Disorder-induced decoupling of attracting identical fermions: transfer matrix approach

We consider a pair of identical fermions with a short-range attractiveinteraction on a finite lattice cluster in the presence of strong sitedisorder. This toy model imitates a low density regime of the stronglydisordered Hubbard model. In contrast to spinful fermions, which cansimultaneously occupy a site with a minimal energy and thus always form a boundstate resistant to disorder, for the identical fermions the probability ofpairing on neighboring sites depends on the relation between the interactionand the disorder. The complexity of `brute-force' calculations (both analyticaland numerical) of this probability grows rapidly with the number of sites evenfor the simplest cluster geometry in the form of a closed chain. Remarkably,this problem is related to an old mathematical task of computing the volume ofa polyhedron, known as NP-hard. However, we have found that the problem in thechain geometry can be exactly solved by the transfer matrix method. Using thisapproach we have calculated the pairing probability in the long chain for anarbitrary relation between the interaction and the disorder strengths andcompletely described the crossover between the regimes of coupled and separatedfermions.