# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## Appendix B Extended Variable Approach to the Generalized Langevin Equation: Two Simple Cases  
In this appendix, we illustrate the extended variable strategy of [PERSON] et al. for practical implementation of the GLE with two simple examples where the number of auxiliary momenta is \(N=1\) and \(N=2\), the matrix \(\mathbf{A_{p}}\) and \(\mathbf{B_{p}}\) are given explicitely as well as the corresponding memory kernel.  
In the \(N=1\) case, we consider following equations of motions for \(x\), \(p\) (the physical position and momentum, respectively) and the auxiliary momentum \(s\):  
\[\dot{x} = p/m\] \[\dot{p} = -\
abla V(x)+\sqrt{\gamma\tau_{F}^{-1}}\,s \tag{10}\] \[\dot{s} = -\,\tau_{F}^{-1}\,s\ -\,\sqrt{\gamma\tau_{F}^{-1}}\,p\,+\,\sqrt{2m \tau_{F}^{-1}k_{B}T}\,R(t)\]  
where \(m\) is the mass of the (physical) degrees of freedom and \(R\) a Gaussian white noise. According to the general expression given in Section 3.2, the memory kernel is then a finite time-range decaying exponential: \(K(\tau)=(\gamma\tau_{F}^{-1})\,\mathrm{e}^{-|\tau|/\tau_{F}}\), \(\tau_{F}\) is the self-correlation time and \(\gamma\) the amplitude of the friction. Both are free parameters of the thermostat and the coupling between the physical momentum \(p\) and the auxiliary momentum \(s\) increases with \(\gamma\), while the limit of white noise is recovered for \(\tau_{F}\to 0\) (more precisely, for all relevant frequencies \(\omega\ll\tau_{F}^{-1}\)). If the coupling terms are dropped (the second term on the right-hand side of the two last equations), the motion reduces to a markovian Brownian dynamics for \(s\) and a Newtonian conservative dynamics for \(p\), both decoupled. The equations of motion (10) obey to the classical FDT (17), and the equilibrium distribution for the extended system can be shown to be \(P\propto\exp\left(-\frac{p^{2}/2m+V(x)}{k_{B}T}-\frac{g^{2}}{2 mk_{B}T}\right)\).  
More versatile functional forms for the memory kernel can be obtained by increasing the number of auxiliary momenta. In particular, for \(N=2\), the following choice of matrices:  
\[\mathbf{A_{p}}=\begin{bmatrix}0&\sqrt{\gamma\tau_{F}^{-1}/2}&\sqrt{\gamma\tau _{F}^{-1}/2}\\ -\sqrt{\gamma\tau_{F}^{-1}/2}&0&-\Omega\\ -\sqrt{\gamma\tau_{F}^{-1}/2}&\Omega&2\tau_{F}^{-1}\end{bmatrix},\qquad \quad\mathbf{B_{p}}=\begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&b\end{bmatrix},\,\text{with}\,\,\Omega=\sqrt{\omega^{2}+\tau_{F}^{-2}}\]  
yields the memory kernel \(K(\tau)=\frac{\gamma}{\tau_{F}}e^{-|\tau|/\tau_{F}}\mathrm{Re}[e^{i\omega\tau}]\). The classical FDT is then simply \(b^{2}=4 mk_{B}T\tau_{F}^{-1}\). The corresponding equations of motion for the extended system describe the coupling between the physical coordinates \((x,p)\) and a damped harmonic oscillator of frequency \(\Omega\), subject to a (Markovian) Langevin thermostat with a friction coefficient \(2\tau_{F}^{-1}\). The coefficient \(\gamma\) controls the coupling between this harmonic oscillator and the physical system. From this simple example, it is easy to understand how, by further increasing the number \(N\) of auxiliary momenta, one can obtain the generic memory kernel \(K(\tau)=\mathrm{Re}\Big{[}c_{k}e^{-a_{k}|\tau|+i\omega_{k}\tau}\Big{]}\) that can be arbitrarily tuned via the coefficient \(c_{k}\), \(a_{k}\) and \(\omega_{k}\). Furthermore, the classical FDT condition can be broken, for example by coupling the physical system with damped harmonic oscillators thermalized at different effective temperatures. In the QT method, the classical FDT is broken in such a way as to thermalize the harmonic modes of the physical system with the quantum equilibrium energy \(\theta(\omega,T)\), instead of enforcing the classical equipartition of energy.