Magnetic materials are generally characterized by atomic moments (or electron spins, in an itinerant picture) all aligned parallel or antiparallel to the same direction in space - the global quantization axis. Even at zero temperature however, many notable exceptions to this rule exist in which the direction of the magnetization varies from point to point in space. There are many examples of coherent helical, spiral and canted spin structures, a classic example being fcc iron. Other straightforward examples are given by topologically frustrated antiferromagnets such as triangular lattices which have physical manifestations in ultrathin films of manganese atoms adsorbed on a copper (111) surface, and in particular phases of condensed molecular oxygen. Even something as simple as the zero-temperature ground state of the homogeneous electron gas is a static helical spin density wave in the Hartree-Fock approximation.

Despite the fact that the spin-polarized density functional theory was originally and correctly formulated in terms of complex spinor wave functions, most implementations of it are done in an approximate way such that the vast majority of modern electronic structure codes are incapable of treating problems involving non-collinear spins. Spin-polarized calculations are of course routine but these deal with the spin density which can be loosely defined as `excess of up-spin electrons compared with down-spin electrons', or, `density of spin angular momentum around the z-axis'. There is no sense in which the spin has a vector nature. To begin to go further we must consider the form of the one-electron spin orbitals used in electronic structure calculations. In the restricted orbital form, all spin orbitals are pure space-spin products occupied in pairs with a common spatial factor. In spin-unpolarized calculations, orbitals of unrestricted form are generally used i.e. they are no longer occupied in pairs and have different spatial factors for different spins but still take the form of pure space-spin products. In the general unrestricted form required to treat non-collinear spins the orbitals are no longer restricted to simple product form - each orbital becomes a complex two-component spinor wave function. In this representation, we can define a vector magnetization density by summing over occupied spinor orbitals. The regular spin density is just one of the three components of this vector field.

Once the orbitals are expressed in this form, the question of writing down the correct expression for the energy in DFT is still an active research topic (conventional exchange correlation functionals are written in terms of the density and spin density only). Efforts have been made for some time to correctly formulate a DFT appropriate for systems with non-collinear spins, beginning with the simple approximations of Kübler and collaborators in the 1980s, in which they made a type of `atomic sphere appromixation' and chose beforehand the desired magnetization direction in each sphere. Using local rotation matrices they then diagonalized the 2x2 spin density matrix in each sphere, allowing local application of the conventional LSDA. The focus since then has been on developing new density functionals which depend explicitly on quantities such as the so-called `staggered density' in addition to the density in a form which explicitly allows non-local spin-dependent correlations. As far as we know, a formulation of QMC in terms of non-collinear spins has never been attempted. In this project I therefore intend to derive the relevant algorithms for doing this in QMC. In the long term I intend to implement this capability in CASINO and to apply it to systems of interest.