BdG Equation in different spaces

X.-Z. CHEN, 17/05/2024, EIT, Ningbo

1. BdG Eq. in type1-Nambu space

Let us consider a two-dimensional electron system with Rashba spin-orbit-coupling (SOC), Dresselhaus SOC, spatialy-varying magnetic texture and proximity-induced superconductivity. The time-independent Schrodinger
describing such an electron is written in real space as

$\begin{pmatrix} H_0 & -i\Delta(\vec{k}) \sigma_y \\ i \Delta(\vec{k}) \sigma_y & -H_0^{*} \\ \end{pmatrix}\hat{\Psi}(\vec{r})=E\hat{\Psi}(\vec{r}),\tag{1}$

where $\hat{\Psi}(\vec{r})=( c_{\uparrow,\vec{r}},c_{\downarrow,\vec{r}},c_{\uparrow,\vec{r}}^{\dagger},c_{\downarrow,\vec{r}}^{\dagger})^{T}$ is the wavefunction in particle-hole and spin spaces, $\Delta$ is the superconductor order parameter, $E$ are the energy eigenvalues and

$H_0 = \left( \frac{\vec{p}^2}{2m}-\mu \right)\sigma_0 -\frac{\alpha}{\hbar}(p_x\sigma_y-p_y\sigma_x)+ \frac{\gamma}{\hbar} p_x\sigma_z + \vec{J}(\vec{r}) \cdot \vec{\sigma},\tag{2}$

where $\vec{J}(\vec{r}) \cdot \vec{\sigma}=J_x\sigma_x + J_y\sigma_y + J_z\sigma_z$,
$\vec{p}=(p_x,p_y,p_z)$ are the momentum operators, with $p_{i}=-i\hbar\partial_{i}$ ($i=x,y,z$), $m$ is the effective mass of electron, $\mu$ is the chemical potential, $\alpha$ is the strength of Rashba SOC, $\gamma$ is the strength of Dresselhaus SOC, $\vec{J}(\vec{r})$ describes the effective exchange field at position $\vec{r}$, $\vec{\sigma}=(\sigma_x,\sigma_y,\sigma_z)$ are the Pauli
spin matricies, and $\sigma_0$ is the $2\times 2$ unit matrix.

$H_0^{*} = \left( \frac{\vec{p}^2}{2m}-\mu \right)\sigma_0 -\frac{\alpha}{\hbar}(p_x\sigma_y+p_y\sigma_x)- \frac{\gamma}{\hbar} p_x\sigma_z + J_x\sigma_x - J_y\sigma_y + J_z\sigma_z,\tag{3}\label{ham_conj}$

so we get

$\begin{split} H_{BdG} &= \left( \frac{\vec{p}^2}{2m}-\mu \right)\sigma_0\tau_z -\frac{\alpha}{\hbar}p_x\sigma_y\tau_z+\frac{\alpha}{\hbar}p_y\sigma_x\tau_0 + \frac{\gamma}{\hbar} p_x\sigma_z\tau_0 \\ & + J_x\sigma_x\tau_z + J_y\sigma_y\tau_0 + J_z\sigma_z\tau_z + \Delta(\vec{r}) \sigma_y \tau_y, \\ \end{split},\tag{4}$

For the one-dimensional case, and ingore the z-component of $\vec{J}$ and Dresselhaus SOC, we get

$H_{BdG} = \left( \frac{\vec{p_x}^2}{2m}-\mu \right)\sigma_0\tau_z -\frac{\alpha}{\hbar}p_x\sigma_y\tau_z+ J_x\sigma_x\tau_z + J_y\sigma_y\tau_0 + \Delta(\vec{r}) \sigma_y\tau_y,\tag{5}$

1.1 $s$-wave and $p$-wave pair

1.1.1 $s$-wave 1d BdG equation

$s$-wave S.C. with a constant pairing $\Delta(\vec{k})=\Delta_s$

$H_{BdG} = \left( \frac{\vec{p_x}^2}{2m}-\mu \right)\sigma_0\tau_z -\frac{\alpha}{\hbar}p_x\sigma_y\tau_z+ J_x\sigma_x\tau_z + J_y\sigma_y\tau_0 + \Delta_s \sigma_y\tau_y,\tag{6}$

which correspond the Hamiltonian given in arXiv:2311.09009.

s-wave Josephson junction in convention space

Left S.C.

$\Delta_{s,l} = \Delta_s \sigma_y \tau_y ,\tag{7}$

Right S.C. with phase $\phi$ arXiv:2311.09009,

$\begin{split} \Delta_{s,r} &= \Delta_s e^{i\phi}\sigma_y(\tau_y-i\tau_x)/2 + h.c. \\ &= \Delta_s e^{i\phi}\sigma_y(\tau_y-i\tau_x)/2 + \Delta_s e^{-i\phi}\sigma_y(\tau_y+i\tau_x)/2 \end{split}\tag{8}$

1.1.2 $p$-wave 1d BdG equation

$p$-wave S.C. with a time-reversal broken p-wave S.C. with $\Delta(\vec{k})=\Delta_p(p_x-ip_y)$ [Ramon:review Eq.46, 2024-PRL-Strkalj]

$H_{BdG} = \left( \frac{\vec{p_x}^2}{2m}-\mu \right)\sigma_0\tau_z -\frac{\alpha}{\hbar}p_x\sigma_y\tau_z+ J_x\sigma_x\tau_z + J_y\sigma_y\tau_0 + \Delta_p p_x \sigma_y\tau_y,\tag{9}$

p-wave Josephson junction in convention space

Left S.C.

$\Delta_{p,l} = \Delta_p p_x \sigma_y \tau_y,\tag{10}$

Right S.C.

$\begin{split} \Delta_{p,r} &= \Delta_p p_x e^{i\phi} \sigma_y (\tau_y-i\tau_x)/2 + h.c.,\\ &=\Delta_p p_x e^{i\phi} \sigma_y (\tau_y -i\tau_x)/2 + \Delta_p p_x e^{-i\phi} \sigma_y (\tau_y+i\tau_x)/2 \end{split}\tag{11}\label{eq11}$

Note that, the Hermitian conjugate (h.c.) seems not work on the $p_x$?

2. BdG Eq. in type2-Nambu space

Same as the Eq. (1) setting but for Nambu space. The Hamiltonian read as

$\begin{pmatrix} H_0(\vec{r}) & \Delta(\vec{r})\\ \Delta^{*}(\vec{r}) & -\sigma_y H_0^{*} \sigma_y\\ \end{pmatrix}\hat{\Psi}(\vec{r}) = E \hat{\Psi}(\vec{r}),\tag{12}$

where $\hat{\Psi}(\vec{r})=(c_{\uparrow,\vec{r}},c_{\downarrow,\vec{r}},-c_{\downarrow,\vec{r}}^{\dagger},c_{\uparrow,\vec{r}}^{\dagger})^{T}$ is the wavefunction in Nambu spaces. Here, we first show the common used for calculation,

$\begin{split} \sigma_y \sigma_0 \sigma_y = \sigma_0,\quad & \sigma_y \sigma_x \sigma_y = -\sigma_x,\\ \sigma_y \sigma_y \sigma_y = \sigma_y,\quad & \sigma_y \sigma_z \sigma_y = -\sigma_z. \end{split},\tag{13}$

Combined with Eq. \eqref{ham_conj}, we can get

$\begin{split} \sigma_y H_0^{*} \sigma_y &= \left( \frac{\vec{p}^2}{2m}-\mu \right)\sigma_0 - \frac{\alpha}{\hbar}(p_x\sigma_y-p_y\sigma_x)\\ &+\frac{\gamma}{\hbar}p_x\sigma_z - J_x\sigma_x -J_y\sigma_y -J_z\sigma_z \end{split},\tag{14}$

so we get

$\begin{split} H_{BdG} &= \left( \frac{\vec{p}^2}{2m}-\mu \right)\sigma_0 \tau_z -\frac{\alpha}{\hbar}p_x\sigma_y\tau_z + \frac{\alpha}{\hbar}p_y\sigma_x\tau_z\\ &+ \frac{\gamma}{\hbar}p_x\sigma_z\tau_z + J_x\sigma_x\tau_0 + J_y\sigma_y\tau_0 + J_z\sigma_z\tau_0 + \Delta(\vec{r}) \sigma_0 \tau_x \end{split},\tag{15}$

For the one-dimensional case, and ignore the z-component of $\vec{J}$ and Dresselhaus SOC, we get

$H_{BdG}=\left( \frac{\vec{p}^2}{2m}-\mu \right) \tau_z -\frac{\alpha}{\hbar}p_x\sigma_y\tau_z + J_x\sigma_x + J_y\sigma_y + \Delta(\vec{r}) \tau_x,\tag{16}$

2.1 $s$-wave and $p$-wave pair

2.1.1 $s$-wave 1d BdG equation

$s$-wave S.C. with a constant pairing $\Delta(\vec{r})=\Delta$

$H_{BdG}=\left( \frac{\vec{p}^2}{2m}-\mu \right) \tau_z -\frac{\alpha}{\hbar}p_x\sigma_y\tau_z + J_x\sigma_x + J_y\sigma_y + \Delta_s \tau_x,\tag{17}$

$s$-wave Josephson junction in Nambu spaces

Left S.C.

$\Delta_{s,l}(x) = \Delta_s \tau_x,\tag{18}$

Right S.C. with phase $\phi$

$\begin{split} \Delta_{s,r}(x) &= \Delta_s e^{i\phi} (\tau_x-i\tau_y)/2 + h.c.\\ & = \Delta_s e^{i\phi} (\tau_x-i\tau_y)/2 + \Delta_s e^{-i\phi}(\tau_x+i\tau_y)/2 \end{split}\tag{19}$

2.1.2 $p$-wave 1d BdG equation

$p$-wave S.C. $\Delta(\vec{r})=\Delta_p (p_x-ip_y)$

$H_{BdG} = \left( \frac{\vec{p}_x^2}{2m} -\mu \right) \tau_z -\frac{\alpha}{\hbar} p_x \sigma_y \tau_z + J_x\sigma_x + J_y\sigma_y + \Delta_p p_x \tau_x, \tag{20}$

$p$-wave Josephson junction in Nambu spaces

Left S.C.

$\Delta_{p,l}(x) = \Delta_p p_x \tau_x,\tag{21}$

Right S.C. with phase $\phi$

$\begin{split} \Delta_{p,r}(x) &= \Delta_p p_x e^{i\phi}(\tau_x-i\tau_y)/2 + h.c. \\ &= \Delta_p p_x e^{i\phi}(\tau_x-i\tau_y)/2 + \Delta_p p_x e^{-i\phi}(\tau_x+i\tau_y)/2 \end{split}\tag{22}$

When we solve BdG equation, we always find some Nambu spaces which just slightly different from the standard one $\hat{\Psi}(\vec{r})=(c_{\uparrow,\vec{r}},c_{\downarrow,\vec{r}},-c_{\downarrow,\vec{r}}^{\dagger},c_{\uparrow,\vec{r}}^{\dagger})^{T}$, i.e. the Nambu space used in Fu’s paper[2008-Fu-PRL] is $\hat{\Psi}(\vec{r})=(c_{\uparrow,\vec{r}},c_{\downarrow,\vec{r}},c_{\downarrow,\vec{r}}^{\dagger},-c_{\uparrow,\vec{r}}^{\dagger})^{T}$, an important question is how to solve the BdG equation in this new Nambu space? The following article answer this question.

W. Tu, X.-Z. CHEN, BdG in slightly different Nambu space