Until this moment multiple concepts related to topological quantum computation have been covered on this blog. Notably, we simulated basic Majorana Zero Modes physics in 1D topology, then we performed topological braids to convert ground state into ground state of different topological phase, we defines and simulated topological qubits and applied them to simulate quantum teleportation.

The idea of the topological gap has been briefly mentioned and came up as source of topological protection. One of the characteristics of the topological regime is the degeneracy of the ground state. This degeneracy gets broken during the topological phase transition as system enters the trivial regime. We have already covered and simulated those phase transitions both adiabatically and using unitary braids, however we did not demonstrate how the gap opens. That is the goal of this tutorial. We will replicate the plot from Topology in Condensed Matter course showing how the gap opens while on-site potential increases.

For this the Bogoliubov transformation can be applied, which is a nifty tool in theoretical physics allowing to find solutions for BCS superconducting systems as ours. Given the original form of the Kitaev Chain Hamiltonian as presented earlier

\begin{aligned} H = -\frac{1}{2} \mu \sum_j^L (2 a_j^\dagger a_j - 1) - \sum_j^{L-1} [w (a_j^\dagger a_{j+1} + a_{j+1}^\dagger a_j) - \Delta a_j a_{j+1} - \Delta^\ast a_{j+1}^\dagger a_j^\dagger)] \end{aligned}

we may construct a mean-field approximation Bogoliubov-de Gennes Hamiltonian as follows

$H_\text{BdG} = \left \vert 0 \rangle \langle 0 \right \vert \otimes H_{\mu, w} - \left \vert 1 \rangle \langle 1 \right \vert \otimes H_{\mu, w}^\ast + \left \vert 0 \rangle \langle 1 \right \vert \otimes H_{\Delta} - \left \vert 1 \rangle \langle 0 \right \vert \otimes H_{\Delta}^\ast .$

The $H$ has a Hilbert space of $2^L$ while $H_\text{BdG}$ is much smaller of size $2L$. The $H_{\mu, w}$ is Hermitian and $H_{\Delta}$ is skew-symmetric. Both are of size $2L$ by $2L$ elements and constructed as follows.

\begin{aligned} H_{\mu, w} &= \mu \sum_j^L \left \vert j \rangle \langle j \right \vert + w \sum_j^{L-1} (\left \vert j \rangle \langle j+1 \right \vert + \left \vert j+1 \rangle \langle j \right \vert) \\ H_{\Delta} &= \sum_j^{L-1} i \Delta (\left \vert j \rangle \langle j+1 \right \vert - i \Delta^\ast \left \vert j+1 \rangle \langle j \right \vert) \end{aligned}

When we write $H^\ast$ we mean element-wise complex conjugation, not Hermitian conjugation. In Python the $H_\text{BdG}$ Hamiltonian can be constructed as follows

def Hbdg(L, mu, w, delta):
# sub-matrices, h-matrix
MH = np.zeros((L, L), dtype=np.complex128)
for j in range(L-1):
MH[j, j] = mu
MH[j, j+1] = np.conjugate(w)
MH[j+1, j] = w
MH[L-1, L-1] = mu

# sub-matrices, D-matrix
MD = np.zeros((L, L), dtype=np.complex128)
for j in range(L-1):
MD[j, j+1] = np.conjugate(1j*delta)
MD[j+1, j] = 1j*delta

# construct Hbdg matrix
Hbdg = np.kron(np.array([[1, 0], [0, 0]]), MH)
Hbdg += np.kron(np.array([[0, 0], [0, 1]]), -np.conjugate(MH))
Hbdg += np.kron(np.array([[0, 1], [0, 0]]), MD)
Hbdg += np.kron(np.array([[0, 0], [1, 0]]), -np.conjugate(MD))

return Hbdg


and here below we apply it to study the system of $L = 20$ sites increasing $\mu$ up to value of $4 w$ where $w = \Delta$. The number of sites is impressive and would be numerically instensive if we wanted to simulate whole system instead of applying the Bogoliubov-de Gennes approach.

Which matches plot from Topology in Condensed Matter course nicely! We can observe the topological gap opening at $\mu = 2 w$.

The energy spectrum obtained from $H_\text{BdG}$ Hamiltonian is exact in a sense that in our case it provides energies of a single fermion in the 1D chain. From those energies of multi-fermion system can be obtained by computing all binomial combinations of those energies. It is exact in a sense that it provides all the necessary information to recreate the full energy spectrum of the multi-fermion system.

The full source code of this simulation is published under MIT licence on GitHub. If you find errors please tweet me and let me know.