# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 1 Introduction
In particular, while these methods deliver remarkable performances in a growing number of applications, they cannot be formally justified except for harmonic systems. This intriguing scenario motivates our approach for this review. In Section 2, we start by stating some fundamental results in the statistical mechanics of classical and quantum systems, the fluctuation-dissipation theorems (FDT). We then introduce the standard and generalized Langevin equations and set the stage for the following developments. This is done, in particular, by recalling how the GLE can be derived starting from the physical picture of a generic system (not necessarily harmonic) bilinearly coupled to an environment modelled as a set of harmonic oscillators. We shall consider both classical and quantum versions of this system-bath model and describe how the quantum version can be used, via an appropriate quasiclassical limit, to motivate the quantum bath algorithms on which we focus in this review. While more general models for the bath are possible (including, in particular, anharmonicity) these are not particularly relevant for this work and will therefore not be discussed, nor included in examples. Section 3 focuses on using the quantum FDT to build GLEs that combine classical deterministic evolution with quantum stochastic behaviour to mimic NQEs. In particular, we introduce two approaches, the Quantum Thermal Bath [12] (QTB) and the Quantum Thermostat [13; 14] (QT) that have recently attracted considerable interest. Section 4 aims at understanding more in depth the capabilities of the quantum thermal bath via the detailed analysis of simple low-dimensional systems that enable a comparison with exact quantum results. In particular, the most important pathology of this type of methods, namely the ZPE leakage, is illustrated on an elementary example. Although this section focuses on the QTB, which is formally simpler and has fewer parameters than the QT, the general observations apply to both frameworks. In Section 5, we review the capabilities of quantum bath methods by summarizing some significant applicationsto realistic models of condensed matter systems. Remarkably, these techniques prove useful not only in the simulation of static equilibrium properties of quantum nuclei, but also in the calculation of vibrational spectra [15], including subtle anharmonic features that elude other trajectory-based approximations to quantum dynamics [16]. In Section 6, we complete the overview of important applications of thermal baths by showing how they can be used to improve the convergence of calculations based on the path integral formalism. We then introduce in Section 7 a recent development that, by exploiting the quantum fluctuation-dissipation theorem, enables to monitor and efficiently compensate ZPE leakage. This approach, known as the adaptive quantum thermal bath [17] (adQTB) extends considerably the domain of application of the QTB, leading, for example, to the successful simulation of liquid water [18]. The review ends with a discussion of remaining limitations, open questions and possible future developments of quantum baths as effective and accurate tools for the simulation of condensed phase systems. We conclude this Introduction by noting that GLEs have a long and rich history, with applications in a very broad range of domains that we do not account for here. The literature on quantum baths is also rapidly increasing. While we have tried to provide appropriate references throughout the text, we have chosen to present the material focusing on results and derivations that seemed more directly related to the methods at the core of this work, intended as a relatively self-contained introduction to a still-developing set of exciting new algorithms for growing classes of quantum problems in physics and chemistry.