Chapter 2

Quantum Hall Effect

Classical Hall Effect Review

When a two-dimensional electron gas (2DEG) is placed in a perpendicular magnetic field $\mathbf{B} = B\hat{z}$, the Lorentz force deflects moving charges transversely. In steady state, charge accumulation at the sample edges creates a transverse electric field that balances the magnetic force.

Drude Model Result

The classical resistivity tensor in a perpendicular magnetic field is:

$$\rho = \begin{pmatrix} \rho_{xx} & \rho_{xy} \\ -\rho_{xy} & \rho_{xx} \end{pmatrix} = \begin{pmatrix} \frac{m}{ne^2\tau} & \frac{B}{ne} \\ -\frac{B}{ne} & \frac{m}{ne^2\tau} \end{pmatrix}$$

The Hall resistance $R_H = \rho_{xy} = B/(ne)$ grows linearly with field, while the longitudinal resistance $\rho_{xx}$ remains constant. The quantum Hall effect fundamentally departs from this classical behavior.

The classical Hall coefficient $R_H = 1/(ne)$ measures the carrier density and sign. The quantum Hall effect fundamentally departs from this linear behavior when$\omega_c\tau \gg 1$.

Landau Levels

Quantum mechanically, a free electron in a uniform magnetic field has its continuous kinetic energy spectrum collapse into discrete, highly degenerate levels -- the Landau levels. In the Landau gauge$\mathbf{A} = (0, Bx, 0)$, the Hamiltonian for a 2D electron is:

$$H = \frac{1}{2m}\left(p_x^2 + (p_y - eBx)^2\right)$$

Since $p_y$ commutes with $H$, we can write $p_y = \hbar k_y$. Substituting gives a shifted harmonic oscillator in $x$, centered at$x_0 = k_y l_B^2$. The resulting energy eigenvalues are:

Landau Level Energies

$$E_n = \hbar\omega_c\left(n + \frac{1}{2}\right), \quad n = 0, 1, 2, \ldots$$

where the cyclotron frequency is $\omega_c = eB/m$. For GaAs ($m^* = 0.067\,m_e$) at $B = 5$ T, the level spacing is$\hbar\omega_c \approx 8.6$ meV.

Magnetic Length

$$l_B = \sqrt{\frac{\hbar}{eB}}$$

This is the characteristic length scale of the problem, representing the spatial extent of the cyclotron orbit in the lowest Landau level. At $B = 5$ T,$l_B \approx 11.5$ nm. Each Landau level state occupies an area$2\pi l_B^2$ in real space.

Degeneracy per Landau Level

$$N_\phi = \frac{BA}{\Phi_0} = \frac{eBA}{h}$$

where $\Phi_0 = h/e$ is the magnetic flux quantum and $A$ is the sample area. Each Landau level can accommodate exactly one electron per flux quantum threading the sample. The filling factor $\nu = n_e/n_B = n_e h/(eB)$ counts how many Landau levels are filled.

Integer Quantum Hall Effect

Discovered by Klaus von Klitzing in 1980 (Nobel Prize 1985), the integer quantum Hall effect (IQHE) occurs when the filling factor $\nu$ is near an integer. The Hall conductivity exhibits exact quantization:

$$\sigma_{xy} = \nu\frac{e^2}{h}, \quad \sigma_{xx} = 0, \quad \nu = 1, 2, 3, \ldots$$

The quantization is exact to parts in $10^{-9}$, independent of sample geometry, material details, or disorder. This remarkable universality has made the quantum Hall resistance $R_K = h/e^2 = 25\,812.807\,\Omega$ the international resistance standard.

Role of Disorder

Paradoxically, disorder is essential for observing the quantum Hall plateaus. Without disorder, the Hall conductivity would jump discontinuously at each integer filling. Disorder broadens the Landau levels into bands of localized and extended states:

- Extended states: exist only at the center of each broadened Landau level. These carry the Hall current.
- Localized states: fill the tails of the broadened levels. Adding electrons to localized states does not change $\sigma_{xy}$, creating the plateaus.
- Mobility gap: when the Fermi level lies in the localized states between Landau bands, $\sigma_{xx} = 0$ and $\sigma_{xy}$ is quantized.

Experimental Signatures

The hallmark of the IQHE is the simultaneous observation of:

Quantized Hall resistance

$$R_{xy} = \frac{h}{\nu e^2} = \frac{R_K}{\nu}$$

Vanishing longitudinal resistance

$$R_{xx} = 0 \quad \text{(dissipationless)}$$

TKNN Formula: Topology of the Hall Conductance

Thouless, Kohmoto, Nightingale, and den Nijs (1982) showed that the quantized Hall conductance has a deep topological origin. For a filled Landau level (or any filled band in a periodic potential plus magnetic field), the Hall conductance is given by a Chern number:

TKNN Invariant

$$\sigma_{xy} = \frac{e^2}{h}\sum_{n\,\text{filled}} C_n$$

where the first Chern number of the $n$-th band is:

$$C_n = \frac{1}{2\pi}\int_{\text{BZ}} \mathcal{F}_{xy}^{(n)}\,dk_x\,dk_y$$

Here $\mathcal{F}_{xy}^{(n)} = \partial_{k_x}\mathcal{A}_y^{(n)} - \partial_{k_y}\mathcal{A}_x^{(n)}$is the Berry curvature of the $n$-th band, and$\mathcal{A}_\mu^{(n)} = -i\langle u_{n\mathbf{k}}|\partial_{k_\mu}|u_{n\mathbf{k}}\rangle$is the Berry connection.

Because $C_n$ is an integer topological invariant (the integral of curvature over a closed manifold), the Hall conductance is inherently quantized. The Chern number cannot change under smooth deformations of the Hamiltonian that do not close the energy gap, explaining the extraordinary robustness of the quantization.

Geometric Interpretation

The TKNN formula is the condensed matter analog of the Gauss-Bonnet theorem: just as integrating Gaussian curvature over a closed surface yields the integer Euler characteristic $\chi$, integrating Berry curvature over the Brillouin zone torus yields the integer Chern number. Different band structures carry different Chern numbers, exactly as spheres ($\chi = 2$) and tori ($\chi = 0$) have different topology.

Edge States and Bulk-Boundary Correspondence

A quantum Hall system with filling factor $\nu$ supports exactly $\nu$ chiral edge modes at each boundary. These edge states are the physical carriers of the quantized Hall current and arise because the confining potential bends the Landau levels upward at the sample edges.

Chiral Edge Modes

At each edge, the bent Landau levels cross the Fermi energy, creating one-dimensional conducting channels. These edge states have remarkable properties:

- Chirality: electrons propagate in one direction only along each edge (clockwise on one side, counterclockwise on the other). The direction is set by the magnetic field.
- No backscattering: since states at opposite edges propagate in opposite directions, backscattering requires tunneling across the entire bulk, which is exponentially suppressed.
- Quantized conductance: each edge mode contributes $e^2/h$ to the Hall conductance, giving $\sigma_{xy} = \nu e^2/h$ for $\nu$ filled Landau levels.

Bulk-Boundary Correspondence

The number of chiral edge modes equals the bulk topological invariant:

$$N_{\text{edge}} = \sum_{n\,\text{filled}} C_n = \nu$$

This is the bulk-boundary correspondence: the topology of the bulk band structure (characterized by the total Chern number) dictates the number of protected edge modes at any boundary. This principle extends far beyond the quantum Hall effect -- it underlies all topological phases of matter and guarantees that the edge/surface states cannot be removed by any perturbation that preserves the bulk gap.

Laughlin's Gauge Argument

Robert Laughlin (1981) provided an elegant argument for the exact quantization of$\sigma_{xy}$ using gauge invariance alone, without reference to microscopic details.

The Thought Experiment

Consider a 2DEG on a cylinder (Corbino geometry) threaded by a solenoid carrying flux $\Phi$ through the hole. As the flux is adiabatically increased by one flux quantum $\Phi_0 = h/e$:

- By gauge invariance, the system returns to its original state after $\Phi \to \Phi + \Phi_0$.
- The changing flux induces an azimuthal EMF: $\mathcal{E} = -d\Phi/dt$.
- The Hall current flows radially: $I = \sigma_{xy}\mathcal{E}$.
- The total charge transferred from one edge to the other is:

$$\Delta Q = \int I\,dt = \sigma_{xy}\int\mathcal{E}\,dt = \sigma_{xy}\,\Phi_0$$

Since the system returns to its original state, $\Delta Q$ must be an integer number of electron charges: $\Delta Q = \nu e$. Therefore:

$$\sigma_{xy} = \frac{\nu e}{\Phi_0} = \nu\frac{e^2}{h}$$

The argument relies only on gauge invariance and the existence of a mobility gap, explaining the universality of the quantization across materials and geometries.

Fractional Quantum Hall Effect

Discovered by Tsui, Stormer, and Gossard in 1982 (Nobel Prize 1998 with Laughlin), the fractional quantum Hall effect (FQHE) occurs at fractional filling factors$\nu = p/q$ (with $q$ odd). Unlike the IQHE, the FQHE is intrinsically a many-body phenomenon driven by strong electron-electron interactions within a partially filled Landau level.

Laughlin Wavefunction

For filling $\nu = 1/m$ (with $m$ an odd integer), Laughlin proposed the celebrated trial wavefunction:

$$\Psi_m(z_1, \ldots, z_N) = \prod_{i<j}(z_i - z_j)^m \,\exp\!\left(-\sum_k \frac{|z_k|^2}{4l_B^2}\right)$$

where $z_k = x_k + iy_k$ is the complex coordinate of the $k$-th electron. Key properties:

- Antisymmetry: the Jastrow factor $(z_i - z_j)^m$ with $m$ odd ensures fermionic statistics.
- Correlations: the probability of finding two electrons at the same point vanishes as $|z_i - z_j|^{2m}$, keeping electrons apart and minimizing Coulomb repulsion.
- Filling: the highest power of any $z_i$ is $m(N-1)$, corresponding to $m(N-1)$ flux quanta, so $\nu = N/N_\phi \to 1/m$.
- Incompressibility: the state has a finite energy gap to all excitations (quasiholes and quasielectrons), explaining the plateau.

Quasiparticle Excitations

The elementary excitations of the Laughlin state carry fractional charge and obey fractional (anyonic) statistics. A quasihole at position $\eta$ is created by:

$$\Psi_{\text{qh}} = \prod_i (z_i - \eta)\,\Psi_m$$

This quasihole carries fractional charge $e^* = e/m$ and fractional statistics parameter $\theta = \pi/m$. For $\nu = 1/3$, the quasiholes have charge$e/3$ and are anyons (neither bosons nor fermions). The fractional charge has been directly observed in shot noise experiments.

Composite Fermions

Jain's composite fermion (CF) theory (1989) provides a unifying framework for both the IQHE and FQHE. The central idea is to attach an even number $2p$ of magnetic flux quanta to each electron, forming a composite particle.

Flux Attachment

An electron at filling $\nu$ is transformed into a composite fermion experiencing a reduced effective magnetic field:

$$B^* = B - 2p\,n_e\Phi_0$$

The composite fermions fill $\nu^*$ effective Landau levels in the reduced field. The relation between electron filling $\nu$ and CF filling $\nu^*$ is:

$$\nu = \frac{\nu^*}{2p\nu^* \pm 1}$$

Jain Sequence

For $p = 1$ (two flux quanta attached) and $\nu^* = 1, 2, 3, \ldots$, the principal Jain sequence is:

$\nu^*$	$\nu = \nu^/(2\nu^+1)$	$\nu = \nu^/(2\nu^-1)$
1	1/3	1
2	2/5	2/3
3	3/7	3/5
4	4/9	4/7
$\infty$	1/2	1

The $\nu = 1/2$ state corresponds to composite fermions at zero effective field ($B^* = 0$), forming a Fermi sea rather than filling Landau levels. This compressible state has been confirmed by measurements of the CF Fermi surface via commensurability oscillations.

Jain Wavefunctions

The CF wavefunctions are constructed by projecting onto the lowest Landau level:

$$\Psi_{\nu} = \mathcal{P}_{\text{LLL}}\left[\prod_{i<j}(z_i - z_j)^{2p}\,\Phi_{\nu^*}\right]$$

For $\nu^* = 1$ with $p = 1$, this reproduces the Laughlin $\nu = 1/3$ state. Numerical overlaps exceed 99% with exact diagonalization ground states.

Detailed Derivations

Derivation 1: Landau Level Quantization from the Hamiltonian

We derive the Landau level spectrum from the 2D Hamiltonian in a perpendicular magnetic field $\mathbf{B} = B\hat{z}$, working in the Landau gauge $\mathbf{A} = (0, Bx, 0)$.

Step 1: The kinetic momentum operator is $\boldsymbol{\pi} = \mathbf{p} + e\mathbf{A}$. The Hamiltonian is:

$$H = \frac{\boldsymbol{\pi}^2}{2m} = \frac{1}{2m}\left[p_x^2 + (p_y + eBx)^2\right]$$

Step 2: Since $[H, p_y] = 0$ (the Hamiltonian has no explicit $y$ dependence), $p_y$ is a good quantum number. Write $p_y = \hbar k_y$ and substitute:

$$H = \frac{p_x^2}{2m} + \frac{1}{2m}(\hbar k_y + eBx)^2 = \frac{p_x^2}{2m} + \frac{m\omega_c^2}{2}(x - x_0)^2$$

where $\omega_c = eB/m$ is the cyclotron frequency and $x_0 = -\hbar k_y/(eB) = -k_y l_B^2$ is the guiding center coordinate, with $l_B = \sqrt{\hbar/(eB)}$.

Step 3: This is a one-dimensional harmonic oscillator centered at $x_0$. Defining the shifted coordinate $\xi = x - x_0$ and the standard ladder operators:

$$a = \sqrt{\frac{m\omega_c}{2\hbar}}\left(\xi + \frac{ip_x}{m\omega_c}\right), \quad a^\dagger = \sqrt{\frac{m\omega_c}{2\hbar}}\left(\xi - \frac{ip_x}{m\omega_c}\right)$$

these satisfy $[a, a^\dagger] = 1$, and the Hamiltonian becomes:

$$H = \hbar\omega_c\left(a^\dagger a + \frac{1}{2}\right)$$

Step 4: The eigenvalues follow immediately:

$$E_n = \hbar\omega_c\left(n + \frac{1}{2}\right), \quad n = 0, 1, 2, \ldots$$

Step 5 (Degeneracy): The energy is independent of $k_y$, which means each Landau level is hugely degenerate. For a rectangular sample of dimensions $L_x \times L_y$, the allowed values of $k_y$ are $k_y = 2\pi n_y/L_y$. The guiding center must lie inside the sample: $0 \leq x_0 \leq L_x$, which constrains $0 \leq k_y \leq eB L_x/\hbar$. The number of states per Landau level is:

$$N_\phi = \frac{L_y}{2\pi}\cdot\frac{eBL_x}{\hbar} = \frac{eBA}{h} = \frac{\Phi}{\Phi_0}$$

Each Landau level accommodates exactly one state per flux quantum $\Phi_0 = h/e$ threading the sample.

Derivation 2: Laughlin's Gauge Argument in Full Detail

We provide a step-by-step derivation of the quantized Hall conductance using Laughlin's thought experiment on a Corbino (annular) geometry.

Step 1: Consider a 2DEG on an annulus (inner radius $r_1$, outer radius $r_2$) in a perpendicular magnetic field $B$. Thread an Aharonov-Bohm flux $\Phi$ through the central hole. By azimuthal symmetry, the vector potential due to the solenoid is:

$$\mathbf{A}_\text{sol} = \frac{\Phi}{2\pi r}\hat{\phi}$$

Step 2: Changing $\Phi$ adiabatically induces an azimuthal EMF by Faraday's law:

$$\mathcal{E} = -\frac{d\Phi}{dt}$$

This EMF drives a Hall current radially: $I_r = \sigma_{xy}\mathcal{E}$.

Step 3: Integrate over one full flux quantum insertion ($\Phi: 0 \to \Phi_0 = h/e$):

$$\Delta Q = \int_0^T I_r\,dt = \sigma_{xy}\int_0^T \mathcal{E}\,dt = -\sigma_{xy}\int_0^T \frac{d\Phi}{dt}\,dt = -\sigma_{xy}\,\Phi_0$$

Step 4 (Key argument): After inserting exactly one flux quantum, the vector potential can be gauged away by the transformation $\psi \to e^{i\phi}\psi$ (where $\phi$ is the azimuthal angle). This gauge transformation is single-valued, so the Hamiltonian returns to its original form. Therefore the spectrum is unchanged and the system is in the same state as before.

Step 5: Since the system returns to an equivalent state, the total charge transferred must be an integer number of electron charges: $|\Delta Q| = \nu e$ where $\nu$ is an integer (the number of filled Landau levels). Combining:

$$\sigma_{xy}\,\frac{h}{e} = \nu e \quad \Longrightarrow \quad \sigma_{xy} = \nu\frac{e^2}{h}$$

The beauty of this argument is that it uses only gauge invariance and the existence of a spectral gap. It does not depend on the details of the disorder potential, the sample geometry, or the electron-electron interactions, explaining the extraordinary universality of the quantization.

Derivation 3: TKNN Formula via Kubo Formalism

We derive the TKNN result by computing the Hall conductivity from the Kubo formula for a filled band.

Step 1: The Kubo formula for the Hall conductivity of non-interacting electrons is:

$$\sigma_{xy} = \frac{ie^2\hbar}{A}\sum_{\mathbf{k}}\sum_{n \neq m} \frac{f(E_n) - f(E_m)}{(E_n - E_m)^2}\,\langle n\mathbf{k}|v_x|m\mathbf{k}\rangle\langle m\mathbf{k}|v_y|n\mathbf{k}\rangle$$

where $f(E)$ is the Fermi function and $v_\mu = \frac{1}{\hbar}\partial_{k_\mu}H$.

Step 2: At zero temperature with $\nu$ completely filled bands, $f(E_n) = 1$ for filled bands and $f(E_m) = 0$ for empty bands. Substituting $\langle n|v_\mu|m\rangle = \frac{1}{\hbar}\langle n|\partial_{k_\mu} H|m\rangle$:

$$\sigma_{xy} = \frac{ie^2}{\hbar A}\sum_{\mathbf{k}}\sum_{\substack{n\,\text{occ}\\m\,\text{unocc}}} \frac{\langle n|\partial_{k_x}H|m\rangle\langle m|\partial_{k_y}H|n\rangle - (x\leftrightarrow y)}{(E_n - E_m)^2}$$

Step 3: Recognize that the summand is exactly the Berry curvature $\Omega_n(\mathbf{k})$ derived from perturbation theory (see the Berry phase chapter). Thus:

$$\sigma_{xy} = -\frac{e^2}{\hbar}\sum_{n\,\text{occ}}\frac{1}{A}\sum_\mathbf{k} \Omega_n(\mathbf{k})$$

Step 4: In the thermodynamic limit, replace $\frac{1}{A}\sum_\mathbf{k} \to \frac{1}{(2\pi)^2}\int_\text{BZ} d^2k$:

$$\sigma_{xy} = -\frac{e^2}{h}\sum_{n\,\text{occ}}\frac{1}{2\pi}\int_\text{BZ}\Omega_n(\mathbf{k})\,d^2k = \frac{e^2}{h}\sum_{n\,\text{occ}} C_n$$

where $C_n = \frac{1}{2\pi}\int_\text{BZ}\Omega_n\,d^2k \in \mathbb{Z}$ is the first Chern number (the sign convention depends on the definition of $\Omega_n$). The integer quantization follows from the fact that the Brillouin zone is a compact manifold (a torus), and the integral of the curvature of a U(1) bundle over a compact surface without boundary is always $2\pi$ times an integer.

Real-World Applications of the Quantum Hall Effect

1. Metrological Resistance Standard

The quantum Hall resistance $R_K = h/e^2 = 25\,812.807\,459\,\Omega$ is now an exact constant in the SI system (since 2019). National metrology institutes worldwide (NIST, PTB, NPL) maintain primary resistance standards based on quantum Hall devices operating at $\nu = 2$in GaAs/AlGaAs heterostructures at temperatures below 1 K and fields around 10 T. The reproducibility exceeds one part in $10^{10}$.

2. Determination of the Fine Structure Constant

The quantum Hall effect provides an independent measurement of the fine structure constant$\alpha = e^2/(4\pi\epsilon_0\hbar c)$ through the relation $R_K = h/e^2 = \mu_0 c/(2\alpha)$. Combined with measurements of the electron anomalous magnetic moment, this provides one of the most precise tests of quantum electrodynamics.

3. Graphene-Based Quantum Hall Standards

Epitaxial graphene on SiC exhibits the quantum Hall effect at lower magnetic fields and higher temperatures than GaAs, thanks to the large cyclotron gap of Dirac fermions ($E_n \propto \sqrt{nB}$rather than $nB$). Graphene QH resistance standards operate at fields as low as 3.5 T and temperatures up to 10 K, making the technology more accessible and cost-effective for calibration laboratories.

4. Topological Quantum Computing (FQHE)

The non-Abelian anyons predicted to exist in the $\nu = 5/2$ fractional quantum Hall state (described by the Moore-Read Pfaffian wavefunction) could serve as the basis for fault-tolerant topological quantum computation. Braiding these anyons performs quantum gates that are inherently protected from local noise. Microsoft and other groups have pursued this approach through the study of GaAs/AlGaAs heterostructures at millikelvin temperatures.

5. Quantum Anomalous Hall Effect in Magnetic TIs

The quantum anomalous Hall effect -- quantized Hall conductance without an external magnetic field -- was observed in magnetically doped topological insulator thin films (Cr-doped (Bi,Sb)$_2$Te$_3$) by Chang et al. in 2013. This opens the possibility of dissipationless chiral edge transport at zero field for low-power electronics and interconnects, and has since been observed at higher temperatures in intrinsic magnetic topological insulators such as MnBi$_2$Te$_4$.

Historical Context

The quantum mechanical treatment of electrons in a magnetic field was first given by Lev Landau in 1930, who showed that the continuous 2D kinetic energy spectrum collapses into discrete, infinitely degenerate levels. The classical Hall effect had been discovered by Edwin Hall in 1879, but the quantum regime required extremely clean two-dimensional electron systems at low temperatures, which only became available in the late 1970s with advances in molecular beam epitaxy (MBE) of GaAs/AlGaAs heterostructures.

On February 5, 1980, Klaus von Klitzing, working at the High Magnetic Field Laboratory in Grenoble, France, observed that the Hall resistance of a silicon MOSFET at liquid helium temperatures showed exact plateaus at values $h/(\nu e^2)$ with $\nu$ integer. He was awarded the Nobel Prize in Physics in 1985 for this discovery. The topological explanation was provided by Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) in 1982, who showed that each plateau corresponds to a topological Chern number -- a result for which David Thouless shared the 2016 Nobel Prize.

In 1982, Daniel Tsui and Horst Stormer, working at Bell Labs with ultra-high-mobility samples grown by Arthur Gossard, discovered fractional quantization at $\nu = 1/3$. Robert Laughlin provided the theoretical explanation through his celebrated trial wavefunction (1983), revealing the fractional charge and fractional statistics of quasiparticle excitations. Tsui, Stormer, and Laughlin shared the 1998 Nobel Prize. Jain's composite fermion theory (1989) subsequently unified the integer and fractional effects, showing that the FQHE of electrons is the IQHE of composite fermions.

Python Simulation: Landau Levels and Hall Conductivity

Landau level spectrum, quantized Hall conductivity with integer and fractional plateaus, and edge state dispersion in a confined quantum Hall system.

Quantum Hall Effect: Landau Levels, Plateaus, and Edge States

Python

Landau level spectrum, Hall conductivity vs filling factor, and chiral edge state dispersion

script.py139 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fig, axes = plt.subplots(1, 3, figsize=(18, 6))
fig.patch.set_facecolor('#0a0a1a')
for ax in axes:
    ax.set_facecolor('#0a0a1a')
    for spine in ax.spines.values():
        spine.set_edgecolor('#334155')

# Panel 1: Landau Level Spectrum
n_levels = 8
omega_c = 1.0  # in units of hbar*omega_c
colors_ll = ['#a855f7', '#8b5cf6', '#7c3aed', '#6d28d9',
             '#c084fc', '#d8b4fe', '#e9d5ff', '#f3e8ff']

for n in range(n_levels):
    E_n = omega_c * (n + 0.5)
    kx = np.linspace(-3, 3, 200)
    E_flat = np.full_like(kx, E_n)
    axes[0].plot(kx, E_flat, color=colors_ll[n], linewidth=2.5,
                 label=f'n={n}' if n < 5 else None)
    axes[0].text(3.2, E_n, f'n={n}', color=colors_ll[n], fontsize=9, va='center')

axes[0].set_xlabel(r'$k_x \, l_B$', color='#94a3b8', fontsize=12)
axes[0].set_ylabel(r'$E \, / \, \hbar\omega_c$', color='#94a3b8', fontsize=12)
axes[0].set_title('Landau Level Spectrum', color='white', fontsize=14, fontweight='bold')
axes[0].tick_params(colors='#94a3b8')
axes[0].set_ylim(-0.2, 8.5)
axes[0].set_xlim(-3.5, 4.5)
axes[0].axhline(y=0, color='#64748b', linewidth=0.5, linestyle='--')

E_fermi = 3.0 * omega_c
axes[0].axhline(y=E_fermi, color='#f59e0b', linewidth=1.5, linestyle='--', alpha=0.8)
axes[0].text(-3.3, E_fermi + 0.2, r'$E_F$ ($\nu=3$)', color='#f59e0b', fontsize=10)
for n in range(3):
    E_n = omega_c * (n + 0.5)
    axes[0].fill_between(np.linspace(-3, 3, 10), E_n - 0.15, E_n + 0.15,
                         color=colors_ll[n], alpha=0.15)

# Panel 2: Hall Conductivity vs Filling Factor
nu_range = np.linspace(0, 6, 2000)
sigma_xy = np.zeros_like(nu_range)

for i, nu in enumerate(nu_range):
    n_nearest = int(np.round(nu))
    if n_nearest == 0:
        sigma_xy[i] = 0
    elif abs(nu - n_nearest) < 0.35:
        sigma_xy[i] = n_nearest
    else:
        if nu > n_nearest:
            n_low, n_high = n_nearest, n_nearest + 1
        else:
            n_low, n_high = n_nearest - 1, n_nearest
        if n_low == 0:
            frac = (nu - 0.35) / (1.0 - 0.7)
            sigma_xy[i] = frac
        else:
            center = (n_low + n_high) / 2.0
            width = 0.3
            frac = 0.5 * (1 + np.tanh((nu - center) / (width * 0.3)))
            sigma_xy[i] = n_low + (n_high - n_low) * frac

frac_plateaus = [(1/3, 1/3), (2/5, 2/5), (3/5, 3/5), (2/3, 2/3)]
for nu_f, sigma_f in frac_plateaus:
    mask = np.abs(nu_range - nu_f) < 0.06
    sigma_xy[mask] = sigma_f

axes[1].plot(nu_range, sigma_xy, color='#a855f7', linewidth=2.5)

for n in range(1, 6):
    axes[1].axhline(y=n, color='#334155', linewidth=0.5, linestyle=':')
    axes[1].plot(n, n, 'o', color='#22c55e', markersize=8, zorder=5)
    axes[1].text(n + 0.08, n + 0.15, f'{n}', color='#22c55e', fontsize=10, fontweight='bold')

for nu_f, sigma_f in frac_plateaus:
    axes[1].plot(nu_f, sigma_f, 's', color='#f59e0b', markersize=6, zorder=5)

axes[1].set_xlabel(r'Filling factor $\nu$', color='#94a3b8', fontsize=12)
axes[1].set_ylabel(r'$\sigma_{xy}$ $(e^2/h)$', color='#94a3b8', fontsize=12)
axes[1].set_title('Hall Conductivity Plateaus', color='white', fontsize=14, fontweight='bold')
axes[1].tick_params(colors='#94a3b8')
axes[1].set_xlim(0, 5.5)
axes[1].set_ylim(-0.3, 5.5)

axes[1].annotate('IQHE', xy=(2, 2), xytext=(3.5, 1.2),
                 color='#22c55e', fontsize=11, fontweight='bold',
                 arrowprops=dict(arrowstyle='->', color='#22c55e', lw=1.5))
axes[1].annotate('FQHE', xy=(1/3, 1/3), xytext=(1.5, 0.3),
                 color='#f59e0b', fontsize=11, fontweight='bold',
                 arrowprops=dict(arrowstyle='->', color='#f59e0b', lw=1.5))

# Panel 3: Edge State Dispersion
y_pos = np.linspace(-1, 1, 500)  # position across sample

for n in range(5):
    E_bulk = omega_c * (n + 0.5)
    edge_bend = 0.8 * (np.cosh((y_pos - 0.65) / 0.15) + np.cosh((y_pos + 0.65) / 0.15) - 2)
    E_edge = E_bulk + np.minimum(edge_bend, 4.0)

axes[2].plot(y_pos, E_edge, color=colors_ll[n], linewidth=2)

E_fermi_edge = 2.8
axes[2].axhline(y=E_fermi_edge, color='#f59e0b', linewidth=1.5, linestyle='--', alpha=0.8)
axes[2].text(0, E_fermi_edge + 0.15, r'$E_F$', color='#f59e0b', fontsize=11, ha='center')

axes[2].annotate('', xy=(-0.75, 1.8), xytext=(-0.75, 0.8),
                 arrowprops=dict(arrowstyle='->', color='#ef4444', lw=2.5))
axes[2].text(-0.85, 1.3, r'$\uparrow$', color='#ef4444', fontsize=14, fontweight='bold')

axes[2].annotate('', xy=(0.75, 0.8), xytext=(0.75, 1.8),
                 arrowprops=dict(arrowstyle='->', color='#3b82f6', lw=2.5))
axes[2].text(0.85, 1.3, r'$\downarrow$', color='#3b82f6', fontsize=14, fontweight='bold')

axes[2].text(-0.82, 5.5, 'Left\nedge', color='#ef4444', fontsize=9, ha='center')
axes[2].text(0.82, 5.5, 'Right\nedge', color='#3b82f6', fontsize=9, ha='center')
axes[2].text(0, 0.15, 'Bulk', color='#94a3b8', fontsize=10, ha='center')

axes[2].axvspan(-0.55, 0.55, alpha=0.08, color='#a855f7')

axes[2].set_xlabel('Position across sample', color='#94a3b8', fontsize=12)
axes[2].set_ylabel(r'$E \, / \, \hbar\omega_c$', color='#94a3b8', fontsize=12)
axes[2].set_title('Edge State Dispersion', color='white', fontsize=14, fontweight='bold')
axes[2].tick_params(colors='#94a3b8')
axes[2].set_ylim(-0.2, 6.5)
axes[2].set_xlim(-1.05, 1.05)

plt.tight_layout()
plt.savefig('quantum_hall.png', dpi=130, bbox_inches='tight',
            facecolor=fig.get_facecolor())
print("Landau level spacing: E_n = hbar*omega_c*(n + 1/2)")
print("Hall conductivity: sigma_xy = nu * e^2/h")
print("Integer plateaus at nu = 1, 2, 3, ... (IQHE)")
print("Fractional plateaus at nu = 1/3, 2/5, 2/3, ... (FQHE)")
print("Edge states are chiral -- backscattering forbidden")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran Implementation: Landau Level Calculator

Compute Landau level energies, degeneracies, filling factor as a function of magnetic field, and the quantized Hall resistance values.

Landau Level Energies and Filling Factors

Fortran

Landau level spectrum, degeneracy per level, filling factor vs B, and quantized Hall resistance

landau_levels.f9060 lines

program landau_levels
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: hbar = 1.0545718e-34_dp, e_ch = 1.60217663e-19_dp
  real(dp), parameter :: m_e = 9.1093837e-31_dp, h_pl = 6.62607015e-34_dp
  real(dp), parameter :: Phi_0 = h_pl / e_ch
  integer, parameter :: n_max = 10, n_B = 15
  real(dp) :: B, B_min, B_max, dB, omega_c, l_B, E_n, N_phi, A_sample, nu
  integer  :: n, iB, n_electrons

A_sample = 1.0e-8_dp
  n_electrons = 50000
  B_min = 1.0_dp; B_max = 15.0_dp
  dB = (B_max - B_min) / real(n_B - 1, dp)

write(*,'(A)') '=== LANDAU LEVEL ENERGIES (B = 5 T) ==='
  B = 5.0_dp
  omega_c = e_ch * B / m_e
  l_B = sqrt(hbar / (e_ch * B))
  N_phi = B * A_sample / Phi_0
  write(*,'(A,ES12.4,A)') 'Cyclotron energy: hbar*wc = ', hbar*omega_c/e_ch*1000, ' meV'
  write(*,'(A,F8.3,A)')   'Magnetic length:  l_B    = ', l_B*1e9_dp, ' nm'
  write(*,'(A,ES12.4)')   'Flux quanta:      N_phi  = ', N_phi
  write(*,'(A)') ''
  write(*,'(A)') '  n     E_n (meV)     Degeneracy/Area (1/m^2)'
  write(*,'(A)') ' ---   ----------    -------------------------'
  do n = 0, n_max
    E_n = hbar * omega_c * (real(n, dp) + 0.5_dp)
    write(*,'(I4, F13.4, ES22.4)') n, E_n/e_ch*1000.0_dp, e_ch*B/h_pl
  end do

write(*,'(A)') ''
  write(*,'(A)') '=== FILLING FACTOR vs MAGNETIC FIELD ==='
  write(*,'(A)') '   B (T)    hbar*wc (meV)  l_B (nm)    N_phi       nu       Status'
  write(*,'(A)') '  ------   ------------   --------   ---------   ------   ---------'
  do iB = 1, n_B
    B = B_min + real(iB - 1, dp) * dB
    omega_c = e_ch * B / m_e
    l_B = sqrt(hbar / (e_ch * B))
    N_phi = B * A_sample / Phi_0
    nu = real(n_electrons, dp) / N_phi
    n = nint(nu)
    if (n >= 1 .and. abs(nu - real(n, dp)) < 0.3_dp) then
      write(*,'(F8.2, F13.4, F11.4, ES12.3, F9.4, A, I2, A)') &
        B, hbar*omega_c/e_ch*1000, l_B*1e9_dp, N_phi, nu, '   nu=', n, ' plateau'
    else
      write(*,'(F8.2, F13.4, F11.4, ES12.3, F9.4, A)') &
        B, hbar*omega_c/e_ch*1000, l_B*1e9_dp, N_phi, nu, '   transition'
    end if
  end do

write(*,'(A)') ''
  write(*,'(A)') '=== QUANTIZED HALL RESISTANCE ==='
  write(*,'(A)') '   nu      R_H (Ohm)       sigma_xy (e^2/h)'
  write(*,'(A)') '  ----   ---------------   -----------------'
  do n = 1, 8
    write(*,'(I5, F16.4, F16.6)') n, h_pl/(real(n,dp)*e_ch**2), real(n,dp)
  end do
  write(*,'(A,F12.4,A)') 'Von Klitzing constant R_K = h/e^2 = ', h_pl/e_ch**2, ' Ohm'
end program landau_levels

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Key Concepts Summary

Landau Quantization

Continuous 2D spectrum collapses into discrete levels $E_n = \hbar\omega_c(n+1/2)$, each with macroscopic degeneracy $N_\phi = BA/\Phi_0$.

Integer QHE

Quantized Hall conductance $\sigma_{xy} = \nu e^2/h$ with$\sigma_{xx} = 0$. Disorder creates localized states that produce plateaus.

Topological Protection

TKNN Chern number = integer topological invariant. Quantization is exact because topology cannot change without closing the gap.

Fractional QHE

Many-body state with fractional charge excitations. Laughlin wavefunction at$\nu = 1/m$; composite fermion theory unifies the hierarchy.

← Chapter 1: Berry Phase Chapter 3: Topological Insulators →