# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 5 Applications of Quantum Baths to Realistic Systems
### Dynamical Properties
#### 5.3.4 Thermal Conductivity of Insulating Crystals  
[PERSON] aimed at simulating heat transport in realistic nanostructures from the ballistic regime at low temperatures to the diffusive regime at high \(T\), via molecular dynamics [64]. He noted that the kinetic theory for phonons provides the thermal conductivity as \(\kappa=\frac{1}{3}cvl\), where \(c\) is the heat capacity, \(v\) the sound velocity and \(l\) the mean free path. Therefore, classical molecular dynamics cannot provide a reliable estimation of heat transport in the low-temperature regime as \(c\) goes to a non-zero constant for \(T\to 0\) in the classical framework, which clearly contradicts the experimental facts. [PERSON] then followed the [PERSON] and [PERSON]'s approach [63], that he extended to anharmonic systems. In order to make the problem tractable in that case, [PERSON] introduces a quasiclassical approximation similar to that of [PERSON] [32], and ended up with a generalized [PERSON] equation. It is formally identical to (26), apart from the fact that the harmonic force term \(-Ku^{C}\) is replaced by a general nonlinear force \(F(u^{C})\). However, the interpretation of this equation is completely modified in the quasiclassical approximation as, instead of quantum position operators, the vector \(u^{C}(t)\) now describes the classical-like stochastic trajectories of the atoms that can be obtained by integrating the GLE using molecular dynamics techniques. [PERSON] named such a method _quantum molecular dynamics_, but in the context of the present review, we refer to this approach as quantum heat bath (QHB): he showed that the QHB allows the recovery of the exact quantum thermal conductivity in the case of harmonic chains, and provides consistent results for both the high- and low-temperature regimes for anharmonic one-dimensional chains [64]. Later on, [PERSON] and coworkers [65] applied the QHB to analyze thermal transport in a two-dimensional realistic model of graphene nanoribbon. They show that the QHB results are more consistent than other schemes that use _a posteriori_ quantum corrections to otherwise classical dynamics. For instance, some authors run classical MD simulations at a \"quantum-equivalent classical temperature\" \(T_{MD}=\int d\omega D(\omega)\theta(\omega,T)/k_{B}\), where \(D(\omega)\)is the phonon density of states [68]. Such _a posteriori_ correction schemes are nevertheless not fully consistent, even for the one-dimensional harmonic chain, as they differ from the exact result by a numerical factor [65]. Interestingly, [PERSON] and coworkers also provide a diagrammatic perturbative expansion of the QHB results for weakly anharmonic systems and analyze their discrepancies with respect to a fully quantum-mechanical nonequilibrium Green functions approach. From this analysis, they argue that QHB is exact in leading order in the anharmonic perturbation for the phonon lifetime. The thermal conductivity of solid argon, modelled via Lennard-Jones interatomic potentials, was also investigated using nonequilibrium QTB molecular dynamics by [PERSON] et al. [51]. In these calculations, the simulation cell is divided into slabs, two of which are thermalized at different temperatures. The ratio between the imposed temperature gradient and the resulting heat flux then yields the thermal conductivity. As was observed for other materials [12; 69], the QTB greatly improves the estimation of the heat capacity with respect to classical simulations. Nonetheless, these authors observed that, at low temperatures (where NQEs are expected to be most relevant), the experimental trends for the thermal conductivity were much better predicted when a classical Langevin equation was used as a thermostat than with the QTB. This result, which contrasts the outcomes by [PERSON] and coworkers [64; 65], was related by [PERSON] et al. to the underestimation of the phonon lifetimes.