Quantum Phase Transitions and Conserved Charges

The constraints on the scaling properties of conserved charge densities in the vicinity of a zero temperature ($T$), second-order quantum phase transition are studied. We introduce a generalized Wilson ratio, characterizing the non-linear response to an external field, $H$, coupling to any conserved charge, and argue that it is a completely universal function of $H/T$: this is illustrated by computations on model systems. We also note implications for transitions where the order parameter is a conserved charge (as in a $T=0$ ferromagnet-paramagnet transition).