Phase transitions in spin systems with frustration

Ferromagnetic spin systems frustrated by random antiferromagnetic bonds are considered. We show that for arbitrary dimensions $d\geqslant2$ the random system without frustration is equivalent to the regular one without antiferromagnetic bonds. In two and three dimensions completely frustrated systems on the square and cubic lattices are shown to be equivalent to the corresponding periodic completely frustrated systems. An explicit form of local gauge transformation which transforms each of systems considered into the equivalent one is pointed out. We prove that there is no phase transition in the completely frustrated two-dimensional Ising model on square lattice. The general case of partially frustrated two-dimensional Ising model on square lattice is also considered and an improved lower bound on the ground-state energy for this model is obtained.