This Perspective presents an overview of the development of the hierarchy of Davydov's Ansätze and a few of their applications in many-body problems in computational chemical physics. Davydov's solitons originated in the investigation of vibrational energy transport in proteins in the 1970s. Momentum-space projection of these solitary waves turned up to be accurate variational ground-state wave functions for the extended Holstein molecular crystal model, lending unambiguous evidence to the absence of formal quantum phase transitions in Holstein systems. The multiple Davydov Ansätze have been proposed, with increasing Ansatz multiplicity, as incremental improvements of their single-Ansatz parents. For a given Hamiltonian, the time-dependent variational formalism is utilized to extract accurate dynamic and spectroscopic properties using Davydov's Ansätze as its trial states. A quantity proven to disappear for large multiplicities, the Ansatz relative deviation is introduced to quantify how closely the Schrödinger equation is obeyed. Three finite-temperature extensions to the time-dependent variation scheme are elaborated, i.e., the Monte Carlo importance sampling, the method of thermofield dynamics, and the method of displaced number states. To demonstrate the versatility of the methodology, this Perspective provides applications of Davydov's Ansätze to the generalized Holstein Hamiltonian, variants of the spin-boson model, and systems of cavity-assisted singlet fission, where accurate dynamic and spectroscopic properties of the many-body systems are given by the Davydov trial states.
The dynamics of a coupled electron–boson system is investigated by employing a multitude of the Davydov D 1 trial states, also known as the multi-D 1 Ansatz, and a second trial state based on a superposition of the time-dependent generalized coherent state (GCS Ansatz ). The two Ansätze are applied to study population dynamics in the spin-boson model and the Holstein molecular crystal model, and a detailed comparison with numerically exact results obtained by the (multilayer) multiconfiguration time-dependent Hartree method and the hierarchy equations of motion approach is drawn. It is found that the two methodologies proposed here have significantly improved over that with the single D 1 Ansatz, yielding quantitatively accurate results even in the critical cases of large energy biases and large transfer integrals. The two methodologies provide new effective tools for accurate, efficient simulation of many-body quantum dynamics thanks to a relatively small number of parameters which characterize the electron–nuclear wave functions. The wave-function-based approaches are capable of tracking explicitly detailed bosonic dynamics, which is absent by construct in approaches based on the reduced density matrix. The efficiency and flexibility of our methods are also advantages as compared with numerically exact approaches such as QUAPI and HEOM, especially at low temperatures and in the strong coupling regime.
We examine the applicability of the numerically accurate method of time dependent variation with multiple Davydov Ansätze (mDA) to non-Hermitian systems. As illustrative examples, three systems of interest have been studied, a non-Hermitian system of dissipative Landau-Zener transitions, a non-Hermitian multimode Jaynes-Cummings model, and a dissipative Holstein-Tavis-Cummings model, all of which are shown to be effectively described by the mDA method. Our findings highlight the versatility of the mDA as a powerful numerical tool for investigating complex many-body non-Hermitian systems, which can be extended to explore diverse phenomena such as skin effects, excited-state dynamics, and spectral topology in the non-Hermitian field.
The Dirac-Frenkel time-dependent variational approach with Davydov Ansätze is a sophisticated, yet efficient technique to obtain an accurate solution to many-body Schrödinger equations for energy and charge transfer dynamics in molecular aggregates and light-harvesting complexes. We extend this variational approach to finite temperature dynamics of the spin-boson model by adopting a Monte Carlo importance sampling method. In order to demonstrate the applicability of this approach, we compare calculated real-time quantum dynamics of the spin-boson model with that from numerically exact iterative quasiadiabatic propagator path integral (QUAPI) technique. The comparison shows that our variational approach with the single Davydov Ansätze is in excellent agreement with the QUAPI method at high temperatures, while the two differ at low temperatures. Accuracy in dynamics calculations employing a multitude of Davydov trial states is found to improve substantially over the single Davydov Ansatz, especially at low temperatures. At a moderate computational cost, our variational approach with the multiple Davydov Ansatz is shown to provide accurate spin-boson dynamics over a wide range of temperatures and bath spectral densities.
We perform benchmark simulations using the time-dependent variational approach with the multiple Davydov Ansatz (mDA) to study real-time nonequilibrium dynamics in a single qubit model coupled to two thermal baths with distinct temperatures. A broad region of the parameter space has been investigated, accompanied by a detailed analysis of the convergence behavior of the mDA method. In addition, we have compared our mD2 results to those from two widely adopted, numerically "exact" techniques: the methods of hierarchical equations of motion (HEOM) and the quasi-adiabatic path integral (QUAPI). It is found that the mDA approach in combination with thermal field dynamics yields numerically accurate, convergent results in nearly all regions of the parameter space examined, including those that pose serious challenges for QUAPI and HEOM. Our results reveal that mDA offers a highly adaptable framework capable of capturing long-time dynamics, even in challenging regimes where other methods face limitations. These findings underscore the potential of mDA as a versatile tool for exploring quantum thermodynamics, energy transfer processes, and non-equilibrium quantum systems.
For the Davydov Hamiltonian several special cases are known which can be solved analytically. Starting from these cases we show that the initial state for a simulation using Davydov's \ensuremath{\Vert}
D1〉 approximation has to be constructed from a given set of initial lattice displacements and momenta in the form of a coherent state with its amplitudes independent of the lattices site, corresponding to Davydov's \ensuremath{\Vert}
D2〉 approximation. The site dependences in the \ensuremath{\Vert}
D1〉 ansatz evolve from this initial state exclusively via the equations of motion. Starting the \ensuremath{\Vert}
D1〉 simulation from an ansatz with site-dependent coherent-state amplitudes leads to an evolution which is different from the analytical solutions for the special cases. Thus also in applications of the \ensuremath{\Vert}
D1〉 ansatz to polyacetylene \ensuremath{\Vert}
D2〉-type initial states always have to be used in contrast to our previous suggestion [W. F\"orner, J. Phys. Condens. Matter 6, 9105 (1994)]. Further we expand the known exact solutions in Taylor series in time and compare expectation values in different orders with the exact results.We find that for an approximation up to third order in time (for the wave function) norm, and total energy, as well as displacements and momenta are reasonably correct for a time up to \ensuremath{\approxeq}0.12--0.14 ps, depending somewhat on the coupling strengh for the transportless case. We performed long-time simulations using the \ensuremath{\Vert}
D1〉 approximation where we computed expectation values of the relevant operators with the state (H^/J)\ensuremath{\Vert}
D1〉 and the deviation \ensuremath{\Vert}\ensuremath{\delta}〉 from the exact solution over long times, namely 10 ns. We found that in the very long-time scale the \ensuremath{\Vert}
D1〉 ansatz is very close to an exact solution. Further we report results from an investigation of the very short-time behavior of the \ensuremath{\Vert}
D1〉 state compared with that of an expansion of the exact solution in powers of time t. Within a time of roughly 0.10--0.15 ps the second- and third-order corrections turned out to be not very important. This is due to the fact that our first-order state contains already some terms of the expansion, summed up to infinite order. We found good agreement of the results obtained with our expansion and those from the corresponding \ensuremath{\Vert}
D1〉 simulations within the time of about 0.10 ps. Altogether we have shown that the \ensuremath{\Vert}
D1〉 state, although of approximative nature, is very close to an exact solution of the Davydov model on time scales from some fs up to ns. \textcopyright{} 1996 The American Physical Society.
Dynamics of the sub-Ohmic spin-boson model is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz. Accuracy in dynamics simulations is improved significantly over the single D1 Ansatz, especially in the weak system-bath coupling regime. The reliability of the multi-D1 Ansatz for various coupling strengths and initial conditions is also systematically examined, with results compared closely with those of the hierarchy equations of motion and the path integral Monte Carlo approaches. In addition, a coherent-incoherent phase crossover in the nonequilibrium dynamics is studied through the multi-D1 Ansatz. The phase diagram is obtained with a critical point sc = 0.4. For sc < s < 1, the coherent-to-incoherent crossover occurs at a certain coupling strength, while the coherent state recurs at a much larger coupling strength. For s < sc, only the coherent phase exists.