Six-loop $\epsilon$ expansion study of three-dimensional $O(n)\times O(m)$ spin models

Abstract

The Landau-Wilson field theory with $O(n)\times O(m)$ symmetry which describes the critical thermodynamics of frustrated spin systems with noncollinear and noncoplanar ordering is analyzed in $4-\epsilon$ dimensions within the minimal subtraction scheme in the six-loop approximation. The $\epsilon$ expansions for marginal dimensionalities of the order parameter $n^H(m,4-\epsilon )$, $n^{-}(m,4-\epsilon )$, $n^{+}(m,4-\epsilon )$ separating different regimes of critical behavior are extended up to ${\epsilon }^5$ terms. Concrete series with coefficients in decimals are presented for $m=\{2,\dots ,6\}$. The \textit{diagram of stability} of nontrivial fixed points, including the chiral one, in $(m,n)$ plane is constructed by means of summing up of corresponding $\epsilon$ expansions using various resummation techniques. Numerical estimates of the chiral critical exponents for several couples $\{m,n\}$ are also found. Comparative analysis of our results with their counterparts obtained earlier within the lower-order approximations and by means of alternative approaches is performed. It is confirmed, in particular, that in physically interesting cases $n=2,m=2$ and $n=2,m=3$ phase transitions into chiral phases should be first-order.

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