# CM6: Spin-Charge Separation & Quantum Magnetism

# Module Title: Spin-Charge Separation & Quantum Magnetism

# Module Convenor: Martin Long

## Module aims:

This course will develop some of the mathematical ideas that explain why the charge and spin of an electron become separate and the system exhibits charge-only and spin-only excitations. The model that will be tackled is the large-U Hubbard model in one-dimension with first- second- and third-neighbour hopping. The mathematical technique employed will be `infinite particle transformations' of which the simplest example is Jordan-Wigner. The method of solution will not extend beyond Hatree-Fock mean-field theory but the solutions will be exact in a variety of limits.

## Module content:

In detail the course will contain

- (1) Ordinary mean-field theory for the Hubbard model and magnetism
- (2) Nagaoka's theorem and extensions (Charge separation)
- (3) The Jordan-Wigner transformation
- (4) The charge-spin analogue to Jordan-Wigner and the Hubbard model
- (5) The extension to longer-range hopping and magnetic phase transitions

The complete understanding can only be obtained by then using the ideas presented in the course "Luttinger Liquid Theory" which is a pre-requisite.