Off-Diagonal Density Matrix Elements

In this chapter, we are going to describe the sampling of off-diagonal matrix elements using the path-integral formalism. Instead of calculating diagonal density matrix elements $\rho({\bf R},{\bf R}'={\bf R};\beta)$, for which the paths return to their starting point, we now include the possibility of open paths. Opening one path allows one the sampling of the single-particle reduced density matrix defined by,

$$\rho^{[1]}({\bf r}_1,{\bf r}_1') = \frac{{V^{\!\!\!\!\!\!\:^... ...{\bf r}_N \, , \,{\bf r}_1',{\bf r}_2,\ldots,{\bf r}_N) \quad,$$ (208)

which is related to the momentum distribution. A PIMC simulation with two open paths samples the two-particle reduced density matrix,

$$\rho^{[2]}({\bf r}_1,{\bf r}_2,{\bf r}_1',{\bf r}_2') = \fra... ..., , \,{\bf r}_1',{\bf r}_2',{\bf r}_3,\ldots,{\bf r}_N) \quad,$$ (209)

which will be used to study natural orbitals. In the following two sections, we discuss the modifications to the sampling procedure in order to deal with the open ends.