Anderson transitions: critical theory and exponents

5 problems

• Conductivity exponent puzzle in doped semiconductors

  Contested

  Adding phosphorus to silicon switches it from insulator to metal at a critical concentration, and the conductivity then rises as a power of the distance from that point. Measured powers differ between kinds of samples and from theory for non-interacting electrons, and the reason is disputed.

• Correlation-length exponent of the 3D Anderson-Mott transition with Coulomb interaction

  Open

  For electrons that do not repel, the transition exponent is known numerically to three digits. For real electrons, which repel through the Coulomb force, no reliable value exists.

• Critical exponents of the quantum Hall transition with Coulomb interaction

  Open

  Real quantum Hall transitions involve repelling electrons, which changes how time and length scale near the transition. Experiments measure a combined exponent of about 0.42, but theory has not computed it with the repulsion included.

• Is the integer quantum Hall transition a conventional critical point?

  Contested

  Between two quantum Hall plateaus, electron states spread out at a single energy, with a size that diverges according to an exponent. Simulations give about 2.6, experiments about 2.4, and the simulated value drifts with system size, so even the type of critical point is in doubt.

• Upper critical dimension of the Anderson transition

  Contested

  For many phase transitions there is a number of spatial dimensions above which the exponents become simple and stop changing. For the localization transition of electrons in a random potential it is unknown whether such a dimension exists or is infinite.