"End Semester Report Anirban Sharma (20121049) April 19, 2016Model 1 If we consider a spin Heisenberg antiferromagnet on a Kagome lattice, the Hamil- 2 tonian will be given by X H = J SS (1) ij i j ij There will be various types of interactions which can be seen in the \u0000gure below. Figure 1: Spin interactions In case of Ising model all the neighbouring interactions were di\u0000erent. However in the BFG model we consider all the interactions to be same i.e. J = J = J = 1 2 3 J.This type of interactions can be mediated by the phonons. The Hamiltonian can be written as X H = JS S (2) 7 7 7 Also sum of the spins in a single hexagon is zero. In the extreme easy axis limit, where the exchange interaction along z axis in spin pace is larger than that in x-y plane,J can be considered in the perturbation term. So the \u0000nal Hamiltonian will ? be \u0000 \u0000 2 X X J ? i i j ^ H =J S + (S S +h:c) (3) z z + \u0000 2 <ij> i27 The main aim was to emulate BFG model on a hexagonal lattice. Studies have been done on two hexagonal plaquette system. We want to extend the same to 3 and possibly 4 hexagonal plaquettes and study the properties of the system. So the \u0000rst part of the project consisted of making hexagonal plaquettes. Calculations To \u0000nd the equilibrium position of the ions, we \u0000rst consider the potential experi- enced by them. N N 2 X X 1 Ze 1 2 2 2 2 2 2 V = M(! x +! y +! z ) + (4) x i y i z i 2 8\u0000\u0000 jr \u0000 rj 0 i j i=1 i;j=1;i6=j 2 2 3 Z e After scaling by l = , we get the potential as 2 x 4\u0000\u0000 M! 0 x N N X X 1 1 1 2 2 2 2 2 V = (x +\u0000 y +\u0000 z ) + (5) i y i z i 2 2l 2l jr \u0000 rj x i j x i=1 i;j=1;i=6 j 1! y ! z where \u0000 = ;\u0000 = . At equilibrium, we have y z ! ! x x @V @V @V = = = 0 (6) x y z i i i Using these conditions, we get coupled equations solving which we get the equilib- rium positions of the ions. Plots Figure 2: ion position N=32,! = 2\u0000 \u0000 3MHz;! = 2\u0000 \u0000 8MHz;! = 2\u0000 \u0000 1MHz x y z Figure 3: ion position for N=31,! = 2\u0000 \u0000 2:75MHz;! = 2\u0000 \u0000 8MHz;! = x y z 2\u0000 \u0000 1MHz 2Figure 4: ion position for N=27,! = 2\u0000 \u0000 2:65MHz;! = 2\u0000 \u0000 8MHz;! = x y z 2\u0000 \u0000 1MHz Results As we can see we get three hexagonal plaquettes from \u0000gure 2 using 32 ions. How- ever, we want to \u0000nd the con\u0000guration with minimum number of ions. This is because as number of ions increases, number of phonon increases three times. So for that we tried to get the same con\u0000guration with lesser number of ions.From g- \u0000 ure 3, we get four hexagonal plaquette with 31 ions. This means it is possible to get our desired system with even lesser ions. So we have to keep on varying the number of ions and frequencies.With 27 ions, we get the three hexagonal plaquettes. Once we get the three hexagonal plaquette system, we will proceed to the BFG model and study the properties of the system. References \u0000 L. Balents, M. P. A. Fisher, and S. M. Girvin Phys. Rev. B 65, 224412 \u0000 Hexagonal plaquette spin{spin interactions and quantum magnetism in a two- dimensional ion crystal by R. Nath et. al 3"