Valence-bond theory of highly disordered quantum antiferro- magnets

We present a large-N variational approach to describe the magnetism of insulating doped semiconductors based on a disorder-generalization of the resonating-valence-bond theory for quantum antiferromagnets. This method captures all the qualitative and even quantitative predictions of the strong-disorder renormalization group approach over the entire experimentally relevant temperature range. Finally, by mapping the problem on a hard-sphere fluid, we could provide an essentially exact analytic solution without any adjustable parameters. The metal-insulator transition (MIT) in doped semiconductors (DS) [1] is one of the most fundamental, yet theoretically less understood problems in condensed matter physics. Even aside from their pivotal technological role, the DS have long been recognized as a bellwether system for the study of quantum criticality at the MIT. Careful transport experiments have revealed sharply defined critical behavior, although with exponents inconsistent with early theoretical predictions [2]. What are the basic physical processes that drive this transition and localize the electrons? Important clues have been provided by the thermodynamic response on the insulating side. Here, no magnetic ordering has been experimentally observed down to the lowest temperatures, while both the spin susceptibility and the specific heat display signatures of randomly interacting localized magnetic moments [1,3]. This puzzling behavior was largely explained by the Bhatt-Lee (BL) theory [4] of random singlet (RS) formation, using a strong-disorder renormalization group (SDRG) approach [5]. The remarkable success of the BL theory provides strong support to the early ideas of Mott [6], who first emphasized that strong Coulomb repulsion may localize the electrons by converting them into localized magnetic moments. According to this picture, the MIT in DS should be viewed as a disordered version of the Mott transition, a phenomenon dominated by strong correlation effects. An appropriate theory should then be able to describe both the local moment magnetism in (a)E-mail: [email protected] the insulator and the transmutation of these local moments into conduction electrons on the metallic side of the MIT. Unfortunately, the SDRG approach of BL, which was so successful in the insulator, is difficult to extend across the transition. The essential challenge, therefore, is to develop an alternative approach to Mott localization in a strongly disordered situation, one that at the very least can reproduce the RS physics of Bhatt and Lee. An attractive avenue to describe strong correlations has emerged in the last twenty years from studies of various Mott systems, based on resonating-valence bond (RVB) ideas of Anderson [7] and others. At the mean field level, these theories provide variational wavefunctions for quasiparticle states, which become exact in appropriate large-N limits [8]. Very recent work has extended similar variational studies to disordered systems, providing a description of phenomena such as disorder-induced non-Fermi liquid behavior [9], but did not address the physics of inter-site spin correlations central to the BL paradigm. In this Letter we examine an appropriate t-J model capable of describing the Mott transition in a disordered environment. While the large-N limit of this model generally reduces to an RVB-like variational problem, here we concentrate on the localized (t → 0) limit in the presence of strong positional disorder modeling the insulating DS. We show that: (i) the large-N formulation quantitatively reproduces all the key features of the RS regime; (ii) an accurate analytic solution of the variational problem can be thus obtained, providing closed form expressions for various physical quantities; and (iii) the approach can