Do holes have wavefunctions?

Question

Do holes (as in the absence of an electron) have wavefunctions?

In my understanding, when we talk about holes, we are implicitly invoking two multiparticle wavefunctions: $$\tag{1} \Psi(x_1,...,x_N)= \left| \begin{matrix} \psi_1(x_1) & ... & \psi_N(x_1) \\ \vdots & & \vdots \\ \psi_1(x_N) & ... & \psi_N(x_N) \end{matrix} \right|$$ and $$\tag{1} \Phi(x_1,...,x_{N-1})= \left| \begin{matrix} \psi_1(x_1) & ... & \psi_{N-1}(x_1) \\ \vdots & & \vdots \\ \psi_1(x_{N-1}) & ... & \psi_{N-1}(x_{N-1}) \end{matrix} \right|$$ Then we can say that $\Phi$ is $\Psi$ with an additional hole at a single-particle orbital $N$. (I am ignoring the fact that not all multiparticle states can be written as a Slater determinant.)

I think I've heard people talk about "hole wavefunctions." How do we define a hole wavefunction? In fact, can we even define a single particle wavefunction in a multiparticle system? If it exists, does the hole wavefunction need to be antisymmetrized with electron states (i.e., for an exciton, writing a Slater determinant for the hole and the electron)?

As a side note, I could also ask similar questions about positrons.

Answer 1

You have everything pretty much correct. If you have a piece of semiconductor with $10^{18}$ electrons, a full valence band would be $$\tag{3} \Psi(x_1,...,x_{10^{18}})= \left| \begin{matrix} \psi_1(x_1) & ... & \psi_{10^{18}}(x_1) \\ \vdots & & \vdots \\ \psi_1(x_{10^{18}}) & ... & \psi_{10^{18}}(x_{10^{18}}) \end{matrix} \right|$$ and then a valence band with five holes in it would be $$\tag{4} \Phi(x_1,...,x_{10^{18}-5})= \left| \begin{matrix} \psi_1(x_1) & ... & \psi_{10^{18}-5}(x_1) \\ \vdots & & \vdots \\ \psi_1(x_{10^{18}-5}) & ... & \psi_{10^{18}-5}(x_{10^{18}-5}) \end{matrix} \right|$$

Those five holes would have the wavefunctions $\psi_{10^{18}-4}(x)$, $\psi_{10^{18}-3}(x)$, $\psi_{10^{18}-2}(x)$, $\psi_{10^{18}-1}(x)$, $\psi_{10^{18}}(x)$.

So a single-particle wavefunction in a multiparticle system is just one of the entries of the Slater determinant.

Most metals and semiconductors can be described in the "single-particle approximation", i.e. there is at least one way to write the Slater determinant so that to a very good approximation you can treat each entry $\psi_i$ as a separate particle, behaving like you would expect a typical particle to behave, and only weakly interacting with the other particles (described by the other $\psi_j$). That is the situation in which people are usually talking about single-electron or single-hole wavefunctions.

Answer 2

Holes are easy to define in second quantization, as the result of removing an electron below the Fermi surface. That is, we defined the ground state $|\emptyset\rangle$ as: $$ a_{\mathbf{k},\sigma}|\emptyset\rangle=0,\text{ if } \epsilon_{\mathbf{k},\sigma}>\epsilon_F,\\ a_{\mathbf{k},\sigma}^\dagger|\emptyset\rangle=0,\text{ if } \epsilon_{\mathbf{k},\sigma}\leq\epsilon_F $$ that is, all the states above the Fermi level are empty, whereas all the states below it are full. Then we can redefine the (true) electrons as electrons (quasiparticles) and holes with creation/annihilation operators: $$ a_{\mathbf{k},\sigma}=\begin{cases} e_{\mathbf{k},\sigma}, \text{ if } \epsilon_{\mathbf{k},\sigma}>\epsilon_F,\\ e_{\mathbf{k},\sigma}, \text{ if } h_{-\mathbf{k},-\sigma}^\dagger\leq\epsilon_F \end{cases},\\ a_{\mathbf{k},\sigma}^\dagger=\begin{cases} e_{\mathbf{k},\sigma}, \text{ if } \epsilon_{\mathbf{k},\sigma}^\dagger>\epsilon_F,\\ e_{\mathbf{k},\sigma}, \text{ if } h_{-\mathbf{k},-\sigma}\leq\epsilon_F \end{cases} $$

The operator creating a true electron at point $\mathbf{x}$ then can be decomposed as $$ \psi_\sigma(\mathbf{x})=\sum_{\mathbf{k},\sigma}a_{\mathbf{k},\sigma}\phi_{\mathbf{k},\sigma}(\mathbf{x})= \sum_{\mathbf{k},\sigma}e_{\mathbf{k},\sigma}\phi_{\mathbf{k},\sigma}(\mathbf{x})\theta(\epsilon_{\mathbf{k},\sigma}-\epsilon_F)+\sum_{\mathbf{k},\sigma}h_{-\mathbf{k},-\sigma}\phi_{\mathbf{k},\sigma}(\mathbf{x})\theta(\epsilon_F-\epsilon_{\mathbf{k},\sigma})=\\ \psi_\sigma^{(e)}(\mathbf{x})+\psi_{-\sigma}^{(h)}(\mathbf{x}) $$ where $\psi_\sigma^{(e)}(\mathbf{x})$, $\psi_{-\sigma}^{(h)}(\mathbf{x})$ are electron and hole annihilation operators at point $\mathbf{x}$ - corresponding to the "wavefunctions" of electron and hole in the first quantization.