Physics Letters B 314 (1993) 381-386 PHYSICS LETTERS B North-Holland Critical exponents of new universality classes B. R o s e n s t e i n , H o i - L a i Y u Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan a n d A. K o v n e r Theory Division, T-8, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 23 July 1993 Editor: M. Dine We calculate critical exponents of chiral Ising, X Y and Heisenberg universality classes up to second order in 4 - expansion. General expressions for any number N of fermionic variables are given and compared with recent ext l/N calculation and lattice simulations. The possibility that these new universality classes describe critical pheno in certain 3D magnetic systems and in finite temperature phase transitions of 3 + l dimensional quantum field th is briefly discussed. A large number of nontrivial conformal field theories (CFT) is known in two dimensions and none is firmly established in four. The dimensionality d = 3 is between these extremes. There are basically two well established families of nontrivial CFT's or equivalently nongaussian universality classes. The first contains the usual Ising, X Y , Heisenberg and other similar critical sigma models. It is well known that critical properties of great majority of phase transitions in various 3D magnetic systems, superconductors etc. are quite accurately described by these CFT's [ 1 ]. Also the finite temperature phase transitions in certain 3 + 1 dimensional QFT were argued to belon to these universality classes [2]. In some other 3D materials like helimagnets or frustrated antiferromagnets on the triangular lattice [3,4], diluted magnets as well as in field theories like QCD with few fermionic flavours the situation is not so clear [5,6]. The second family of CFT's contains various critical four Fermion models [7]. In these models the chiral symmetry is broken. Even when the chiral symmetry breaking pattern is the same as for a scalar model, the critical properties like critical exponents (calculated for the chiral order parameter -ff~, ) turn out to be different. The fact that the anomalous dimensions and critical exponents of the Z2 - - 1 chiral universality class (these conformal models will be called the chiral Ising, X Y etc.) are distinct from the standard values for critical Ising, was first observed using the 1IN expansion where N is the number of flavors [8], and then using Monte Carlo simulations [9,10]. The physical reason is very transparent: on the chirally invariant side of the phase transition there are massless fermions which do not decouple at the infrared fixed point. Although the l / N expansion was pushed to quite high orders (see [9,1 1 ] and especially [ 12] ), the physically interesting models are of course those with small N. One can easily see that unlike some other expansions, th 1 IN, unfortunately, does not have a small "helping" factor like l / 4 n and turns out to be useless even for not very small N. For example in the case of critical exponent u measured in [10] for N = 4 (we count two components Elsevier Science Publishers B.V. 3 81