Excitons in a Magnetic Field: Hamiltonian and Pseudomomentum

1. The Hamiltonian for an Electron-Hole Pair in a Magnetic Field

Our starting point is a system of two interacting particles: an electron (charge $-e$, effective mass $m_e^{\ast }$, position ${\mathbf{r}}_e$) and a hole (charge $+e$, effective mass $m_h^{\ast }$, position ${\mathbf{r}}_h$). They are in a uniform magnetic field $\mathbf{B}$ and interact via a Coulomb potential $V(|{\mathbf{r}}_e-{\mathbf{r}}_h|)$.

From your studies in quantum mechanics and electromagnetism, you know that we incorporate a magnetic field via the vector potential $\mathbf{A}$ using the principle of minimal coupling, which replaces the canonical momentum $\mathbf{p}$ with the kinetic momentum $\pi =\mathbf{p}-q\mathbf{A}$. The total Hamiltonian is the sum of the kinetic energies plus the potential energy:

For a uniform magnetic field, a convenient choice is the symmetric gauge, $\mathbf{A}(\mathbf{r})=\frac{1}{2}(\mathbf{B}\times \mathbf{r})$. Expanding the squared terms introduces terms linear and quadratic in $\mathbf{B}$.

2. Center-of-Mass and Relative Coordinates

The next logical step is to transform to center-of-mass (CM) and relative coordinates, which simplifies two-body problems. We define:

• Relative coordinate: $\mathbf{r}={\mathbf{r}}_e-{\mathbf{r}}_h$
• CM coordinate: $\mathbf{R}=\frac{m_e^{\ast }{\mathbf{r}}_e+m_h^{\ast }{\mathbf{r}}_h}{m_e^{\ast }+m_h^{\ast }}$

We also define the associated total mass $M=m_e^{\ast }+m_h^{\ast }$ and reduced mass $\mu =\frac{m_e^{\ast }{m}_h^{\ast }}{m_e^{\ast }+m_h^{\ast }}$.

Rewriting the Hamiltonian in these new coordinates is a standard but algebraically intensive task. The following resource provides a detailed, step-by-step derivation.

After this transformation, the Hamiltonian (Eq. S-7 in the resource) contains terms like $\frac{1}{2M}{\hat{\mathbf{P}}}^2$ and $\frac{1}{2\mu }{\hat{\mathbf{p}}}^2$, which look familiar. However, it also contains problematic coupling terms, such as:

These terms mix the CM ($\mathbf{R},\hat{\mathbf{P}}$) and relative ($\mathbf{r},\hat{\mathbf{p}}$) degrees of freedom. This means the Hamiltonian is not separable, and we cannot write the wavefunction as a simple product $\Psi (\mathbf{R},\mathbf{r})=\varphi (\mathbf{R})\psi (\mathbf{r})$. Physically, the Lorentz force on the composite object depends not just on its overall velocity but also on its internal structure and motion, and vice versa.

3. The Conserved Pseudomomentum

The obstacle to separability is the breaking of simple translational invariance. A spatial translation $\mathbf{R}\to \mathbf{R}+\mathbf{a}$ changes the vector potential, so the Hamiltonian is not invariant, and the total canonical momentum ${\hat{\mathbf{P}}}_{total}={\hat{\mathbf{p}}}_e+{\hat{\mathbf{p}}}_h$ is not conserved.

However, a different symmetry exists. It corresponds to a "magnetic translation," which is a spatial translation combined with a specific gauge transformation that leaves the physics unchanged. The generator of this symmetry transformation is a conserved quantity known as the pseudomomentum.

For a neutral two-body system like an exciton, the pseudomomentum is defined in several ways. The paper we are following defines it in relation to the CM canonical momentum $\hat{\mathbf{P}}$ and the relative coordinate $\mathbf{r}$:

The most important property of ${\hat{\mathbf{P}}}_0$ is that it commutes with the full exciton Hamiltonian, $[{\hat{\mathbf{P}}}_0,{\hat{H}}_{ex}]=0$. This means ${\hat{\mathbf{P}}}_0$ is a constant of motion, and we can find simultaneous eigenfunctions of both operators.

The proof of this commutation relation involves carefully calculating the commutator of ${\hat{\mathbf{P}}}_0$ with each term in the Hamiltonian from Eq. (S-7).

The physical interpretation of the pseudomomentum is rich. In the classical limit, for a single charged particle, it is related to the "guiding center" of its cyclotron orbit. For our neutral exciton, the pseudomomentum $\mathbf{K}$ (the eigenvalue of ${\hat{\mathbf{P}}}_0$) is proportional to the vector connecting the guiding centers of the electron and the hole. It represents the momentum of the exciton's collective motion through the magnetic field.

4. Pseudoseparation of the Hamiltonian

Since ${\hat{\mathbf{P}}}_0$ is a conserved quantity with eigenvalue $\mathbf{K}$, we can seek a solution of the form:

This is not yet a full separation. The key insight is to use an eigenfunction of the pseudomomentum operator itself. The correct ansatz, which is an eigenfunction of ${\hat{\mathbf{P}}}_0$ with eigenvalue $\mathbf{K}$, is:

Notice that the CM part of the wavefunction (the exponential) contains the relative coordinate $\mathbf{r}$. This is the mathematical embodiment of the coupling.

Substituting this form into the Schrödinger equation ${\hat{H}}_{ex}{\Psi }_{\mathbf{K}}=E_{total}{\Psi }_{\mathbf{K}}$ allows us to eliminate the $\mathbf{R}$ dependence and obtain an effective Schrödinger equation for the relative-motion wavefunction ${\psi }_{\mathbf{K}}(\mathbf{r})$. This procedure is called pseudoseparation.

The total energy is $E_{total}=\frac{K^2}{2M}+E_{rel}$, where the first term is the kinetic energy of the CM motion. The relative motion is governed by the Hamiltonian derived in Eq. (S-21):

where $\sigma =m_e^{\ast }/m_h^{\ast }$, $\hat{\mathbf{p}}$ is the relative momentum, and ${\hat{l}}_z$ is the z-component of the relative angular momentum.

Let's dissect this crucial result:

1. $\frac{{\hat{\mathbf{p}}}^2}{2\mu }+V(|\mathbf{r}|)$: The standard hydrogen-like Hamiltonian for the relative motion.
2. $\frac{e^2{B}^2}{8\mu }{r}^2$: The diamagnetic term. This quadratic potential arises from the ${\mathbf{A}}^2$ term in the original Hamiltonian and is responsible for the quadratic energy shift of exciton states in a magnetic field.
3. $\frac{1-\sigma }{1+\sigma }\frac{eB}{2\mu }{\hat{l}}_z$: A Zeeman-like term. It is linear in the magnetic field $B$ and splits states with different orbital angular momentum $m_l$. Note that this is not the spin Zeeman effect (which we have neglected so far) but an orbital effect modified by the mass asymmetry ($\sigma$).
4. $-\frac{e}{M}(\mathbf{B}\times \mathbf{K})\cdot \mathbf{r}$: The motional Stark term. This is perhaps the most interesting term. It describes how the exciton's CM motion (via pseudomomentum $\mathbf{K}$) in the magnetic field $\mathbf{B}$ creates an effective electric field, ${\mathbf{E}}_{eff}=\frac{1}{M}(\mathbf{K}\times \mathbf{B})$, which then interacts with the exciton's electric dipole moment $e\mathbf{r}$.

Conclusion

Today, we have constructed the essential theoretical framework for analyzing excitons in magnetic fields. The apparent complexity of the problem led us to a deeper physical principle: the conservation of pseudomomentum.

Key Takeaways:

• The Hamiltonian for a neutral exciton in a uniform magnetic field is non-trivial because it couples the center-of-mass and relative motions.
• The total canonical momentum is not a good quantum number, but the pseudomomentum, $\mathbf{K}$, which generates magnetic translations, is conserved.
• By using $\mathbf{K}$ as a quantum number, we can perform a pseudoseparation of the Hamiltonian. The total energy splits into a CM kinetic energy term, $\frac{K^2}{2M}$, and a term for the relative motion.
• The resulting relative Hamiltonian contains the familiar Coulomb and kinetic terms, plus a quadratic diamagnetic term, an orbital Zeeman-like term, and a motional Stark term that couples the CM motion to the internal structure.

Preview of the Next Lesson

For many experiments, especially optical absorption and PL at low temperature, excitons are created with very small center-of-mass momentum ($\mathbf{K}\approx 0$). In this common and important limit, the motional Stark term vanishes, simplifying the Hamiltonian considerably.

In our next lesson, we will analyze this $\mathbf{K}=0$ Hamiltonian in the weak-field regime. Using perturbation theory, we will calculate the diamagnetic shift and the linear Zeeman splitting of exciton energy levels, providing the theoretical basis for the first observable effects in magneto-spectroscopy.