# How can a Bogoliubov transformation be implemented numerically?

I'm working with an antiferromagnetic spin model on a bipartite lattice, which I am trying to analyze via Holstein-Primakoff spin-wave theory. Because the theory has two sublattices, there are naturally two boson flavors $$a,b$$. In Fourier space, I've found an exact form for the Hamiltonian in terms of the bosons:

$$\sum_{\vec{k}}\begin{pmatrix} a_{k}^{\dagger} && b_{k}^{\dagger} && a_{-k} && b_{-k} \end{pmatrix} \begin{pmatrix} c && w^{*} && 0 && x^{*}\\ w && c && x && 0\\ 0 && x^{*} && c && w^{*}\\ x && 0 && w && c \end{pmatrix} \begin{pmatrix} a_{k}\\ b_{k}\\ a_{-k}^{\dagger}\\ b_{-k}^{\dagger} \end{pmatrix}$$

I have exact expressions for the parameters $$c, x$$, and $$w$$, and they depend on $$k$$. The trouble is that I'm not aware of a simple way to implement a Bogoliubov transformation for Hamiltonians of this type - there are some papers on spin-wave theory which address similar problems, but the examples I have seen do not have the same structure of couplings, so those closed forms do not apply.

For my purposes, I don't need a closed form - just a numerical procedure for extracting the band structure. Is there such an approach, in general? Of course, I'm happy to get a closed form for the band structure, but all I want is a well-defined numerical procedure. The difficulty is that Bogoliubov transformations are non-unitary, so brute-forcing the relevant equations seems computationally expensive. Besides, it would be useful to know how to do this in general.

The central results are as follows. If you can write your Hamiltonians in the form you have, i.e. $$H = \vec{\Psi}_k^\dagger h(k) \vec{\Psi}_k$$ where $$\vec{\Psi}^\dagger_k=\left( a_1^\dagger(k), \dots, a_n^\dagger(k), a_1(-k),\dots, a_n(-k)\right)$$, and $$h(k)$$ is a $$2n\times 2n$$ positive semi-definite matrix, then you can generally perform a so-called paraunitary diagonalization by (numerically) diagonalizing $$\sigma_3 h(k)$$, where $$\sigma_3 = \begin{pmatrix} I_{n\times n} && 0\\ 0 && -I_{n\times n}\\ \end{pmatrix}$$ and $$I_{n\times n}$$ is the $$n\times n$$ identity-matrix. The structure of $$\sigma_3$$ ensures that bosonic commutation relations are preserved. (In the fermionic case, $$\sigma_3$$ can be replaced by the $$2n\times 2n$$ unit matrix.) Because of particle-hole symmetry in the enlarged space (in your case $$4\times 4$$ rather than $$2\times 2$$), eivenvalues show up in pairs $$\pm \epsilon_n$$. The physical eigenvalues are likely the positive ones. At least that tends to be the case in spin-wave treatments where we're interested in excitations.
So in a numerical approach, you can simply diagonalize $$\sigma_3 h(k)$$ for each $$k$$ and select eigenstates with positive eigenvalues. Before plotting the band structure it may be useful to sort the eigenvalues.