A new critical exponent koppa and its logarithmic counterpart koppa-hat

Abstract

It is well known that standard hyperscaling breaks down above the upper critical dimension d_c, where the critical exponents take on their Landau values. Here we show that this is because, in standard formulations in the thermodynamic limit, distance is measured on the correlation-length scale. However, the correlation-length scale and the underlying length scale of the system are not the same at or above the upper critical dimension. Above d_c they are related algebraically through a new critical exponent koppa, while at d_c they differ through logarithmic corrections governed by an exponent koppa-hat. Taking proper account of these different length scales allows one to extend hyperscaling to all dimensions.