The $\mathfrak{su}(2)$ spin $s$ representations via $\mathbb{C}P^{2s}$ sigma models

We establish and analyze a new relationship between the matrices describing an arbitrary component of a spin $s$, where $2s\in \mathbb{Z}^+$, and the matrices of $\mathbb{C}P^{2s}$ two-dimensional Euclidean sigma models. The spin matrices are constructed from the rank-1 Hermitian projectors of the sigma models or from the antihermitian immersion functions of their soliton surfaces in the $\mathfrak{su}(2s+1)$ algebra. For the spin matrices which can be represented as a linear combination of the generalized Pauli matrices, we find the dynamics equation satisfied by its coefficients. The equation proves to be identical to the stationary equation of a two-dimensional Heisenberg model. We show that the same holds for the matrices congruent to the generalized Pauli ones by any coordinate-independent unitary linear transformation. These properties open the possibility for new interpretations of the spins and also for application of the methods known from the theory of sigma models to the situations described by the Heisenberg model, from statistical mechanics to quantum computing.


INTRODUCTION
The simplest CP 2s sigma models, 2s ∈ Z + , were invented by Gell-Mann in 1960 [12] and later developed by Callan et al. [5,6] to explain pion lifetime. They have found many other applications since then, such as [8,24,30,31,36,41]. In this paper we consider another possible application of these models. Namely, an appropriate combination of rank-1 projectors, which are the basic building blocks of the models, may be a representation of the su(2) algebra in C N and thus describe spin (or isospin) matrices corresponding to the maximal component of a spin s = (N − 1)/2. Our objective is to analyze the conditions which make it possible and suggest its applications.
In Section 2 we summarize the basic information about the CP 2s models. In Section 3 we show that some linear combination of the projectors behaves like a component of a spin vector and is always congruent to the generalized Pauli matrix of the appropriate size.
On the other hand, we present a counterexample, demonstrating that these combinations of projectors cannot always be combinations of the generalized Pauli matrices. Finally we show that those matrices which actually are such combinations (or are congruent to them by a coordinate-independent unitary transformation) satisfy a propagation equation of a stationary two-dimensional (2D) Heisenberg model. In Section 4 we recall the results of [7] which state that projectors mapping on the directions of the Veronese vectors always yield spins when combined in the above way. Possible applications of these results, which include quantum computing, are mentioned in Section 5.

BASICS OF THE CP 2s SIGMA MODELS
The main feature of the nonlinear sigma models in field theory is that the transformed field admits a very simple effective Lagrangian density, defined in C, assuming values in some manifold [38] with appropriate algebraic constraints on the field φ. This way, the complexity of their dynamics relies on the geometry of the target space. Such an approach has found many applications, such as [5,6,8,24,30,31,36,41]. Even for the very simple CP N −1 models, where the target is a single complex N − dimensional sphere, their properties are highly nontrivial [9, 10, 13-15, 17, 19, 20, 38, 39]. These models are the starting point of our research.
As a rule, the domain is parametrized in terms of the complex variables ξ = x + iy ∈ C, while the target manifold variables are either vectors z of a complex unit sphere, z † ·z = 1, embedded in C N , or the Grassmannian homogeneous variables f such that z = f /(f † ·f ) 1/2 (the dagger superscript denotes the Hermitian conjugation). Another convenient choice of the variables may be projectors P ∈ GL N (C) mapping on the directions of z (and f ), namely where ⊗ is the tensor product. The last description proves to be simple and fruitful. The action corresponding to the Lagrangian density (1) integrated over the Riemann sphere S 2 (with a constant factor for convenience) becomes under the idempotency condition where ∂ and∂ are the derivatives with respect to ξ andξ respectively. The Euler-Lagrange (E-L) equations are simply [38] [P, ∂∂P ] = 0, where the square bracket denotes the commutator. Their solutions satisfying the condition (4) may be obtained by a recurrence procedure. Namely, it was proven in [9,10] that all solutions corresponding to the finite action (3), expressed in terms of the homogeneous variables f k , result from a holomorphic solution f 0 (any) by consecutive application of a raising operator where I N is the N × N unit matrix.
The last nontrivial vector f k+1 is the antiholomorphic solution f 2s (the action of P + on an antiholomorphic vector obviously yields a zero vector). Similarly all solutions can be obtained from an antiholomorphic solution by an analogous lowering operator P − .
The CP 2s sigma models with finite action are completely integrable [10]. Furthermore, the E-L equations can be written in the form of a conservation law which shows that a total differential can be constructed out of the commutators (9). The integral of the total differential over any contour γ (with the constant of integration ensuring tracelessness) is an antihermitian immersion function X k (ξ,ξ) of a two-dimensional (2D) surface in a su(N) Lie algebra. Moreover, these immersion functions can explicitly be expressed as linear combinations of the projectors P k , k = 0, ..., 2s, [20]. The definition and the explicit form of the immersion functions are They describe 2D soliton surfaces whose conditions of immersion are the E-L equations (5).
These surfaces have no common points, except for CP 1 , where the only two surfaces X 0 and X 1 coincide [16].
The immersion function matrices X k span a Cartan subalgebra of the su(N) algebra.
From the fact that the P k are mutually orthogonal projectors of rank 1, it follows that each of them has only one nonzero eigenvalue equal to 1 and together they constitute a partition of unity. Hence an appropriate linear combination of these projectors may have any required set of eigenvalues. A matrix corresponding to a component of a spin vector, say S z , has eigenvalues −s, −s + 1, ..., s with s being a positive integer or half-integer (2s ∈ Z + ), with the exception of the trivial case s = 0. This way, a spin matrix may be constructed from CP 2s projectors as Up to a constant factor, this combination is equal to the sum of the immersion functions of the disjoint soliton surfaces X k which can provide an interpretation of the spin as a composite phenomenon.
We will refer to the matrices which may describe components of spins as spin matrices.

SPIN MATRICES
Before proceeding to our main results, we summarize some properties of the spin matrices associated with the CP 2s models expressed in terms of rank-1 Hermitian projectors P k

Properties of the spin matrices
Note that all the discussed properties of the spin matrices S z will follow from the defining relation (11) and from the fact that the Hermitian matrices P k map onto one-dimensional subspaces of C N .
Property 1 . If a Hermitian rank-1 projector P k maps onto a one-dimensional subspace of C N , N = 2s + 1, then the trace and the rank of the spin matrix S z (11) are The fact that, for even N, the ranks of the matrices S z are equal to N, while for odd N, the ranks of the matrices S z are equal to N − 1 is due to the presence of a zero eigenvalue corresponding to the eigenvector P k , where k = s = 1 2 (N − 1). Property 2 . The quadratic form corresponding to the Killing form in su(N) for the matrix S z is constant and may be defined as which corresponds to the expected value for the length of one (say z) component of a quantum-mechanical spin vector.
Property 3 . The spin matrix S z satisfies an algebraic condition determined by the characteristic polynomial corresponding to the eigenvalue problem has the form where I N is the N × N identity matrix. As all eigenvalues are different, (16) is the minimal polynomial.
Property 4 . According to our earlier result (eq. (47) of [15]) if rank-1 projectors P k satisfy the E-L equations (5), then any linear combination of these projectors is also a solution of (5) (this fact is nontrivial, because the E-L equations are nonlinear). Hence the spin matrices S z (11) also satisfy the same E-L equations (5), except that the constraint is Property 5 . It follows from Property 4 that the spin matrices S z are conditional stationary points of the same action integral (3) as the projectors, but the condition is (16) (rather than P 2 = P ).
S z matrices as spins A 3-dimensional (3D) basis for spin matrices describing a system of spin s = (N − 1)/2 are three N × N generalized Pauli Hermitian matrices whose elements read [27] ( where 0 ≤ m, n ≤ N − 1 These matrices generate an irreducible representation of su(2) in We know from the proof of [7] that a special role is played by the solutions of the CP 2s sigma models which stem from the Veronese sequence of holomorphic functions The functions f 0 (18) and f k , k = 1, ..., 2s, obtained from f 0 by the recurrence formulae (6), define the corresponding rank-1 projectors by means of (2). The spin matrix S z obtained from these projectors (11) is tridiagonal and can be uniquely decomposed into a linear combination of matrices σ x , σ y , σ z (17). These solutions will be discussed in detail in the next section. Unfortunately, not all spin matrices (11) have such a property.
The resulting spin matrix S z is obtained from the projectors P k (2) as their linear combination (11) where the projectors follow from the recurrence formulae (7) applied to P 0 = . This matrix does not have to be tridiagonal, whereas any combination of the diagonal σ z and tridiagonal σ x , σ y has to be tridiagonal as its components are. In the simplest case of CP 2 , the 3 × 3 S z matrix has a nonzero element Obviously, (S z ) 31 is also nonzero as its complex conjugate. Hence the matrix is not tridiagonal.
On the other hand, it is evident that any diagonalizable N × N matrix having the proper eigenvalues is congruent to σ z /2 (and also to σ x /2 or σ y /2, by different congruency transformations). The spin-like linear combination of projectors (11) is Hermitian, so it is diagonalizable by a unitary matrix. Let U be the unitary diagonalizing matrix for S z (11), both S z and U being functions of (ξ,ξ). Then If U acts on σ x and σ y in the same way, we obtain three matrices which constitute a basis for another irreducible representation of su(2) in C N (it is straightforward to show that they span a Lie subalgebra of su(N)).
In the context of (21b), the whole dynamics of the spin matrix lies in the unitary transformation U. On the other hand, in some situations, we can analyze the dynamics of spin matrices without referring to the transformation (21).
The coefficients in (22) are the coordinates of the spin vector in the basis (17). For such matrices S z we have Proposition 1. Let be a vector whose components are the α x , α y , α z of equation (22).
Then it satisfies the equation where × denotes the usual vector product in C 3 . The vector α is subject to the normalization condition 4 α · α − 1 = 0.
Equation (23) is a counterpart of the E-L equations (5) in terms of the vector α.
Proof. According to Property 4, the spin matrices S z satisfy the E-L equations (5). Substituting (22) into those equations, we obtain the coordinates of (23) in the basis (17).
Equation (23) The Lagrange multiplier µ = µ † ∈ Aut(C) in the action integral has been introduced to Proof. The proof is straightforward if we take the Frechet derivative of (25) with respect to α(ξ,ξ). This yields On the vector multiplication of both sides by α, we obtain (23).
Corollary. It is evident from the above Proof that equation (23) is merely the necessary condition for α to follow spin dynamics. It has to be supplemented by the normalization condition (24). For the action (25), we can get the complete E-L equations by the scalar multiplication of (26) with α, which allows us to calculate the multiplier µ explicitly from the α-normalization condition (24), thus getting µ = α · α ξξ . Substituting this value into (26), we obtain where I 3 is the 3D identity tensor.
This result can be generalized to the matrices congruent to σ x , σ y , σ z by a coordinateindependent unitary transformation, namely where α k ∈ R, the Euclidean norm |α| = 1/2, while s k are congruent to σ k by a constant Then if the commutator [S z , (S z ) ξξ ] vanishes, the vector α satisfies equation (23).
A simple proof follows from the fact that all commutators [s k , s m ], k, m ∈ {x, y, z} are congruent to [σ k , σ m ] by the same transformation matrix U.
Remark 1. Proposition 3 is trivial for s = 1 2 (i.e. N = 2) due to the isomorphism of SU(2) with SO(3), which makes the transformation a rotation of the vector α by a constant angle.
It is nontrivial for higher spins (N > 2) as the set of constant U transformations is much richer.
Remark 2. In the general case, the Proposition 3 is not true if the transformation matrix depends on the coordinates. In all the proven cases, the S z matrix depends on ξ,ξ through the coefficients α x , α y , α z . Only two of these coefficients are algebraically independent (note the normalization condition (24)). In general, the system (5) can have more degrees of freedom.

SPINS FROM THE VERONESE VECTORS
The E-L equations (5) expressed in terms of the homogeneous variables f k , k = 0, ..., 2s have the form One of the most useful holomorphic solutions (i.e. for k = 0) is given by Starting from this solution, a sequence of solutions for k = 1, ..., 2s may be obtained from the recurrence relations (6). The sequence {f 0 , f 1 , ..., f 2s } is called the Veronese sequence [3].
In this section we consider the CP 2s models in which the vectors f 0 , ..., f N −1 make a Veronese sequence. All these vectors f 0 , ..., f N −1 , N = 2s + 1, may be expressed in terms of the Krawchouk orthogonal polynomials [7].
with the stereographic projection variable p Here (f k ) j is the jth component of the vector f k ∈ C 2s+1 \ {∅} and K j (k; p, 2s) are the Krawtchouk polynomials for which we use the convention that for k = 0 K j (0; p, 2s) = 1.
The element in the i-th row and j-th column of the rank-1 Hermitian projector P k as given by (2) has the form [7] (P k ) ij = 2s k where we have omitted the dependence of K i on p and 2s.
For the Veronese sequence solutions of the CP 2s models, the analytic recurrence relations can be replaced by simpler algebraic ones. It is convenient to use the combinations of the x and y components of the spin matrices S ± = S x ± i S y and σ ± = σ x ± i σ y , rather than the components themselves. The spin matrix S z may be simply represented by a combination of the diagonal σ z and tridiagonal σ + , σ − , namely [7] S z = 1 2(1 + ξξ) The su(2) commutation relations (identical to the relations satisfied by the respective combinations σ ± ), suggest the following form of the components S + and S − S + = 1 2(1 + ξξ) 2ξ σ z +ξ 2 σ + − σ − , (our σ z and σ ± are twice as large as those of [7] to comply with their notion as the generalized Pauli matrices (17)).
It is easy to check that the components of the spin S z , S ± indeed satisfy the commutation relations (38). Moreover [7], S ± play the role of the creation and annihilation operators for where by convention f −1 = f 2s+1 = 0 (see [7] for the proof ).
Consequently, the algebraic recurrence relations for the immersion functions X k satisfy the algebraic conditions The algebraic recurrence relations allow us to recursively construct the Veronese sequence of solutions f k (or rank-1 projectors P k ) from the holomorphic solution f 0 (or P 0 ) in a simpler way than the analytic relations (6).
Thus the angle between the vector α and the z-direction depends on the modulus |ξ| only, while a change in the phase of ξ yields an identical rotation of the spin vector about the z axis.

POSSIBLE APPLICATIONS
The results presented here complement our previous studies [13][14][15][16][17] on the theory of 2D Euclidean sigma models. The inclusion of the su(2) spin-s representations may have a significant impact on many problems with physical applications. The equations (23) describe stationary states of the 2D Heisenberg model (see Appendix). The connection which we have found describes this model of ferro-, antiferro-and ferrimagnets in terms of CP 2s sigma models, interpreting it as their special case. The complete integrability of the CP 2s sigma models provides a useful tool for solving problems of these magnetic materials, e.g. [25]. In particular, the Veronese solutions of the CP 2s sigma models allow us to explicitly build the corresponding spin fields.
Our description of the spin-s matrix in terms of the CP 2s projectors is an example of representing an array of functions C into C as a finite sum of orthogonal vectors. In the special case of the Veronese solutions, the vectors were sequences of Krawtchouk polynomials. Such a representation using the Fourier-Krawtchouk transformation was recently introduced in [32] to achieve a quantum information processing in constant time. Moreover, this constanttime signal-evolution analysis works on finite strings with arbitrary length [1,32]. The transformation represents the transformed function limited to a finite interval in terms of the Krawtchouk polynomials multiplying the Fourier variable exp[−i π 2 (l − k)], k, l ∈ Z. It has its 2D counterpart in splitting the spin component S z into the rank-1 projectors proportional to products of two Krawtchouk polynomials with ξξ in consecutive powers. Our more general scheme encompasses such a possibility. It is very likely that a discretization (sampling), followed by an appropriate transformations of this kind, may be suitable for efficient quantum computations. Further it is promising for possible applications to digital image processing, in medical image reconstruction and recognition [11,29,32,37,40]. perpendicularity of the spins to their first derivatives. The continuous Hamiltonian, up to a constant, reads H = a 2 (J 1 − J 2 − 2J 3 )α x 2 + b 2 (−J 1 + J 2 − 2J 3 )α y 2 + (J 1 + J 2 + 2J 3 )α · a 2 α xx + b 2 α yy , (48) (where the subscripts x, y denote differentiation).