Geometry of Landau Level without Galilean or Rotational Symmetry

The integer quantum Hall effect is usually modeled using Galilean-invariant or rotationally-invariant Landau levels. However, these are not generic symmetries of electrons moving in a crystalline background. We explicitly break both symmetries by considering a inversion-symmetric Hamiltonian with quartic terms. We carry out exact diagonalization numerically with a truncated Hilbert space, and define an emergent metric $g^n_{ab}$ for each Landau level as the expectation value of a bilinear form in momentum. With an appropriate choice of the guiding center coherent state, the Landau level wavefunctions are holomorphic functions of $z^*$ times a Gaussian (this is distinct from a well-known property of rotationally-invariant lowest-Landau-level wavefunctions). We show that the zeroes of the wavefunction define a ``topological spin $s_n$'', with its original definition as an ``intrinsic angular momentum'' no longer valid without rotational symmetry. This is now related to the number of zeroes $n$ encircled by the classical orbit by $s_n=n+\frac{1}{2}$. Finally we introduce a mass tensor $m^n_{ab}$ for each Landau level using a Lagrangian formalism. We conclude that topological and geometric information can be extracted without resort to Galilean or Rotational symmetries.