Thermodynamics and quantum criticality in cuprate superconductors

We will present elementary scaling arguments focused on the thermodynamics in the proximity of the quantum critical point in the cuprate superconductors. Extending the analysis centered on the Gr\"uneisen parameter by Rosch, Si, and co-workers to the cuprates, we demonstrate that a combination of specific-heat and chemical potential measurements can reveal the nature of the zero temperature singularity. From the known specific-heat data it follows that the effective number of time dimensions has to equal the number of space dimensions, while we find a total of six scaling laws governing the temperature and density dependence of the chemical potential, revealing directly the coupling constant scaling dimension.