NEFROPATIA CRONICA DEL TRASPLANTE

5.2.1.1 Crystal phase

Our crystal phase many-body trial wave functions are Slater determinants of Gaus- sian orbitals, arranged on stripes of a triangular lattice, multiplied by Jastrow factors. A particular lattice point is associated with a single-particle orbital of the form ψPσ(ri) = exp −g_|ri−Pσ|2 , (5.19)

where we denote the Wigner crystal site associated with the ith electron (having spinσ) asPσ, and the (optimisable) parameter controlling the width of the Gaus-

sian orbital g. This removes any possible confusion over the underlying lattice vectors for our TMD host material (_{R_}), which now play no role in our contin- uum model for conduction charges. The Pσ are distributed as shown in Fig. 5.3.

The Jastrow factors in our crystal calculations are initially comprised of isotropic two-body, and plane-wave expansions in simulation cell reciprocal lattice vectors (u, pterms respectively), which are optimised by variance minimisation [36,38,182]. We then include a backflow function comprised solely of a two-body (η) term, and re-optimise the Jastrow factor and backflow function in tandem with energy min- imisation [39,40]. Throughout the optimisation process, we allow the Gaussian parameterg to vary, setting its value initially to equal the “DMC-optimised” value

∼1/√2g

Figure 5.3: Lattice points for the striped configuration of electrons in our crystal phase trial function. Black squares (blue circles) denote up(down)-spin, K(0)-valley electrons (i.e. the _{P↑}({P↓}) set). The grey shaded region denotes one standard

deviation of an individual electron orbital, as related to the orbital parameterg. reported for the two-dimensional HEG in Ref. [262]. Having tested the strategy of optimising g at the DMC level, then optimising a backflow on top of a fixed g

value, we find that we are able to obtain better VMC energies by utility of our aforementioned strategy: let all parameters vary, allow backflow parameters and the Gaussian parameter to (ultimately) vary together.

The success of this approach is likely related with the additional freedom it allows in the description of the Gaussian orbital itself. On its own, the backflow function enters the r-dependence of the individual Gaussian orbitals, and hence acts as a means to control the width of the Gaussian, in a somewhat restrictive way. By also allowing the parameter that directly controls the width to vary (g), we effectively create variational freedom in the long-range part of the Gaussian orbital. Such freedom is clearly important in lowering the energy of the crystal phase, at least near the phase boundary, where the de-localised fluid phase competes with the crystal.

In principle,g can be spin-dependent, but we have neglected to include this de- pendence here, as we find no significant lowering of the VMC energy (no symmetry breaking) when allowing this additional freedom.

5.2.1.2 Fluid phase

Our fluid phase many-body trial wave functions are simply Slater determinants of plane-wave orbitals, ψk(ri). The grid of k points on which we perform our

calculations is formed as described in 4.1.1. An important difference in this case, however, is that the twist ks is of critical importance. Finite-size effects in fluid- phase calculations are extreme, and linked directly with the varying occupation of single-particle states as one changes system size, and/or twist angle ks. We describe our strategy for twist averaging our fluid-phase total energies later in this chapter.

The Jastrow factors in our fluid calculations are initially comprised of isotropic two-body, and plane-wave expansions in simulation cell reciprocal lattice vectors (u, pterms respectively), which are optimised by variance minimisation [36,38,182]. We then include a backflow function comprised solely of a two-body (η) term, and again re-optimise the Jastrow factor and backflow function in tandem by energy minimisation [39,40].

Finally, we note that we only consider the paramagnetic case where an equal number of electrons of each spin and valley degree of freedom are present. In the case of the two-dimensional HEG, it has been observed [262] that there is phase competition between the paramagnetic and ferromagnetic fluids for an intermedi- ate density regime around the crystallisation density, but that the ferromagnetic fluid is never stable at any density. At high density, the paramagnetic fluid is always the most stable. However, in the present case, the full inclusion of physics of this variety would also require an additional treatment of inter-valley scattering, and a description of the spin splitting of the conduction bands. Whilst desirable in general, such a study is beyond the scope of the present work.

We can make some basic comments on the likely crystallisation prospects, and those of itinerant ferromagnetism in metallic two-dimensional systems. The Keldysh interaction suppresses the strong Coulomb repulsion, especially at short range. Short-range interactions are most important for the paramagnetic fluid

(unlike spins don’t experience an exchange interaction), followed by the ferromag- netic fluid, followed by the Wigner crystal phase, which is largely insensitive to short-range interactions where it is relevant (low density). We expect, then, that starting from the two-dimensional HEG, switching on a finite polarisability leads to, predominantly, an energy lowering of the paramagnetic fluid phase. This acts to move the crystallisation density lower (to a higher rs value). Assuming the ef- fects of the softened interaction stabilise the paramagnetic fluid the most, as stated above, the transition from paramagnetic to ferromagnetic fluid in a HEG with po- larisability (not necessarily under the restricted model we describe) becomes even less likely, and it is therefore unreasonable to expect itinerant ferromagnetism to be any more likely in doped two-dimensional semiconductor systems as compared to the two-dimensional HEG.