# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 8 Quantumness from Baths: State of the Art and Open Issues  
Quantum baths rely on a kind of blend of classical and quantum features: the system degrees of freedom follow a classical dynamics in which quantum effects are introduced via a generalized Langevin equation. Langevin dynamics turns out to be remarkably versatile and enables various choices for the bath characteristics and the strength of the coupling to the system. In quantum baths methods, it is used to reproduce zero-point energy (ZPE) effects by thermalizing the modes of the system at a well-chosen frequency-dependent effective temperature. These approaches are particularly suited for the simulation of the statistical properties of systems that are at the borderline between the classical and the quantum worlds. This is typically the case of light nuclei in molecules and condensed matter, such as H, He, or Li, or even heavier nuclei at low temperatures. In these cases, ZPEs can dramatically change the energetic landscape of the system, especially for highly anharmonic systems and shallow potential energy profiles; at low temperatures (much smaller than the ZPE) some fluctuations can be frozen, which impacts the computed response functions, such as the heat capacity; when energy barriers are present between several distinct configurations, the quantum description can greatly differ from the classical one, whenever the extension of the nuclear wavefunction is comparable to the width of the energy barrier. Isotope effects are another important example of NQEs, as purely classical averages are mass-independent. Therefore, the variation under isotope substitution of the thermodynamical properties (geometric isotope effect) or the reaction rates (kinetic isotope effect) are often clear experimental markers of the importance of quantum mechanics. In addition, within the realm of condensed matter physics, simulations involve a large number of degrees of freedom subject to non trivial potentials that are often expensive to calculate. When NQEs cannot be neglected--in many cases even at room temperature and pressure--methods to account for them must therefore be adapted to these specific constraints. Quantum baths attempt to use a classical formalism in which the energy distribution is altered to follow the Bose-Einstein distribution via a stochastic process included into a generalized Langevin equation. Apart from some technical subtleties that come along with the use of the generalized Langevin formalism, these methods do not involve increasing the computational load substantially, which make them highly desirable for the simulation of condensed matter systems, with thousands of atoms and more, that could be unfeasible otherwise. The idea of enforcing the correct quantum energy distribution in realistic molecular dynamics simulations through a generalized Langevin equation is rather recent. As for many other approaches [86], this quantum-classical blend is clearly an approximation, which must be assessed. An important shortcoming of these approaches, which is common also to some other mixed quantum-classical methods, is the zero-point energy leakage, that is, the trend to transfer a significant part of the excess ZPE from high to low frequencies. In this respect, the use of a generalized fluctuation-dissipation theorem (FDT) to monitor and regulate energy fluxes between the bath and the distinct degrees of freedom of the system has provided an elegant and easily implementable solution to this problem. The last-generation adaptive QTBs enforce the first-kind FDT to every frequency separately rather than the second-kind FDT, as used in previous quantum baths methods, and, by keeping the ZPE leakage below a certain threshold via an effective built-in feedback mechanism, is affirming as a quite safe mixed quantum-classical approximate scheme to appreciate NQEs in many systems.