Landau quantization

From the possible solutions for A, a gauge fixing introduced by Lev Landau is often used for charged particles in a constant magnetic field.[2]

When ${\displaystyle \mathbf {B} ={\begin{pmatrix}0\\0\\B\end{pmatrix}}}$ then ${\displaystyle \mathbf {A} ={\begin{pmatrix}0\\B\cdot x\\0\end{pmatrix}}}$ is a possible solution[3] in the Landau gauge.

In this gauge, the Hamiltonian is

$${\displaystyle {\hat {H}}={\frac {{\hat {p}}_{x}^{2}}{2m}}+{\frac {1}{2m}}\left({\hat {p}}_{y}-qB{\hat {x}}\right)^{2}+{\frac {{\hat {p}}_{z}^{2}}{2m}}.}$$

The operator ${\displaystyle {\hat {p}}_{y}}$ commutes with this Hamiltonian, since the operator ŷ is absent by the choice of gauge. Thus the operator ${\displaystyle {\hat {p}}_{y}}$ can be replaced by its eigenvalue ħk_y. Since ${\displaystyle {\hat {z}}}$ does not appear in the Hamiltonian and only the z-momentum appears in the kinetic energy, this motion along the z-direction is a free motion.

The Hamiltonian can also be written more simply by noting that the cyclotron frequency is ω_c = qB/m, giving

$${\displaystyle {\hat {H}}={\frac {{\hat {p}}_{x}^{2}}{2m}}+{\frac {1}{2}}m\omega _{\rm {c}}^{2}\left({\hat {x}}-{\frac {\hbar k_{y}}{m\omega _{\rm {c}}}}\right)^{2}+{\frac {{\hat {p}}_{z}^{2}}{2m}}.}$$

This is exactly the Hamiltonian for the quantum harmonic oscillator, except with the minimum of the potential shifted in coordinate space by x₀ = ħk_y/mω_c .

To find the energies, note that translating the harmonic oscillator potential does not affect the energies. The energies of this system are thus identical to those of the standard quantum harmonic oscillator,[4]

$${\displaystyle E_{n}=\hbar \omega _{\rm {c}}\left(n+{\frac {1}{2}}\right)+{\frac {p_{z}^{2}}{2m}},\quad n\geq 0~.}$$

The energy does not depend on the quantum number k_y, so there will be a finite number of degeneracies (If the particle is placed in an unconfined space, this degeneracy will correspond to a continuous sequence of ${\displaystyle p_{y}}$). The value of ${\displaystyle p_{z}}$ is continuous if the particle is unconfined in the z-direction and discrete if the particle is bounded in the z-direction also. Each set of wave functions with the same value of n is called a Landau level.

For the wave functions, recall that ${\displaystyle {\hat {p}}_{y}}$ commutes with the Hamiltonian. Then the wave function factors into a product of momentum eigenstates in the y direction and harmonic oscillator eigenstates ${\displaystyle |\phi _{n}\rangle }$ shifted by an amount x₀ in the x direction:

$${\displaystyle \Psi (x,y,z)=e^{i(k_{y}y+k_{z}z)}\phi _{n}(x-x_{0})}$$

where ${\displaystyle k_{z}=p_{z}/\hbar }$. In sum, the state of the electron is characterized by the quantum numbers, n, k_y and k_z.

The derivation treated x and y as asymmetric. However, by the symmetry of the system, there is no physical quantity which distinguishes these coordinates. The same result could have been obtained with an appropriate interchange of x and y.

A more adequate choice of gauge, is the symmetric gauge, which refers to the choice

$${\displaystyle {\hat {\mathbf {A} }}={\frac {1}{2}}\mathbf {B} \times {\hat {\mathbf {r} }}={\frac {1}{2}}{\begin{pmatrix}-By\\Bx\\0\end{pmatrix}}.}$$

In terms of dimensionless lengths and energies, the Hamiltonian can be expressed as

$${\displaystyle {\hat {H}}={\frac {1}{2}}\left[\left(-i{\frac {\partial }{\partial x}}-{\frac {y}{2}}\right)^{2}+\left(-i{\frac {\partial }{\partial y}}+{\frac {x}{2}}\right)^{2}\right]}$$

The correct units can be restored by introducing factors of ${\displaystyle q,\hbar ,\mathbf {B} }$ and ${\displaystyle m}$.

Consider operators

$${\displaystyle {\begin{aligned}{\hat {a}}&={\frac {1}{\sqrt {2}}}\left[\left({\frac {x}{2}}+{\frac {\partial }{\partial x}}\right)-i\left({\frac {y}{2}}+{\frac {\partial }{\partial y}}\right)\right]\\{\hat {a}}^{\dagger }&={\frac {1}{\sqrt {2}}}\left[\left({\frac {x}{2}}-{\frac {\partial }{\partial x}}\right)+i\left({\frac {y}{2}}-{\frac {\partial }{\partial y}}\right)\right]\\{\hat {b}}&={\frac {1}{\sqrt {2}}}\left[\left({\frac {x}{2}}+{\frac {\partial }{\partial x}}\right)+i\left({\frac {y}{2}}+{\frac {\partial }{\partial y}}\right)\right]\\{\hat {b}}^{\dagger }&={\frac {1}{\sqrt {2}}}\left[\left({\frac {x}{2}}-{\frac {\partial }{\partial x}}\right)-i\left({\frac {y}{2}}-{\frac {\partial }{\partial y}}\right)\right]\end{aligned}}}$$

These operators follow certain commutation relations

$${\displaystyle [{\hat {a}},{\hat {a}}^{\dagger }]=[{\hat {b}},{\hat {b}}^{\dagger }]=1.}$$

In terms of above operators the Hamiltonian can be written as

$${\displaystyle {\hat {H}}=\hbar \omega _{\rm {c}}\left({\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}\right),}$$

where we reintroduced the units back.

The Landau level index ${\displaystyle n}$ is the eigenvalue of the operator ${\displaystyle {\hat {N}}={\hat {a}}^{\dagger }{\hat {a}}}$.

The application of ${\displaystyle {\hat {b}}^{\dagger }}$ increases ${\displaystyle m_{z}}$ by one unit while preserving ${\displaystyle n}$, whereas ${\displaystyle {\hat {a}}^{\dagger }}$ application simultaneously increase ${\displaystyle n}$ and decreases ${\displaystyle m_{z}}$ by one unit. The analogy to quantum harmonic oscillator provides solutions

$${\displaystyle {\hat {H}}|n,m_{z}\rangle =E_{n}|n,m_{z}\rangle ,}$$

where

$${\displaystyle E_{n}=\hbar \omega _{\rm {c}}\left(n+{\frac {1}{2}}\right)}$$

and

$${\displaystyle |n,m_{z}\rangle ={\frac {({\hat {b}}^{\dagger })^{m_{z}+n}}{\sqrt {(m_{z}+n)!}}}{\frac {({\hat {a}}^{\dagger })^{n}}{\sqrt {n!}}}|0,0\rangle .}$$

One may verify that the above states correspond to choosing wavefunctions proportional to

$${\displaystyle \psi _{n,m_{z}}(x,y)=\left({\frac {\partial }{\partial w}}-{\frac {\bar {w}}{4}}\right)^{n}w^{n+m_{z}}e^{-|w|^{2}/4}}$$

where ${\displaystyle w=x-iy}$.

In particular, the lowest Landau level ${\displaystyle n=0}$ consists of arbitrary analytic functions multiplying a Gaussian, ${\displaystyle \psi (x,y)=f(w)e^{-|w|^{2}/4}}$.

The effects of Landau levels may only be observed when the mean thermal energy kT is smaller than the energy level separation, kT ≪ ħω_c, meaning low temperatures and strong magnetic fields.

Each Landau level is degenerate because of the second quantum number k_y, which can take the values

$${\displaystyle k_{y}={\frac {2\pi N}{L_{y}}},}$$

where N is an integer. The allowed values of N are further restricted by the condition that the center of force of the oscillator, x₀, must physically lie within the system, 0 ≤ x₀ < L_x. This gives the following range for N,

$${\displaystyle 0\leq N<{\frac {m\omega _{\rm {c}}L_{x}L_{y}}{2\pi \hbar }}.}$$

For particles with charge q = Ze, the upper bound on N can be simply written as a ratio of fluxes,

$${\displaystyle {\frac {ZBL_{x}L_{y}}{(h/e)}}=Z{\frac {\Phi }{\Phi _{0}}},}$$

where Φ₀ = h/e is the fundamental magnetic flux quantum and Φ = BA is the flux through the system (with area A = L_xL_y).

Thus, for particles with spin S, the maximum number D of particles per Landau level is

$${\displaystyle D=Z(2S+1){\frac {\Phi }{\Phi _{0}}}~,}$$

which for electrons (where Z = 1 and S = 1/2) gives D = 2Φ/Φ₀, two available states for each flux quantum that penetrates the system.

The above gives only a rough idea of the effects of finite-size geometry. Strictly speaking, using the standard solution of the harmonic oscillator is only valid for systems unbounded in the x-direction (infinite strips). If the size L_x is finite, boundary conditions in that direction give rise to non-standard quantization conditions on the magnetic field, involving (in principle) both solutions to the Hermite equation. The filling of these levels with many electrons is still[5] an active area of research.

In general, Landau levels are observed in electronic systems. As the magnetic field is increased, more and more electrons can fit into a given Landau level. The occupation of the highest Landau level ranges from completely full to entirely empty, leading to oscillations in various electronic properties (see De Haas–Van Alphen effect and Shubnikov–de Haas effect).

If Zeeman splitting is included, each Landau level splits into a pair, one for spin up electrons and the other for spin down electrons. Then the occupation of each spin Landau level is just the ratio of fluxes D = Φ/Φ₀. Zeeman splitting has a significant effect on the Landau levels because their energy scales are the same, 2μ_BB = ħω_c. However, the Fermi energy and ground state energy stay roughly the same in a system with many filled levels, since pairs of split energy levels cancel each other out when summed.

Moreover, the above derivation in the Landau gauge assumed an electron confined in the z-direction, which is a relevant experimental situation — found in two-dimensional electron gases, for instance. Still, this assumption is not essential for the results. If electrons are free to move along the z direction, the wave function acquires an additional multiplicative term exp(ik_zz); the energy corresponding to this free motion, (ħ k_z)²/(2m), is added to the E discussed. This term then fills in the separation in energy of the different Landau levels, blurring the effect of the quantization. Nevertheless, the motion in the x-y-plane, perpendicular to the magnetic field, is still quantized.

Each Landau level has degenerate orbitals labeled by the quantum numbers ${\displaystyle m_{z}}$ in symmetric gauge. The degeneracy per unit area is the same in each Landau level.

The z component of angular momentum is

$${\displaystyle {\hat {L}}_{z}=-i\hbar {\frac {\partial }{\partial \theta }}=-\hbar ({\hat {b}}^{\dagger }{\hat {b}}-{\hat {a}}^{\dagger }{\hat {a}})}$$

Exploiting the property ${\displaystyle [{\hat {H}},{\hat {L}}_{z}]=0}$ we chose eigenfunctions which diagonalize ${\displaystyle {\hat {H}}}$ and ${\displaystyle {\hat {L}}_{z}}$, The eigenvalue of ${\displaystyle {\hat {L}}_{z}}$ is denoted by ${\displaystyle -m_{z}\hbar }$, where it is clear that ${\displaystyle m_{z}\geq -n}$ in the ${\displaystyle n}$th Landau level. However, it may be arbitrarily large, which is necessary to obtain the infinite degeneracy (or finite degeneracy per unit area) exhibited by the system.