Surface hopping with a manifold of electronic states. II. Application to the many-body Anderson-Holstein model

Abstract

We investigate a simple surface hopping (SH) approach for modeling a single impurity level coupled to a single phonon and an electronic (metal) bath (i.e., the Anderson-Holstein model). The phonon degree of freedom is treated classically with motion along--and hops between--diabatic potential energy surfaces. The hopping rate is determined by the dynamics of the electronic bath (which are treated implicitly). For the case of one electronic bath, in the limit of small coupling to the bath, SH recovers phonon relaxation to thermal equilibrium and yields the correct impurity electron population (as compared with numerical renormalization group). For the case of out of equilibrium dynamics, SH current-voltage (I-V) curve is compared with the quantum master equation (QME) over a range of parameters, spanning the quantum region to the classical region. In the limit of large temperature, SH and QME agree. Furthermore, we can show that, in the limit of low temperature, the QME agrees with real-time path integral calculations. As such, the simple procedure described here should be useful in many other contexts.

Full counting statistics of vibrationally assisted electronic conduction: Transport and fluctuations of thermoelectric efficiency

“…(22). By further ignoring off-diagonal coherence elements for the reduced density matrix, we obtain the population dynamics for the vibrational states (written here for an arbitrary number of levels, m = 0, 1, 2, ...), (29) where p λ m (t) = m|ρ vib λ (t)|m and |m denotes the m-th vibrational level. k d and k u are rates evaluated at λ = 0.…”

“…The celebrated Anderson-Holstein model, with a single electronic orbital coupled to a local phonon mode, exposes an intricate interplay between the electronic and nuclear degrees of freedom. The model has been extensively studied to reveal the behavior of the current and its fluctuations in different regimes of electron-phonon coupling, see for example [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] . An extension of the Anderson-Holstein model with a secondary phonon bath was examined in many studies, see e.g.…”

Lindblad equation and its semiclassical limit of the Anderson-Holstein model

“…The semiclassical limit of the Redfield equation, known as the classical master equation (CME), has been considered in [5], which also proposed a numerical method based on surface hopping. The CME perspective has then been used to study various physical aspects of Anderson-Holstein model, e.g., broadening, Marcus rate [17][18][19]. In all these works, the focus has been on Redfield equation (or Redfield generator).…”

“…Anderson-Holstein model, since introduced, has been widely studied using various theoretical and numerical approaches, for instance, the Green's function approach [6,7], equation-of-motion method [8,9], quantum Monte Carlo method [10], semi-classical approximation [11,12], non-crossing approximation [13] and by using quantum master equations [5,[14][15][16][17][18][19]. In the perspective of quantum master equation, which is mostly related to the current work, the quantum master equation in Redfield flavor for Anderson-Holstein model has been derived in [14][15][16].…”

Reconciling perturbative approaches in phonon-assisted transport junctions

“…Nuclear motion in conducting molecules and the surroundings governs effects such as local heating of the junction, which may lead to instabilities 11 , and incoherent tunnelling processes, responsible for the development of ohmic conduction 12 . These effects can be captured within simple models: The celebrated Anderson-Holstein model includes a single electronic site embedded between metals, further coupled to a single vibration, or a harmonic bath [13][14][15][16][17][18][19][20][21][22][23][24][25][26] . A different class of problems, relevant as well to photovoltaic devices 27 , concerns donor-acceptor type molecular systems, with two electronic sites coupled to molecular vibrations 11,[28][29][30][31][32][33][34] .…”