The semicircle law was first described theoretically in Dykhne and Ruzin's analysis of the quantum Hall effect as a mixture of 2 phases: a free electron gas, and a free hole gas. Mathematically, it states that σ σ x x 2 + ( σ σ x y − − σ σ x y 0 ) 2 = ( σ σ x x 0 ) 2 where σ is the mean-field Hall conductivity, and σ is a parameter that encodes the classical conductivity of each phase. A similar law also holds for the resistivity.