Bogoliubov–de Gennes Numerics

Bogoliubov–de Gennes (BdG) numerics represents a quadratic pairing Hamiltonian in a doubled particle–hole space. The doubled eigenproblem is useful computationally, but it does not double the number of independent physical fermionic modes. Correct densities and self-consistency require a consistent Nambu convention and careful handling of the positive- and negative-energy partners.

Helpful background. Bogoliubov quasiparticles supplies the physical transformation, and matrix diagonalization supplies Hermitian solver checks.

Fermionic Nambu matrices

For $M$ fermion orbitals, use $\Psi=(c_1,\ldots,c_M,c_1^\dagger,\ldots,c_M^\dagger)^T$ and

$$H=\frac12\Psi^\dagger\mathcal H\Psi+\text{constant}, \qquad \mathcal H= \begin{pmatrix} h&\Delta\\ \Delta^\dagger&-h^T \end{pmatrix},$$

where $h=h^\dagger$ includes the chemical potential and $\Delta^T=-\Delta$. The constant compensates normal ordering and any mean-field double counting; it matters for comparing free energies even though it does not change eigenvectors.

With $C=\tau_x K$, where $K$ complex-conjugates and $\tau_x$ exchanges particle and hole blocks,

$$C\mathcal H C^{-1}=-\mathcal H.$$

Thus an eigenvector $(u,v)^T$ at energy $E$ has partner $(v^*,u^*)^T$ at $-E$. Check this matrix identity before diagonalization and compare paired energies afterward. A wrong transpose or pairing antisymmetry often breaks it.

At exact zero energy, individual eigenvectors can mix arbitrarily within the zero subspace. Construct a canonical quasiparticle basis or use projectors and correlation matrices consistently; an arbitrary choice of “positive” zero modes is not an invariant prescription.

Reconstructing densities without double counting

For a complete set of independent positive-energy modes away from zero-mode ambiguity,

$$c_i=\sum_{n>0}(u_{in}\gamma_n+v_{in}^*\gamma_n^\dagger).$$

With thermal occupation $f_n=[e^{E_n/(k_BT)}+1]^{-1}$,

$$n_i=\sum_{n>0}\left[ |u_{in}|^2f_n+|v_{in}|^2(1-f_n)\right].$$

Summing this same expression over both energy partners would count the degrees of freedom twice. Check canonical anticommutation identities of the reconstructed transformation and compare the density trace with the intended particle number. Pairing generally fixes particle number only in expectation; adjust the chemical potential when that constraint is required.

A self-consistent calculation updates $\Delta$ and any Hartree fields from the anomalous and normal correlations using the sign convention of the chosen interaction. Iterate these fields and the number equation together. Residual convergence is separate from selecting the thermodynamically preferred branch; compare the correctly constructed free energy.

A uniform pairing block

A reduced spin-singlet block at a given momentum takes the form

$$\mathcal H_k= \begin{pmatrix}\xi_k&\Delta\\\Delta^*&-\xi_k\end{pmatrix}, \qquad E_k=\sqrt{\xi_k^2+|\Delta|^2}.$$

Its eigenvalues are $\pm E_k$, with coherence weights $|u_k|^2=(1+\xi_k/E_k)/2$ and $|v_k|^2=(1-\xi_k/E_k)/2$. For nonzero $\Delta$ at $\xi_k=0$, both weights are $1/2$. As $\Delta\to0$ away from the Fermi level, the occupations approach the normal-state Fermi–Dirac distribution; it becomes a step at zero temperature. This reduced block must be embedded with the proper spin partner when applying the full Nambu particle–hole identity.

Converge the spatial basis, momentum mesh, temperature, and any interaction cutoff independently. For contact models, changing a cutoff while keeping an unrenormalized coupling fixed can change the physical model.

Bosonic BdG is a different eigenproblem

For quadratic bosons, commutation preservation introduces the metric $\Sigma_z=\operatorname{diag}(I,-I)$. If $\mathcal H_B$ has energy units, the angular frequencies are the eigenvalues of $\Sigma_z\mathcal H_B/\hbar$. This dynamical matrix is generally non-Hermitian in the Euclidean metric even when $\mathcal H_B$ is Hermitian. Eigenvectors have paraunitary normalization. Feeding $\mathcal H_B$ to an ordinary Hermitian fermionic solver does not solve the bosonic dynamics.

Complex bosonic frequencies can indicate dynamical instability, subject to numerical verification. Fermionic $\mathcal H$ as defined above must have real eigenvalues; significant imaginary parts signal a violated assumption or inappropriate solver.

Exercises

For $\xi=3$ and real $\Delta=4$, compute the positive energy and coherence weights.

Solution

$E=5$, $|u|^2=(1+3/5)/2=0.8$, and $|v|^2=0.2$. Their sum is one. The paired spectrum and these weights test both diagonalization and density reconstruction.

References

• Colpa, J. H. P., “Diagonalization of the quadratic boson Hamiltonian,” Physica A 93, 327–353 (1978), doi:10.1016/0378-4371(78)90160-7 — the distinct bosonic metric problem.
• de Gennes, P. G., Superconductivity of Metals and Alloys, W. A. Benjamin, 1966 — fermionic BdG equations and coherence factors.