Topological Catastrophe and Isostructural Phase Transition in Calcium

We predict a quantum phase transition in fcc Ca under hydrostatic pressure. Using density functional theory, we find, at pressures below 80 kbar, the topology of the electron charge density is characterized by nearest neighbor atoms connected through bifurcated bond paths and deep minima in the octahedral holes. At pressures above 80 kbar, the atoms bond through non-nuclear maxima that form in the octahedral holes. This topological change in the charge density softens the $C^{\prime }$ elastic modulus of fcc Ca, while $C_{44}$ remains unchanged. We propose an order parameter based on applying Morse theory to the charge density, and we show that near the critical point it follows the expected mean-field scaling law with reduced pressure.

Figure 1

Atoms and bond paths in the low pressure topology of fcc Ca are shown left. Contours of the total charge density are shown for the (200) plane (center). The shaded regions represent minima in $\rho (\overrightarrow{r})$. The bifurcated bond paths and arrows indicating their motion under pressure are shown far right.

Figure 2

The atoms and bond paths in the high-pressure topology of fcc Ca (left). Contours of the total charge density are shown for the (200) plane (right). The shaded regions represent local minima in $\rho (\overrightarrow{r})$.

Figure 3

Plot of $C_{44}$ (crosses) and $C^{\prime }$ (circles) in Mbar as a function of pressure in kbar. The $C^{\prime }$ points are also connected by a dashed line to help guide the eye, and the lattice constant corresponding to the topological change is indicated with a vertical dashed line.

Figure 4

Strains with only nonvanishing ($A$) ${ϵ}_{xy}$ and ($B$) ${ϵ}_{xx}=-{ϵ}_{yy}$. The dashed line indicates the strained square. Non-nuclear maxima in the high-pressure phase are denoted by open circles, while filled circles denote Ca atoms.

Figure 5

Plot of $\ln s/\ln p$ vs $p$. Extrapolation to $p=0$ gives a limiting value of $1/\delta =0.33$, consistent with the mean-field scaling law.