Darboux transformations of supersymmetric Heisenberg magnet model

The Darboux transformation of supersymmetric Heisenberg magnet model is investigated. We calculate the exact superfield multisoliton solutions of the HM model by iteration of Darboux transformation. Explicit soliton solutions are also constructed for the SU(2) case.


Introduction
Supersymmetry (SUSY) is a symmetry that relates the particles having different statistical properties and spins or we can say that it is a symmetry in which we treat equally with the fermions and bosons. Mathematically, it is applied by introducing anti-commuting variables called the Grassman variables along with the usual commuting variables 2 . The purpose of introducing supersymmetry in integrable systems of classical and quantum theories is to calculate the fermionic extensions of known integrable systems and also develope understanding about their geometric structures, interactions and physical content in the form of soliton solutions [1,2].
During the past few years, there has been great interest in the field of classical and quantum integrability of Heisenberg magnet (HM) model [3][4][5][6][7][8][9][10][11][12]. The integrability of the HM model based upon SU(2) through the inverse scattering method is studied in [3,4]and its SU(n) is reported in [6]. The HM model based upon Hermitian symmetric spaces has been investigated in [9][10][11][12]. The Darboux transformation of the generalized Heisenberg magnet GHMmodel and its soliton solutions in terms of quasideterminants were presented in [13].
In this paper, we extend the earlier results obtained in [13] for the case of supersymmetric GHM model. We propose the supersymmetric generalization of the HM model and express the superfield Lax representation of the system. Further, we investigate the Darboux transformation by introducing a superfield Darboux matrix for HM model and calculate the superfield multisoliton solutions of the supersymmetric HM model. We then present the Darboux transformation on the component form. In the last section, we take the system based on Lie group SU(N) and derive the SU(2) based explicit solutions. In order to construct the SUSY field equations, we have to extend the bosonic system having spacetime variables (x, t) to one with super-spacetime variables (x, t, θ , ξ ) where θ and ξ are anti-commuting Grassmann variables 3 The Hamiltonian of the supersymmetric Heisenberg magnet model is defined as Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 2 The theory that combines fermions and bosons. 3 i.e θ 2 =0, ξ 2 =0. In superspace, the covariant derivatives are defined by and D x anticommutes with D t . The SUSY transformations are generated by the given operators where Q x and Q t anticommutes with covariant derivatives D x and D t respectively. 4 Note that here ( ) n is a general linear group over the field of real numbers therefore it is a real Lie group of dimensions n 2 .
where the bracket {,} represents an anti-commutator. By substituiting equation (1.1) in (1.2), we get , , , is corresponding to  transformation is  = -GTG 1 , where T is the N×N constant matrix. The superfield equation of motion (1.3) can also be expressed as the zerocurvature condition The above zero-curvature condition (1.4) is equivalent to the compatibility condition of the following supersymmetric Lax pair is given as where λ is the real (or complex) parameter, , , , is the matrix superfield.

Darboux transformation of supersymmetric HM model and multisoliton solutions
In this section, we construct the Darboux transformation for the supersymmetric HM model. In order to construct the multisoliton solution of the SUSY HM model, we define the Darboux transformation on the superfields that provides the generalization of Darboux transformation to the supersymmetric case. In our case, the Darboux transformation is defined by N×N superfield matrix  q x l ( ) x t , , , , called the superfield Darboux matrix (for detail discussion on the Darboux transformation we can see [14][15][16][17][18][19][20][21][22][23][24][25][26]). Now, let us define the superfied matrix  q x l ( ) x t , , , ; , such that superfield  of the Lax pair transforms as , , is the Darboux matrix which is written as where  is the N×N matrix superfield, λ is spectral parameter and I be the N×N identity matrix. Here it is noted that in order to calculate the superfield Darboux matrix we have to only find the value of superfield  because superfield Darboux matrix is linear in λ. The new solution [ ] 1 satisfies the Lax pairs are given as Now substituting (2.1) and (2.2) in (2.3) and (2.4) respectively, we get the following Darboux transformation on superfield  , i.e: x and the superfield  is required to satisfy the following conditions We can solve the above equations (2.6), (2.7) to calculate the explicit solution for the superfield matrix function  . An explicit expression for  can be found as follows. Let us construct a matrix superfield from the particular column superfield solutions of the superfield Lax pair at a particular values of spectral parameter λ i . Take N distinct real (or complex) constant parameters λ 1 , λ 2 , the N×N superfield matrix generalization of the super Lax pair (2.9), (2.10) can be written as Now the superfield matrix  is selected such that when it is inserted in equation (2.2), then it will give the Darboux transformation. So the choice of superfield matrix  is as: 2 . 1 3 1 We can now check whether the equation (2.13) be the solution of equation (2.6) and (2.7) by apply D x and D t on equtaion (2.13). Which shows that the choice is the good one. We can also say that is the solution of super Lax pair (1.5) and (1.6) and the matrix  is defined by equation (2.13) 1 , 1 defined by (2.1) and (2.5) respectively, are also the solutions of the same super Lax pair. Therefore, we can write as the required Darboux transformation on the solution  to the super Lax pair. Now we will express the superfield multisoliton solutions of the supersymmetric Heisenberg magnet model in terms of quasideterminants over a noncommutative ring of matrices whose entries are Grassmann even superfields. We can write (2.1) as 1 and the superfield matrix  can be expressed as Now we write the K-fold Darboux transformation on the superfield  as

Supersymmetric HM model in component form
In this section, we calculate the component fermionic and bosonic expressions by expanding the superfields , , , can be written as

Now the compatibility condition for the Lax pairs (3.3), (3.4) gives the expressions
where the bosonic component is U=U − ih + and the expression (3.7) is the equation of motion for the bosonic limit. where D ans S are the leading bosonic components of the superfields matrix  and , respectively. The matrix D must be defined such that the solution V of the Lax pair (3.3), (3.4 ) transforms as

Darboux trasnsformation on component SUSY HM model
satisfying the equations (3.5) and (3.6). This implies that are required to satisfy the transformed Lax pair , . 3.11 x t 2 2 2 Now following the previous section's argument. Let  where the conditions on S are given by ¶ -= - x --= -+ ¶ - Our next step is to verify that whether the equation (3.14) is solution of equations (3.16), (3.17). For this we operate ∂ x , ¶ t on (3.14), we get ¶ = -+ -----

The explicit solutions of the SU(2) system
Before starting with SU(2) case, we first discuss the generalized HM model based on SU(N). The matrix U takes values in the Lie algebra SU (N) i.e. we can write U as where the generator T a of the group SU(N) are the anti-Hermitian N×N matrices satisfying the condition Tr These generators satisfy the following commutation relation The seed solution will have the form where δ is some non-zero real parameter. Also we have Now for the SU(2) case equations (4.7), (4.8), (4.9) and (4.12) becomes The corresponding solution of the V(λ ) can be written as Note that in equation (4.15), we used The direct calculations give us  Let us consider μ=e iθ and simplifying, we have  From the arbitrary seed solution, we can generate the new solutions. In the asymptotic limit  ¥  ¥ t u r , , , the solution (4.23) reduces to the bosonic model. When the fermions are set to zero then the results we obtained here for the supersymmetric Heisenberg model becomes reduce to the results which are presented in [13]. By using the Darboux transformation we get the explicit solutions for the U, h + of the SUSY HM model. When we substitute (4.25) into (4.23) and (4.24), we get that TrU [1]=Trh + [1]=0. Therefore U [1] and h + [1] satisfies the additional constraints for gäSU(N).

Conclusions
In this paper, we have composed the supersymmetric Heisenberg magnet model and calculated the multisoliton solutions with the help of Darboux transformation. We started from the superspace formalism and expressed the Lax pair of the HM model in the form of component fields, then expressed the solutions in the form of quasideterminants. We have also explicitly calculated the expressions of multisolitons for the case of SU(2) for the given model. When we set the fermions equal to zero, these results reduced to the bosonic model. The work can be extended by analyzing the soliton solutions obtained. We can also bilinearize the system. Our next goal is to find the Poisson bracket algebra for the SUSY HM model. We will then extend it to Dirac brackets of the system. It may lead to some interesting results.