Simulation and understanding of atomic and molecular quantum crystals

Quantum crystals abound in the whole range of solid-state species. Below a certain threshold temperature the physical behavior of rare gases (${}^4\mathrm{He}$ and Ne), molecular solids (${\mathrm{H}}_2$ and ${\mathrm{CH}}_4$), and some ionic (LiH), covalent (graphite), and metallic (Li) crystals can be explained only in terms of quantum nuclear effects (QNE). A detailed comprehension of the nature of quantum solids is critical for achieving progress in a number of fundamental and applied scientific fields such as planetary sciences, hydrogen storage, nuclear energy, quantum computing, and nanoelectronics. This review describes the current physical understanding of quantum crystals formed by atoms and small molecules, as well as the wide palette of simulation techniques that are used to investigate them. Relevant aspects in these materials such as phase transformations, structural properties, elasticity, crystalline defects, and the effects of reduced dimensionality are discussed thoroughly. An introduction to quantum Monte Carlo techniques, which in the present context are the simulation methods of choice, and other quantum simulation approaches (e.g., path-integral molecular dynamics and quantum thermal baths) is provided. The overarching objective of this article is twofold: first, to clarify in which crystals and physical situations the disregard of QNE may incur in important bias and erroneous interpretations. And second, to promote the study and appreciation of QNE, a topic that traditionally has been treated in the context of condensed matter physics, within the broad and interdisciplinary areas of materials science.

Figure 1

de Boer parameter estimated in a series of crystals spanning over the entire range of solid-state species. The cases in which ${\mathrm{\Lambda }}^{*}$ adopts a value larger than 0.012, namely, C and above, are indicated with red (dark) dots.

Figure 2

Sketch of the main differences between quantum (left) and classical (right) solids in the zero-temperature limit. (a) Radial pair distribution function $g(r)$. (b) Localization of the atoms around their equilibrium lattice positions; in the quantum case, lines represent the evolution of the atomic positions along time and show the occurrence of atomic quantum exchanges. (c) Compton profile of the longitudinal momentum distribution $J(y)$.

Figure 3

Shear modulus in an extremely pure ${}^4\mathrm{He}$ crystal expressed as a function of temperature. Three different regimes are observed that can be explained in terms of dislocation dynamics (see text). The unsolved problem about which are the interactions between dislocations and isotopic impurities is noted schematically. Adapted from [263].

Figure 4

Sketch of the key idea behind machine learning for the modeling of atomic interactions. Ab initio results (solid line), which are obtained at high computational cost, are approximated by interpolating with ML potentials (dashed line) between selected reference configurations (blue dots).

Figure 5

Phase diagram of ${}^4\mathrm{He}$ at low pressures and temperatures.

Figure 6

Energy per particle in solid ${}^4\mathrm{He}$ expressed as a function of density. Open triangles represent experimental data from [175]. Other symbols and lines correspond to the DMC results. Solid circles and line represent results in which the bias introduced by finite-size effects has been reduced significantly. Solid triangles and dashed line represent results which have been corrected only partially for the same type of bias. Adapted from [560].

Figure 7

One-body density matrix calculated in hcp ${}^4\mathrm{He}$ at density $\rho =0.0294{\AA }^{-3}$. Circles, squares, and diamonds stand for results obtained at $T=0$, 1, and 2 K, respectively. Adapted from [484].

Figure 8

Zero-temperature equation of state of hcp ${}^4\mathrm{He}$ at high pressures calculated with several effective three-body interaction models fitted to reproduce ab initio data, and the DFT-D method (see text). Experimental data from [356] are shown for comparison. Inset: $P(V)$ curves in the high-$P$ region are magnified in order to better appreciate the differences. From [92].

Figure 9

DMC energy per ${\mathrm{H}}_2$ molecule as a function of density. Squares and circles correspond to the liquid and solid phases, respectively. Solid and dashed lines are polynomial fits to the DMC energies for the liquid and solid phases, respectively. The diamond represents the experimental energy of hcp molecular hydrogen from [503]. From [429].

Figure 10

Equation of state of liquid (solid line) and solid (dashed line) ${\mathrm{H}}_2$. Experimental points for the solid phase are represented with solid circles ([166]). From [429].

Figure 11

Quantum nuclear effects in solid neon at low temperatures. (Top panel) Excess kinetic energy expressed as a function of temperature. Open up triangles: experimental data from [536]. Solid circles: measurements from [435]. Solid up trianges: PIMC calculations from [535]. Solid down triangle: DMC ground-state calculations from [88]. The lines in the plot are linear fits to the experimental data. (Bottom panel) Ground-state momentum distribution in solid Ne calculated with the DMC method (green dots). The solid line (red) is a Gaussian fit to the results, the width of which represents its uncertainty. Adapted from [88].

Figure 12

Elastic constants in solid ${}^4\mathrm{He}$ at moderate pressures. Solid up triangles: experimental data from [143]; open down triangles: [242]. Solid down triangles: $C_{44}$ measurements from [529]. Open up triangles: VMC calculations from [445]. Solid circles: DMC ground-state calculations from [97]. The dashed lines (green) represent linear fits to the DMC results. The freezing pressures in the crystals are marked with vertical (magenta) lines. Adapted from [97].

Figure 13

Common types of defects in crystals. (a) Representation of point defects; $V$, $Im$, and $I$, respectively, stand for vacancy, impurity, and interstitial. (b) Representation of an edge dislocation, a class of line defect, in a solid with cubic symmetry.

Figure 14

The measured distribution $N(L_N)$ of lengths $L_N$ between dislocation network nodes in a ${}^4\mathrm{He}$ crystal at $T=27\mathrm{mK}$; the contribution to the shear modulus from each dislocation length is also indicated. From [189].

Figure 15

Energy per particle in two-dimensional solid ${\mathrm{D}}_2$ (solid line and filled circles) and two-dimensional solid ${\mathrm{H}}_2$ (dotted line and empty triangles). From [89].

Figure 16

Luttinger parameter $K$ in one-dimensional ${}^3\mathrm{He}$ expressed as a function of the linear density $\rho$. The corresponding speed of sound, as extracted from the phononic part of the static structure factor (symbols) and thermodynamic compressibility (line), is also shown. Adapted from [15].

Figure 17

Optimal distribution of equilibrium sites in solid ${\mathrm{H}}_2$ clusters with $N=18$, 19, and 20 molecules at zero temperature. From [520].

Figure 18

High-$P$ phase diagram of solid molecular hydrogen and deuterium. Open and filled circles are Raman measurements for hydrogen and deuterium, respectively, and open squares are IR data for the hydrogen ([224]). Triangles and the dashed lines indicate inferred boundaries for the recently proposed phase IV ([286]). The dashed line and the existence of phase ${\mathrm{I}}^{\prime }$ are still matters of debate (see text). From [387].

Figure 19

The classical phase diagram of water based on the TIP4PQ/2005 force field (dashed blue lines) as compared to the quantum phase diagram obtained with path-integral simulations (solid red lines). Roman numerals label different crystal structures with hexagonal symmetry (${\mathrm{I}}_h$), rhombohedral (II), tetragonal (III and IV), and monoclinic (V). Adapted from [385].

Figure 20

The influence of QNE in normal systems and materials with a shallow multiwell potential-energy surface (PES). (a) The free-energy barrier separating two local PES minima is too large as compared to the atomic zero-point motion (ZPE), hence the system remains indifferent. (b) The free-energy barrier separating two different states is similar in magnitude to the ZPE; hence the system transits from one to the other. (c) Sketch of the phase diagram in a quantum paraelectric; $\delta$ represents a particular tuning field (e.g., pressure).

Figure 21

Phase diagram of ${\mathrm{BaTiO}}_3$ calculated with classical (open circles and small labels) and quantum (solid circles and large labels) simulation methods. From [294].