It is remarkable how the Froehlich polaron, one of the simplest examples of a Quantum Field Theoretical problem, as it basically consists of a single fermion interacting with a scalar Bose field of ion displacements, has resisted full analytical or numerical solution at all coupling since 1950, when its Hamiltonian was first written. The field has been a testing ground for analytical, semi-analytical, and numerical techniques, such as path integrals, strong-coupling perturbation expansion, advanced variational, exact diagonalisation (ED), and quantum Monte Carlo (QMC) techniques. This article reviews recent developments in the field of continuum and discrete (lattice) Froehlich (bi)polarons starting with the basics and covering a number of active directions of research.
It is remarkable how the Froehlich polaron, one of the simplest examples of a Quantum Field Theoretical problem, as it basically consists of a single fermion interacting with a scalar Bose field of ion displacements, has resisted full analytical or numerical solution at all coupling since 1950, when its Hamiltonian was first written. The field has been a testing ground for analytical, semi-analytical, and numerical techniques, such as path integrals, strong-coupling perturbation expansion, advanced variational, exact diagonalisation (ED), and quantum Monte Carlo (QMC) techniques. This article reviews recent developments in the field of continuum and discrete (lattice) Froehlich (bi)polarons starting with the basics and covering a number of active directions of research.
The Fröhlich polaron model describes a ubiquitous class of problems concerned with understanding properties of a single mobile particle interacting with a bosonic reservoir. Originally introduced in the context of electrons interacting with phonons in crystals, this model found applications in such diverse areas as strongly correlated electron systems, quantum information, and high energy physics. In the last few years this model has been applied to describe impurity atoms immersed in Bose-Einstein condensates of ultracold atoms. The tunability of microscopic parameters in ensembles of ultracold atoms and the rich experimental toolbox of atomic physics should allow to test many theoretical predictions and give us new insights into equilibrium and dynamical properties of polarons. In these lecture notes we provide an overview of common theoretical approaches that have been used to study BEC polarons, including Rayleigh-Schrödinger and Green's function perturbation theories, self-consistent Born approximation, mean-field approach, Feynman's variational path integral approach, Monte Carlo simulations, renormalization group calculations, and Gaussian variational ansatz. We focus on the renormalization group approach and provide details of analysis that have not been presented in earlier publications. We show that this method helps to resolve striking discrepancy in polaron energies obtained using mean-field approximation and Monte Carlo simulations. We also discuss applications of this method to the calculation of the effective mass of BEC polarons. As one experimentally relevant example of a non-equililbrium problem we consider Bloch oscillations of Bose polarons and demonstrate that one should find considerable deviations from the commonly accepted phenomenological Esaki-Tsu model. We review which parameter regimes of Bose polarons can be achieved in various atomic mixtures.
Employing the recently developed self-consistent variational basis generation scheme, we have investigated the bipolaron-bipolaron interaction within the purview of Holstein-Hubbard and the Froehlich-Hubbard model on a discrete one-dimensional lattice. The density-matrix renormalization group (DMRG) method has also been used for the Holstein-Hubbard model. We have shown that there exists no bipolaron-bipolaron attraction in the Holstein-Hubbard model. In contrast, we have obtained clear-cut bipolaron-bipolaron attraction in the Froehlich-Hubbard model. Composite bipolarons are formed above a critical electron-phonon coupling strength, which can survive the finite Hubbard U effect. We have constructed the phase diagram of Froehlich-Hubbard polarons and bipolarons, and discussed the phase separation in terms of the formation of composite bipolarons.
This review describes the field of Bose polarons, arising when mobile impurities are immersed into a bosonic quantum gas. The latter can be realized by a Bose-Einstein condensate (BEC) of ultracold atoms, or of exciton polaritons in a semiconductor, which has led to a series of experimental observations of Bose polarons near inter-species Feshbach resonances that we survey. Following an introduction to the topic, with references to its historic roots and a presentation of the Bose polaron Hamiltonian, we summarize state-of-the-art experiments. Next we provide a detailed discussion of polaron models, starting from the ubiquitous Fröhlich Hamiltonian that applies at weak couplings. We proceed by a survey of concurrent theoretical methods used for solving strongly interacting Bose polaron problems. The subsequent sections are devoted to the large bodies of work investigating strong coupling Bose polarons, including detailed comparisons with radio-frequency (RF) spectra obtained in ultracold atom experiments; to investigations of universal few-body and Efimov states associated with a Feshbach resonance in atomic mixtures; to studies of quantum dynamics and polarons out of equilibrium; Bose polarons in low-dimensional; induced interactions among polarons and bipolaron formation; and to Bose polarons at non-zero temperatures. We end our review by detailed discussions of closely related experimental setups and systems, including ionic impurities, systems with strong light-matter interactions, and variations and extensions of the Bose polaron concepts e.g. to baths with topological order or strong interactions relevant for correlated electrons. Finally, an outlook is presented, highlighting possible future research directions and open questions in the field as a whole.
This review describes the field of Bose polarons, arising when mobile impurities are immersed into a bosonic quantum gas. The latter can be realized by a Bose-Einstein condensate of ultracold atoms, or of exciton polaritons in a semiconductor, which has led to a series of experimental observations of Bose polarons near inter-species Feshbach resonances that we survey. Following an introduction to the topic, with references to its historic roots and a presentation of the Bose polaron Hamiltonian, we summarize state-of-the-art experiments. Next we provide a detailed discussion of polaron models, starting from the ubiquitous Fröhlich Hamiltonian that applies at weak couplings. Already this highly simplified model allows insights into ultra-violet divergencies, logarithmic and power-law, that need to be properly regularized. To capture the physics near a Feshbach resonance, two-phonon scattering terms on the impurity as well as phonon-phonon interactions need to be included. We proceed by a survey of concurrent theoretical methods used for solving strongly interacting Bose polaron problems, ranging from Lee-Low-Pines mean-field theory, Chevy-ansatz, Gross-Pitaevskii-equation to diagrammatic Monte Carlo approaches. The subsequent sections are devoted to the large bodies of work investigating strong coupling Bose polarons, including detailed comparisons with radio-frequency spectra obtained in ultracold atom experiments; to investigations of universal few-body and Efimov states associated with a Feshbach resonance in atomic mixtures; to studies of quantum dynamics and polarons out of equilibrium; Bose polarons in low-dimensional 1D and 2D quantum systems; induced interactions among polarons and bipolaron formation; and to Bose polarons at non-zero temperatures. We end our review by detailed discussions of closely related experimental setups and systems, including ionic impurities, systems with strong light-matter interactions, and variations and extensions of the Bose polaron concepts e.g. to baths with topological order or strong interactions relevant for correlated electrons. Finally, an outlook is presented, highlighting possible future research directions and open questions in the field as a whole.
We present results for the solution of the large polaron Fr\"ohlich Hamiltonian in 3 dimensions (3D) and 2 dimensions (2D) obtained via the diagrammatic Monte Carlo (DMC) method. Our implementation is based on the approach by Mishchenko [A. S. Mishchenko et al., Phys. Rev. B 62, 6317 (2000)]. Polaron ground state energies and effective polaron masses are successfully benchmarked with data obtained using Feynman's path integral formalism. By comparing 3D and 2D data, we verify the analytically exact scaling relations for energies and effective masses from $3\mathrm{D}\ensuremath{\rightarrow}2\mathrm{D}$, which provides a stringent test for the quality of DMC predictions. The accuracy of our results is further proven by providing values for the exactly known coefficients in weak- and strong-coupling expansions. Moreover, we compute polaron dispersion curves which are validated with analytically known lower and upper limits in the small-coupling regime and verify the first-order expansion results for larger couplings, thus disproving previous critiques on the apparent incompatibility of DMC with analytical results and furnishing useful reference for a wide range of coupling strengths.
This article reviews recent theoretical developments in the ab initio study of polarons in materials. The polaron is an emergent quasiparticle that arises from the interaction between electrons and phonons in solids, and consists of an electron or a hole accompanied by a distortion of the crystal lattice. Recent advances in experiments, theory, and computation have made it possible to investigate these quasiparticles with unprecedented detail, reigniting the interest in this classic problem of condensed matter physics. Recent theoretical and computational advances include ab initio calculations of polaron spectral functions, wavefunctions, lattice distortions, and transport and optical properties. These developments provide new insight into polaron physics, but they have evolved somewhat independently from the earlier effective Hamiltonian approaches that laid the foundation of the field. This article aims to bridge these complementary perspectives by placing them within a single unified conceptual framework. To this end, we start by reviewing effective Hamiltonians of historical significance in polaron theory, ab initio techniques based on density functional theory, and many-body first-principles approaches to polarons. After this survey, we outline a general field-theoretic framework that bridges between these diverse approaches to polaron physics. For completeness, we also review recent progress in the study of exciton polarons and self-trapped excitons and their relations to polarons. Beyond the methodology, we discuss recent applications to several classes of materials that attracted attention in the context of polaron physics.