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Institute of Physics eBooks

Extensions to many-body configurations and additional degrees of freedom

Jonas Larson, Th. K. Mavrogordatos

Published Sep 1, 2024• 0 citations
Publisher

Abstract

Chapter 8 is devoted to extensions and truly challenging quantum many body systems and problems: Jaynes–Cummings–Hubbard models, many-body cavity, mean-field explorations, critical phenomena for bosons and fermions, and, last but not least, polaritonic chemistry, using the Born–Oppenheimer approximation and deriving the molecular Jaynes–Cummings Hamiltonian.

DOI
10.1088/978-0-7503-6452-2ch8

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Institute of Physics eBooks•Mar 1, 2024•2 citations
Chapter 8 is devoted to extensions and truly challenging quantum many body systems and problems: Jaynes–Cummings–Hubbard models, many-body cavity, mean-field explorations, critical phenomena for bosons and fermions, and, last but not least, polaritonic chemistry, using the Born–Oppenheimer approximation and deriving the molecular Jaynes–Cummings Hamiltonian.
In the previous chapters DMFT has been developed as a methodology for calculation of dynamical and thermodynamic properties of many-body systems, with applications to model systems. In order for DMFT to be a general method for quantitative calculations, there must be systematic procedures to derive all the ingredients from first principles. In this chapter we consider the most prominent examples, materials with partially filled 3 d , 4 f , or 5 f localized, atomic-like states, and set up the characteristic form of the hamiltonian. This is the basis for DFT+DMFT methods discussed here and used in the examples in Ch. 20, and for the combination of DMFT with many-body perturbation theory in Ch. 21. Simpler techniques, including self-interaction correction, DFT+U, and DFT+Gutzwiller, are also useful and are summarized in the last sections of this chapter. Up to this point, dynamical mean-field theory has been developed for model systems with Hubbard-type hamiltonians, where the hopping matrix elements and interactions are considered as parameters. For problems with strong interactions, DMFT leads to renormalized bands, satellites in the one-particle spectra, magnetic phase transitions and local moments, metal–insulator transitions, and many other phenomena observed in real materials, as brought out in Ch. 2. The purpose of this chapter is to provide the framework for a quantitative theory of these materials. These are difficult problems, experimentally, theoretically, and computationally, and the first step is to identify the aspects of the materials that must be treated explicitly as the local strongly interacting system, and the “rest” that are treated by a different method. This division of the problem is useful because a real material involves many degrees of freedom, i.e., the electronic states must be described in a basis sufficiently large to describe all the relevant states. However, the part that can be treated by non-perturbative solvers (Ch. 18) is limited by the computational cost that scales rapidly with the number of states included in the embedded site or cluster. Thus one must choose a small number of localized orbitals that inevitably are non-unique. This may seem like a step backward for quantitative methods; however, it opens a window for methods to treat these interesting problems and it presents a challenge to overcome the limitations.
This chapter presents a brief account on both approaches, analytic and synthetic, within chemistry. It reviews the different levels at which molecular systems are traditionally studied. The chapter explores the Born–Oppenheimer approximation and then revisiting the methodologies available to deal with both electronic configuration and dynamics, distinguishing in this latter case between molecular dynamics for a few degree-of-freedom systems and statistical mechanics. It focuses on some fundamental aspects of Bohmian mechanics and describes some specific illustrations of the application of Bohmian mechanics to problems typical of the levels mentioned. Molecular systems typically include a rather large number of degrees of freedom, which makes unaffordable their direct study. In the wave model each atom is assumed to be a point-like, positively charged nucleus surrounded by an electronic cloud, which is associated with a certain orbital and represents the electron probability distribution in configuration space.
Analog quantum simulation---the technique of using one experimentally well-controlled physical system to mimic the behavior of another---has quickly emerged as one of the most promising near-term strategies for studying strongly correlated quantum many-body systems. In particular, systems of interacting photons, realizable in solid-state cavity and circuit QED frameworks, for example, hold tremendous promise for the study of nonequilibrium many-body phenomena, in part due to the capability to locally create and destroy photons. These systems are typically modeled using a Jaynes-Cummings-Hubbard (JCH) Hamiltonian, named due to similarities with the Bose-Hubbard Hamiltonian. While comparisons between the two are often made in literature, the JCH Hamiltonian comprises both bosonic and psuedospin operators, leading to physical deviations from the Bose-Hubbard model for particular parameter regimes. Here, we present a nonperturbative procedure for transforming the Jaynes-Cummings Hamiltonian into a dressed operator representation that, in its most general form, admits an infinite sum of bosonic kkk-body terms where kkk is bound only by the number of excitations in the system. We closely examine this result in both the dispersive and resonant coupling regimes, finding rapid convergence of this sum in the former and contributions from $k\ensuremath{\gg}1$ in the latter. Through extension to the simple case of a two-site JCH system, we demonstrate that this approach facilitates close inspection of the analogy between the JCH and Bose-Hubbard models and its breakdown for resonant light-matter coupling. Finally, we use this framework to survey the many-body character of a two-site JCH for general system parameters, identifying four unique quantum phases and the parameter regimes in which they are realized, thus highlighting phenomena realizable with finite JCH-based quantum simulators beyond the Bose-Hubbard model. More broadly, this paper is intended to serve as a clear mathematical exposition of bosonic many-body interactions underlying Jaynes-Cummings-type systems, often postulated either through analogy to Kerr-like nonlinear susceptibilities or by matching coefficients to obtain the appropriate eigenvalue spectrum.
Recent progress in the theory and computation of electronic structure is bringing an unprecedented level of capability for research. Many-body methods are becoming essential tools vital for quantitative calculations and understanding materials phenomena in physics, chemistry, materials science and other fields. This book provides a unified exposition of the most-used tools: many-body perturbation theory, dynamical mean field theory and quantum Monte Carlo simulations. Each topic is introduced with a less technical overview for a broad readership, followed by in-depth descriptions and mathematical formulation. Practical guidelines, illustrations and exercises are chosen to enable readers to appreciate the complementary approaches, their relationships, and the advantages and disadvantages of each method. This book is designed for graduate students and researchers who want to use and understand these advanced computational tools, get a broad overview, and acquire a basis for participating in new developments.