could call them on shell RTT relations).
Using (4.14) it is already possible to see for instance that (and clearly the same with both +∞ and both −∞), which is one of the simplest RTT relations in [11] evaluated on shell. Similarly, we can prove where we have used the fact that the transfer matrix A − D happens to be independent of θ 0 for two physical spaces. This is also another RTT relation from [11].
If we take a look at the C operators on shell, we discovered that they translate into They do satisfy (and clearly the same with both +∞ and both −∞), however they mix the two types of annihilation operators, which suggests that even more at higher N one should expect the map between the operators of the Algebraic Bethe Ansatz and the free oscillators to be progressively more and more complicated.
In this respect, the operators A and D separately are already rather complicated at this stage: Nevertheless, we expect that at higher N the relationship between B and η † i will always be one-to-one, since the B operators create eigenstates as much as η † i do. What might happen is for some function f of all the number operators, while for the C operators we will rather have 22) and for the A and D operators Let us now make some remarks on the generalization to higher values of N . Can we see the additive pattern of eigenvalues which a free fermion description seems to suggest? The answer is of course yes.
Strong in the knowledge of the general formula we have found in the previous section, we now proceed to analyse how this applied to our particular case.
The generic eigenvalue λ of the transfer matrix T N reads [11] , (4.24) where the β m are M values chosen amongst the N possible solutions to the auxiliary Bethe equations where we restrict to the strip β ∈ [0, iπ] in order not to over count.
The factor Ψ N is a complicated function of the rapidities, but it is common to all eigenvalues. What changes is the number of excitations M , specifically the different choices of M solutions out of the N possible solutions of (4.25), corresponding to how many magnons have been turned on and with which magnon-rapidities β m . From the nature of (4.25) it is clear that solutions with finite β come in pairs, the second solution always corresponding to inverting all the factors of b, equivalently sending β → β + iπ.
There is always then β = ∞ solution, and finally there is the solution β = −∞ for even N .
This is consistent with the following result for the transfer matrix that we encountered in the previous section 26) with no need of Baker-Campbell-Haussdorff given the mutual commutativity of all the different N i ≡ η † i η i . Clearly the complication is buried in the explicit form of the ψ N , ω i and η i , η † i . These quantities will depend on θ 0 and θ 1 , .., θ N , except the creation and annihilation operators which only depend on θ 1 , .., θ N (this is because the eigenstates cannot depend on θ 0 by the mutual commutativity of the transfer matrix taken at different values of the spectral parameter).
Conforming to our expectations from the structure of the Algebraic Bethe Ansatz, we can argue that one of the ω i will always be zero, corresponding to the β = ∞ auxiliary rapidity of the Algebraic Bethe Ansatz. We could have decided to always make this to be ω N = 0 and truncate the sum to N − 1. For even N , another one of the ω i will be equal to π. The remaining ones will come in pairs of the type The N = 2 case is clearly recovered very easily: given the fermionic nature of the operators, we have the standard Fermi property from which we obtain, at N = 2 and with the choice ω 2 = 0, ω 1 = π, that Setting ψ 2 = −i log cosh θ12 2 reproduces the desired result. In fact we can also write in a remarkably compact way As a further test of this idea, we can check the N = 3 case. From [11], once the appropriate common factor has been taken out, we have three eigenvalues corresponding to the following three different ω i in our ansatz: The pattern is exactly what we expected from the general arguments we just outlined, however the true test is whether the ω i are all real. Indeed, we can see that (4.32) which is a pure phase. The transfer matrix (4.26) is then manifestly unitary (when suitably normalised).
Finally, a rather non-trivial check is whether the additive pattern is reproduced. According to our ansatz, the 8 eigenstates of the transfer matrix on two physical spaces should have eigenvalues respecting the following scheme: {e 0 , e 0 , e ω , e −ω , e ω , e −ω , e 0 , e 0 }, (4.33) obtained by creating at most three free fermions and adding the energies (4.30). This is exactly the pattern one observes in [11] once the common factor is taken out.
By the general commutativity of the N i , we can then write which is a dramatic simplification of the general result (3.24) for the case of the RR AdS 3 model.

The R-matrix and its symmetries
Remarkably, in the new variables we also have (4.35) Considering once again that we are counting fermions, hence then it is easy to recover [11,24,26] 12 2 is both hermitian and unitary for real rapidities, by virtue of (4.36). This is also straightforward to realise in the form We find the expression (4.38) probably to be the simplest possible way we can write the massless AdS 3 R-matrix. It makes most of its properties completely evident, included the fact that Hermitian unitary matrices are diagonalisable and have eigenvalues ±1. Projectors onto the corresponding eigenspaces can be obtained by considering P ± = 1±R 2 . In Appendix B we elaborate on how such a (perhaps deceptively) simple form of the R-matrix, where the complication is buried in the definition of the operators η i and η † i , may be used to generate a recursion relation for the transfer matrix. Here we focus instead on how the symmetries of the R-matrix manifest itself in the free fermion language.
It is easy to see that one has the following supersymmetry property: (Q † being the hermitian conjugate of Q for real rapidities), which can be proven using the free fermion anti-commutation relations. In the traditional Hopf-algebra language we were recapitulating at the beginning of the paper, this is nothing but the coproduct of the supercharges: where we have introduced the momenta where the opposite coproduct ∆ op is obtained by permuting (in a graded fashion) the two factors in the tensor product of ∆. One also uses the fact that ∆(q) = ∆ op (q) and ∆(q † ) = ∆ op (q † ) to reduce these relations to simple commutators.
The operation of taking the opposite coproduct is simply translated in free fermion language by the rule: "swap the indices of particle 1 and 2". Because of the fermionic nature of the creation and annihilation operators, this rule automatically accounts for any fermionic sign, as the following example shows: 42) which perfectly reproduces the graded rule This supersymmetry respects the difference form of the R-matrix, since it reduces to the basic condi- via eliminating by a scalar function of the rapidities (namely, factoring out e θ 2 2 ). In the language of the η i and η † i operators, it is immediate to see that It is therefore straightforward to check the invariance under supersymmetry, since the R-matrix commutes with (in fact any function of) η 2 and η † 2 by virtue of the commutation relations of these operators. This provides a noticeable reinterpretation of the supersymmetry coproduct: it is given by those canonical combinations of creation and annihilation operators on which the R-matrix trivially does not depend.
The op transformation has a consequence on the variable α as well, since it exchanges the labels 1 and 2. The effect is encoded in the following map: This means that The relation Q op = Q is easily seen by considering that Using (4.47) it is immediate to see thatR op =R [11,26], and in fact also R op = R. Since we havẽ R 2 = 1, this implies braiding unitarity: A longer calculation along the same lines reproduces the result of [11,26]: which allows the re-writing (cf. [11]) Π s being the graded permutation acting on states. In verifying these formulas one has to be careful to the fact that the operators η i and η † i depend themselves on the rapidities via their very definition. Equation (4.51) does not represent a symmetry of the R-matrix but rather a constraint. The boost symmetry in the Hopf-algebra sense has been discussed in [23,69]. There is of course the standard relativistic boost symmetry of the R-matrix which reads . (4.53) In the case of non-relativistic massless excitations it took a non-trivial series of steps [27,28] to recast the R-matrix in such a way that, in the appropriate variables, such symmetry became completely manifest.
Combinations of the conditions (4.51) and (4.53) can be taken to produce derivative conditions separately w.r.t. θ 1 and θ 2 in a variety of fashions [26,27]. We speculate that these two conditions, together with the other symmetries displayed in this section, are in fact a basis for all the (differential) relations satisfied by this R-matrix.
There is a more involved supersymmetry, which however is the natural one from the string theory viewpoint. In fact, as discovered in [27,28], the remarkable property of the massless (right-right and left-left moving) R-matrix of pure Ramond-Ramond AdS 3 string theory is that it looks exactly the same both in the non-relativistic and in the relativistic (BMN) limit, provided one simply adjusts the rapidity variable. In particular, difference form is always there in the appropriate variables. We can recover this fact here: if we take the combinations where we notice the appearance of the familiar non-local braiding of the AdS coproduct [70], we can impose that they satisfy This condition is consistent with difference form, since it simply amounts to the two separate conditions It is now extremely easy to see that requiring [27] θ i = log tanp i 4 (4.58) solves (4.57). We can then simplify Q ± and obtain where g 12 is a symmetric function under the exchange 1 ↔ 2. If we revisit the condition of invariance of the R-matrix under this symmetry, we notice that we can drop the symmetric function in front of , we see that this condition is nothing else then 61) or equivalently which completely manifests the difference-form. It is also immediate to recover from this property that as it is clearly visible by inspection. The two types of supersymmetry that have been appearing in the literature are therefore in free fermion language essentially reduced to the more or less manifest symmetry-properties of the R-matrix R ∝ 1 − 2N 1 w.r.t the η i and η † i operators, i = 1, 2. From a strict quantum-group viewpoint, we can interpret the appearance of the second supersymmetry as the presence of a Yangian symmetry at level 1 with zero evaluation parameterû = 0 in Drinfeld's first realisation, which is rather typical of a conformal field theory [54]. In fact, (4.60) can be read as By simply taking the Hermitian conjugate of (4.55) we find where reproducing again the string theory result [24]. In terms of the conformal field theory Yangian we have We see that setting the evaluation parameterû = 0 is only one of the many possible choice, since adding any multiple of the level-0 supercharges by an even function of θ would still produce a symmetry. It is however the natural choice of organizing the symmetries according to the Yangian structure.
The fermionic number operator for one particle is given by and on two-particle states one has therefore the local coproduct From this observation it is immediate to see that, from R ∝ 1 − 2N 1 , we have 70) which is clear since the AdS 3 R-matrix preserves the fermionic number.
Another longer calculation reproduces also the following known result [19]: where the secret symmetry B 1 [71] is given in our new language by 72) with f any even function of the variable θ = θ 1 − θ 2 implementing the difference form. In particular, since n 1 + n 2 is basically proportional to B 0 apart from a trivial shift, it is obvious that the first part is a symmetry, and we see that the core of the secret symmetry lies in the relation This can also be interpreted in terms of the conformal field theory Yangian [54], with the level 1 secret symmetry being evaluated at 0 corresponding to setting f (θ) = 0, which is again only one of the possible choices.
One final remark of this section is related to crossing symmetry. Despite the great simplification of using the free fermion variables, the operators η and η † do not appear to have nice transformation properties under crossing θ 1 → θ 1 + iπ. Therefore, one does not seem to learn much more than what is already known [11] in translating the crossing relations in the new language.

AdS 3 with mixed flux
The R-matrix studied in [28] is another example which satisfies the free fermion condition: if we use the notations of [28], we can see that Here the index M refers to the notation of [59].
The transfer matrix T 2 is definitely more complicated than the previous case, and it can be efficiently treated by means of computer algebra. The change of variables engineered in this case is however the same as in the previous section, with the only difference that we simply have now (4.75) With this transformation we obtain, with a suitable choice of normalisation of the R-matrix, that where 5 (4.77) As befitting the general formula, the same additive pattern we have described for the pure Ramond-Ramond case repeats here as well: the auxiliary Bethe equations are the same when written in terms of the function b(θ, k), and the eigenvalue of the transfer matrix for general N , when written in terms of the function Y defined in [28], has the exact same structure we observed in the previous section. This means that the ansatz (4.26), mutatis mutandis, works in exactly the same way.
On of the specifics which is slightly different is the fact that already at N = 2 we have a pair of finite solutions of the auxiliary Bethe equations, replacing the ±∞ of the pure Ramond-Ramond case. This means that the specific pattern of ω i is different. In particular, we have at N = 2 the two auxiliary roots

The massive AdS 3 pure Ramond-Ramond case
Let us see now how the general formalism applies to the massive AdS 3 pure Ramond-Ramond case. The free fermion condition is easily seen to hold equally well: one has in fact from [4] that the massive-sector where the functions A, B, C, D, E, F are explicitly given in Appendix M of [4]. One can explicitly verify that these functions satisfy the condition which produce exactly the free fermion condition of [59] (see also comments in [7]). Our formulas apply therefore, although the technical complication resides in the difficulty in getting further simplifications.
The same type of transformation we have been using for the massless case, with just a different choice of the parameter α, does work for the massive case as well. Starting with the R-matrix, one can verify with computer algebra that choosing . (4.80) It is interesting to note that the expression (4.80) behaves as a a 0 0 in the BMN limit, and in fact tends to a finite non-trivial limit 6 . Most importantly, this limit is real, which might give some reassurance of a real angle α at least in a neighborhood of the physical region sufficiently close to the BMN point.
We also notice that the expression for R after the transformation characterised by (4.80) is purely in terms of N 1 and N 2 , although the coefficients are rather complicated expressions of the representation variables. It is possible to see however that the coefficient of the term N 1 N 2 vanishes in the massless kinematics, which is another consistency check.
One can explicitly recast the two-site transfer matrix T 2 in free fermion form with a canonical transformation. One can again check this with computer algebra by choosing where we have indicated by the indices 01 and 02 the assignment of variables according to (4.82) 6 As a consistency check, we have also verified that in massless right-right kinematics (4.80) tends to expression cot 2α = sinh θ 12 2 in (4.7) in the BMN limit.
In particular, the complication arises when solving the auxiliary Bethe equations in terms of x ± explicitly.
Even for just one magnon, the expressions quickly become very cumbersome.
Besides being only dependent on N 1 and N 2 , we have not been able to explicitly simplify T 2 much further, in fact even expression (4.81) remains rather difficult to reduce to small compact expressions in terms of the representation variables. We remain nevertheless firm in the knowledge that the formula we have derived for general 6-vertex models holds and, modulo the slight difficulty in containing the volume of the explicit expressions, achieves the complete diagonalisation and free fermion form for this case as well. We can hope that solving the Bethe equations for the momentum carrying roots can help simplify the expressions further.

8-vertex B model and application to AdS 2 /CF T 1
As opposed to 6-vertex models, we do not have a general formula displaying the free fermion form for 8-vertex models. We can in any case make progress and show that on two sites we can recast the transfer matrix and the Hamiltonian in the desired structure. This is particular relevant to the AdS 2 string theory models, which share the 8-vertex nature of the R-matrix.

Free oscillators
The main new complication that arises for models of 8-vertex type is that there are terms that violate the fermion number. Because of this, the simple free fermion transformation that we used previously will not work and we need to add terms that mix c and c † . Thus, we will need to consider a general transformation of the form 1) or in components where sum in repeated indices is assumed. In order to identify c † with the conjugate of c we take A * to be the complex conjugate of A. Imposing that the new oscillator basis η i and η † i also satisfies we find a set of consistency conditions on the matrices A, B. In particular, they need to satisfy Notice that A * B † + B * A † = 0 as well. In case of the 6-vertex model we can set B = 0 and the transformation is basically described by elements of SU (N ).

8-vertex model and pseudo -pseudo vacuum
The R-matrix for 8-vertex B presented in section 2 can be written in terms of oscillators as where r i ≡ r i (u, v) and are given by (2.10)-(2.14) 7,8 .
We then compute the monodromy for two sites T 2 = str 0 R 01 R 02 (5.6) and obtain For this simple case we can solve the conditions (5.4) so that we have a canonical transformation explicitly and we find together with their starred versions.
We then proceed to fix the remaining A ij and B ij by diagonalizing T 2 in equation (5.7). Requiring that the nondiagonal terms vanish yields the following constraints
At this point we notice that, although these models do not possess a pseudovacuum, we can find, thanks to the particle-hole transformation we have performed, a state which behaves in a similar way, and allows us to construct the spectrum in the usual fashion a genuine pseudovacuum would allow us to do. We call this state the pseudo pseudo vacuum 9 . It is given by such that η 1 |vac = 0 = η 2 |vac and a is an arbitrary function.
The eigenvalues λ i of the transfer matrix are given by 9 While we have found it for N = 2, we do not know whether it would actually exist for higher values of N .

The AdS 2 transfer matrix
We shall finally apply our formalism to the R-matrix dubbed Solution 3 in [44], corresponding to a particular situation in massless relativistic AdS 2 integrable superstrings. This R-matrix still satisfies the free fermion condition, as was noticed in [44,54]. This fact was in fact crucial for the purposes of [44,54], since it was only thanks to this condition that a procedure devised in [65] and used in [50] for N = 1 supersymmetric Sine-Gordon could be repeated in the AdS 2 case. The AdS 2 transfer matrix shares with those other models the same feature of the absence of a pseudo-vacuum state for the algebraic Bethe Ansatz, invalidating its applicability. The method which [44] used was instead to obtain the auxiliary Bethe equations via Zamolodchikov's inversion relations, and then [54] brute-force computed the transfer-matrix eigenvalues for a few number of sites and extrapolated the expression for the eigenvalues at any N . This is only viable thanks to the free fermion condition, which allows the very first step to be implemented.
With an appropriate normalisation of the R-matrix (and again, as always in this paper, neglecting for these purposes the overall dressing factor), the transfer matrix T 2 , which has been computed in [54], translates here into As we can see, this case is significantly different and more akin to an 8-vertex model, due to the non conservation of the fermion number. This also implies that we ought to change our transformation, and rather impose One can easily check that this transformation is still canonical.
It turns out that the choice α = π 4 (5.26) again eliminates unwanted terms, and allows the rewriting (5.27) As in the previous subsection, it can be verified that there is a state annihilated by both η 1 and η 2 , which will effectively play the substitute of the pseudovacuum on two physical spaces. The absence of global pseudovacua for all N forced us to employ alternative techniques to the Algebraic Bethe Ansatz in [44,54], however here we saw that there is a way to simulate the existence of a pseudovacuum for N = 2.
In this N = 2 case this pseudo -pseudo vacuum is given by The spectrum is then populated in the usual fashion.
The paper [54] extended the derivation of the auxiliary Bethe equations via the method of inversion relations to the massive AdS 2 R-matrix as well. It is now no surprise that this could be achieved (although not much further progress could be made due to technical complication of the R-matrix entries) once again because of the general result we have proven at the beginning of this paper. The massive AdS 2 R-matrix satisfies equally well the free fermion condition following those very general arguments.
One peculiar difference between the massive and the massless AdS 2 case is that R(p, p ′ ) equals the graded permutation only for the massive R-matrix, not the massless one. This is due to a noncommutativity of limits (equal arguments vs massless limit). The massless limit for the AdS 2 case is especially subtle and it has been discussed in detail in [38]. As a consequence, the massive R-matrix sits inside the classification of [16] (and admits in principle a nearest-neighbour Hamiltonian at equal inhomogeneities), while the massless one does not. It is still true that the same form of the massless R-matrix is valid for the BMN limit as well as for the full non-relativistic case, with the same change of variable as in AdS 3 , as shown in [27]. It is also still true that the massless R-matrix equally satisfies the free fermion condition.

Free fermion condition for AdS 5 sector
In this section we will finally apply our approach discussed in Section 2 to obtain a free fermion condition to the case of a 4-dimensional Hilbert space. We will focus on the class of model whose Hamiltonian and R-matrix exhibit su(2) ⊕ su (2) symmetry [17,56]. This class of models is particularly important because it contains the R-matrices of the AdS 5 × S 5 superstring sigma model and the one-dimensional Hubbard model.
In order to achieve this, we follow the procedure outlined in Section 2.3. In matrix form, the Hamiltonian is given by where h i s are dependent on the spectral parameter θ. The R-matrix takes the same form, but with coefficients r i (u, v).
As explained in Section 2.3, we should first substitute the Hamiltonian and the R-matrix in the Sutherland equations (2.24). Then, we solved them for the derivativesṙ i and r ′ i . This was done formally considering the derivativesṙ i and r ′ i as independent variables from the r i . It is worth mentioning that in order to avoid divergences it is important to properly chose the order of solutions of the equations.
In our case, we used the list of models in [17] as test. In this way we were able to keep the divergences under control and to chose the correct branch of solutions 10 . In particular, we find the following very simple conditions h 3 (v)r 10 r 5 = h 10 (v)r 3 r 7 , (6.2) h 3 (u)r 10 r 7 = h 10 (u)r 3 r 5 . (6.3) At this point, we need to distinguish two cases r 3 = 0 and r 3 = 0. In the next subsection the physical meaning of this choice will be clear.

Case r 3 = 0
The one-dimensional Hubbard model and the S-matrix of AdS 5 × S 5 superstring sigma model fall in this class of models. Similar to the Baxter condition, we find We remark that h 3 = 0 does not cause divergences, since r 3 = 0 implies that h 3 is also non-zero.
By using the regularity condition R(u, u) = P and from the definition of the Hamiltonian, we can derive that (6.8)-(6.11) impose some very simple constraints on the Hamiltonian 11 (6.14) Furthermore, differentiating two times and combining (6.8)- (6.11) in order to get rid of the second derivatives, we also got 10 Some equations in fact can be factorized in the form AB = 0, with A and B functions of the rs. We used the test to select whether the solution was A = 0 or B = 0. If one chose the models classified in [56] as test, some choices would have been different. 11 (6.12) and (6.13) were derived differentiating (6.10) and (6.11), (6.14) differentiating (6.8) two times.

6.2
Case r 3 = 0 One can check that the case r 3 = 0 and r 10 = 0 actually satisfies (6.8)-(6.11) as well. Hence, we can restrict to the case where r 10 = 0. Analyzing this case leads to a set of factorized equations . (6.16) Plugging the solutions for the hs into the Sutherland equations, it can be noticed that it is not possible to find conditions that hold in all the cases. One should consider four possible subcases separately: r 1 = 0 r 8 = 0, r 1 = 0 r 8 = 0, r 1 = 0 r 8 = 0 and r 1 = 0 r 8 = 0. The numbers of subcases can be reduced using the transformations on the R-matrix that preserve integrability, [16,17,56]. In particular, we will use 1 Local basis transformations: The subcases to be considered are then Subcase r 1 = 0, r 8 = 0 We obtain r 5 r 7 = r 2 r 9 , (6.17) r 4 r 6 r 9 = r 2 r 2 8 , (6.18) r 2 1 r 2 9 = r 2 2 r 2 8 . (6.19) These imply (6.12) on the entries of the Hamiltonian together with h 2 2 = h 2 9 , h 2 5 h 2 7 = h 2 9 .

And on the Hamiltonian
We can notice that the subcase r 1 = 0, r 8 = 0 can be recovered from r 1 = 0, r 8 = 0 by performing an off-diagonal basis transformation 12 on the R-matrix and on the Hamiltonian.
In the following section we would like to show how to rewrite the Hamiltonian to make the free fermion nature explicit.

Towards a free fermion Hamiltonian
Here we will mainly follow Section 3 and [63,64]. As already mentioned, the Hilbert space is four dimensional and is spanned by two bosons |φ 1,2 and two fermions |ψ 1,2 . We introduce two sets of canonical fermionic creation and annihilation operators c † α,j , c α,j where α = ↑, ↓ is the spin and j is the site of the chain (running from 1 to chain length L). If we denote the vacuum by |0 such that c α,j |0 = 0, then our local Hilbert space is spanned by Those oscillators satisfy the usual anti-commutation relations The su(2) ⊕ su(2) R-matrix is given by
Furthermore, we can notice that also the Hamiltonians of model 8 and 12 of [56] verify the conditions (6.12) and (6.13) and with a diagonal local basis transformation the coefficient of H (3pt) can be set to zero. In these two cases the Hamiltonians are also of free fermion type. For model 12 this is not surprising since it corresponds to the free 14 Hubbard model, [72].

Conclusions
In this paper we have shown how the new models classified in [16] satisfy the free fermion condition.
We collected some of the relevant literature which motivates how this condition relates to the fact that 14 It only contains the kinetic term. all these models should be connected to theories of free fermions. We have then elucidated the power of this condition in reformulating the algebraic structure of lower-dimensional AdS/CF T R-matrices in a novel and drastically simplified language of free fermion creation and annihilation operators, achieved by means of a suitable array of Bogoliubov transformations. Given that one of our examples is not nearest-neighbour, we conjecture that the free fermion condition must hold in an even more general setting.
We explicitly subdivided the models classified in [16] into two classes (A and B), based on the single assignment of a constant, which is non-zero for class A satisfying the so-called Baxter condition, and zero for class B satisfying the free fermion condition. Such separation was reproduced by analysis and resolution of the Sutherland equation. Moreover, we have shown how one can apply the free fermion condition to obtain a substitute to the algebraic Bethe ansatz in order to find the spectrum of 6-vertex class B models, and proceed instead with generalised inhomogeneous ansatz, from which one can also obtain homogeneous limit. We explicitly found the Bogoliubov transformation that always puts in all these cases the Hamiltonian in the manifest free fermion form, letting ourselves be guided by the intuition that comes from the traditional idea of the coordinate Bethe ansatz.
We have then applied these methods to the supersymmetric AdS 3 and AdS 2 cases in string theory, and found how they permit to rewrite known results in much more compact and suggestive forms. For the particular case of the 8-vertex B model, which is associated to the AdS 2 R-matrix, on two sites it was found what we call the pseudo-pseudo vacuum, by which we mean a vector which works just the same as the pseudo-vacuum (which such chains do not possess) and that exists only thanks to the particle-hole transformation which the free fermion condition has unveiled as possible.
As a next step, we raised the study of 16 × 16 models, which contain the AdS 5 integrable model and the Hubbard chain. We showed how a free fermion type condition for such generic models arises. We then try to recast the Hamiltonian in a form as close as possible to free fermions. One further direction, is to investigate the free fermion condition analogues for generalised AdS 4,5 integrable models and their deformations, that could also potentially be useful for understanding the higher dimensional construction of Korepanov [53,52] and its special limits.
Our findings might shed some light for example on the result obtained in [11] for the central charge of the CF T described by massless relativistic left-left and right-right R-matrices of AdS 3 with pure RR flux. Such TBA was solved exactly and revealed a mysterious inner simplicity to the model, which might be related to its intimate free fermion nature. Likewise, the relative simplicity of the scalar products of Bethe states [21] may naturally be related to the existence of a free fermion realisation 15 . Similarly, the results of [44,54] are here contextualized within the free fermion general framework, and a new light is provided on the physical significance of their occurrence.
No data beyond those presented and cited in this work are needed to validate this study.

A Coordinate Bethe Ansatz for inhomogeneous chains
The main advantage of the coordinate Bethe Ansatz is that it provides explicit wave functions, which are needed to map the system to free fermions. However, in order to deal with inhomogeneous spin-chains we need to use a slightly different version of the coordinate Bethe Ansatz that uses some of the properties of the algebraic Bethe Ansatz. In particular, it uses the R-matrix to derive the explicit form of the Bethe vectors.

Formalism
The idea is to build explicit eigenvectors of the transfer matrix using a generalization of a plane-wave type Ansatz. Consider a general transfer matrix build up out of an R-matrix R(u, v) We are interested in computing the explicit eigenvectors of T N . For inhomogeneous spin-chains, all the commuting charges that T generates are generically of range N and hence the usual coordinate Bethe Ansatz approach does not work. In order to work around this, we will derive a different nearest neighbour condition that uses the R-matrix.
Suppose we have an eigenvector |v of the transfer matrix with eigenvalue Λ By using the Yang-Baxter equation, it is easy to see that the R-matrix R i,i+1 acts like a permutation operator on the transfer matrix In other words, commuting the transfer matrix with R i,i+1 results in permuting sites i, i + 1 on the spin-chain. Now, since Λ needs to be completely symmetric in the inhomogeneities u i , we find that In particular, R i,i+1 (u i , u i+1 ) |v is an eigenvector of the permuted transfer matrix.
But, on the other hand, the permuted eigenvector |v (i,i+1) is an eigenvector of the permuted transfer matrix by definition. Thus we are naturally lead to the conditions that eigenvectors have the same transformation properties as the transfer matrix under the R-matrix and that R i,i+1 |v and |v (i,i+1) need to be proportional for some function R 0 . By then assuming a factorized Bethe-type Ansatz for the wave function we see that (A.5) actually allows us to fix the Bethe vectors.
We will now work out this procedure explicitly for a general R-matrix of 6-vertex type (3.10).
Vacuum The ferromagnetic vacuum |0 trivially satisfies (A.5), namely using the explicit form of the R-matrix (3.10) we find 6) which shows that in this case R 0 = r 1 (u i , u i+1 ).
One magnon The next step is to consider a state with a flipped spin, which takes the form This state has to satisfy (A.5) for i = 1, . . . N . Clearly, when the R-matrix does not act on the flipped spin, the state should behave like the vacuum. The easiest way to incorporate this is by making a factorized Ansatz of the form This is an obvious generalization of the usual coordinate Bethe Ansatz. Indeed setting all u i equal and taking S = e iv reproduces the well-known plane-wave form of the Bethe vectors.
We have suppressed the dependence of r i on u 1,2 . These give us a set of functional equations whose solution will depend on the explicit form of r i . In particular, the solution can be read off from the Yang-Baxter equation to be The constant γ simply determines the overall normalization of the vector and can be set to 1 without loss of generality.

Multiple magnons
In the case of multiple magnons we need to introduce an S-matrix S that deals with the exchange of two magnons. Thus, inspired by the Bethe Ansatz, we make the following Ansatz We can again restrict to two sites and derive the following equation from (A.5) where the − sign on the right hand side comes from the fact that the permutation is graded. This gives a simple linear equation for S which is easy to solve . (A.14) We see upon using the free fermion condition this simply reduces to −1 as expected. .15) In the homogeneous limit this reduces to the normal Bethe equations. If we interpret the inhomogeneities as particles with momenta, then we can further impose .16) and total momentum conservation

Bethe equations Finally, imposing periodicity leads to the following Bethe equations
However, the last two conditions are not needed for the diagonalization of the transfer matrix.
Covectors Following the exact same reasoning as above we can also determine the covectors of the transfer matrix. These are needed in the canonical transformation to free fermions. Thus, we consider the dual vacuum 0| and excited states of the form 0|c i so that For covectors we should use R −1 instead of R in our considerations. By using braiding unitarity, we see that .19) and hence we find R 0 = 1/r 1 . Let us denote the coefficients of the Bethe covectors by f * , S * and S * then an analogous computation shows that As expected, we see that the Bethe equations are the same for vectors and covectors.

B Recursion formulas for the transfer matrix
In this appendix, we shall produce recursion formulas for the transfer matrix with a generic number of sites, exploiting the (though deceptively) simple form of the R-matrix where we have renamed the operators η 1 and η † 1 in (4.35) as η 0i and η † 0i , respectively, to display the fact that the complication of the R-matrix is in fact hidden in their definition 16 .
The transfer matrix is therefore simply given by 16 Let us point out that these operators differ from the η i and η † i operators introduced in the main text, which should be the result of a global transformation involving all sites of the chain of particles, and should directly diagonalise the transfer matrix.
where we have denoted by |0 0 the state |φ in the auxiliary space 0. It is clear that one needs to find a way to compute the generic expressions 0 0 |N 01 ...N 0m |0 0 and 0 0 |c 0 N 01 ...N 0m c † 0 |0 0 (B .3) for generic m = 1, .., N to be able to calculate T N .
For these quantities we are able to find a combined system of recursion relations. Let us begin with the first one, and let us define two objects: One can easily see that the following holds true: where we have used the fact that |0 0 is a bosonic state, and we have defined This results in the following recursion relation Similar manipulations performed starting from 0 0 |N 01 ...N 0m c † 0 |0 0 bring to another relation: Given the starting points 17 with (B.6) and (B.7) one generates all the terms in (B.3) 18 . The next few terms are for instance straightforwardly obtained: By resolving the recursion system (B.6)-(B.7) in terms of suitable operators and performing reindexing one can obtain a closed system for X m and Y m , which results in for m ≥ 0, Ω 1,q = α q β q c q , Ω 2,q = β 2 q n q , Ω 3,q = α q β q c q † , Ω 4,q = α 2 q , (B.10) 17 The recursion could actually be set to start from m = 0 with initial values X 0 = 1 and Y 0 = 0, and it would automatically generate (B.8). 18 with additional 0 0 |Õ|0 0 redefinitions associated to single and mixed terms where α i , β i are constant prefactors defined in (B.5), also all terms that develop on the right hand side of (B.10) have ordered action : · :, i.e. operators c q and n q with lower q appear to the left. Alternatively, by eliminating Y m using (B.7) as (B.11) and substituting back into (B.6), we obtain a recursion only for X m (which are the objects ultimately relevant to the transfer matrix): By defining we can rewrite the recursion more compactly as Experimenting with up to m = 5 leads us to conjecture that the general solution is obtained by sprinkling block-molecules Γ ij = ψ † i ξ ij in a sea of ν's. For instance, It is also easy to see that, by further defining 16) we have that X m -which is what ultimately enters the expansion (B.2) -is obtained by sprinkling blockmolecules Γ ij in a sea of ν's. This way we have conveniently reabsorbed all the factor of −2 and the expansion (B.2) will have all coefficients equal to 1.
With analogous reasoning we can obtain the recursion relations for the other part of the supertrace.
We simply state the result: by defining we can derive (B.18) and resolving closed ordered recursions we deduce U -Z system in analogy with B.10 By again substituting into the first formula in (B.18), we obtain a recursion only for Z m which is what matters for the transfer matrix: We can therefore see that by defining we can rewrite this recursion more compactly as well in the form This is virtually the same recursion as for the X m mutatis mutandis, hence the solution is again conjectured to be obtained by sprinkling block-molecules Ψ ij ≡ ζ i ρ † ij is a sea of µ's in much the same way. Likewise, by further defining

C Examples of Hamiltonians
In this appendix, we provide two examples from AdS 3 which exemplify the general procedure outlined in the main text.

C.1.1 Open chain
In this short section we show how to diagonalise the Hamiltonian with free fermion operators for an open chain. Although for AdS 3 purposes this is not directly applicable, it is interesting to note that in the pure Ramond-Ramond case the Hamiltonian reads [26] We have verified this result on two sites using the new variables (η i , η † i ), where one needs to pay particular attention to the θ-dependence of the new operators themselves. In particular, one immediately notices that the transformation which diagonalises T 2 is not suitable to diagonalise H 2 . This can be seen in the new variables, since taking the derivative w.r.t. θ breaks up N 1 into something which ceases to be diagonal in the new basis.
Nevertheless, the Hamiltonian is Hermitian and can be diagonalised, one simply needs a different unitary transformation. It is easy to see that the map transforms the Hamiltonian on two sites to The structure of the 1-particle sub-block of the Hamiltonian is always the following: The diagonalisation of this sub-block always appears to produce the result where we have defined We conjecture that (C.8) is in fact the complete Hamiltonian: 2 sin q n 2 ψ † n ψ n , (C.10) We can make a connection with gapless spin-chains and spinon-excitations as noticed in [26] -here with quantised momenta on a finite chain, see also [63]. The form of the free fermion operators ψ n grows complicated with N , and is determined using the explicit form of the eigenvectors. The cases N = 2 and N = 3 fit into this framework, with an appropriate name-redefinition of the generators. We have verified the one-particle eigenvalue pattern up to N = 7, and the form (C.10) explicitly for the N = 4 case as well with the help of Mathematica 19 .
Notice that roughly half of the states have negative energy according to (C.10) -for instance, for even N this occurs for n = N 2 + 1, ..., N . For these states, we need to perform a particle-hole transformation in the standard fashion to have all particles and antiparticles with positive massless dispersion relation ǫ(q) = 2 sin q 2 . (C.12)

C.1.2 Closed chain
It is easy to provide an example of the closed-chain treatment performed in the text in the AdS 3 case as well. The closed chain has two extra terms corresponding to H N,1 , and the appropriate transformation is exposed in the text. For instance, for three sites The unitary map for general N is given by (C.14) 19 In the case of N = 4, the unitary transformation reads We have used the commutation relations of the fermionic operators to combine the two terms into one, with the contribution from the 1 in the commutator vanishing due to a complete sum of roots of unity.
The result can be generalised to arbitrary N based on the properties of the roots of unity as exploited in the main text, revealing again a spectrum similar to the one of the open chain: We have explicitly tested these particular formulas up to N = 4.

C.2.1 Open chain
The open-chain mixed flux case seems to work as well, thanks to the fact that R(0) equals the graded permutation in this case too, however the complication of the functional form of the R-matrix forces us to stop much earlier in N . Let us just treat the two cases which we can manage with Mathematica at the moment.
For N = 2 the Hamiltonian, obtained with the same method as in the previous section (and with a convenient normalisation), reads where we have defined ω ≡ π 2k .
(C. 20) We remind that the external parameter k = 2, 3, ..., is a fixed natural number which measures the mixture of the fluxes in the string theory 20 . The Hamiltonian H 2 is Hermitian and more similar to the one studied in [63], except for the cos 2ω term.
In order to get some feeling of how the Hamiltonian work we can show the matrix in the one-particle sector: 24) In fact, in general the one particle sector will look like this: In the case of N = 3 we can show with the help of the computer that the orthogonal transformation Proceeding with general k appears to be rather involved. Only the case k = 2 (which is closely related to the properly-regularised k = 1 case [28]) simplifies drastically. In fact, with a suitable normalisation, the full mixed-flux Hamiltonian H N at k = 2 reduces to a special case of the one diagonalised in [63] (it corresponds to setting the parameter γ of [63] to 0, and mapping N here to N − 1 in [63]).
General N Solving it for general N goes as follows. First, since the Hamiltonian is quadratic in the fermionic oscillators c and only contains terms of the form c † c, it is enough to restrict to the one-particle sector. Diagonalizing the Hamiltonian then gives a linear map between the standard basis vectors c † i |0 and the eigenvectors which we will denote by η † i |0 . The same holds true for the covectors 0|c i and 0|η i . By construction, the Hamiltonian then takes the form 27) where τ i correspond to the solutions of P τ = sin(N + 1)τ sin τ − 2 cos 2ω sin N τ sin τ + cos 2 2ω sin(N − 1)τ sin τ = 0. (C. 28) It is not hard to see that this can be rewritten as a polynomial equation in cos τ which has exactly N solutions. In fact, by definition of the Chebyschev polynomials of the second kind U m (cos τ ) ≡ sin(m + 1)τ sin τ , (C. 29) we see that (C.28) identifies the zeroes of a combination of Chebyschev polyomials with highest degree N : P τ = U N (cos τ ) − 2 cos 2ω U N −1 (cos τ ) + cos 2 2ω U N −2 (cos τ ) = 0. (C.30) Using the recursion between different U m 's we may also write the above as The minus sign which implies negative energy for half the states can again be dealt with by a standard particle-hole transformation [63].

C.2.2 Closed chain
In the case of the closed chain the problem simplifies drastically and one is reduced to the general theory we have developed in the main text, since the Hamiltonian simply becomes The very first term is proportional to the identity and simply shifts the energies of 2 cos 2ω. The remaining non-diagonal part is easily diagonalised if we simply notice that it is very similar to the pure Ramond-Ramond case of section C.1.2, except for a relative minus sign between the two main terms (and no multiplying factor of i). Namely, if we take for instance the N = 3 case, we now have, with the very same eigenvectors, that with the exact same map (C.14) still holding.