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Quantum critical scaling and the Gross-Neveu model in 2+1 dimensions

The quantum critical behavior of the 2+1 dimensional Gross--Neveu model in the vicinity of its zero temperature critical point is considered. The model is known to be renormalisable in the large $N$ limit, which offers the possibility to obtain expressions for various thermodynamic functions in closed form. We have used the concept of finite--size scaling to extract information about the leading temperature behavior of the free energy and the mass term, defined by the fermionic condensate and determined the crossover lines in the coupling ($\g$) -- temperature ($T$) plane. These are given by $T\sim|\g-\g_c|$, where $\g_c$ denotes the critical coupling at zero temperature. According to our analysis no spontaneous symmetry breaking survives at finite temperature. We have found that the leading temperature behavior of the fermionic condensate is proportional to the temperature with the critical amplitude $\frac{\sqrt{5}}3π$. The scaling function of the singular part of the free energy is found to exhibit a maximum at $\frac{\ln2}{2π}$ corresponding to one of the crossover lines. The critical amplitude of the singular part of the free energy is given by the universal number $\frac13[\frac1{2π}ζ(3)-\mathrm{Cl}_2(\fracπ3)]=-0.274543...$, where $ζ(z)$ and $\mathrm{Cl}_2(z)$ are the Riemann zeta and Clausen's functions, respectively. Interpreted in terms the thermodynamic Casimir effect, this result implies an attractive Casimir "force". This study is expected to be useful in shedding light on a broader class of four fermionic models.