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arXiv:2109.00028 · Physical review. B./Physical review. B

Anharmonic lattice dynamics from vibrational dynamical mean-field theory

Petra Shih, Timothy C. Berkelbach

Published Oct 13, 2022• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.106.144307

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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.
Since the first investigation of the Hubbard model in the limit of infinite dimensions by Metzner and Vollhardt, dynamical mean-field theory (DMFT) has become a very powerful tool for the investigation of lattice models of correlated electrons. In DMFT the lattice model is mapped on an effective quantum impurity model in a bath which has to be determined self-consistently. This approach led to significant progress in our understanding of typical correlation problems such as the Mott transition; furthermore, the combination of DMFT with ab-initio methods now allows for a realistic treatment of correlated materials. The focus of these lecture notes is on the relation between quantum impurity physics and the physics of lattice models within DMFT. Issues such as the observability of impurity quantum phase transitions in the corresponding lattice models are discussed in detail.
Dynamical mean-field theory (DMFT) provides an optimal local approximation for correlated lattice systems by mapping the lattice onto a self-consistent effective impurity model. To account for the missing long-range correlations, we propose a novel extended approach, which we term fluctuating dynamical mean-field theory (fDMFT). It incorporates collective fluctuations of auxiliary impurity models across different sites via functional integration. Technically, this method involves obtaining a family of DMFT solutions on a grid for a self-consistent auxiliary classical field applied to the lattice. While the result can, in principle, be improved diagrammatically, we find that the minimal version of the theory already yields accurate results, with lowest-order diagrammatic corrections offering only minor improvements. This consistent framework, based on our fluctuating local field concept, demonstrates superior performance for the nearly half-filled Hubbard model compared to other known diagrammatic extensions of DMFT.
We present an efficient ab initio dynamical mean-field theory (DMFT) implementation for quantitative simulations in solids. Our DMFT scheme employs ab initio Hamiltonians defined for impurities comprising the full unit cell or a supercell of atoms and for realistic quantum chemical basis sets. We avoid double counting errors by using Hartree-Fock as the low-level theory. Intrinsic and projected atomic orbitals (IAO + PAO) are chosen as the local embedding basis, facilitating numerical bath truncation. Using an efficient integral transformation and coupled-cluster Green's function impurity solvers, we are able to handle embedded impurity problems with several hundred orbitals. We apply our ab initio DMFT approach to study a hexagonal boron nitride monolayer, crystalline silicon, and nickel oxide in the antiferromagnetic phase, with up to 104 and 78 impurity orbitals in the spin-restricted and unrestricted cluster DMFT calculations and over 100 bath orbitals. We show that our scheme produces accurate spectral functions compared to both benchmark periodic coupled-cluster computations and experimental spectra.
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Numerical studies on strongly correlated fermionic systems are very complicated and still provide essential problems. The main reason is the exponential growth of the un- derlying Hilbert state space with the system size and the fermionic sign problem for Monte Carlo studies. Among the most widely employed numerical techniques for study- ing two-dimensional quantum many-body systems are cluster extensions of the dynamical mean-field theory (DMFT), e.g. dynamical cluster approximation (DCA). They map an infinitely large multi-dimensional lattice problem to a one-dimensional impurity problem. In 2015 it was shown that the density matrix renormalisation group (DMRG) used as an impurity solver for DMFT (DMFT+DMRG) on the imaginary-frequency axis allows to solve multi-site and multi-band problems extremely fast compared to other solvers. Within this thesis, we further develop this DMRG+DMFT approach to apply the method on real material settings. The step from artificial, completely degenerate multi-band mod- els with simple dispersion relations on a Bethe lattice, studied in 2015, to systems with realistic band structures and lifted degeneracies involves more challenges than originally suspected. In this thesis, we will first recapitulate relevant methods for our approach like matrix prod- uct states, the density matrix renormalisation group and several time evolution methods. In this context we will present several improvements ranging from optimised time evo- lutions to entanglement based optimisations of tensor networks. Second, we will present a very detailed description of the dynamical mean field theory. We will focus on both methodological aspects and implementation details. This chapter is intended to allow other researcher to implement their own DMFT code using DMRG as an impurity solver. Third, we will discuss three different models to show the extent of problems DMRG+ DMFT is able to solve. We will focus on multi-site DCA calculations in the case of the two-dimensional Hubbard model and show that DMRG allows to tackle systems with intermediate interaction strengths at low temperatures, which are unsolvable with other solvers. In the second case, the real material Sr2VO4, we will show the first two-site DCA results for a realistic three-band model. In contrast to assumptions, partly reintroducing the momentum dependence of the self-energy does not improve agreement between exper- imental observations and theoretical results. Finally, we will move on to another realistic three-band model, which describes Sr2RuO4, to show how to deal with the influence of spin-orbit coupling on DMFT. We will present the first low-temperature results for this material and will confirm previous results of simplified model calculations.
It is shown that a minimum realization of the dynamical mean-field theory (DMFT) can be achieved by mapping a correlated lattice model onto an impurity model in which the impurity is coupled to an uncorrelated bath that consists of a single site only. The two-site impurity model can be solved exactly. The mapping is approximate. The self-consistency conditions are constructed in a way that the resulting ``two-site DMFT'' reduces to the previously discussed linearized DMFT for the Mott transition. It is demonstrated that a reasonable description of the mean-field physics is possible with a minimum computational effort. This qualifies the simple two-site DMFT for a systematic study of more complex lattice models which cannot be treated by the full DMFT in a feasible way. To show the strengths and limitations of the new approach, the single-band Hubbard model is investigated in detail. The predictions of the two-site DMFT are compared with results of the full DMFT. Internal consistency checks are performed which concern the Luttinger sum rule, other Fermi-liquid relations and thermodynamic consistency.