# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 2 The Langevin Equation as a Thermal Bath
### Coupling to a Harmonic Bath  
The generalized Langevin equation (6) appears in a variety of contexts. In particular, a slightly different form of GLE can be derived under very general assumptions using projection operator techniques [20; 23], an approach that has recently lead to interesting developments [24; 25]. However, the damping and random forces obtained in this formalism are generally not explicit. In this section, we focus on an alternative, more practical and explicit approach: we derive the GLE from the description of a system bilinearly coupled to a bath of independent harmonic oscillators. In passing, we note that, in principle, the bath can be constructed in different ways, for instance by including non-harmonic oscillators. However, the bath of harmonic oscillators is at the same time easy-to-treat analytically and can describe all features that are relevant for the physical system. The extended Hamiltonian for the bath and the physical system, which can be harmonic or not, (the latter case being the most physically relevant one) is then:  
\[H=\frac{p^{2}}{2m}+V(x)+\sum_{j}\Biggl{[}\frac{p_{j}^{2}}{2m_{j}}+\frac{1}{2}m _{j}\omega_{j}^{2}(q_{j}-x)^{2}\Biggr{]} \tag{9}\]  
where \(x\) and \(p\) are the system coordinate and momentum, while \(q_{j}\), \(p_{j}\) are an ensemble of harmonic oscillators that constitute the thermal bath. The coupling to the system arises from the fact that the equilibrium position of the bath oscillators is taken at \(q_{j}-x\) and therefore changes in time with the variations of \(x\). It is then easily shown that the equation of motion for the system coordinate can be recast into a GLE of the form of (6), where the memory kernel \(K(\tau)\) is defined as:  
\[ \begin{split} K(\tau)&=\sum_{j}m_{j}\omega_{j}^{2} \cos(\omega_{j}\tau),\\ \text{or equivalently, }\tilde{K}(\omega)&=\pi\sum_{j}m_{j} \omega_{j}^{2}\bigl{[}\delta(\omega-\omega_{j})+\delta(\omega+\omega_{j}) \bigr{]}\end{split}  \tag{10}\]  
The damping force is therefore generally non-Markovian with a kernel that depends on the spectral density of the bath oscillators. In particular, in the limit of an infinite number of bath oscillators with a spectral density proportional to \(1/\omega^{2}\), \(\tilde{K}(\omega)\) becomes constant and the Markovian limit is recovered. The random force \(R(t)\) is related to the initial configuration of the bath at an initial time \(t_{0}\):  
\[R(t)=-K(t-t_{0})x(t_{0})+\sum_{j}\Bigl{[}m_{j}\omega_{j}^{2}q_{j}(t_{0})\cos[ \omega_{j}(t-t_{0})]+\omega_{j}p_{j}(t_{0})\sin[\omega_{j}(t-t_{0})]\Bigr{]} \tag{11}\]  
If \(t_{0}\) is in a distant past, the first term vanishes. Moreover, if the bath oscillators are assumed to be initially (i.e., at \(t_{0}\)) in thermal equilibrium, the random force autocorrelation function can be shown to obey the second-kind FDT of Equations (7) and (8). Interestingly, if the system-bath Hamiltonian (9) is treated quantum-mechanically, a GLE formally identical to Equation (6) can be derived for the time-dependent operator \(x\) in the Heisenberg picture. The expression of the memory kernel \(K(\tau)\) and the random force \(R(t)\) are the same as in the classical case (only \(R(t)\) is now an operator), but the second-kind FDT becomes (compare to (8)):  
\[C_{RR}^{s}(\omega)=\int_{-\infty}^{\infty}dt\ \frac{1}{2}\bigl{\langle}R(t)R(t+ \tau)+R(t+\tau)R(t)\bigr{\rangle}\ \text{e}^{-i\omega\tau}=\theta(\omega,T)\tilde{K}(\omega) \tag{12}\]  
where, similarly to what was reported for the first-kind FDT (3), the symmetrised correlation function is used and the classical average thermal energy \(k_{B}T\) is replaced by its quantum counterpart \(\theta(\omega,T)\). In the developments above, we followed mostly the work and formalism of [PERSON] and coworkers in [26; 27]. In their studies, the authors point out that the quite simple harmonic bath model is more general than it appears. First, they show that, for the GLE (6) to be physically meaningful, the memory kernel \(\tilde{K}(\omega)\) should be positive for all frequencies.