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Through the RG method, one can obtain the hyperscaling relation between the critical exponents of classical second-order phase transitions: \begin{equation} 2-\alpha=\nu d \end{equation} 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.

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I've thought about this for a while more and came up with what may be an answer.

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.

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