Topological Superconductors

『この可憐な魔女は誰でしょう。ーーそう、Majoranaです。』

Classification of Topological Superconductors

• Topological superconductors are those superconductors with topologically nontrivial wavefunctions.

• BCS (s-wave) superconductors are topologically trivial.

• P-wave superconductors are basically topologically nontrivial.

Candidates of Topological Superconductors

• \(\ce{Cu_xBi2Se3}\) (doped \(\ce{Bi2Se3}\)).

• \(\ce{Sn_{1-x}In_xTe}\) (doped topological crystalline insulator \(\ce{SnTn}\)).

  • Strong spin-orbit interaction couples electrons of different parities into Cooper pairs.

Majorana Fermions

• Majorana condition: \(\gamma\) denoting the annihilation operator of Majorana fermions,

  \[\gamma^\dagger = \gamma.\]

• Pairing electrons of the same spin:

  \[\gamma = uc_\uparrow + u^*c_\uparrow^\dagger.\]

• Existence of Majorana fermions (as Bogoliubov quasiparticles) in superconductors implies nontrivial topology, but not vice versa.

Kitaev Model

• Hamiltonian (spinless):

  \[H = -\mu \sum_i c^\dagger_i c_i - \frac{1}{2} \sum_i (tc^\dagger_i c_{i+1} - \Delta c_{i} c_{i+1} + \mathrm{h.c.}),\]

  where \(\mu\) is the chemical potential, \(t\ge 0\) is the hopping amplitude, and \(\Delta \ge 0\) is the strength of p-wave pairing.

Kitaev Model: Periodic Boundary Condition

• Pre-diagonalization

  \[H = \frac{1}{2} \sum_k C^\dagger_{k} \mathcal{H}_k C_k,\]

  where

  \[\begin{split}\mathcal{H}_k &= \begin{pmatrix} \epsilon(k) & \Delta^*_k \\ \Delta_k & -\epsilon(k) \end{pmatrix}, \\ \epsilon(k) &= -t\cos k - \mu, \\ \Delta_k &= -i\Delta \sin k,\end{split}\]

  and

  \[C^\dagger_k = (c^\dagger_k, c_{-k}).\]

• BdG (Bogoliubov-de Gennes) transformation:

  \[\begin{split}a_k &= u_k c_k + v_k c^\dagger_{-k}, \\ u_k &= \frac{\Delta_k}{\abs{\Delta_k}} \frac{\sqrt{E_{\mathrm{b}} + \epsilon(k)}}{\sqrt{2E_{\mathrm{b}}}}, \\ v_k &= \qty(\frac{E_{\mathrm{b}} - \epsilon(k)}{\Delta_k}) u_k.\end{split}\]

• Diagonalization:

  \[H = \sum_k E_{\mathrm{b}} a^\dagger_k a_k,\]

  where

  \[E_{\mathrm{b}} = \sqrt{\epsilon(k)^2 + \abs{\Delta_k}^2}.\]

  • Nonzero except for \(k = \pm \pi\) for \(\mu = t\) and \(k = 0\) for \(\mu = -t\).

  • Finite gap of excitation exists in most cases.

• \(\mathbb{Z}_2\) index:

  \[(-1)^\nu = \frac{\epsilon(0)}{\abs{\epsilon(0)}} \frac{\epsilon(\pi)}{\abs{\epsilon(\pi)}}.\]

  • Trivial (\(\nu = 0\)) for \(\abs{\mu} > t\).

  • Non-trivial (\(\nu = 1\)) for \(\abs{\mu} < t\).

Kitaev Model: Open Boundary Condition

• Hamiltonian with Majorana operators:

  \[H = -\frac{\mu}{2} \sum_{i} (1+i \gamma_{\mathrm{B}, i} \gamma_{\mathrm{A}, i}) - \frac{i}{4} \sum_i \qty[(\Delta + t) \gamma_{\mathrm{B}, i} \gamma_{\mathrm{A}, i+1} + (\Delta - t) \gamma_{\mathrm{A}, i} \gamma_{\mathrm{B}, i+1}],\]

  where

  \[\begin{split}\begin{cases} \displaystyle c_i = \frac{1}{2}(\gamma_{\mathrm{B}, i} + i\gamma_{\mathrm{A}, i}), \\ \displaystyle c^\dagger_i = \frac{1}{2} (\gamma_{\mathrm{B}, i} - i\gamma_{\mathrm{A}, i}), \end{cases} \Leftrightarrow \begin{cases} \gamma_{\mathrm{B}, i} = c^\dagger_i + c_i, \\ \gamma_{\mathrm{A}, i} = i(c^\dagger_i - c_i). \end{cases}\end{split}\]

• Every site now resides two kinds of particles \(\mathrm{A}\) and \(\mathrm{B}\).

• Topologically trivial case \(\mu<0\) and \(t=\Delta = 0\): only on-site terms.

• Topologically non-trivial case \(\mu=0\) and \(t=\Delta \neq 0\):

  \[H = -i\frac{t}{2} \sum_i \gamma_{\mathrm{B},i} \gamma_{\mathrm{A},i}.\]

  • Bonding between adjacent sites.

  • Diagonalization:

    \[H = t \sum_i (\tilde{c}^\dagger_i \tilde{c}_i - \frac{1}{2})\]

    where

    \[\tilde{c}_i = \frac{1}{2} (\gamma_{\mathrm{A},i} + i \gamma_{\mathrm{B},i}).\]

  • Zero-energy Majorana fermion at the edge: Majorana zero mode

    \[f = \frac{1}{2} (\gamma_{\mathrm{A}, i} + i\gamma_{\mathrm{B},N}).\]

  • Ground states with (\(\ket{1}\)) or without (\(\ket{0}\)) Majorana zero mode at the edge:

    \[\ket{1} = f^\dagger \ket{0}.\]

• Such two-fold degeneracy occurs in p-wave topological superconductors.

Spinless P-Wave Chiral Superconductor

• Hamiltonian:

  \[H = \int \dd{^2 \vb{r}} \qty{{ \psi^\dagger \qty(-\frac{\hbar^2 \nabla^2}{2m} - \mu) \psi + \frac{\Delta}{2}\qty[e^{i\phi} \psi \qty(\pdv{}{x} + i\pdv{}{y}) \psi + \mathrm{h.c.}] }},\]

  where \(\phi\) denotes the phase of superconductivity.

• Pre-diagonalization:

  \[H = \frac{1}{2} \int \frac{\dd{^2 \vb{k}}}{(2\pi)^2} \Psi^\dagger(\vb{k}) \mathcal{H}(\vb{k}) \Psi(\vb{k}),\]

  where

  \[\begin{split}\mathcal{H}(\vb{k}) &= \begin{pmatrix} \epsilon(k) & \Delta^*(\vb{k}) \\ \Delta(\vb{k}) & -\epsilon(k) \end{pmatrix}, \\ \epsilon(k) &= \frac{\hbar^2 k^2}{2m} - \mu, \\ \Delta(\vb{k}) &= i\Delta e^{i\phi} (k_x + ik_y),\end{split}\]

  and

  \[\Psi^\dagger(\vb{k}) = (\psi^\dagger(\vb{k}), \psi(-\vb{k})).\]

• BdG (Bogoliubov-de Gennes) transformation:

  \[a(\vb{k}) = u(\vb{k}) \psi(\vb{k}) + v(\vb{k}) \psi^\dagger(-\vb{k}),\]

  and \(u\) and \(v\) takes the same form as in the Kiatev model.

• Diagonalization:

  \[H = \int \frac{\dd{^2\vb{k}}}{(2\pi)^2} E_{\mathrm{b}}(\vb{k}) a^\dagger(\vb{k}) a(\vb{k}),\]

  where

  \[E_{\mathrm{b}}(\vb{k}) = \sqrt{\epsilon(k)^2 + \abs{\Delta(\vb{k})}^2}.\]

  • Gap closed at \(\mu = 0\) and \(\vb{k} = 0\).

  • Otherwise the gap remains finite.

• Ground state

  \[\ket{\mathrm{GS}} = \prod_{k_x \ge 0,k_y} \qty[u(\vb{k}) + v(\vb{k}) \Psi^\dagger(-\vb{k}) \Psi^\dagger(\vb{k})]\ket{0}.\]

Topological Invariance of Spinless P-Wave Superconductor

• Topological if \(\mu>0\).

• Trivial if \(\mu<0\).

• Hamiltonian as in the two-band topological system

  \[\mathcal{H}(\vb{k}) = \vb{h}(\vb{k})\cdot \vb*{\sigma},\]

  where

  \[h_x(\vb{k}) = \Re[\Delta(\vb{k})],\quad h_y(\vb{k}) = \Im[\Delta(\vb{k})],\quad h_z(\vb{k}) = \epsilon(k).\]

• Chern number:

  \[C = \int \frac{\dd{^2 \vb{k}}}{4\pi} \qty[\hat{\vb{h}}(\vb{k}) \cdot \qty({ \pdv{}{k_x} \hat{\vb{h}}(\vb{k}) \times \pdv{}{k_y} \hat{\vb{h}}(\vb{k}) })].\]

  • Counting how many times \(\hat{\vb{h}}\) covers \(S^2\).

• Evaluation of Chern number:

  • \(\hat{\vb{h}}(\infty) = +\hat{\vb{z}}\) for any \(\mu\).

  • \(\hat{\vb{h}}(\infty) = +\hat{\vb{z}}\) for \(\mu < 0\) and therefore topological trivial in this case.

  • \(\hat{\vb{h}}(\infty) = -\hat{\vb{z}}\) for \(\mu > 0\) and therefore topological nontrivial in this case since \(C=-1\).

• Majorana fermions exist at the edge in the topological phase.

  • Majorana zero mode exists with magnetic field applied.

Topological Superconductors in Hybrid Systems

• Contacting an s-wave BCS SC with TI may induce SC states at the surface of TI due to superconducting proximity effect.

• Hamiltonian of TI surface states:

  \[\begin{split}\mathcal{H}_0(\vb{k}) &= \psi^\dagger [-i\hbar v_{\mathrm{F}}(\sigma_x \partial_x + \sigma_y \partial_y)] \psi \\ &= \psi^\dagger(-i\hbar v_{\mathrm{F}} \vb*{\sigma} \cdot \nabla - \mu) \psi.\end{split}\]

• Hamiltonian item added by the proximate SC:

  \[V = \Delta_0 e^{i\phi} \psi^\dagger_\uparrow \psi^\dagger_\downarrow + \mathrm{h.c.}.\]

• Pre-diagonalization:

  \[H = \frac{1}{2} \Psi^\dagger \mathcal{H} \Psi,\]

  where

  \[\begin{split}\mathcal{H} = \begin{pmatrix} -i\hbar v_{\mathrm{F}} \vb*{\sigma} \cdot \nabla - \mu & \Delta_0 (\cos\phi - i \sin\phi) \\ \Delta_0 (\cos\phi + i\sin\phi) & i\hbar v_{\mathrm{F}} \vb*{\sigma} \cdot \nabla + \mu \end{pmatrix},\end{split}\]

  and the Nambu basis is given by

  \[\Psi = ((\psi_\uparrow, \psi_\downarrow), (\psi^\dagger_\downarrow, -\psi^\dagger_\uparrow))^T.\]

• As p-wave super conductor: for \(\hbar v_{\mathrm{F} k_0 \approx \mu\),

  \[H = \sum_{\vb{k}} \qty[(\hbar v_{\mathrm{F}} k_0 - \mu) c^\dagger_{\vb{k}} c_{\vb{k}} + \frac{1}{2} \qty({ \Delta_0 e^{i(\phi + \theta_{\vb{k}})} c^\dagger_{\vb{k}} c^\dagger_{-\vb{k}} + \mathrm{h.c.} })],\]

  where

  \[c_{\mathrm{k}} = \frac{1}{\sqrt{2}} (\psi_{\uparrow\vb{k}} + e^{i\theta_{\vb{k}}} \psi_{\downarrow \vb{k}})\]

  and

  \[\vb{k} = k_0(\cos \theta_{\vb{k}}, \sin \theta_{\vb{k}}).\]