Find the quadratic bosonic theory that describes a 1-dimensional Heisenberg ferromagnet


 * 1) Identify the Hamiltonian that describes the Heisenberg ferromagnet: \(\mathcal{H} = -J\sum_i\hat{\mathbf{S}}_i\hat{\mathbf{S}}_{i+1}\)
 * 2) Expand in terms of the projections of spin: \(\mathcal{H} = -J\sum_i\left(\hat{S}_i^z\hat{S}_{i+1}^z+ \frac{1}{2}\left(\hat{S}^+_i\hat{S}^-_{i+1} + \hat{S}^-_i\hat{S}^+_{i+1}\right)\right)\)
 * 3) Substitute in the creation operator identities of the spin operators: \(\mathcal{H} = -J\sum_i\left(\left(S - \hat{b}^{\dagger}_i\hat{b}_i\right)\left(S - \hat{b}^{\dagger}_{i+1}\hat{b}_{i+1}\right)+ S\hat{b}_i\hat{b}^{\dagger}_{i+1} + S\hat{b}^{\dagger}_i\hat{b}_{i+1}\right)\)
 * 4) Factorise: \(\mathcal{H} = -JNS^2 +JS\sum_i\left(\hat{b}^{\dagger}_i\hat{b}_i + \hat{b}^{\dagger}_{i+1}\hat{b}_{i+1} - \hat{b}_i\hat{b}^{\dagger}_{i+1} - \hat{b}^{\dagger}_i\hat{b}_{i+1}\right)\)
 * 5) Take the Fourier transform: \(\mathcal{H} = -JNS^2 +JS\sum \limits_{n, k, k\prime}\left(e^{inl(k-k\prime}\hat{b}^{\dagger}_i\hat{b}_i + e^{i(n+1)l(k-k\prime)}\hat{b}^{\dagger}_{i+1}\hat{b}_{i+1} - e^{-inlk +i(n+1)lk\prime)}\hat{b}_i\hat{b}^{\dagger}_{i+1} - e^{i(n+1)lk -inlk\prime}\hat{b}^{\dagger}_i\hat{b}_{i+1}\right)\)
 * 6) Expand the exponentials: \(\mathcal{H} = -JNS^2 +JS\sum \limits_{n, k, k\prime}\left(e^{inl(k-k\prime}\hat{b}^{\dagger}_k\hat{b}_{k\prime} + e^{i(n+1)l(k-k\prime)}\hat{b}^{\dagger}_{k}\hat{b}_{k\prime} - e^{-inl(k-k\prime)}e^{ilk\prime)}\hat{b}_k\hat{b}^{\dagger}_{k\prime} - e^{inl(k-k\prime)}e^{-ilk\prime}\hat{b}^{\dagger}_k\hat{b}_{k\prime}\right)\)
 * 7) Write in terms of delta functions: \(\mathcal{H} = -JNS^2 +JS\sum \limits_{k, k\prime}\left(\delta\left(k-k\prime\right)\hat{b}^{\dagger}_k\hat{b}_k\prime + \delta\left(k-k\prime\right)\hat{b}^{\dagger}_{k}\hat{b}_{k\prime} - \delta\left(k-k\prime\right)e^{ilk\prime)}\hat{b}_k\hat{b}^{\dagger}_{k\prime} - \delta\left(k-k\prime\right)e^{-ilk\prime}\hat{b}^{\dagger}_k\hat{b}_{k\prime}\right)\)
 * 8) Use the delta functions to simplify the summation: \(\mathcal{H} = -JNS^2 +JS\sum_k\left(\hat{b}^{\dagger}_k\hat{b}_k + \hat{b}^{\dagger}_k\hat{b}_k - e^{ilk}\hat{b}_k\hat{b}^{\dagger}_k - e^{-ilk}\hat{b}^{\dagger}_k\hat{b}_k\right)\)
 * 9) Simplify: \(\mathcal{H} = -JNS^2 + 2JS\sum_k \hat{b}^{\dagger}_k\hat{b}_k\)