Some algebraic aspects of the inhomogeneous six-vertex model

The inhomogeneous six-vertex model is a 2D multiparametric integrable statistical system. In the scaling limit it is expected to cover different classes of critical behaviour which, for the most part, have remained unexplored. For general values of the parameters and twisted boundary conditions the model possesses ${\rm U}(1)$ invariance. In this paper we discuss the restrictions imposed on the parameters for which additional global symmetries arise that are consistent with the integrable structure. These include the lattice counterparts of ${\cal C}$, ${\cal P}$ and ${\cal T}$ as well as translational invariance. The special properties of the lattice system that possesses an additional ${\cal Z}_r$ invariance are considered. We also describe the Hermitian structures, which are consistent with the integrable one. The analysis lays the groundwork for studying the scaling limit of the inhomogeneous six-vertex model.

field [12][13][14]. Perhaps the simplest inhomogeneous six-vertex model corresponds to a staggering of the variables associated with the vertical lines of the lattice such that η 1 = η 3 = . . . , η 2 = η 4 = . . . , while η 1 = η 2 and a similar staggering for the variables assigned to the horizontal lines. In the case when η 1 /η 2 > 0 and |q| = 1, with a properly defined scaling limit where η 1 /η 2 → 0 + (or +∞) and N → ∞, the lattice model exhibits universal behaviour that is described by the massive Thirring/sine-Gordon model [15]. The situation is different if the ratio of η 1 to η 2 is negative. For η 1 /η 2 = −1 and |q| = 1 the system is critical and its important feature is the presence of a Z 2 symmetry. For this reason the model is sometimes referred to as the Z 2 invariant inhomogeneous six-vertex model. Two types of universal behaviour occur depending on whether arg(q 2 ) ∈ (0, π 2 ) or ( π 2 , π). The former case was the subject matter of the works [16][17][18][19][20][21][22]. It was originally observed in refs. [16,17] that there is a continuous component in the spectrum of conformal dimensions for the model. In ref. [22] it was pointed out that the algebra of extended conformal symmetry underlying the critical behaviour is the W ∞ algebra.
The identification of the critical surfaces in the space of couplings of the inhomogeneous six-vertex model as well as a description of the corresponding universality classes is, without doubt, an interesting problem. However, apart from the few cases mentioned above not much work has been done in this direction. An evident starting point to approach the problem would be to explore the restrictions on the parameters which give rise to extra symmetries in the model. The latter would include translational invariance as well as global involutions such as C, P and T , which are common attributes of a local QFT. There may also be other global symmetries, e.g., the Z 2 invariance of the staggered model with η J /η 1 = (−1) J−1 . Of course, the identification of the universality classes requires more than just an analysis of the symmetries. The integrability of the model makes possible a detailed quantitative study of the scaling limit. The goal of this work is to describe the integrable structure underlying the general inhomogeneous six-vertex model as well as its interplay with the global symmetries.
The paper is organized as follows. In sec. 2 we set-up the notation and give a short summary of the results of [8]. We also present the construction of the transfer matrix as well as its eigenstates (Bethe states) within the framework of the QISM. The commuting family for the inhomogeneous six-vertex model, apart from T(ζ), contains other important members -the so-called Baxter Q -operators [9]. Their construction along the lines of refs. [23,24] is presented in sec. 3.
Sec. 4 is devoted to a discussion of the "charge" conjugation C, parity inversion P and time reversal transformation T . We describe the conditions under which these conjugations are consistent with the integrable structure in the sense that they preserve the family of commuting operators.
For a generic set of the inhomogeneities the transfer matrix and Q -operators are not Hermitian w.r.t. the standard matrix Hermitian conjugation. In sec. 5 a family of Hermitian structures is introduced, which are also consistent with the integrable structure of the model. The norm of the Bethe states associated with these Hermitian structures can be computed via the formula originally conjectured by Gaudin, McCoy and Wu [25] for the homogeneous case and then extended and proven, using the formalism of the QISM, for the general inhomogeneous model by Korepin [26].
Further restrictions on the parameters of the model, which lead to the presence of additional global symmetries, are discussed in sections 6 and 7. Among these is the invariance w.r.t. lattice translations. The latter appears when the number of columns of the lattice is divisible by some integer r ≥ 1 and the inhomogeneities η J satisfy the r-site periodicity conditions η J+r = η J . In this case the commuting family of the model contains r Hamiltonians, which are given by a sum of local operators. By the latter we mean that they are built from the local spin operators acting on r + 1 consecutive sites of the lattice. Sec. 7 discusses the lattice system with the {η J } being further restricted such that η J /η 1 = e 2πi(J−1)/r . Then the model possesses an extra Z r cyclic group symmetry. The last section contains a collection of explicit formulae for the model with r = 1 (homogeneous) and r = 2 (Z 2 invariant). They lay the groundwork for a detailed analysis of the scaling limit for these two cases, which is performed in ref. [27].

The inhomogeneous six-vertex model
Consider the square lattice with N vertical columns. Each edge of the lattice can be in one of two states, labeled as ±1. Introduce the R-matrix of the six-vertex model, whose non-zero entries are given by where ζ and q are arbitrary parameters. Let a 1 , a 2 , . . . , a N denote the states of a row of N vertical edges of the lattice and b 1 , b 2 , . . . , b N be the states on the row above. The elements of the row-to-row transfer matrix of the integrable inhomogeneous six-vertex model is defined as Here we have imposed twisted boundary conditions depending on the parameter ω, while η 1 , η 2 , . . . , η N are arbitrary parameters controlling the "inhomogeneity" of the model. The prefactor q − N 2 has been introduced for convenience, to ensure the normalization condition (2.14) below. With the following graphical representation for the R-matrix the transfer matrix is depicted in fig. 1.
The transfer matrix is an operator acting in the "quantum" space, which is formed by the direct product of N two-dimensional spaces Its diagonalization problem was solved by Baxter [8] via the coordinate Bethe ansatz. The eigenvectors with M down spins have the form where the (unnormalized) Bethe ansatz wave function reads as with suitably chosen coefficients AP and functions φ m (x). The vector | 0 denotes the state with all spins up, and σ z J , σ ± J ≡ 1 2 (σ x J ±iσ y J ) stand for the Pauli matrices that act in the J-th factor of the tensor product (2.4). The summation is taken over all M ! permutationsP of the integers (1, 2, . . . , M ). Note that the number of down spins M is simply related to the eigenvalue S z of the z-component of the total spin operator which commutes with the transfer matrix.
To within an overall normalization of the wave function (2.6) the results of [8] can be summarized as follows: where the set of complex numbers {ζ j } M j=1 satisfies the system of algebraic equations and the functions φ m (x) in (2.6) have the form The corresponding eigenvalue of the transfer matrix, is given by 1 It is easy to see from the definition (2.2) that the transfer matrix is an N -th order polynomial in the variable ζ satisfying the conditions Since a multiplication of all the η J by the same factor can be absorbed into a redefinition of the spectral parameter ζ, we will always assume The Bethe state (2.5) may also be constructed within the framework of the QISM [10,11]. Introduce the so-called monodromy matrix where R m is given by (2.1), but regarded as a two by two matrix whose elements act in the m-th factor of the tensor product (2.4). It is convenient to denote the entries of the monodromy matrix as With the above notations, the transfer matrix (2.2) is given by Note that the following remarkable formula holds true: where we use the notation This formula with all η J = 1 was originally conjectured by Gaudin, McCoy and Wu in [25]. Its generalization and proof for the inhomogeneous case was worked out by Korepin in [26].

Q -operators
The eigenvalues of the transfer matrix are polynomials in the variable ζ. Indeed, its matrix elements defined through (2.2) and (2.1) are polynomials of ζ and its eigenvectors (2.5)-(2.11) do not depend on this variable. The latter also implies that the matrices T(ζ) with different values of ζ mutually commute. It turns out that the model possesses a larger commuting family which, together with the transfer matrix, contains other important members -the so-called Baxter Q -operators [9]. This section gives an overview of their construction as well as their main properties.
First note that, introducing the functions one can rewrite the expression for the eigenvalues T (ζ) (2.13) in the form Since T (ζ) is a polynomial, the r.h.s. of the last relation must vanish at the zeroes of A + (ζ), i.e., when ζ = ζ m , so that These are precisely the Bethe ansatz equations (2.10). The functional relation (3.2) implies the existence of an operator A + (ζ), with eigenvalues A + (ζ) (3.1), that commutes with the transfer matrix T(ζ) and satisfies the operator version of (3.2). 2 Actually there are two such operators, which we denote as A ± (ζ), that satisfy the operator relations of this type where for convenience we have used the notation Note that eq.(3.4) can be rewritten in a way so that it has the same form for the "+" and "−" cases. Introducing the operators Q ± (ζ), which differ from A ± (ζ) by simple factors involving fractional powers of ζ, The latter is the famous Baxter "T Q -relation" [9]. We prefer to work with the operators A ± (ζ) as their matrix elements and eigenvalues are polynomials in ζ.
Similar to T(ζ), which can be expressed as a trace of a 2 × 2 matrix (2.20), the operators A ± (ζ) are also constructed as traces of certain monodromy matrices. However, this time the trace is taken over infinite dimensional representations of the so-called q -oscillator algebra. The latter arises as an evaluation representation of the Borel subalgebra of the quantum affine algebra U q ( sl 2 ). The required R-matrices (see (3.9) below) are obtained by a suitable specialization of the universal R-matrix [29]. Originally these calculations were performed in the context of the quantum KdV theory [23,24], but the same procedure can be readily applied to the inhomogeneous six-vertex model [30].
The q -oscillator algebra is generated by the elements H and E ± satisfying the commutation rela- (3.8) The basic building blocks for the construction of A ± are the 2 × 2 matrices which act in the m-th component of the tensor product (2.4) and whose entries involve the formal generators (3.8). Let ρ ± be representations of the q -oscillator algebra such that the traces exist and are non-vanishing. Then, using the notation (3.5) and assuming that the twist parameter ω is such that one can introduce the operators that act in the quantum space V N (2.4). With these preparations, define It is easy to see that such operators commute with S z and, therefore, with Z ± . As was first pointed out in [23], the (normalized) trace in (3.13) is completely determined by the commutation relations (3.8) and the cyclic property of the trace, so that the specific choice of the representations ρ ± is not significant as long as the traces (3.12) exist and are non-vanishing. This way it is possible to define A ± (ζ) ∈ End(V N ) for all complex ω (except some discrete set of isolated points, see below) through analytic continuation, despite that the definition (3.13) applies literally to the case (3.11) only.
Following the line of [24], one can show that the operators A ± (ζ) commute with the transfer matrix and among themselves for different values of the spectral parameter: (3.14) Furthermore they satisfy a number of important operator valued relations, which are derived algebraically from the decomposition properties of products of representations of the q -oscillator algebra. This includes the quantum Wronskian relation as well as As a simple corollary of the above two formulae, the operators A ± satisfy (3.4).
From the definition (3.13) it follows that the eigenvalues of A ± (ζ) are polynomials Their roots ζ (±) m are determined by the Bethe ansatz equations where the operators A (∞) ± act in V N (2.4) and their eigenvalues on the Bethe state (2.21) read as It should be pointed out that, defined through eq. (3.13), the operators A ± (ζ) obey the normalization condition A ± (0) =1. However, in the subspace with fixed S z and when ω = ±q −S z +m with m = ±1, ±2 . . . the Bethe roots {ζ m } for some of the eigenvalues of A ± may become zero. In this case, as it follows from (3.17), the normalization imposed by (3.13) is not suitable and that formula requires modification. As our considerations are not sensitive to this subtlety, we will continue to use (3.13) and assume that (3.17) always holds.
Finally note that for S z = 0 both A ± (ζ) tend to the same operator A(ζ) as ω → 1. A careful taking of this limit in eq. (3.13) leads to a generalization of Baxter's formula for the matrix elements of the Q -operator of the zero-field six-vertex model in the sector with S z = 0: In the homogeneous case, when all the η J = 1, the last formula reduces to eq. (101) of ref. [31]. Note that the operator (3.23) is still normalized as A(0) = 1, which may not be immediately obvious.

C, P and T conjugations
So far we have briefly reviewed the formal algebraic aspects of the diagonalization problem for the commuting family of operators in the general inhomogeneous six-vertex model. All the parameters {η J }, q and ω were assumed to be generic complex numbers. Though the constraint (2.15) was imposed, it does not reduce the generality, as it can always be achieved via a redefinition of the spectral parameter ζ. In this section we introduce three global conjugations, which are similar to the charge conjugation C, parity inversion P, and time reversal transformation T from quantum field theory. We describe the restrictions on the parameters such that the global involutions are consistent with the integrable structure, in the sense that they preserve the family of commuting operators.
Let us first consider parity inversion. This transformation may be defined via the following adjoint action on the local spin operatorŝ where | 0 stands for the pseudovacuum (2.7). 3 Its matrix elements read as In order forP 2 =1, one must require that Then the action of the parity conjugation on the transfer matrix and A ± (ζ) is given bŷ where we explicitly indicate the dependence on the twist parameter ω. These relations may be deduced directly from the definition of T(ζ) and A ± (ζ). For instance, the first equality follows from (2.20) and (2.16) as well as the simple property where R m (q ζ) t stands for the transpose of the 2 × 2 operator valued matrix (2.17). Since T(ζ | ω) and A ± (ζ | ω) do not commute amongst themselves for different values of the twist parameter ω, formula (4.5) shows that unless ω 2 = 1, the parity inversion does not preserve the commuting family of operators.
The situation is similar for the charge conjugation. Its generator may be introduced aŝ whose adjoint action on the local spin operators is given bŷ For arbitrary complex values of the inhomogeneities one haŝ The proof of the first equality is based on the relation Though the C and P transformations do not respect the integrable structure themselves for ω 2 = 1, by combining them together one getŝ provided that the conditions (4.4) are imposed. Here all the original and transformed operators correspond to the same value of ω so that they commute with each other and have the same set of eigenvectors. This has an important consequence. Since the CP conjugation intertwines the sectors with S z and −S z it is always possible to focus on the case with S z ≥ 0. Further, the second line of eq. (4.11) implies that for the construction of the Bethe states (2.21) in the sector with arbitrary S z it is sufficient to consider the operator A + (ζ). For these reasons we focus on the zeroes of the eigenvalues of A + and use the notation Up to this point there was no essential need to impose any reality conditions on the parameters of the model. However, for the time reversal transformation, the reality conditions become crucial. Having in mind applications to the Z 2 invariant model studied in [16][17][18][19][20][21][22]27], we will assume that q and ω are unimodular and parameterize them as and The R -matrix then obeys the property (4.14) As for the inhomogeneities, three cases will be considered: (i) All η J are unimodular complex numbers satisfying (4.4): (ii) All the inhomogeneities are real: In the third case we take the number of columns in the lattice to be even and divide the inhomogeneities into two groups: J=1 are taken to have the same absolute value Λ > 0, while for the other group |η N/2+1 | = . . . = |η N | = Λ −1 . The arguments of η J and η N/2+J are chosen to be the same J=1 are unimodular complex numbers. Notice that the constraint (4.4), which is always assumed, implies The time reversal transformation is realized as an anti-unitary operator acting on an arbitrary state ψ ∈ V N asT ψ =Û ψ * , (4.19) where the asterisk ( * ) stands for complex conjugation. The matrixÛ satisfies the condition which ensures thatT 2 =1. For the first case (4.15) we set the matrixÛ to be Then it follows thatT i.e., defined in this way the time reversal transformation preserves the integrable structure of the model similar to CP. Combining all three transformations (4.9), (4.5) and (4.22) together, yieldŝ For the Bethe states, the above equation together with (3.1) implies that the sets {ζ m } corresponding to Ψ andĈPT Ψ are complex conjugate to each other. Notice that the phase assignment in (2.11) and consequently in (2.19) has been chosen in such a way that In the second case (ii) it is also possible to introduce the time-reversal symmetry in such a way that relations (4.22) are preserved. To proceed we will need to establish how the transfer matrix and, more generally, the monodromy matrix M (ζ | η N , η N −1 , . . . , η 1 ) (2.16), behaves under an interchange of the inhomogeneities. The eigenvalues of the transfer matrix (2.13) depend on {η J } both explicitly and implicitly via the Bethe roots which satisfy eq. (2.10). In both cases the inhomogeneities enter only through symmetric combinations, so that the eigenvalues are symmetric functions of them. Therefore two transfer matrices obtained from each other by a permutation of the same set of inhomogeneities must be connected by a similarity transformation in the quantum space V N . In fact, the same holds true for all the entries of the monodromy matrix as well. The group of permutations of the ordered set (η J ) is generated by elementary permutations of neighbouring pairs: (η N , . . . , η n+1 , η n , . . . , η 1 ) → (η N , . . . , η n , η n+1 , . . . , η 1 ) .

Hermitian structure
There are many ways to introduce the Hermitian structure, i.e., the sesquilinear form for the states and the Hermitian conjugation of operators in the 2 N dimensional linear space V N (2.4). Here we discuss the Hermitian structures that are consistent with the integrable structure of the model.
Consider each factor in the product in the r.h.s. of eq. (2.16). Under the standard matrix ("dagger") conjugation in the quantum space V N , they transform as In turn, taking the † -conjugation of the monodromy matrix results in Now suppose that the (non-ordered) set of inhomogeneities coincides with the complex conjugated set: In particular, this property holds for all the three cases (4.15), (4.16) and (4.17) considered above. Then there exists a similarity transformation such that Combining the latter with eq. (5.2), yields with the Hermitian matrixX given bŷ Formula (5.6) implies that, in general, the transfer matrix is not Hermitian w.r.t. the † -conjugation. However, for an arbitraryÔ ∈ End(V N ), one can introduce the non-standard conjugation aŝ Since the matrixX is Hermitian (5.7), it is guaranteed that Ô ‡ ‡ =Ô so that (5.8) indeed defines a conjugation. Then eq. (5.6) can be rewritten as the ‡ -conjugation condition for the operator valued entries of the monodromy matrix (2.18): In turn, these relations imply T(ζ) ‡ = T(ζ * ) .
In a similar way, one can show that As a matter of fact, there is a more general version of (5.8) for which the condition similar to (5.10) holds. Indeed consider the conjugation of the form In order to fulfill the requirement Ô ⋆ ⋆ =Ô, the operator Y should be such that If, in addition, Y, A ± (ζ) = 0 , (5.13) then it follows from (5.10) that 14) The ‡ -conjugation is a special case of the ⋆ -conjugation with Y =1.
For a given Y ∈ End(V N ) satisfying (5.12) and (5.13) there is a unique sesquilinear form ψ 2 , ψ 1 ⋆ ( ψ 1,2 ∈ V N ) that is compatible with the ⋆ -conjugation (5.11). It is defined by the requirement and the overall normalization condition for the pseudovacuum Ψ 0 ≡ | 0 (2.7): The second equality in (5.14), together with the fact that the spectrum of the operators A ± (ζ) is expected to be non-degenerate for generic values of k, implies that the Bethe states (2.21) satisfy an orthogonality condition of the form where Y (ζ 1 , . . . , ζ M ) stands for the eigenvalue of the operator Y on the Bethe state Ψ.
One can construct a variety of matrices Y obeying eqs. However, though the Hermitian structures corresponding to Y and Y ′ (5.21) are equivalent in the formal linear algebra sense, the action of the conjugation (5.11) on the local spin operators could be radically different. In particular, a conjugation that acts locally on σ A J may become highly non-local upon the substitution Y → Y ′ .
The above discussion is valid for any set of inhomogeneities provided that (5.4) is satisfied. However, the Hermitian matrixX (5.7), which was left unspecified, depends on the precise form of the reality conditions imposed on {η J }. For the three cases from the previous section one has: (i) For unimodular inhomogeneities η * J = η −1 J = η N +1−J , the matrixŜ (5.5) coincides withŜ 2 from (4.28). HenceX is given by (ii) In the case with η J real (4.16), the matrixŜ is the identity andX reads as Then the matrixX is given by

Lattice translation symmetry
Translational invariance is a fundamental symmetry of a local quantum field theory. Having in mind the study of the universality classes for the inhomogeneous six-vertex model, it is natural to focus on the lattice, where the number of columns N is divisible by some integer r, while the inhomogeneities satisfy the r -site periodicity conditions (here by definition we take η J+N ≡ η J ). With these restrictions, the commuting family contains a set of operators that play a central rôle for the description of the critical behaviour. The aim of this section is to introduce these operators and discuss their basic properties.

r -site translation operator
The one -site translation operator may be defined without any restrictions imposed on the parameters of the model. Taking into account the quasi-periodic boundary conditions with twist parameter ω = e iπk , its matrix elements are given by 3) The action of the one -site translation on the local spin operators reads aŝ In turn, the adjoint action of K on the transfer matrix (2.20) results in a cyclic permutation of the inhomogeneities:K This relation shows that with the conditions (6.1) and (6.2) imposed, the r -site translation operator belongs to the commuting family, In fact K not only commutes, but it can be expressed in terms of the transfer matrix. Assuming the normalization (2.14) for T(ζ) and taking into account (2.15), it's straightforward to show It follows that the eigenvalue of the r -site translation operator on the Bethe state (2.21) is given by where A + (ζ) is the same as in (3.1).

Quasi-shift operators
In ref. [18], which was devoted to the study of the Z 2 invariant model, the authors introduced the so-called quasi-shift operator. It turns out to be a key player for the description of the scaling limit. Similar operators may be defined for the general model provided the restrictions (6.1) and (6.2) are imposed. They are likewise expected to be important for analyzing the critical behaviour.
The construction of the quasi-shift operators is based on the following observation. When the restrictions (6.2) on the η J are imposed, the cyclic permutation of the inhomogeneities within each group (η J+r , η J+r−1 , . . . , η J+1 ) → (η J+1 , η J+r , . . . , η J+2 ) with J = ℓ (mod r) is equivalent to an overall cyclic permutation of the ordered set (η N , η N −1 , . . . η 1 ). In view of eq. (4.25), this implies that there exists r inequivalent operatorsD (ℓ) , such that Explicitly, they are given bŷ where the outer product is unordered since the factors inside it commute with each other. It is worth mentioning that the operatorD Combining eqs. (6.5) and (6.9) it is easy to see that the operators K (ℓ) =D (ℓ)K (ℓ = 1, 2, . . . , r) (6.13) belong to the commuting family, K (ℓ) , T(ζ) = 0 . (6.14) Moreover, their product coincides with the r -site translation: We will refer to K (ℓ) as the quasi-shift operators. 4 Similar as in eq. (6.7), it is possible to express them in terms of the transfer matrix as Their eigenvalues for the Bethe state (2.21) are given by with A + (ζ) from (3.1).

Hamiltonians
In the case of the homogeneous six-vertex model the transfer matrix commutes with the spin 1 2 Heisenberg XXZ Hamiltonian. A similar important property holds true for the model, where the parameters η J satisfy the periodicity conditions (6.2) with any r ≥ 1. Namely the corresponding commuting family contains spin chain Hamiltonians that are given by a sum of terms, each of which is built out of local spin operators from r + 1 consecutive sites of the lattice. There are r such Hamiltonians and, in terms of the row-to-row transfer matrix, they are expressed as A calculation shows that whereĤ (ℓ) ∈ End V N are built from the σ A J with J = ℓ, ℓ + 1, . . . , ℓ + r. Assuming that L ≥ 2 so that the sites ℓ and ℓ + r are not identified with one another, the operatorsĤ (ℓ) are given bŷ (6.20) Here we use the notationĴ n+1,n (ζ) = i ζ ∂ ζ log Ř n+1,n (ζ) (6.21) withŘ n+1,n as in (4.26).

Interplay with CP, T and Hermitian conjugation
In order to incorporate the C, P and T conjugations for the model with translational invariance, extra constraints need to be imposed on the parameters in addition to (6.2). For the three cases considered in sec. 4, a brief examination shows that the first and second ones .
The Hermitian conjugation (5.11) that is consistent with the integrable structure involves the matrixX, which depends on the reality conditions imposed on the set η * J N J=1 = η J N J=1 . In the case (i), the expression (5.23), (4.28) can be simplified if the r -site periodicity conditions (6.2) are taken into account. It turns out that Notice that each of the terms K mX(1) K −m entering into the product expression (6.25) acts nontrivially only on sites mr + 1, mr + 2, . . . , mr + r. For case (ii) from sec. 4,X is the diagonal matrix given by eq. (5.24).
The formula for the ⋆ -conjugation also contains the operator Y, subject to the conditions (5.12) and (5.13). Independently of its choice T(ζ) ⋆ = T(ζ * ) from which it follows that case (i) :

Z r -invariance
The operatorsD (ℓ) ∈ End V N , acting on the transfer matrix, result in a cyclic shift of the inhomogeneities as in eq. (6.9). Taking into account that η J+r = η J , after r consecutive applications of this transformation the inhomogeneities will return to their original order. Thus D (ℓ) r commutes with T(ζ). It turns out that if the inhomogeneities are specified to be then further D (ℓ) r =1 ( ℓ = 1, 2, . . . , r ) .
In this case all the operatorsD (ℓ) can be expressed in terms of m+1,m e −2iπm/r (7.3) and the one -site translation operator K (6.3). Namely, When the condition (7.1) is imposed, both operatorsD andK preserve the commuting family. Their adjoint action on the transfer matrix and A ± (ζ) is given by the similar formulaê Among others, these imply that With H (6.28) taken as the Hamiltonian, the system possesses Z r invariance (in the usual quantum mechanical sense) generated by the operatorD :D r = 1. It deserves to be mentioned that in this case eq. (6.12) becomesD −1 M (ζ)D = M e +2πi/r ζ . (7.9) With the latter at hand and taking into account thatD | 0 = | 0 , it is straightforward to check that the action ofD on the Bethe state (2.21) is given bŷ To summarize, when the {η J } are fixed to be as in (7.1) and with |q| = |ω| = 1, the inhomogeneous six-vertex model, together with the U(1) symmetry generated by the operator S z , possesses a set of discrete global symmetries whose generators areD,K,ĈP andT . The latter obey the commutation relationsĈPDĈP (7.11) Having discussed some general aspects of the lattice system, we now turn to the two simplest cases -the homogeneous and Z 2 invariant models. The accent will be placed on the properties specific to these models.

Homogeneous six-vertex model
In the homogeneous case all the parameters {η J } are equal and can be set to one: Then it turns out that the operators A ± (ζ) with ζ real are Hermitian w.r.t. the standard Hermitian matrix conjugation: The latter follows from the general relation (5.10b) and the fact that the matrixX (6.25), entering into the conjugation condition (5.8), becomes the identitŷ Since the sesquilinear form corresponding to the † -conjugation is positive definite, the eigenvalues of A ± (ζ) are polynomials in ζ with real coefficients. In consequence the set of zeroes of A + (ζ), The normalization of the Bethe state Ψ (2.21) is different to that from ref. [25]. In the latter, the authors consider with the wavefunction being Here AP is the same as in (2.9) and e ipm = 1 + qζ m q + ζ m . (8.9) Notice that the function φ m (x) (2.11) with η J = 1 may be written as This way the relation between the Bethe state defined through (2.5)-(2.11) and Ψ ′ reads The norm of the Bethe state Ψ ′ (8.7) w.r.t. the positive definite inner product is given by The remarkable formula for this norm, originally conjectured by Gaudin, McCoy and Wu in [25] and proven in the work of Korepin [26] can be written as At the same time Ψ ′ 2 ≡ Ψ ′ , Ψ ′ ⋆ for the ⋆ -conjugation given by (5.11) with Y =1. This way, with Y as in eq. (8.15), when written explicitly in terms of the local spin operators, would result in a highly cumbersome expression.
Finally in the case of the homogeneous model, which corresponds to r = 1 in (6.2), there is one quasi-shift operator that coincides with the one -site lattice translation: The Hamiltonian H (1) (6.18) is just the XXZ spin chain Hamiltonian subject to the quasi-periodic boundary conditions The model corresponds to the case when the number of lattice columns N is even and the inhomogeneities are given by eq. (7.1) with r = 2:

Bethe ansatz equations
For the Z 2 invariant model, the algebraic system (2.10) reduces to The parameters q and ω are assumed to be unimodular, i.e., In refs. [22,27] the parameter γ is swapped for n, such that

Hamiltonians
Let's denote by H (+) = H (1) and H (−) = H (2) the Hamiltonians defined by eq. (6.18) with r = 2. It is straightforward to derive the following expressions: and 28) 5 The transfer matrix T(β) and the Hamiltonian defined in eqs. (5) and (2) of ref. [22], respectively, coincide with where T(ζ) is given by (2.2), H is as in (8.31) below, the matrixV = N m=1 exp iπ 4 σ z 2m−1 , while ζ = e −2β . Figure 2: A graphical representation of the matrix elements (B) bN bN−1...b1 aN aN−1...a1 of the quasi-shift operator (8.34) for the chain of length N = 6. Summation over the spin indices assigned to internal edges is assumed. The black and white dots correspond to ω c and ω −a6 , respectively. Note that, since quasi-periodic boundary conditions are imposed, c and c ′ should be identified. where The eigenvalues of these operators for the Bethe state (2.21) are expressed through the set {ζ m } as .

(8.30)
The Hamiltonian H = H (+) + H (−) (6.28), in terms of the local spin operators, takes the form Notice that H (±) and H are given by a sum of terms, where the local spin operators couple with their nearest and next-to-nearest neighbours only.

Quasi-shift operators
For the Z 2 invariant model there are two quasi-shift operators K (+) ≡ K (1) and K (−) ≡ K (2) (6.16), whose product is equal to the two -site translation operator In this case it is convenient to define (see also footnote 4) A computation based on eqs. (6.10) and (6.13) shows that This implies that B coincides with the transfer matrix of a homogeneous six-vertex model on the 45 •rotated square lattice with quasi-periodic boundary conditions (see fig. 2). Its eigenvalues are given by

Discrete symmetries
The generators of the C, P and T conjugations are given by the formulae (4.7), (4.3) and (4.21), respectively, with η J = i (−1) J−1 . Their adjoint action on the Hamiltonians H (±) and quasi-shift operators K (±) reads asĈP while the similar relations for H and K were already quoted in eq. (6.29). Also it is easy to see that The lattice system possesses an additional Z 2 symmetry, whose generator is defined by eq. One can show that the adjoint action ofD on the local spin operators is given bŷ Here Σ z stands for the diagonal matrix Σ z = σ z N ⊗ σ z N −2 ⊗ · · · ⊗ σ z 2 . (8.44) Note that the Hermiticity ofX follows from the relation Σ zD † Σ z = e iπ(S z −N/2)D , (8.45) which can be easily verified using (8.38).
Unlike the homogeneous case, the matrixX is non-trivial. Among others, this implies that the operators T(ζ) and A ± (ζ) are not Hermitian w.r.t. the standard matrix Hermitian conjugation. Of the Hermitian structures consistent with the integrable structure, a special rôle is played by the one defined through the ⋆ -conjugation (5.11) with

Conclusion
The subject matter of this paper is the integrable inhomogeneous six-vertex model on the square lattice. The discussion is focused on various algebraic properties of the model, which are important for studying the scaling limit.
We summarized the results concerning the diagonalization problem for the transfer matrix. Explicit formulae for the Baxter Q -operators were presented as special transfer matrices associated with the infinite dimensional representations of the q -oscillator algebra.
The C, P and T conjugations were introduced, and we described the constraints on the parameters of the model such that these are consistent with the integrable structure. The special features of the model possessing translational invariance were discussed. Among these, it was pointed out that the commuting family contains a set of quasi-shift operators K (ℓ) as well as Hamiltonians H (ℓ) . The latter are distinguished in that they are given by a sum of "local" operators. We also formulated the conditions for which the translationally invariant model possesses an extra Z r global symmetry and described its basic properties.
Both H (ℓ) and K (ℓ) are expected to play a key rôle for the study of the critical behaviour. The Hamiltonians are sparse matrices (most of their elements are vanishing) and there exist efficient algorithms for finding the eigenvectors and eigenvalues belonging to the low energy part of the spectrum. For a given eigenvector, it is straightforward to compute the eigenvalue of the Q -operators and, in turn, obtain the corresponding solution of the Bethe ansatz equations. In our subsequent paper [27] this approach is used for analyzing the scaling limit of the homogeneous and Z 2 invariant six-vertex models. The explicit formulae collected in the last section form the starting point of that work.
For the case when both the anisotropy and twist parameters are unimodular, |q| = |ω| = 1, as well as when the set of inhomogeneities {η J } coincides with the complex conjugated set {η * J }, a family of Hermitian structures were introduced, which are consistent with the integrable one. Again, this analysis forms the preliminary set up for the study of the scaling limit of the Hermitian structures performed in [27].
Finally note that here we have only considered the square lattice models. More generally, one could define solvable inhomogeneous models on arbitrary planar lattices formed by intersecting straight lines [32]. Some of these are related to the quantizations of circle patterns [33], arising in the context of a discrete counterpart of the Riemann mapping theorem [34]. It would be interesting to explore the scaling limit of these constructions and their connection to CFT.