# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 1 Introduction  
This paper reviews methods that use generalized thermal baths, that we shall indicate generically as quantum baths. Other expressions, such as generalized or quasiclassical baths, are used in the literature to designate a family of methods revolving around various flavors of dynamics aiming at mimicking quantum properties via generalized Langevin equations. We adopt here the generic expression quantum bath, which is clear enough to specify the field without entering into the technical details separating these schemes to approximate nuclear quantum effects (NQEs) and, in particular, atomic zero-point energy (ZPE). It is well-recognised that NQEs cannot be neglected for systems at low temperatures or at high pressure, where they can have, for example, a strong influence on phase transitions [1; 2; 3]. Nuclear quantum effects, however, also often come into play at ambient conditions, typically when the thermal energy of chemical bonds in molecules is comparable or smaller than the associated vibrational ZPE or when characteristic interatomic distances in the system are comparable to the de Broglie wavelength of the nuclei. These conditions are met, in particular, in systems containing light atoms and most notably hydrogen [4; 5], which is a ubiquitous constituent of inorganic compounds and a basic elementin biological systems. Thus, nuclear quantum effects influence, for example, the stability of crystal polymorphs of pharmaceutical interest, the spectroscopy of ice and water, or enzymatic reactions in living organisms [6; 7; 8; 9]. NQEs are also required to reveal the effect of isotopic substitutions on equilibrium properties, as a classical description of the nuclei predicts identical statistical properties for all isotopes. The exact simulation of these effects is, however, a challenging problem due to the large (sometimes unachievable) computational cost of available methods. Time-independent statistical properties of quantum nuclei at thermal equilibrium can be obtained using the so-called classical isomorphism of path integrals (see Section 6), in which a quantum particle is represented as a collection of beads interacting via an effective classical-like potential. While this approach guarantees theoretical convergence to the exact quantum result in the limit of an infinite number of beads, it necessitates in practice to simulate a number of degrees of freedom that can become prohibitive, in particular when first-principles methods are used for the interatomic interactions. The situation is even more problematic for time-dependent properties for which exact simulation methods scale exponentially with the number of degrees of freedom. Different approaches have been proposed, such as in [10; 11], each with their own strengths and weaknesses; but, in spite of several interesting results, no clear, practical and general reference method has emerged. In the following, we focus on the family of approximate methods for simulating quantum nuclear effects that rely on the use of quantum baths. Although these methods can substantially differ from one another in their practical implementation, they all aim at introducing quantum statistical properties into classical trajectory-based dynamics. This is achieved via a stochastic evolution that takes the form of a generalized Langevin Equation (GLE), in which the quantum Bose-Einstein energy distribution is enforced instead of equipartition, by means of an appropriate tuning of the random and friction forces. The main advantage of these approaches is that the number of degrees of freedom in the system and the deterministic part of the propagation remain classical, thus enabling them to mimic quantum properties at a numerical cost directly comparable to that of standard Langevin dynamics. Contrary to standard Langevin, however, quantum baths exhibit a complex non-Markovian dynamics whose formal properties are not obvious.