Supersymmetric Hubbard operators: Formalism, application and inspiration gained from their usage

In this thesis a supersymmetric Hubbard operator formalism is introduced as a candidate for a microscopic theory of how the Fermi surface could break down close to an antiferromagnetic quantum critical point. This U(1)xSU(1|1) gauge theory hopes to describe the physics of heavy fermions about such a point where purely magnetic or paramagnetic descriptions break down, extending work by Coleman et al. [1, 2] to include charge fluctuations. Within this formalism, an SP(N) t-J model has been written down which may allow one to investigate the possibility of spin and charge separation in the cuprate superconductors. For a simple uncoupled atomic model these operators are found to admit a phase diagram bearing a superficial resemblance to that seen experimentally, in the uncontrolled limit of a single charged SU(2) spin. Exact results for this model are derived and shown to agree well with the field theoretic description as the number of spin degrees of freedom N becomes large. Various approximation schemes have been compared at both a mean field and gaussian corrections level: Abrikosov fermions and Schwinger bosons with susy spins; slave bosons and slave fermions with susy Hubbard operators. A spin degenerate single impurity Anderson model has been treated in the T → 0 limit using the slave fermion and supersymmetric representations. The local magnetization as a function of applied field is found to reproduce the two-stage Kondo effect seen by Coleman et al. [1, 2] for E*f << -Delta0. For E*f ≈ -Delta0 the mixed valent region, the second Kondo quenching process occurs at increasing separation from the first quenching process; while for E*f≥-D0 2 the second quenching effectively disappears altogether. Insight gained from this description led to an effective model for LiV2O 4, a mixed valent frustrated system, whose strongly Hund's coupled spins appear to undergo a two-stage spin quenching to form a heavy Fermi liquid.