Landau Levels, Projection, and Magnetic Translations

A magnetic field separates electron motion into a quantized cyclotron oscillator and a noncommuting guiding center. Projecting to one Landau level freezes the former but leaves the latter fully quantum mechanical. Interactions are thereby reorganized into a problem with no ordinary kinetic dispersion and a distinctive magnetic-translation algebra.

Required background. Integer quantum Hall matter supplies Hall response and filled Landau levels; canonical quantization supplies operator commutators.

Helpful background. Parallel transport and holonomy supplies the flux phase behind magnetic translations.

Cyclotron and guiding-center coordinates

For an electron of charge $-e$ in $B>0$, $\boldsymbol\pi=\mathbf p+e\mathbf A$ obeys $[\pi_x,\pi_y]=-i\hbar eB$. Define $\ell_B=\sqrt{\hbar/(eB)}$ and decompose $\mathbf r=\mathbf R+\boldsymbol\eta$ so that

$$[\eta_x,\eta_y]=-i\ell_B^2, \qquad [R_x,R_y]=+i\ell_B^2, \qquad [R_i,\eta_j]=0.$$

The sign is tied to electron charge and the chosen plane orientation; sources that orient with $qB$ often print the opposite guiding-center sign. The Hamiltonian is

$$H_0=\frac{\boldsymbol\pi^2}{2m} =\hbar\omega_c\left(a^\dagger a+\frac12\right), \qquad \omega_c=\frac{eB}{m},$$

and is independent of $\mathbf R$. A Landau level therefore contains $N_\phi=A/(2\pi\ell_B^2)$ orbitals on area $A$ when the flux is compatible with the boundary conditions.

Magnetic translations

Ordinary translations must be supplemented by gauge phases. Guiding-center translations satisfy a central extension: translating around a parallelogram produces the Aharonov–Bohm phase of its enclosed flux. On a torus, flux quantization makes the full-cycle boundary translations compatible, while elementary guiding-center translations retain their projective algebra in an $N_\phi$-dimensional irreducible representation; enlarging the representation does not make those elementary generators commute Zak 1964.

This algebra explains why many-body momentum sectors and flux insertion are central finite-size diagnostics. The momenta are magnetic-translation quantum numbers, not those of zero-field Bloch electrons.

Projected density algebra

Within the lowest Landau level,

$$P_0\rho_{\mathbf q}P_0=e^{-q^2\ell_B^2/4}\,\bar\rho_{\mathbf q}, \qquad \bar\rho_{\mathbf q}=\sum_j e^{i\mathbf q\cdot\mathbf R_j}.$$

Because $[R_x,R_y]=+i\ell_B^2$, Baker–Campbell–Hausdorff gives

$$[\bar\rho_{\mathbf q},\bar\rho_{\mathbf k}] =-2i\sin\!\left(\frac{\ell_B^2}{2}\,\mathbf q\times\mathbf k\right) \bar\rho_{\mathbf q+\mathbf k}.$$

The minus sign reverses with the guiding-center convention. The physical content is the oriented noncommutative area. Projected two-body interactions become $H=\frac12\sum_{\mathbf q}V(q)e^{-q^2\ell_B^2/2}\bar\rho_{\mathbf q}\bar\rho_{-\mathbf q}$ after self-interaction and background terms are handled Girvin, MacDonald, and Platzman 1986.

Projection is controlled when interaction and disorder scales are small compared with the cyclotron gap. Landau-level mixing introduces corrections of order $\kappa\sim(e^2/\epsilon\ell_B)/(\hbar\omega_c)$ and can break idealized particle–hole symmetry.

Exercise

How many one-particle states are in a Landau level on a torus of area $A=20\pi\ell_B^2$?

Solution

$N_\phi=A/(2\pi\ell_B^2)=10$. Flux compatibility is essential: a torus cannot support an arbitrary noninteger $N_\phi$ with periodic magnetic boundary conditions.

References

• S. M. Girvin, A. H. MacDonald, and P. M. Platzman, “Magneto-Roton Theory of Collective Excitations in the Fractional Quantum Hall Effect,” Physical Review B 33 (1986) 2481–2494, doi:10.1103/PhysRevB.33.2481.
• Joshua Zak, “Magnetic Translation Group,” Physical Review 134 (1964) A1602–A1606, doi:10.1103/PhysRev.134.A1602.