Theoretical Condensed Matter

Prototypical Quantum Spin Hamiltonians in the Language of Jordan-Wigner Fermions
Jordan Wigner Presentation

\[Abstract\]

We present here the Jordan-Wigner solutions to three prototypical spin 1/2 Hamiltonians. In particular, we consider the 1D transverse Ising model, XY model, and Kitaev honeycomb model. After transforming the Hamiltonian to spinless fermions via Jordan-Wigner transformation, we employ Fourier transform then Bogoliubov transformation to diagonalize the Hamiltonian exactly. Moreover, the spectrum of elementary excitations as well as the ground-state energy are examined. In addition, some correlation functions of the solved models are covered briefly. Lastly, the research problem of an extension to the Kitaev model is introduced.

Background

I have been exposed to theoretical condensed matter as part of my undergraduate research course, under the supervision of Dr. Michael Vogl. I developed a a special interest for using Jordan-Wigner transformations to solve Kitaev honeycomb model. Look at these transformations, they’re beautiful!

\[\sigma_{ij}^+=2\left[\prod_{j'<j}\prod_{i'}\sigma_{i'j'}^z\right]\left[\prod_{i'<i}\sigma_{i'j}^z\right]c^\dagger_{ij}\] \[\sigma_{ij}^-=2c_{ij}\left[\prod_{j'<j}\prod_{i'}\sigma_{i'j'}^z\right]\left[\prod_{i'<i}\sigma_{i'j}^z\right]\]