Factorizing $F$-matrices and the XXZ spin-1/2 chain: A diagrammatic perspective

Using notation inherited from the six-vertex model, we construct diagrams that represent the action of the factorizing $F$-matrices associated to the finite length XXZ spin-1/2 chain. We prove that these $F$-matrices factorize the tensor $R^{\sigma}_{1... n}$ corresponding with elements of the permutation group. We consider in particular the diagram for the tensor $R^{\sigma_c}_{1... n}$, which cyclically permutes the spin chain. This leads us to a diagrammatic construction of the local spin operators $S_i^{\pm}$ and $S_i^{z}$ in terms of the monodromy matrix operators.


Introduction
The idea of twisting in quantum groups [1,2,3] was applied in [4] in the context of the algebraic Bethe Ansatz, where in this case the twist is represented by an F -matrix F 1...n which acts in the tensor product of quantum spaces V 1 ⊗ · · · ⊗ V n and satisfies the factorizing equation where σ ∈ S n is an arbitrary permutation, and R σ 1...n represents a tensor product of R-matrices associated to σ. Since F σ(1)...σ(n) is invertible we have hence F 1...n is referred to as a factorizing F -matrix. One of the main results in [4] was an explicit formula for F 1...n when specializing to representations of the quantum affine algebra U q ( sl 2 ). This expression for the F -matrix is entirely in terms of products of the corresponding trigonometric R-matrix.
The aim of this paper is to reproduce several of the results from [4], using diagrammatic tensor notation along the lines of [5]. In particular we give a diagrammatic representation of the F -matrix F 1...n , using conventions from the six-vertex model to draw it as a lattice of vertices. This representation allows us to establish identities, such as (1), virtually by inspection.
A key aspect of our approach is a notational trick concerning the crossing of lattice lines. To illustrate this device, we consider the simplest factorizing equation which is obtained from (1) by setting n = 2 and σ{1, 2} = {2, 1}, where R 12 ∈ End(V 1 ⊗ V 2 ) represents the trigonometric R-matrix associated to U q ( sl 2 ). As was noticed in [4], a solution of this equation is given by where I 12 ∈ End(V 1 ⊗ V 2 ) is the identity tensor, while e (ij) k ∈ End(V k ) is the 2 × 2 matrix with entry (i, j) equal to 1 and all remaining entries equal to 0. 1 A suitable diagrammatic representation of the F -matrix (4) is obtained as follows. Using the standard vertex notation for the components of the R-matrix, (5) where i 1 , i 2 , j 1 , j 2 take values in {+, −}, the line labelled by i k , j k can be said to correspond with the vector space V k . The crossing of the two lines indicates the action of R 12 in V 1 ⊗ V 2 . On the other hand, the components of the identity tensor I 12 ∈ End(V 1 ⊗ V 2 ) can be represented as Now the lines are considered to be decoupled, indicating the trivial action of I 12 in V 1 ⊗ V 2 . Based on the notations (5) and (6), we represent the components of the F -matrix (4) by The dot • which has been placed at the center of the vertices (7) denotes that the crossing of the lines 1 and 2 is sensitive to the value of the index next to the open-faced arrows ▽, △. In the vertex on the left, when i 1 = + the lines do not cross and when i 1 = − they do cross. In the vertex on the right, when i 2 = + the lines do cross and when i 2 = − they do not cross. 2 It is easy to check that the components recovered from (7) match the components obtained from equation (4). This crossing/uncrossing device underpins the rest of our work, and will be expanded upon in §3.1.
The role of Drinfel'd twists in the algebraic Bethe Ansatz was further considered in [6]. There it was noticed that the calculation of certain objects in the XXZ spin- 1 2 chain, such as the domain wall partition function and scalar product, is significantly simplified when the objects in question have been transformed appropriately using F -matrices. In the same paper the authors also found formulae for the local spin operators S ± i and S z i entirely in terms of global, monodromy matrix operators. This was achieved using the total symmetry of the monodromy matrix in the F -basis, and equation (1) in the case of a cyclic permutation σ c . The second aim of this paper is to reproduce these formulae from a diagrammatic standpoint. It turns out that knowledge of F -matrices is not necessary to perform this calculation, a fact which was also observed in [7], [8].
In §2 we recall the six-vertex notation for the U q ( sl 2 ) R-matrix. A number of standard identities such as the unitarity, Yang-Baxter and intertwining equations are presented in diagrammatic form. We also give a diagrammatic interpretation of R σ 1...n as a bipartite graph. In §3 we provide a diagrammatic exposition of the paper [4]. We use the notational device (7) to construct partial F -matrices, and give proofs of some basic identities in [4]. Attaching the partial F -matrices in a systematic fashion, we obtain the full F -matrix and show that it satisfies (1). In §4 we list the formulae obtained in [6] for the reconstruction of the local spin operators S ± i and S z i . Starting from the diagrammatic representation of the cyclic propagator R σc 1...n , we outline a purely diagrammatic proof of these results.

2.
Vertex notation for trigonometric R-matrix 2.1. Trigonometric R-matrix R 12 . An essential object in the quantum inverse scattering/algebraic Bethe Ansatz scheme is the quantum R-matrix. The R-matrix associated to the XXZ spin-1 2 chain is given by where the entries of (8) are the trigonometric functions and ξ 1 , ξ 2 are free variables, while η is the crossing parameter of the model. The matrix (8) is a representation of the U q ( sl 2 ) R-matrix on the tensor product V 1 ⊗ V 2 , where V 1 , V 2 are copies of C 2 . In order to denote this fact the matrix (8) is given the subscript 12, and its variables are accordingly ξ 1 , ξ 2 . Often, we will abbreviate R 12 (ξ 1 , ξ 2 ) to R 12 when the variables of the R-matrix are clear from context. The R-matrix is a rank 4 tensor whose components have been arranged in a two-dimensional array for convenience. Correspondingly, four indices are needed to label the components of (8): For all i 1 , i 2 , j 1 , j 2 ∈ {+, −} we can then define The object on the right hand side of (11) is called an R-vertex. 3 We call these R-vertices to distinguish them from the P -vertices and I-vertices that we define below. The indices placed on the four bonds of the vertex are called state variables. The state variable +(−) is interchangeable with an arrow pointing up (down) the page. In this notation, the non-zero entries of (8) get represented by vertices possessing arrow conservation. This means that the number of upward arrows entering the base of the vertex is equal to the number of upward arrows leaving the top of the vertex. Only six configurations of arrows obey this property, leading to the equations while all remaining vertices are equal to zero. This achieves the well known correspondence between the XXZ spin-1 2 chain and the six-vertex model of statistical mechanics. Further, the notation (11) allows a succinct representation of tensor multiplication (contraction). For example, the components of the rank 6 tensor R 13 (ξ 1 , ξ 3 )R 12 (ξ 1 , ξ 2 ) are represented by the diagram where summation is implied over the repeated index k 1 . The label k 1 in the diagram (15) is superfluous if we adopt the convention that all internal bonds are summed over. In the sequel, we represent tensors of rank 2n by diagrams possessing 2n external bonds, while all internal bonds are implicitly summed. 3 We have drawn two vertices on the right hand side of (11), because we use both notations interchangeably. In situations where we employ the latter notation, the reader should remember that each number actually stands for an index subscripted by that number. Numbers at the base of a diagram stand for indices of type i, while numbers at the top of a diagram stand for indices of type j. Notice that each label also encodes the rapidity flowing through that particular line. Also note that the orientation of each vertex is fixed by setting the direction of rapidity flow to be up the page; each vertex will have two lines which exit at the top of the diagram and two which exit at the bottom.

2.2.
Permutation matrix P 12 . In the case where the variables ξ 1 , ξ 2 are equal, the R-matrix (8) satisfies where P 12 is the permutation matrix on the spaces V 1 , V 2 . We shall represent the components of P 12 by the P -vertex where i 1 , i 2 , j 1 , j 2 take values in {+, −}. Interchanging these indices with their appropriate arrows, the non-zero entries of (16) get paired with P -vertices (17) having arrow conservation along each line. In other words, while all remaining P -vertices are equal to zero.
2.3. Identity matrix I 12 . The identity matrix I 12 receives a similar treatment to the preceding matrices. Although its action is trivial, we wish to write it as a vertex in order to place it alongside its companions. Writing we represent the components of I 12 by the I-vertex where i 1 , i 2 , j 1 , j 2 take values in {+, −}. Like the P -vertex, the lines in the I-vertex do not genuinely cross. Swapping the indices with their appropriate arrows, the non-zero entries of (19) get paired with I-vertices (20) having arrow conservation along each line. That is, while all remaining I-vertices are equal to zero. At times we shall prefer to represent the identity matrix in the more simple form in which the lines have been explicitly uncrossed. Our preference for a particular notation (20), (22) will depend largely on context.

Unitarity relation. The R-matrix (8) satisfies the unitarity condition
. Alternatively, using diagrammatic tensor notation we can write (23) as .

(24)
Note that here and subsequently, acting on the left in the symbolic notation corresponds to acting on the bottom in the diagrammatic notation. Equation (24) is established by explicitly checking that all 16 components of the tensors agree.
The monodromy matrix is a rank 2n + 2 tensor, acting in V a ⊗ V 1 ⊗ · · · ⊗ V n . Here V a is another copy of C 2 , called the auxiliary space to distinguish it from the quantum spaces V 1 , . . . , V n . We reserve the variable λ a to accompany the space V a , while the variables ξ 1 , . . . , ξ n accompany V 1 , . . . , V n as usual. Often, the V a dependence of R a,1...n is exhibited explicitly by writing where the entries of the previous matrix are tensors of rank 2n acting in V 1 ⊗ · · · ⊗ V n . The transfer matrix t(λ a ), which plays an important role in the algebraic Bethe Ansatz, is defined as Let us now rephrase these definitions using diagrammatic tensor notation. Firstly, the monodromy matrix (27) is represented by the string of R-vertices R a,1...n (λ a |ξ 1 , . . . , ξ n ) .
A particular component of R a,1...n (λ a |ξ 1 , . . . , ξ n ) is obtained by selecting appropriate values for the indices i a , j a ∈ {+, −}. For example, to obtain A(λ a |ξ 1 , . . . , ξ n ) one should select i a = j a = +. Hence we have the diagrammatic representations Finally, we notice that the transfer matrix (29) is obtained by identifying the ends of the string of vertices (30). It follows that where the marked lines in (32) are taken to be identified.
Before moving on, let us introduce another monodromy matrix whose ordering of spaces is the reverse of (27). Explicitly speaking, we define and represent it diagrammatically as where I a,1...n = I an . . . I a1 and I 1...n,a = I 1a . . . I na both act identically in V a ⊗ V 1 ⊗ · · · ⊗ V n . Using diagrammatic notation, (35) and (36) may be written as which are clearly true by iteration of (24).

Intertwining equation.
Let R a,1...n (λ a ), R b,1...n (λ b ) be monodromy matrices of the form (27). By virtue of the Yang-Baxter equation (25) they satisfy This is known as the intertwining equation. It is important in the algebraic Bethe Ansatz scheme because it generates all commutation relations between the entries of the monodromy matrix (28). It has the diagrammatic equivalent The proof of equation (39) is simplified if one instead proves its diagrammatic version (40). To be precise, by applying the diagrammatic equation (26) to the left hand side of (40) n times successively, the right hand side is obtained. The aim of this paper is to extend such efficient diagrammatic proofs to more equations.
We will revisit this identity in §3.3.  (1), . . . , σ(n)} be an arbitrary permutation of the set of integers {1, . . . , n}. A standard device is to represent this permutation as a bipartite graph. This is achieved by writing down two rows of integers {n, . . . , 1} and {σ(n), . . . , σ(1)}, one row directly above the other, and connecting each integer i in the top row with i in the bottom row. The manner in which the lines cross is, for the moment, not important. We denote the resulting graph by G(σ).
Using the diagrammatic representation (11) of the R-matrix, we define R σ 1...n to be the rank 2n tensor corresponding to the graph G(σ). In order to make this identification, the crossing of lines must now be considered. All ways of drawing G(σ) are equivalent up to applications of the unitarity (24) and Yang-Baxter equation (26). This means that we can prove complicated equations involving products of R-matrices by drawing both sides as bipartite graphs and checking that they have the same connectivity from top to base.
To demonstrate this point, consider the example σ{1, 2, 3, 4, 5} = {3, 5, 2, 1, 4} which is represented by the graphs We obtain the tensor R σ 12345 = R 25 R 15 R 45 R 12 R 13 R 23 from the graph on the left, whereas the graph on the right yields R σ 12345 = R 25 R 41 R 45 R 15 R 14 R 23 R 13 R 12 . These expressions are shown to be equal by application of (23) and (25).
When studying the tensors R σ 1...n , whose form can be rather complicated, it is helpful to have a canonical way of breaking the permutation σ into elementary steps. To this end, let {a 1 , . . . , a n } be an arbitrary set of distinct integers and define σ c {a 1 , a 2 , . . . , a n } = {a 2 , . . . , a n , a 1 } (44) which cyclically permutes the set, and σ p {a 1 , a 2 , a 3 , . . . , a n } = {a 2 , a 1 , a 3 , . . . , a n } since R σ 1...n is generally formed by a large number of such diagrams, up to overall labelling of the lines. We will use this fact frequently throughout the next section.
3. Graphical construction of F -matrices 3.1. Elementary factorizing matrix F 12 . The simplest factorizing problem consists in finding a ma- where R 12 (ξ 1 , ξ 2 ) is the R-matrix (8). As can be verified by direct calculation of all 16 components in equation (49), a solution is given by or equivalently, for k ∈ {1, 2}. Our first goal is to provide a diagrammatic notation for the matrix (50), which can be extended to more complicated factorizing tensors. For all i 1 , i 2 , j 1 , j 2 ∈ {+, −} we define The object on the right side of (54) is similar to an R-vertex (11). The only distinguishing features are the dot • and the triangle ▽ which have been placed on this vertex. The meaning of these symbols is as follows.
1. When i 1 = +, corresponding to a state arrow that points opposite ▽, the two lines of the vertex (54) do not genuinely cross. In this case, (54) behaves as an I-vertex (20): .
2. When i 1 = −, corresponding to a state arrow that points with ▽, the two lines of the vertex (54) cross as they normally would. In this case, (54) behaves as an R-vertex (11): .
The diagram (54) is a representation of the equation (51). Observe that it is also possible to define in accordance with equation (52). Either of the representations (54), (57) is valid and we will use them interchangeably, often within the same equation. For example, we propose the equation (58) in which both representations of the F -matrix (50) appear. We can verify that (58) is true by checking the two cases i 1 = + and i 1 = − separately. In the case i 1 = + we have (59) while in the case i 1 = − we find that .
(60) Thus (58) provides an immediate proof that (50) satisfies (49). The aim of this section is to simplify the proof of other, more complicated identities by using our adopted notation.
Let us now extend the diagrammatic notation from §3.1 to the tensors (61), (62). We make the identifications  .
The diagrams (63), (64) closely resemble their monodromy matrix counterparts (30), (34). The only differences are the arrows ▽, △ and the dots • assigned to every vertex. For the diagram (63), this notation has the following interpretation: 1. When i 1 = +, corresponding to a state arrow that points opposite ▽, the lines in (63) do not genuinely cross. The diagram behaves as a string of I-vertices. .
2. When i 1 = −, corresponding to a state arrow that points with ▽, the lines in (63) cross normally.
The diagram behaves as a string of R-vertices; that is, as a monodromy matrix. .

(66)
An analogous interpretation applies to (64). It is straightforward to check that these conventions are consistent with the algebraic definitions (61) and (62).  in End(V 1 ⊗ · · · ⊗ V n ). We prove this identity by writing it diagrammatically as follows, .
Here and in subsequent calculations, each row of dots • is associated to its own triangle. For example, studying the left hand side of (68), the top row is sensitive to the state variable in ▽, while the bottom row is sensitive to the state variable in △.
In order to prove (68)
It is possible to construct an alternative expression for F 1...n (ξ 1 , . . . , ξ n ). For all n ≥ 3 we define the tensor F ′ 1...n (ξ 1 , . . . , ξ n ) ∈ End(V 1 ⊗ · · · ⊗ V n ) by the recursive equation For the inductive step, we start with the diagram (92) for F 1...n and again use the alternative representation for F 12 to obtain Then by repeated application of the cocycle relation (68), we obtain The equivalence of (92) and (95) will be important in §3.6.
We then reposition the base of line 1 so that all I-vertices are removed from the lattice, in the sense of our earlier remark (22). The result of this elementary transformation is shown below: .
Finally, we use the equivalence between the F -matrix representations (92), (95) to transform the remainder of the diagram (104) into the right hand side of (102), and the proposition is established.
Secondly, applying the relation (58) to the vertices at the base of (107) we produce the equivalent diagram .
(108) Using (22) to remove the I-vertex at the base of (108), we then apply the equivalence between elementary F -matrices (54), (57) to obtain the right hand side of (106). This proves the proposition.
The two cases (101) and (105) are sufficient to prove (100), since any R σ 1...n (ξ 1 , . . . , ξ n ) can be decomposed into a product of tensors of the form R σc a1...an (ξ a1 , . . . , ξ an ) and R σp a1...an (ξ a1 , . . . , ξ an ). In conclusion, let us remark that for 2 ≤ i ≤ n the relation requires a substantially more involved proof than the two cases above. If we were to prove (109) directly, we would deconstruct permutations σ in terms of pairwise swaps, as is more traditional. However, the simplest proof of (109) consists of writing R ii+1 as a product of R σc a1...an and R σp a1...an tensors. Hence for our purposes σ c and σ p comprise the most expedient permutation basis. 4. Local spin operators in XXZ spin-1 2 chain 4.1. Formulae for local spin operators. To begin this section, we list the four formulae which we intend to prove. Recalling the definition of the monodromy matrix operators (28) and the transfer matrix (29), we claim that The local spin operators are given by the equations ). These formulae were originally derived in [6]. Since they embed local spin operators in the algebra of monodromy matrix entries, they have proved useful in the calculation of correlation functions [9,10].
Proof. The proof is accomplished by verifying (114) for the particular permutations σ = σ c , σ p . In fact we have already done this in the proof of identity 1, when considering the (i a = −) case. For σ = σ c we have R 1,2...n (ξ 1 |ξ 2 , . . . , ξ n )R a,1...n (λ a ) = R a,2...n1 (λ a )R 1,2...n (ξ 1 |ξ 2 , . . . , ξ n ) (115) which in diagrammatic notation may be written while for the σ = σ p case we have R 12 (ξ 1 , ξ 2 )R a,1...n (λ a ) = R a,213...n (λ a )R 12 (ξ 1 , ξ 2 ) (116) which in diagrammatic notation may be written The significance of (114) is demonstrated by using the factorization (100) of R σ 1...n . Making this substitution, we find that acts identically in V 1 ⊗ · · · ⊗ V n , or equivalently, Once again, this statement is obvious when the diagrams are considered as bipartite graphs. In either diagram, for all 1 ≤ i ≤ n, line i attaches to the top and base in the position which is i steps from the right. Hence the connectivity of these graphs is the same, and they are necessarily equal.

4.4.
Monodromy matrix under spin chain propagation. We prove, diagrammatically, the equation for the spin-chain propagation of the monodromy matrix. Identity 7. Let R a,1...n denote the monodromy matrix (27), and for all 1 ≤ i ≤ n let be propagators from site 1 to site i, and from site i to site 1 mod n, respectively. They satisfy the equation .
Comparing the diagrams on either side of (128), it is clear that they have the same connectivity from top to base. This shows that they are equal, due to the reasons already discussed. 4.5. U q p as product of transfer matrices. The next step of our calculations is the conversion of U q p into a product of transfer matrices (29). To do this, we require the following result.
Identity 8. For all 1 ≤ i ≤ n we claim that where t(ξ i ) is the transfer matrix (29) evaluated at λ a = ξ i .
Proof. We start from the diagram (32) for the transfer matrix, and set λ a = ξ i . This produces a P -vertex at the intersection of lines a, i: Recall from §2.2 that all non-zero P -vertices exhibit arrow conservation along their lines. It follows that the P -vertex may be deleted from the diagram (130), leaving it invariant: Finally, by performing a trivial rotation of the right hand side of (131) we obtain the equivalent relation t(λ a ) λa=ξi = R i,i+1...n1...i−1 (ξ i |ξ i+1 , . . . , ξ n , ξ 1 , . . . , ξ i−1 ). where the ordering of the product (133) is rendered irrelevant, since the transfer matrices t(λ a ), t(λ b ) commute for all λ a , λ b . 4.6. Construction of local spin operators. We are now in a position to prove the formulae listed in §4.1, using diagrammatic tensor notation. We begin by proving identity 9 and then show that it specializes to each of (110)-(113). Identity 9. For all 1 ≤ j ≤ n, let t(ξ j ) be the transfer matrix (29) evaluated at λ a = ξ j . Similarly let R a,1...n (ξ i ) denote the monodromy matrix (27) evaluated at λ a = ξ i . We have the identity  which holds in End(V a ⊗ V 1 ⊗ · · · ⊗ V n ).
Proof. Using the result of the previous subsection, we know that This means that the left hand side of (134) can be cast into the following diagrammatic form: Extracting the End(V a ) dependence from (134), we recover each of the equations (110)-(113).

Discussion
Following the algebraic methods of [4], we have outlined a diagrammatic treatment of the factorizing F -matrices. The main feature of our work is the diagrammatic depiction of the partial F -matrices F a,1...n and F 1...n,a in §3.2, which parallel the standard representation of the XXZ monodromy matrix. In §3.4 we gave diagrammatic proofs of a number of identities involving partial F -matrices. These proofs are quite transparent in our notation, which allows the components of all tensors to be extracted automatically. In §3.5 we built the full F -matrix F 1...n by stacking partial F -matrices together and proved the factorizing equation (1) in the sufficient cases σ = σ c and σ = σ p . Our proofs are inductive in nature, since they only require iterations of the more basic identities derived in §3.4.
We also considered the problem of constructing the local spin operators S ± i and S z i in terms of monodromy matrix elements, which was solved in [6]. Using the diagrammatic representation of U i 1 , in §4.4 we derived an equation for the monodromy matrix under spin chain propagation. In §4.6 the reconstruction formulae for the local spin operators were derived diagrammatically. Factorizing matrices do not appear explicitly in any of our calculations in §4.6, so our construction of the local spin operators is slightly more direct than that of [6].