Path-integral simulation of solids

Abstract

The path-integral formulation of the statistical mechanics of quantum many-body systems is described, with the purpose of introducing practical techniques for the simulation of solids. Monte Carlo and molecular dynamics methods for distinguishable quantum particles are presented, with particular attention to the isothermal-isobaric ensemble. Applications of these computational techniques to different types of solids are reviewed, including noble-gas solids (helium and heavier elements), group-IV materials (diamond and elemental semiconductors), and molecular solids (with emphasis on hydrogen and ice). Structural, vibrational, and thermodynamic properties of these materials are discussed. Applications also include point defects in solids (structure and diffusion), as well as nuclear quantum effects in solid surfaces and adsorbates. Different phenomena are discussed, as solid-to-solid and orientational phase transitions, rates of quantum processes, classical-to-quantum crossover, and various finite-temperature anharmonic effects (thermal expansion, isotopic effects, electron-phonon interactions). Nuclear quantum effects are most remarkable in the presence of light atoms, so that especial emphasis is laid on solids containing hydrogen as a constituent element or as an impurity.

Quantum effects in graphene monolayers: Path-integral simulations

“…(6)] also vanishes at low T . This is similar to the results of earlier path-integral simulations of 3D solids, 40,48 where the resulting data were in agreement with the basic laws of thermodynamics. A more subtle question appears for the thermal expansion of graphene.…”

“…An additional chain of four barostats was coupled to the area of the simulation box to yield the required constant pressure (here P = 0). 48,60 The equations of motion were integrated by employing the reversible reference system propagator algorithm (RESPA), which allows us to define different time steps for the integration of the fast and slow degrees of freedom. 63 The time step ∆t associated to the interatomic forces was taken in the range between 0.5 and 1 fs, which was found to be adequate for the interatomic interactions, atomic masses, and temperatures considered here, and provided appropriate convergence for the studied magnitudes.…”

“…For motion in the (x, y) plane, both C 2 x and Q 2 x converge fast with increasing system size to their asymptotic value, similarly to the behavior known for vibrational motion in 3D solids. 48 …”

Simulation and understanding of atomic and molecular quantum crystals

“…Nevertheless, in the case of quantum crystals it is well-known that the role of quantum statistics is secondary at moderate and high temperatures (e. g., T > 100 K in hydrogen at P ∼ 100 GPa, see McMahon et al, 2012). In those situations, PIMD can be used to compute, for instance, quantum timecorrelation functions and transition state rates in a very efficient manner (Gillan, 1990;Habershon et al, 2013;Herrero and Ramírez, 2014). The key idea behind PIMD is to formulate a Hamiltonian framework in which new space coordinates and momenta, (u ij , p ij ), are introduced for sampling the integral in Eq.…”

“…Next, we briefly review the knowledge of the phase diagram of these compounds and highlight the aspects that remain contentious. Due to their intrinsically rich and complex nature, it is not possible to provide here a detailed description of H 2 , H 2 O, N 2 , and CH 4 , hence we address the interested reader to other recent and more specialised articles (see, for instance, McMahon et al, 2012;Goncharov, Howie, and Gregoryanz, 2013;Herrero and Ramírez, 2014). For the sake of focus, only those aspects related to the crystalline phases are considered in this section.…”