Volume 228, number 3 PHYSICS LETTERS B 21 September 1989 ON THE DISORDERED FERMION COUPLINGS M. BERNASCHI IBM-ECSEC. via Gtorgtone 159, 1-00147 Rome. Italy S. CABASINO, E. MARINARI I,VFN - Seztone di Roma. Gruppo Collegato dt Roma IL and Diparttmento di Ftsica. Untversttgz dt Roma IL 1-00173 Rome. Ita& and R. SARNO INI"N - Sezione dt Roma. and Dtparttmento dt Ftstca. Universtta dz Roma La Sapwnza. 1-00185 Rome. Italy Received 5 July 1989 We study the possibility of avoiding the fermion doubling problem by using a random coupling We use numerical simulations in order to study the theory in the strong disorder region. We find a sharp crossover as a function of the strength of the disorder. For weak quenched disorder we find that the species doubling survives, while for strong quenched disorder only with a particular choice of the random term (antihermttian) it is possible to get a theory that seems to avoid fermion doubling. At present we do not have a completely satisfac- tory way to regularize fermions on the lattice. In sim- ple terms we can say that we can consider several choices: we can gwe up either chiral invariance (loosing on the lattice a crucial symmetry of the con- tinuum system), or locality (making the system quite unsuitable for numerical stmulauons), or we can try to coexist with at least part of the species doubling, or we can try something more origmal (as we will do m the following). In more precise terms the Ntelsen- Nmomiya theorem [ 1 ] states that it is impossible to avoid the doubling of fermlonic species in a lattice formulauon ofa fermionic theory if one demands lo- cahty, hermit~ctty, translational and chtral mvari- ance of the acuon. in refs. [2,3] a formulation based over a random couphng of fermions has been proposed. The idea is to add a formally irrelevant term (very much like in the Wilson case), in which a random field (that breaks translational invanance) couples to the for- ward and backwark covariant first derivatwe opera- tors. The random scalar field modifies the free naive theory on the lattice m the following way: s= E ~ ¢(n)y. 2 n ,u (~(n+~)-~(n-~)) +m ~ ~(n)~,(n)+ Y, Y. ~(n)zu(n) tl n I~ XTu[~(n+/~) + v/(n-/~)- 2q/(n) ]. (l) The real field z(x) has a gausstan distribution, with zero expectation value and a dispersion a (the cor- relation of two z fields at different points is zero); a plays the role of the coupling constant. One can consider both the quenched and the an- nealed version of such a theory: we will see that in many cases the same considerations apply to the two theories. Such a theory has the correct formal contin- uum limit, but it is not translationally invartant, and the action ~s not hermltian. We hope to escape the N- N theorem m this way. Parisi and Zhang [ 2 ] studied the theory for weak disorder, by using perturbation theory. They found that the doubling is not removed from the random couphng, and they stud they believed the procedure is not likely to be working at any coupling. Their con- 383