Gauge theories of spin systems

Abstract

In this dissertation, I discuss the gauge theory description of interacting spin systems, which results from the application of slave-particle approach. In particular, I discuss three types of gauge theory description. Starting from the Abrikosov fermion representation of spin, I review the effective SU(2) gauge theory of the Heisenberg model on the mean-field level. I then move on to study another types of spin representation, the Majorana fermion representation. After a discussion on the relationship between the three types of Majorana representation, namely the SO(3) Majorana representation, the SO(4) chiral representation and the Kitaev representation, I focus on the SO(3) Majorana representation and show that its non-local nature makes it equivalent to the Jordan-Wigner transformation of spin in both one-dimensional and two-dimensional space. To apply the SO(3) Majorana representation, I discuss three two-dimensional spin models, namely the Kitaev honeycomb model, the quantum XY model on honeycomb lattice and the $90^{\circ}$ compass model on square lattice. Using the SO(3) Majorana representation, I demonstrate how to map the spin Hamiltonians into $Z_{2}$ lattice gauge theories with standard Gauss-law constraint. The mapping differs from the mean-field approach in that the resulting gauge theories are exact. In the third part of the dissertation, I discuss the application of non-local spin representations to some specific spin models. In particular, I review the effective U(1) lattice gauge theory for the spin ice model on pyrochlore lattice and discuss the potential application of staggered Abrikosov fermion representation in spin ice model and kagome antiferromagnetic model.