# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 5 Applications of Quantum Baths to Realistic Systems
### Dynamical Properties
#### 5.3.3 From Equilibrium to Nonequilibrium  
In this section, we first introduce the approach adopted by [PERSON] and [PERSON] [63] to compute the thermal conductivity within a fully quantum formalism in the case of a one-dimensional harmonic chain of atoms. We then shortly describe how this method was generalized to anharmonic models by [PERSON] [64], by introducing a quasiclassical approximation in which the thermal conductivity is evaluated through molecular dynamics simulations with a quantum heat bath (QHB). This approach has enabled the study of two realistic systems [65] and it presents striking formal similarities with the QTB (and QT) methods. We conclude this section by briefly reviewing a few studies that exploit the QTB formalism to evaluate similar nonequilibrium properties. We briefly recall the connection between equilibrium properties and time-dependent propagators via atomic Green's functions, which is the subject of a vast amount of literature [66]. Here, we follow the formalism adopted by [PERSON] and [PERSON] for a harmonic system [63]. They considered a one-dimensional harmonic chain of \(N\) particles of mass \(M\), with displacement \(u_{j}\) (\(j=1,\ldots,N\)) with respect to their equilibrium position. The chain is linearly coupled at its left (L) and right (R) sides with two semi-infinite harmonic lattices that act as heat reservoirs at two different temperatures \(T_{L}\) and \(T_{R}\). Substituting the formal solution of the quantum propagation for these reservoirs, one can eliminate the bath degrees of freedom and obtain the quantum Langevin equation for the displacement vector: \(u^{C}=(u_{1},\ldots,u_{N})\):  
\[\ddot{u}^{C}=\ -Ku^{C}\ -\int_{t_{0}}^{t}ds\ \Sigma^{+}(t,s)\ u^{C}(s)\ +\ \xi_{L}(t)\ +\ \xi_{R}(t) \tag{26}\]  
where \(K\) is the \(N\times N\) coupling matrix describing the harmonic forces within the chain, and \(\Sigma^{+}=\Sigma_{L}^{+}+\Sigma_{R}^{+}\) is the self-energy matrix due to the combined interaction with the left and right reservoirs. Since the baths are assumed to be harmonic, \(\Sigma^{+}(t,s)\) can be expressed analytically in the frequency domain. The noise vectors \(\xi_{L}(t)\) and \(\xi_{R}(t)\) can then be related to the self-energies by assuming that the reservoirs are at thermal equilibrium and therefore obey Bose-Einstein statistics, which yields the following fluctuation-dissipation relation for the symmetrized correlation function in the frequency domain:  
\[\frac{1}{2}\langle\xi(\omega)\xi^{T}(\omega^{\prime})+\xi(\omega^{\prime})\xi ^{T}(\omega)\rangle\ =\ \delta(\omega+\omega^{\prime})\ \frac{\hbar\Gamma(\omega)}{2\pi}\coth \bigg{(}\frac{\hbar\omega}{2k_{B}T}\bigg{)} \tag{27}\]  
where \(\Gamma(\omega)=\text{Im}[\Sigma^{+}(\omega)]\) and \(\tilde{\xi}(\omega)\) refer either to the left or the right lead, with the appropriate temperature \(T=T_{L},T_{R}\). Here the bath self-energy provides the connection between the equilibrium correlation function of the bath and the time propagator of the system that is needed to evaluate transport and nonequilibrium properties. The quantum Langevin Equation (26) is formally very similar to the GLE (6), that was also derived under the assumption of bilinear coupling with an harmonic bath and extended by [PERSON] and [PERSON] to a nonequilibrium case with two baths at different temperatures. The atomistic Green's function for the chain can then be expressed as:  
\[G^{\pm}(\omega)=\Big{[}-M\omega^{2}+K+\Sigma_{L}^{\pm}(\omega)+\Sigma_{R}^{\pm }(\omega)\Big{]}^{-1} \tag{28}\]  
In this expression, the harmonic frequencies of the isolated chain are renormalized by the self-energies of the interaction with the right and left bath leads.