Quantum Hooke's Law to Classify Pulse Laser Induced Ultrafast Melting

Ultrafast crystal-to-liquid phase transition induced by femtosecond pulse laser excitation is an interesting material's behavior manifesting the complexity of light-matter interaction. There exist two types of such phase transitions: one occurs at a time scale shorter than a picosecond via a nonthermal process mediated by electron-hole plasma formation; the other at a longer time scale via a thermal melting process mediated by electron-phonon interaction. However, it remains unclear what material would undergo which process and why? Here, by exploiting the property of quantum electronic stress (QES) governed by quantum Hooke's law, we classify the transitions by two distinct classes of materials: the faster nonthermal process can only occur in materials like ice having an anomalous phase diagram characterized with dTm/dP < 0, where Tm is the melting temperature and P is pressure, above a high threshold laser fluence; while the slower thermal process may occur in all materials. Especially, the nonthermal transition is shown to be induced by the QES, acting like a negative internal pressure, which drives the crystal into a “super pressing” state to spontaneously transform into a higher-density liquid phase. Our findings significantly advance fundamental understanding of ultrafast crystal-to-liquid phase transitions, enabling quantitative a priori predictions.


First principle calculation of hole-induced quantum electronic stress (QES)
The first-principles calculations are based on the density functional theory plane-wave method as implemented in Vienna ab initio simulation package (VASP) 1 . In calculating bulk stress induced by charge carriers, the projector augmented wave (PAW) method 2 and the PBE exchange-correlation functionals 3 are used. Tests have been done with respect to planewave energy cutoff and k-point sampling for each system to ensure convergence. The QES are obtained by calculating the difference between the Martin-Nielsen mechanical stress tensor 4 at the excited or perturbed electron density and that at the ground-state density 5 . The values we show in Fig. 1 in the main text are the average of the three diagonal components of the stress tensor, i.e.  = ( xx +  yy +  zz )/3. For cubic lattice, the QES induced by hole is isotropic, i.e.  xx = yy = zz ; while for other lattice, the QES could be anisotropic.
Similarly, figure S1 shows the hole induced QES in the materials that we proposed in Table II in the main text.

QES induced by a homogeneous electron gas at T = 0 K
For a homogeneous electron gas, QES is equal to the degeneracy pressure (DP) coming from the Pauli exclusion of electrons 6 . The DP can be calculated from the derivative of total energy of the homogeneous electron gas with respect to the volume change. We use a numerical model to quantify the QES of homogeneous electron gas. The total energy of homogeneous gas can be expressed as where V is volume,  is the electron density and () is the total energy per electron of homogeneous electron gas with density . It contains terms of kinetic energy, exchange energy and correlation energy, The kinetic energy and exchange energy at T = 0 K can be expressed analytically: so the total energy per electron can be expressed as In general, there is no analytic expression for  c (r s ), so it is calculated numerically.
We use the VWN interpolation formula for  c (r s ), which gives very good results in the density range we are interested in 7 : The negative sign convention means that a positive tensile QES (equivalent to a negative internal pressure) tends to shrink the volume, while a negative compressive QES (a positive internal pressure) tends to expand the volume.  Figure S2 shows the DP induced by different energy terms as well as the total pressure. One sees that the kinetic energy always induces positive pressure (P T ), tending to expand the volume; while both the exchange and correlation energy induce negative pressure (P x and P c ), tending to shrink the volume. At very low electron density, the exchange and correlation terms dominate (increasing faster with the increasing density); at high density, the kinetic term dominates. This gives rise to a minimum total pressure within the electron density range of our interest (0 -20 nm -3 as shown in Fig. 1 in the main text).
The inset shows the total pressure in the density range of 0 -20 nm -3 , which has a minimum at relatively low electron density, and its magnitude stays less than 1 GPa. This means that the QES of electron gas in the electronhole plasma is essentially negligible compared to the QES induced by the localized valence holes. Therefore, we can approximate the QES induced the electron-hole plasma by just the QES induced by holes as shown in Fig. 1 in the main text.

QES of a free electron gas at finite temperature
When the electron are excited by the pulse laser, it possess high kinetic energy (several eV), the temperature of the electron system could be very high, tens of thousands kelvin.
At finite temperature, the kinetic energy, exchange and correlation energy vary with temperature, and so does the DP. Here, for simplicity, we only consider the kinetic term, i.e. we calculate the temperature dependent DP of a free electron gas, to see how the temperature may affect the QES.
For a free electron gas, the density of states is at finite temperature, the electrons obey Fermi-Dirac distribution, the total energy of the free electron gas can be expressed as the electron number can also be expressed by the integral as where V is the volume,  is the chemical potential of the free electron gas, which varies with temperature. To calculate the total energy vs. T, we need to calculate (T) first.
At T = 0 K, the electron number where  F is Fermi energy.
Since the electron number does not change with temperature, we have Equation (12) can be expressed in a dimensionless form as Using Eq. (9), (13) and (14), we can calculate the total energy of free electron gas at given electron density and temperature, numerically. The DP of free electron gas at finite temperature can be calculated by (15) Figure S4 shows the calculated DP vs. electron density at different temperature. We can see that only at very high temperature ( >10 4 K), the DP differs notably from that at zero temperature; the DP decreases with increasing temperature, which is probably due to the decreasing of chemical potential with increasing temperature, so that adding electron into the system (increasing electron density) increases less energy to the total energy. Another way we may understand such behavior is that at higher temperature, the electrons distribute quasi-equally in all energy states(<k B T), so the Pauli exclusion effect becomes weak, and so does the DP.