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arXiv:1902.00224 · Physical Review A

Stochastic Hamiltonians for correlated electron models

Frederick Green

Published Jun 25, 2019• 2 citations
arXivHTMLPDF

Abstract

Microscopically conserving reduced models of many-body systems have a long, highly successful history. Established perturbative theories of this type are the random-phase approximation for Coulomb fluids and the particle-particle ladder model for nuclear matter. There are also more physically comprehensive diagrammatic approximations, such as the induced-interaction and parquet models. Notwithstanding their explanatory power, some theories have lacked an explicit Hamiltonian from which all significant system properties, static and dynamic, emerge canonically. This absence can complicate evaluation of the conserving sum rules, essential consistency checks on the validity of any theory. In a series of papers Kraichnan introduced a stochastic embedding procedure to generate explicit Hamiltonians for common approximations for the full many-body problem. Existence of a Hamiltonian greatly eases the task of securing fundamental identities in such studies. I revisit Kraichnan's method to apply it to correlation theories for which such a canonical framework has been missing. I exhibit Hamiltonians for more elaborate correlated models incorporating both long-range screening and short-range scattering phenomena. These are relevant to the study of strongly interacting electrons and condensed quantum systems broadly.

DOI
10.1103/physreva.99.062118

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Microscopically conserving reduced models of many-body systems have a long, highly successful history. Established perturbative theories of this type are the random-phase approximation for Coulomb fluids and the particle-particle ladder model for nuclear matter. There are also more physically comprehensive diagrammatic approximations, such as the induced-interaction and parquet models. Notwithstanding their explanatory power, some theories have lacked an explicit Hamiltonian from which all significant system properties, static and dynamic, emerge canonically. This absence can complicate evaluation of the conserving sum rules, essential consistency checks on the validity of any theory. In a series of papers Kraichnan introduced a stochastic embedding procedure to generate explicit Hamiltonians for common approximations for the full many-body problem. Existence of a Hamiltonian greatly eases the task of securing fundamental identities in such studies. I revisit Kraichnan's method to apply it to correlation theories for which such a canonical framework has been missing. I exhibit Hamiltonians for more elaborate correlated models incorporating both long-range screening and short-range scattering phenomena. These are relevant to the study of strongly interacting electrons and condensed quantum systems broadly.
We derive a widely applicable first-principles approach for determining two-body, static effective interactions for low-energy Hamiltonians with quantitative accuracy. The algebraic construction rigorously conserves all instantaneous two-point correlation functions in a chosen model space at the level of the random phase approximation, improving upon the traditional uncontrolled static approximations. Applied to screened interactions within a quantum embedding framework, we demonstrate these faithfully describe the relaxation of local subspaces via downfolding high-energy physics in molecular systems, as well as enabling a systematically improvable description of the long-range plasmonic contributions in extended graphene.
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Quantum embedding theories are promising approaches to investigate strongly correlated electronic states of active regions of large-scale molecular or condensed systems. Notable examples are spin defects in semiconductors and insulators. We present a detailed derivation of a quantum embedding theory recently introduced, which is based on the definition of effective Hamiltonians. The effect of the environment on a chosen active space is accounted for through screened Coulomb interactions evaluated using density functional theory. Importantly, the random phase approximation is not required, and the evaluation of virtual electronic orbitals is circumvented with algorithms previously developed in the context of calculations based on many-body perturbation theory. In addition, we generalize the quantum embedding theory to active spaces composed of orbitals that are not eigenstates of Kohn-Sham Hamiltonians. Finally, we report results for spin defects in semiconductors.
Dynamical screening is a key property of charged many-particle systems. Its theoretical description is based on the GWGWGW approximation that is extensively applied for ground-state and equilibrium situations but also for systems driven out of equilibrium. The main limitation of the GWGWGW approximation is the neglect of strong electronic correlation effects that are important in many materials as well as in dense plasmas. Here we derive the dynamically screened ladder (DSL) approximation that self-consistently includes, in addition to the GWGWGW diagrams, also particle-particle and particle-hole TTT-matrix diagrams. The derivation is based on reduced-density-operator theory and the result is equivalent to the recently presented G1-G2 scheme [N. Schl\"unzen et al., Phys. Rev. Lett. 124, 076601 (2020); J.-P. Joost et al., Phys. Rev. B 101, 245101 (2020)]. We perform extensive time-dependent DSL simulations for finite Hubbard clusters and present tests against exact results that confirm excellent accuracy as well as total-energy conservation of the approximation. At strong coupling and for long simulation durations, instabilities are observed. These problems are solved by enforcing contraction consistency and applying a purification approach.
In this second work in a series of papers the coupled-cluster or exp(S)\mathrm{exp}(S)exp(S) formalism is applied to the problem of ground-state correlations in the one-component Fermi plasma. The main approximation employed is the so-called SUB2 approximation for the two-body subsystem correlation operator S2{S}_{2}S2​ which provides a measure of the two-particle---two-hole component in the true ground-state wave function, and in which the coupling to larger subsystems is neglected. The SUB2 equations for Fermi systems are brought into tractable form by a "state-averaging" procedure which we develop by analogy with the comparable Bose equations, and which is shown by comparison with earlier exact results to be accurate in the metallic density regime at about the 1% level. Our final results for metallic densities include the completely integrated and self-consistent effects of the terms which by themselves generate (i) the random-phase approximation (RPA) and its associated exact long-range screening effects, (ii) the extra random-phase approximation exchange terms necessary to keep the RPA explicitly antisymmetric, (iii) the self-consistent particle-particle ladders (LAD) that describe two-particle scattering within the many-body medium and which describe the exact short-range behavior, (iv) a class of particle-hole ladder terms, and (v) the self-consistent hole-potential terms. Particular attention is paid to the important effects caused by the interference at intermediate separations of the long-range RPA and the short-range LAD effects. By comparison with recent and essentially exact stochastic simulations of the many-body Schr\"odinger equation, our results are seen to be accurate to about 1% for metallic densities, and hence to provide what is probably the currently best microscopic description available for this system. We show also that even in the low-density limit the SUB2 approximation provides a good "translationally-invariant-solid" description of both charged Fermi and Bose systems in this exact Wigner crystal regime.
Correlated electron physics is intrinsically a multiscale problem, since high-energy electronic states screen the interactions between the correlated electrons close to the Fermi level, thereby reducing the magnitude of the interaction strength and dramatically shortening its range. Thus the handling of screening is an essential ingredient in the first-principles modeling of correlated electron systems. Screening is an intrinsically dynamic process and the corresponding downfolding methods such as the constrained random phase approximation indeed produce a dynamic interaction. However, many low-energy methods require an instantaneous interaction as input, which makes it necessary to map the fully dynamic interaction to an effective instantaneous interaction strength. It is not clear if and when such an effective model can capture the physics of the one with dynamic interaction and how to best perform the mapping. Here, we provide a systematic benchmark relevant to correlated materials, in the form of the Anderson impurity model. Overall, we find that a static approximation can be valid and that the moment-based approach recently proposed by Scott and Booth can be a good tool to find the value of the static interaction. We also identify physical regimes, especially under doping, where an instantaneous interaction cannot capture all of the relevant physics.