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arXiv:2010.15725 · Physical Review Research

Gaussian time-dependent variational principle for the finite-temperature anharmonic lattice dynamics

Jae-Mo Lihm, Cheol-Hwan Park

Published Jul 23, 2021• 25 citations
arXivHTMLPDF

Abstract

The anharmonic lattice is a representative example of an interacting, bosonic, many-body system.The selfconsistent harmonic approximation has proven versatile for the study of the equilibrium properties of anharmonic lattices.However, the study of dynamical properties therein resorts to an ansatz, whose validity has not yet been theoretically proven.Here we apply the time-dependent variational principle, a recently emerging useful tool for studying the dynamic properties of interacting many-body systems, to the anharmonic lattice Hamiltonian at finite temperature using the Gaussian states as the variational manifold.We derive an analytic formula for the position-position correlation function and the phonon self-energy, proving the dynamical ansatz of the self-consistent harmonic approximation.We establish a fruitful connection between time-dependent variational principle and the anharmonic lattice Hamiltonian, providing insights in both fields.Our work expands the range of applicability of the time-dependent variational principle to first-principles lattice Hamiltonians and lays the groundwork for the study of dynamical properties of the anharmonic lattice using a fully variational framework.

DOI
10.1103/physrevresearch.3.l032017

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The anharmonic lattice is a representative example of an interacting, bosonic, many-body system.The selfconsistent harmonic approximation has proven versatile for the study of the equilibrium properties of anharmonic lattices.However, the study of dynamical properties therein resorts to an ansatz, whose validity has not yet been theoretically proven.Here we apply the time-dependent variational principle, a recently emerging useful tool for studying the dynamic properties of interacting many-body systems, to the anharmonic lattice Hamiltonian at finite temperature using the Gaussian states as the variational manifold.We derive an analytic formula for the position-position correlation function and the phonon self-energy, proving the dynamical ansatz of the self-consistent harmonic approximation.We establish a fruitful connection between time-dependent variational principle and the anharmonic lattice Hamiltonian, providing insights in both fields.Our work expands the range of applicability of the time-dependent variational principle to first-principles lattice Hamiltonians and lays the groundwork for the study of dynamical properties of the anharmonic lattice using a fully variational framework.
The equilibrium state of a system consisting of a large number of strongly interacting electrons can be characterized by its density operator. This gives a direct access to the ground-state energy or, at finite temperatures, to the free energy of the system as well as to other static physical quantities. Elementary excitations of the system, on the other hand, are described within the language of Green's functions, i.e. time- or frequency-dependent dynamic quantities which give a direct access to the linear response of the system subjected to a weak time-dependent external perturbation. A typical example is angle-revolved photoemission spectroscopy which is linked to the single-electron Green's function. Since usually both, the static as well as the dynamic physical quantities, cannot be obtained exactly for lattice fermion models like the Hubbard model, one has to resort to approximations. Opposed to more ad hoc treatments, variational principles promise to provide consistent and controlled approximations. Here, the Ritz principle and a generalized version of the Ritz principle at finite temperatures for the static case on the one hand and a dynamical variational principle for the single-electron Green's function or the self-energy on the other hand are introduced, discussed in detail and compared to each other to show up conceptual similarities and differences. In particular, the construction recipe for non-perturbative dynamic approximations is taken over from the construction of static mean-field theory based on the generalized Ritz principle. Within the two different frameworks, it is shown which types of approximations are accessible, and their respective weaknesses and strengths are worked out. Static Hartree-Fock theory as well as dynamical mean-field theory are found as the prototypical approximations.
We propose a variational framework for the efficient simulation of stochastic quantum dynamics in interacting bosonic systems. Our method generalizes the time-dependent variational principle to accommodate quantum state diffusion processes and is tailored for describing quantum trajectories unraveled from a master equation. We adopt a wavefunction ansatz composed of a coherent superposition of Gaussian wavepackets, which allows the fully analytical assembly of the variational equations of motion while capturing both the quantum trajectory dynamics and the Lindblad evolution beyond semiclassical approximations. The method is carefully benchmarked on a Bose-Hubbard dimer, and applied to extended Bose-Hubbard lattices in both 1D and 2D. In particular, we show the emergence of a symmetry-breaking phase transition in 2D lattices, which is absent in 1D chains.
In this work, we lay down the ab initio many-body quantum theory of electrons and phonons in equilibrium as well as in steady-state or time-varying settings.Our focus is on the harmonic approximation, but the developed tools can readily incorporate anharmonic effects.We begin by showing the necessity of determining the ab initio Hamiltonian in a self-consistent manner to ensure the existence of an equilibrium state.We then identify the correct partitioning into a "noninteracting" and an "interacting" part to carry out diagrammatic expansions in terms of dressed propagators and screened interactions.The final outcome is the finite-temperature nonequilibrium extension of the Hedin equations, showcasing the emergence of the coupling between electrons and coherent phonons through the time-local Ehrenfest diagram.The Hedin equations have limited practical utility for real-time simulations of systems driven out of equilibrium by external fields.To overcome this limitation, we leverage the versatility of the diagrammatic approach to generate a closed system of differential equations for the dressed propagators and nuclear displacements.These are the Kadanoff-Baym equations for electrons and phonons.The formalism naturally merges with the theory of conserving approximations, which guarantee the satisfaction of the continuity equation and the conservation of total energy during time evolution.As an example, we show that the popular Born-Oppenheimer approximation is not conserving whereas its dynamical extension is conserving, provided that the electrons are treated in the Fan-Migdal approximation with a dynamically screened electronphonon coupling.We also derive the formal solution of the Kadanoff-Baym equations for nonequilibrium steady states, which is useful for studies in photovoltaics and optoelectronics.Interestingly, the expansion of the phononic Green's function around the quasiphonon energies points to a correlation-induced splitting of the phonon dispersion in materials with no time-reversal invariance.
We lay down the {\em ab initio} many-body quantum theory of electrons and phonons in- and out-of-equilibrium at any temperature. We begin by addressing a fundamental issue concerning the {\em ab initio} Hamiltonian in the harmonic approximation, which we show must be determined {\em self-consistently} to avoid inconsistencies. After identifying the most suitable partitioning into a ``noninteracting'' and an ``interacting'' part we embark on the Green's function diagrammatic analysis. We single out key diagrammatic structures to carry on the expansion in terms of dressed propagators and screened interaction. The final outcome is the finite-temperature nonequilibrium extension of the Hedin equations, featuring the appearance of the time-local Ehrenfest diagram in the electronic self-energy. The Hedin equations have limited practical utility for real-time simulations of driven systems. We leverage the versatility of diagrammatic expansion to generate a closed system of integro-differential equations for the Green's functions and nuclear displacements. These are the Kadanoff-Baym equations for electrons and phonons. Another advantage of the diagrammatic derivation is the ability to use conserving approximations, which ensure the satisfaction of all fundamental conservation laws during the time evolution. As an example we show that the adiabatic Born-Oppenheimer approximation is not conserving whereas its dynamical extension is conserving provided that the electrons are treated in the Fan-Migdal approximation with a dynamically screened electron-phonon coupling. We also derive the formal solution of the Kadanoff-Baym equations in the long time limit and at the steady state. The expansion of the phononic Green's function around the quasi-phonon energies points to a possible correlation-induced splitting of the phonon dispersion in materials with no time-reversal invariance.
The phonon is discussed as an example of a collective mode which restores a symmetry property (in this case, translational invariance) to a system whose Hamiltonian is invariant under the symmetry operation, but in whose ground state the symmetry is broken. The crystal lattice is first studied within the time-independent Hartree approximation. It is then shown that allowing small time-dependent changes in the Hartree field generates an equation for the normal modes of vibration. The three k=0 modes with $\ensuremath{\omega}=0$ are shown to represent uniform translations of the solid, as expected. The $\mathbf{k}\ensuremath{\ne}0$ modes are analyzed to extract those three which may be identified as one-phonon modes, and the contribution of these modes to the free energy is computed. The recent theory of Brout is found to be equivalent to the Hartree approximation with the further assumption that the atoms are infinitely heavy. No restrictions are made here on the interatomic potential other than that a hard core is absent. The reduction of the present theory to well-known results in the case of harmonic forces is demonstrated. An extension to include hard cores, analogous to the Brueckner theory, is discussed.
We present a vibrational dynamical mean-field theory (VDMFT) of the dynamics of atoms in solids with anharmonic interactions. Like other flavors of DMFT, VDMFT maps the dynamics of a periodic anharmonic lattice of atoms onto those of a self-consistently defined impurity problem with local anharmonicity and coupling to a bath of harmonic oscillators. VDMFT is exact in the harmonic and molecular limits, nonperturbative systematically improvable through its cluster extensions, usable with classical or quantum impurity solvers (depending on the importance of nuclear quantum effects) and can be combined with existing low-level diagrammatic theories of anharmonicity. When tested on models of anharmonic optical and acoustic phonons; we find that classical VDMFT gives good agreement with classical molecular dynamics, including the temperature dependence of phonon frequencies and lifetimes. Using a quantum impurity solver, signatures of nuclear quantum effects are observed at low temperatures. We test the description of nonlocal anharmonicity via cellular VDMFT and the combination with self-consistent phonon (SCPH) theory, yielding the powerful SCPH +++ VDMFT approach.