# "Quantum" hyperscaling relation from a Renormalization Group (RG) viewpoint

Through the RG method, one can obtain the hyperscaling relation between the critical exponents of classical second-order phase transitions: $$$$2-\alpha=\nu d$$$$ In the case of a quantum phase transition, however, the dimensionality $$d$$ is substituted by an effective one $$d_{eff}=d+z$$, with $$z$$ being the dynamic critical exponent. In many cases $$z=1$$, consistently with the idea that quantum, $$d$$-dimensional systems can be mapped onto classical, $$(d+1)$$-dimensional ones.

What I'm looking for is a justification for the presence of the effective dimensionality $$d_{eff}$$ inside the RG theory. I could only find one reference that treats this, which is the first chapter of Mucio Continentino's book "Quantum Scaling in Many-Body Systems", but I found it a bit sloppy. From reading that chapter it seems that the reason why $$z$$ comes out is that at $$T=0$$ the free energy is equal to the Hamiltonian and thus $$J$$ can be "factored out". Since $$J'=b^{-z}J$$ under an RG step then an extra $$z$$ is added to $$d$$ (eqs (1.4) to (1.7)). This explanation, however, doesn't give much physical insight in my opinion and I don't understand what would happen for quantum systems at non-zero temperature.

Please note that my background in quantum field theory and path integral is limited, what I know of RG comes from studying Goldenfeld's treatment and chapter 3 of Ortiz-Nishimori's.

The fact that $$J$$ can be "factored out" reflects the fact that at $$T=0$$ there is only one energy scale and it is determined by $$J$$. Because of this, we expect that the free energy per degree of freedom scales not only with the dimensionality but also with the dynamic exponent $$z$$. As soon as $$T\neq0$$ there is a competition between energy scales, namely $$J$$ and $$k_B T$$ and we cannot say that the free energy scales with $$z$$ as well as with $$d$$.
Of what follows I'm not sure, but I would say that as soon as $$T\neq 0$$ the system is governed by a finite-temperature fixed point with critical exponent different than those related to the $$T=0$$ fixed point and for which a "classical" hyperscaling relation holds.