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arXiv:2107.00617 · The Journal of Chemical Physics

A thermofield-based multilayer multiconfigurational time-dependent Hartree approach to non-adiabatic quantum dynamics at finite temperature

Eric W. Fischer, Peter Saalfrank

Published Oct 4, 2021• 15 citations
arXivHTMLPDF

Abstract

We introduce a thermofield-based formulation of the multilayer multiconfigurational time-dependent Hartree (MCTDH) method to study finite temperature effects on non-adiabatic quantum dynamics from a non-stochastic, wave function perspective. Our approach is based on the formal equivalence of bosonic many-body theory at zero temperature with a doubled number of degrees of freedom and the thermal quasi-particle representation of bosonic thermofield dynamics (TFD). This equivalence allows for a transfer of bosonic many-body MCTDH as introduced by Wang and Thoss to the finite temperature framework of thermal quasi-particle TFD. As an application, we study temperature effects on the ultrafast internal conversion dynamics in pyrazine. We show that finite temperature effects can be efficiently accounted for in the construction of multilayer expansions of thermofield states in the framework presented herein. Furthermore, we find our results to agree well with existing studies on the pyrazine model based on the ρMCTDH method.

DOI
10.1063/5.0064013

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We introduce a thermofield-based formulation of the multilayer multiconfigurational time-dependent Hartree (MCTDH) method to study finite temperature effects on non-adiabatic quantum dynamics from a non-stochastic, wave function perspective. Our approach is based on the formal equivalence of bosonic many-body theory at zero temperature with a doubled number of degrees of freedom and the thermal quasi-particle representation of bosonic thermofield dynamics (TFD). This equivalence allows for a transfer of bosonic many-body MCTDH as introduced by Wang and Thoss to the finite temperature framework of thermal quasi-particle TFD. As an application, we study temperature effects on the ultrafast internal conversion dynamics in pyrazine. We show that finite temperature effects can be efficiently accounted for in the construction of multilayer expansions of thermofield states in the framework presented herein. Furthermore, we find our results to agree well with existing studies on the pyrazine model based on the ρMCTDH method.
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Nonequilibrium dynamical mean-field theory (DMFT) solves correlated lattice models by obtaining their local correlation functions from an effective model consisting of a single impurity in a self-consistently determined bath. The recently developed mapping of this impurity problem from the Keldysh time contour onto a time-dependent single-impurity Anderson model (SIAM) [C. Gramsch et al., Phys. Rev. B 88, 235106 (2013)] allows one to use wave-function-based methods in the context of nonequilibrium DMFT. Within this mapping, long times in the DMFT simulation become accessible by an increasing number of bath orbitals, which requires efficient representations of the time-dependent SIAM wave function. These can be achieved by the multiconfiguration time-dependent Hartree (MCTDH) method and its multilayer extensions. We find that MCTDH outperforms exact diagonalization for large baths in which the latter approach is still within reach and allows for the calculation of SIAMs beyond the system size accessible by exact diagonalization. Moreover, we illustrate the computation of the self-consistent two-time impurity Green's function within the MCTDH second quantization representation.
The multilayer multiconfigurational time-dependent Hartree (MCTDH) in optimized second quantization representation (oSQR) approach combines the tensor contraction scheme of the multilayer MCTDH approach with the use of an optimized time-dependent orbital basis. Extending the original work on the subject [U. Manthe and T. Weike, J. Chem. Phys. 146, 064117 (2017)], here MCTDH-oSQR propagation in imaginary time and properties related to particle number conservation are studied. Differences between the orbital equation of motion in real and imaginary time are highlighted and a new gauge operator, which facilitates efficient imaginary time propagation, is introduced. Studying Bose-Hubbard models, particle number conservation in MCTDH-oSQR calculations is investigated in detail. Interesting properties of the single-particle functions used in the multilayer MCTDH representation are identified. Based on these results, a tensor contraction scheme, which explicitly utilizes particle number conservation, is suggested.
The improved relaxation approach of Meyer and co-workers is extended by employing the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory. Within this approach the trial wave function is expressed by a recursive, multilayered expansion that is the same as in a normal ML-MCTDH simulation. Calculation of an energy eigenstate then proceeds by iteratively diagonalizing the Boltzmann operator in the top layer representation using a Lanczos/Arnoldi method while relaxing the single particle functions of all layers using the ML-MCTDH imaginary time propagation. Numerical iteration is stopped until the relative energy and/or density of the target state meet the convergence criteria. The application of this multilayer improved relaxation method is illustrated by computing the energy splitting for the lowest pair of eigenstates as well as the spin observables and entanglement entropy of the spin-boson model. A large number of degrees of freedom, up to 10,000 bath modes, can be treated, and thus the appropriate phase transition at a critical system-bath coupling strength may be predicted via appropriate extrapolations.
In this paper, the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method is derived for the case of NNN identical bosons with internal degrees of freedom. The theory for bosons with internal degrees of freedom constitutes a generalization of the MCTDHB method that substantially enriches the many-body physics that can be described. We demonstrate that the numerically exact solution of the time-dependent many-body Schr\"odinger equation for interacting bosonic particles with internal degrees of freedom is now feasible. We report on the MCTDHB equations of motion for bosons with internal degrees of freedom and their implementation for a general many-body Hamiltonian with one-body and two-body terms, both of which may depend on the internal states of the considered particles and time. To demonstrate the capabilities of the theory and its software implementation integrated in the mctdh-x software, we apply MCTDHB to the emergence of fragmentation of parabolically trapped bosons with two internal states: we study the ground state of N=100N=100N=100 bosons as a function of the separation between the state-dependent minima of the two parabolic potentials. To quantify the coherence of the system, we compute its normalized first-order correlation function. We find that the coherence within each internal state of the atoms is maintained, while it is lost between the different internal states. This is a hallmark of a kind of fragmentation absent in bosons without internal structure. We term the emergent phenomenon ``composite fragmentation.''