Field-induced spin-density-wave and butterfly spectrum in three dimensions

Landau’s quantization for incompletely nested Fermi surfaces is known to give rise to magnetic-field-induced spin-density waves (FISDW’s) in two-dimensional organic metals. Here we show that three-dimensional (3D) systems can have 3D-specific series of FISDW phases as energetically stable states, for which we clarify how and why they appear as the magnetic field is tilted. Each phase is characterized by a quantized Hall effect for each ${\sigma }_{\mathrm{xy}}$ and ${\sigma }_{\mathrm{zx}}$ that reside on a fractal-like Hofstadter’s butterfly.