Critical thermodynamics of three-dimensional MN-component field model with cubic anisotropy from higher-loop RG expansions

The critical behavior of an MN-component order parameter Ginzburg-Landau model with isotropic and cubic interactions describing antiferromagnetic and structural phase transitions in certain crystals with complicated ordering is studied in the framework of the four-loop renormalization group (RG) approach in $(4-2\text{ve})$-dimensions. Using dimensional regularization and the minimal subtraction scheme, the perturbative expansions for RG functions are deduced for generic M and N and resummed by the Borel transformation combined with a conformal mapping. Investigation of the global structure of RG flows for the physically significant cases M=2 and N=2, N=3 shows that the model has a three-dimensionally stable fixed point different from the Bose one. The critical dimensionality is proved to be exactly two times smaller than its counterpart in the real cubic model: N_c^C ={1/2} N_c^R. The numerical value $N_c^C=1.447\pm 0.020$ is obtained from resumming the known five-loop $\text{ve}$-series for N_c^R. Since N_c^C < 2, the critical thermodynamics of the model relevant to the phase transitions in real substances should be governed by the complex cubic fixed point with a new set of critical exponents: $\text{gm}=1.404(25)$, $\nu =0.715(10)$, $\text{et}=0.0343(20)$ for N=2 and $\text{gm}=1.390(25)$, $\nu =0.702(10)$, $\text{et}=0.0345(15)$ for N=3.