Trial Density Matrix

The exact density matrix is only known in a few cases. In practice, applications have approximated the fermionic trial density matrix $\rho_T({\bf R},{\bf R}';\beta)$, most commonly by a Slater determinant of single particle density matrices,

$$\rho_T({\bf R},{\bf R'};\beta)=\left\vert \begin{array}{ccc}... ...({\bf r}_{N},{\bf r}'_{N};\beta) \end{array}\right\vert \quad.$$ (91)

More generally, we now discuss systems of spin $\frac{1}{2}$ fermions, for which the Hamiltonian is spin-independent because we do not consider magnetic fields nor relativistic effects including spin-orbit interactions. There, the spin component in $\hat{z}$ direction $S_z$ can be quantized leading to a good quantum number $m$, which describes the magnetization of the systems. For ensembles with fixed magnetization $m$, the density matrix can be written as a product of two determinants,

$$\rho_T({\bf R},{\bf R'};\beta) = \left\Vert \rho_1({\bf r}_... ...{\bf r}_i,{\bf r}'_j) \right\Vert _{i,j \in \downarrow} \quad.$$ (92)

If one applies an operator that antisymmetrizes completely and then projects out states of magnetization $m$ one finds only configurations that can again be written as such a product of two determinants but with relabelled particles. From now on, we only consider spin unpolarized systems ( $m=0, N_\uparrow=N_\downarrow$). Enforcing the nodes means that each determinant stays positive all along the paths because configurations where both determinants flip signs simultaneously have zero measure.

Extensions of this type of nodes are necessary to describe a pairing mechanism that permits the formation of Cooper pairs in super conductors and electron-hole pairs in semi-conductors, which then can Bose condense (Bouchard et al., 1988). Simulations with pairing nodes have been done at zero temperature by Gilgien (1997) and Zhu et al. (1996) and at finite temperature by Shumway and Ceperley (1999).

The above trial density matrix has been extensively applied using the free particle (FP) nodes (Ceperley, 1996) including applications to dense hydrogen (Militzer et al., 1999; Pierleoni et al., 1994; Magro et al., 1996). In this case, the density matrix of a single FP in a periodically repeated box given by Eq. 2.11 is used in Eq. 2.83. It can be shown that for high temperatures, the interacting nodal surface approaches the FP nodal surface. In addition, in the limit of low density, exchange effects are negligible, the nodal constraint has a small effect on the paths and therefore, its precise shape is not important. The FP nodes also become exact in the limit of high density when kinetic effects dominate over the interaction potential. However, for high densities and high degeneracy, interactions could have a significant effect on the fermionic density matrix. To gain some quantitative estimate of the possible effect of the nodal restriction on the thermodynamic properties, it is necessary to try an alternative. In addition to FP nodes, a variational density matrix (VDM) is derived in chapter 3 that already includes interactions and atomic and molecular bound states. The effects on the thermodynamic properties from using those as nodes will be discussed in section .