Critical behavior of the 3D Ising model

3 problems

• Closed-form free energy or exponents of the 3D Ising model

  Open

  Can the free energy of the 3D Ising model, or at least its critical exponents, be written as an exact formula? Numerics and the conformal bootstrap (a method that constrains critical points from symmetry and consistency) give the exponents to six or more digits, but no formula is known.

• Dynamic critical exponent z of the 3D Ising model

  Partially resolved

  Near the critical point a magnet relaxes very slowly, and the exponent z sets how the relaxation time grows with the size of correlated regions. For purely relaxational dynamics in three dimensions, z is known only approximately and has no exact theory.

• Dimension where dimensional reduction fails in the random-field Ising model

  Contested

  Adding a random magnetic field to the Ising model was once predicted to make it behave like the pure model in two fewer dimensions. That prediction is wrong in three dimensions, and it is unsettled at which dimension it starts to fail and why.