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Diagonalize grand-dynamical matrix following the method of Colpa [1].

Algorithm itself is described in section 3, Remark 1 of [1].

Solves the Bogoliubov Hamiltonian of the form:

\[\hat{H} = \sum_{r^{\prime}, r = 1}^m \hat{\alpha}_{r^{\prime}}^{\dagger}\boldsymbol{\Delta}_1^{r^{\prime}r}\hat{\alpha}_r + \hat{\alpha}_{r^{\prime}}^{\dagger}\boldsymbol{\Delta}_2^{r^{\prime}r}\hat{\alpha}_{m+r}^{\dagger} + \hat{\alpha}_{m+r^{\prime}}^{\dagger}\boldsymbol{\Delta}_3^{r^{\prime}r}\hat{\alpha}_r + \hat{\alpha}_{m+r^{\prime}}^{\dagger}\boldsymbol{\Delta}_4^{r^{\prime}r}\hat{\alpha}_{m+r}^{\dagger}\]

In a matrix form the Hamiltonian is:

\[\hat{H} = \boldsymbol{\hat{a}}^{\dagger} \boldsymbol{D} \boldsymbol{\hat{a}}\]


\[\begin{split}\boldsymbol{\hat{a}} = \begin{pmatrix} \hat{\alpha}_1 \\ \cdots \\ \hat{\alpha}_m \\ \hat{\alpha}_{m+1} \\ \cdots \\ \hat{\alpha}_{2m} \\ \end{pmatrix}\end{split}\]

After diagonalization the Hamiltonian is:

\[\hat{H} = \boldsymbol{\hat{c}}^{\dagger} \boldsymbol{E} \boldsymbol{\hat{c}}\]
D(2N, 2N) array-like

Grand dynamical matrix. Must be Hermitian and positive-defined.

\[\begin{split}\boldsymbol{D} = \begin{pmatrix} \boldsymbol{\Delta_1} & \boldsymbol{\Delta_2} \\ \boldsymbol{\Delta_3} & \boldsymbol{\Delta_4} \end{pmatrix}\end{split}\]
E(2N,) numpy.ndarray

The eigenvalues, each repeated according to its multiplicity. First N eigenvalues are sorted in descending order. Last N eigenvalues are sorted in ascending order. In the case of diagonalization of the magnon Hamiltonian first N eigenvalues are the same as last N eigenvalues, but in reversed order. It is an array of the diagonal elements of the diagonal matrix \(\boldsymbol{E}\) from the diagonalized Hamiltonian.

G(2N, 2N) numpy.ndarray

Transformation matrix, which change the basis from the original set of bosonic operators \(\boldsymbol{\hat{a}}\) to the set of new bosonic operators \(\boldsymbol{\hat{c}}\) which diagonalize the Hamiltonian:

\[\boldsymbol{\hat{c}} = \boldsymbol{G} \boldsymbol{\hat{a}}\]


Let \(\boldsymbol{E}\) be the diagonal matrix of eigenvalues E, then:

\[\boldsymbol{E} = (\boldsymbol{G}^{\dagger})^{-1} \boldsymbol{D} \boldsymbol{G}^{-1}\]


[1] (1,2)

Colpa, J.H.P., 1978. Diagonalization of the quadratic boson hamiltonian. Physica A: Statistical Mechanics and its Applications, 93(3-4), pp.327-353.