Zero Energy Wavefunctions in 1D superconductor

In summary, the researcher is trying to derive the wavefunctions of the zero energy solutions of the Schrodinger equation in a 1D p-wave superconductor (Kitaev model). They started with the Hamiltonian$$\begin{equation}H = \left[\begin{array}{cc}\epsilon_k & \Delta^{\ast}_k\\\Delta_k & -\epsilon_k\end{array}\right]\end{equation}and assumption that \epsilon_k is linear and approximate it in terms of Fermi velocity, v_F. They
  • #1
DeathbyGreen
84
16
[itex]\bf{Setup}[/itex]
Hi! I am trying to derive the wavefunctions of the zero energy solutions of the Schrodinger equation in a 1D p-wave superconductor (Kitaev model). I am starting with the Hamiltonian
$$
\begin{equation}
H =
\left[\begin{array}{cc}
\epsilon_k & \Delta^{\ast}_k\\
\Delta_k & -\epsilon_k
\end{array}
\right]
\end{equation}
$$
where [itex]\epsilon_k = -\mu - t\cos{k}[/itex] ([itex]\mu[/itex] is the chemical potential, [itex]t[/itex] the hopping parameter, and [itex]k[/itex] the crystal momentum) and [itex]\Delta = -i\Delta\sin{k}[/itex] ([itex]\Delta[/itex] is a pairing parameter from the superconductivity). Zero energy modes are thought to exist near the gap closing at the Fermi energy.
[itex]\bf{Solution\quad Attempt}[/itex]
We can define
$$
\begin{equation}
H\Psi = 0
\end{equation}
$$
It was suggested that I assume [itex]\epsilon_k[/itex] is linear and approximate it in terms of Fermi velocity, [itex]v_F[/itex]. I had some difficulty determining how to define the Fermi velocity without taking the partial derivative of the full dispersion relation, [itex]E(k) = \sqrt{\epsilon_k^2 + \Delta_k^2}[/itex]. So I decided to Taylor expand around the Fermi momentum (the zero modes are thought to occur near this point)
$$
\begin{equation}
\epsilon_k = -\mu - t\cos{k} \rightarrow \mu - t\cos{k_F} + t\sin{k_F}(k-k_F)
\\ \Delta = -i\Delta\sin{k} \rightarrow -i\Delta\sin{k_F} -i\Delta\cos{k_F}(k-k_F)
\end{equation}
$$
If I define Fermi velocity as [itex]\frac{\partial\epsilon_k}{\partial k}[/itex] then [itex]v_F = t\sin{k_F}[/itex], and also that [itex]\frac{dv_F}{dk} = 0[/itex] (my justification being that the "acceleration" at the Fermi level would be 0 since it is the maximum momentum value), these become
$$
\begin{equation}
\epsilon_k = -\mu - t\sin{k_F}(k-k_F)
\\ \Delta = -i\Delta\sin{k_F}
\end{equation}
$$
I assume [itex]\mu[/itex] is [itex]0[/itex] because we are looking at the Fermi level. So
$$
\begin{equation}
\epsilon_k = t\sin{k_F}(k-k_F) \rightarrow v_Fp\\
\Delta = -i\Delta_{k_F}
\end{equation}
$$
where [itex]p=(k-k_F)[/itex] is the momentum with respect to the Fermi level, which I replace by [itex]p \rightarrow -i\partial_x[/itex] (justification that the momentum near the Fermi level follows the same form everywhere) and [itex]\Delta_{k_F} = \frac{v_F\Delta}{t}[/itex]. So my simplified Hamiltonian becomes
$$
\begin{equation}
H =
\left[\begin{array}{cc}
v_Fp & i\Delta_{k_F}\\
-i\Delta_{k_F} & -v_Fp
\end{array}
\right] = v_Fp\sigma_z - \Delta_{k_F}\sigma_y
\end{equation}
$$
([itex]\sigma[/itex]'s are the Pauli Matrices) I then multiply from the left by [itex]\sigma_z[/itex], assume a wavefunction form of [itex]\Psi = \chi_{\nu}\phi[/itex], with [itex]\chi_{\nu}[/itex] an eigenvector of [itex]\sigma_x[/itex] (the first term in the simplified Hamiltonian is the identity matrix after multiplying by [/itex]\sigma_z[/itex]) and [itex]\phi \propto e^{\lambda x}[/itex].
[itex]\bf{Question}[/itex]
Are my assumptions for the simplifications of the Hamiltonian matrix elements correct? Is my understanding of Fermi velocity and its derivative okay?
Thank you for your time!
 
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  • #2
Sounds mostly right to me. However ##\mu## isn't 0 but rather ##\epsilon_{k_F}##, at least at T=0. Furthermore, I don't believe that acceleration vanishes at the Fermi energy. This is rather an artifact of you breaking the Fourier sum after the linear term.
Btw. I thought there is no superconductivity in one-dimensional systems?
 
  • #3
Thank you! The system is theoretical regarding the SC in 1D question. There are attempted experimental realizations of this using a 1D quantum wire placed on an s wave SC to gain a pairing term from the proximity effect. So if I instead have [itex] \epsilon_k \rightarrow \epsilon_{k_F} + v_Fp[/itex], could I drop the [itex] \epsilon_{k_F} [/itex] term?
 

1. What is a zero energy wavefunction in a 1D superconductor?

A zero energy wavefunction in a 1D superconductor refers to a state in which the energy of the particle or system is equal to zero. In superconductors, this state is characterized by a gap in the energy spectrum, where all states below the gap have zero energy.

2. How does a zero energy wavefunction contribute to superconductivity?

The presence of zero energy wavefunctions in a superconductor is essential for the phenomenon of superconductivity. These wavefunctions allow for the formation of Cooper pairs, which are responsible for the zero resistance and perfect conductivity observed in superconductors.

3. Can zero energy wavefunctions exist in other types of materials?

Yes, zero energy wavefunctions can exist in materials other than superconductors. They can also be found in topological insulators, which are materials that have insulating bulk states but conductive surface states. In these materials, the zero energy wavefunctions are protected by topology and can have unique properties.

4. How do scientists study zero energy wavefunctions in 1D superconductors?

Scientists can study zero energy wavefunctions in 1D superconductors using various techniques, such as tunneling spectroscopy, transport measurements, and scanning tunneling microscopy. These methods allow for the characterization and visualization of the energy spectrum and wavefunctions in superconducting materials.

5. What are the potential applications of zero energy wavefunctions in 1D superconductors?

Zero energy wavefunctions in 1D superconductors have potential applications in quantum computing and information processing. They can also be used in the development of more efficient and high-performance electronic devices, such as superconducting transistors and sensors. Additionally, the unique properties of zero energy wavefunctions in topological insulators could lead to new types of electronic devices and technologies.

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