We solve the problem of polaron localization on an attractive impurity by means of direct-space diagrammatic Monte Carlo implemented for the system in the thermodynamic limit. In particular we determine the ground-state phase diagram in dependence on the electron-phonon coupling and impurity potential strength for the whole phonon frequency range. Including the quantum phonon effects we find and characterize a phase where self-trapped polarons are not localized at shallow impurities, which is missing in the adiabatic approximation. We show that near the localization transition a region with a mixture of weak- and strong-coupling spectral responses is realized.
We solve the problem of polaron localization on an attractive impurity by means of direct-space diagrammatic Monte Carlo implemented for the system in the thermodynamic limit. In particular we determine the ground-state phase diagram in dependence on the electron-phonon coupling and impurity potential strength for the whole phonon frequency range. Including the quantum phonon effects we find and characterize a phase where self-trapped polarons are not localized at shallow impurities, which is missing in the adiabatic approximation. We show that near the localization transition a region with a mixture of weak- and strong-coupling spectral responses is realized.
We study the bound states of a three-dimensional Holstein polaron near various kinds of single impurities, using the momentum-average approximation. We show that the electron-phonon coupling is responsible for a strong renormalization of the impurity potential, resulting in an effective potential with significant retardation effects, which describes essential physics ignored by ``instantaneous'' approximations. The accuracy of our approximation is gauged by comparison with results from diagrammatic Monte Carlo for the case of an impurity that modifies the on-site energy of the electron. We also discuss impurities that modify the local strength of the electron-phonon coupling as well as isotope substitutions that change both the electron-phonon coupling and the phonon frequency, and contrast and highlight the difference between these cases.
We consider the effects of impurities on polarons in three dimensions using a continuous time quantum Monte Carlo algorithm. An exact treatment of the phonon degrees of freedom leads to a very efficient algorithm, and we are able to compute the polaron dynamics on an infinite lattice by developing an auxiliary weighting scheme. The magnitude of the impurity potential, the electron-phonon coupling, and the phonon frequency are varied. We determine the magnitude of the impurity potential required for polaron trapping. For small electron-phonon coupling the number of phonons increases dramatically on trapping combined with a sharp decrease in kinetic energy. The polaron binding diagram is computed, showing that intermediate-coupling low-phonon-frequency polarons are localized by exceptionally small impurities.
A dynamical mean-field theory of the small polaron problem is presented, which becomes exact in the limit of infinite dimensions. The ground-state properties and the one-electron spectral function are obtained for a single electron interacting with Einstein phonons by a mapping of the lattice problem onto a polaronic impurity model. The one-electron propagator of the impurity model is calculated through a continued fraction expansion, at both zero and finite temperature, for any electron-phonon coupling and phonon energy. In contrast to the ground-state properties, such as the effective polaron mass, which show a continuous behavior as the coupling is increased, spectral properties exhibit a sharp qualitative change at low enough phonon frequency: beyond a critical coupling, one energy gap and then more open in the density of states at low energy, while the high-energy part of the spectrum is broad and can be qualitatively explained by a strong coupling adiabatic approximation. As a consequence, narrow and coherent low-energy subbands coexist with an incoherent featureless structure at high energy. The subbands denote the formation of quasiparticle polaron states. Also, divergencies of the self-energy may occur in the gaps. At finite temperature such an effect triggers an important damping and broadening of the polaron subbands. On the other hand, in the large phonon frequency regime such a separation of energy scales does not exist and the spectrum always has a multipeaked structure.
We discuss impurities in a one-dimensional Bose gas with arbitrary boson-boson and boson-impurity interactions. To fully account for quantum effects, we employ numerical simulations based on the density-matrix renormalization group (DMRG) and—in the regime of strong boson-boson interactions—the mapping to weakly interacting fermions. A mean-field description of the Bose polaron based on coupled Gross–Pitaevskii Schrödinger equations predicts the existence of a self-localized polaron. We here show that such a solution does not exist and is an artifact of the underlying decoupling approximation. To this end we consider a mobile impurity in a box potential. Our work demonstrates that correlations between the impurity position and the bosons are important even in the limit where mean-field approaches are expected to work well. Furthermore, we derive analytical approximations for the energy of a single polaron formed by a heavy impurity for arbitrary interaction strengths and large but finite boson-boson couplings which accurately reproduce DMRG results. This demonstrates that the polaron problem of a heavy impurity in a one-dimensional Bose gas can be accurately approximated by a proper mean-field description plus a linearized treatment of quantum fluctuations for arbitrary boson-boson and impurity-boson couplings. Finally, we determine the polaron-polaron interaction potential V ( r ) in the Born-Oppenheimer approximation for small and intermediate distances r , which in the Tonks gas limit is oscillatory due to Friedel oscillations in the Bose gas.
We calculate the low-temperature properties of an acoustic polaron in three dimensions in thermal equilibrium at a given temperature using a specialized path-integral Monte Carlo method. In particular we find numerical evidence that the chosen Hamiltonian for the acoustic polaron describes a phase transition from a localized state to an unlocalized state for the electron as the phonon-electron coupling constant decreases. The phase transition manifests itself with a jump discontinuity in the potential energy as a function of the coupling constant. In the weak coupling regime the electron is in an extended state whereas in the strong coupling regime it is found in a self-trapped state.
This dissertation aims at improving the current understanding of the physics of mobile impurities in highly correlated liquid-like phases of matter. Impurity problems pose challenging and intricate questions in different realms of many-body physics. For instance, the problem of ''solvation'' of charged solutes in polar solvents, has been the subject of longstanding debates among chemical physicists. The significant role of quantum fluctuations of the solvent, as well as the break down of linear response theory, render the ordinary treatments intractable. Inspired by this complicated problem, we first attempt to understand the role of non-specific quantum fluctuations in the solvation process. To this end, we calculate the dynamic structure factor of a model polar liquid, using the classical Molecular Dynamics (MD) simulations. We verify the failure of linear response approximation in the vicinity of a hydrated electron, by comparing the outcomes of MD simulations with the predictions of linear response theory. This nonlinear behavior is associated with the pronounced peaks of the structure factor, which reflect the strong fluctuations of the local modes. A cavity picture is constructed based on heuristic arguments, which suggests that the electron, along with the surrounding polarization cloud, behave like a frozen sphere, for which the linear response theory is broken inside and valid outside. The inverse radius of the spherical region serves as a UV momentum cutoff for the linear response approximation to be applicable.The problem of mobile impurities in polar liquids can be also addressed in the framework of the ''polaron'' problem. Polaron is a quasiparticle that typically acquires an extended state at weak couplings, and crossovers to a self-trapped state at strong couplings. Using the analytical fits to the numerically obtained charge-charge structure factor, a phenomenological approach is proposed within the Leggett's influence functional formalism, which derives the effective Euclidean action from the classical equation of motion. We calculate the effective mass of the polaron in the model polar liquid at zero and finite temperatures. The self-trapping transition of this polaron turns out to be discontinuous in certain regions of the phase diagram.In order to systematically investigate the role of quantum fluctuations on the polaron properties, we adopt a quantum field theory which supports nearly-critical local modes: the quantum Landau-Brazovskii (QLB) model, which exhibits fluctuation-induced first order transition (weak crystallization). In the vicinity of the phase transition, the quantum fluctuations are strongly correlated; one can in principle tune the strength of these fluctuations, by adjusting the parameters close to or away from the transition point. Furthermore, sufficiently close to the transition, the theory accommodates ``soliton'' solutions, signaling the nonlinear response of the system. Therefore, the model seems to be a promising candidate for studying the effects of strong quantum fluctuations and also failure of linear response theory, in the polaron problem. We observe that at zero temperature, and away from the Brazovskii transition where the linear response approximation is valid, the localization transition of the polaron is discontinuous. Upon enhancing fluctuations|of either thermal or quantum nature|the gap of the effective mass closes at distinct second-order critical points. Sufficiently close to the Brazovskii transition where the nonlinear contributions of the field are significantly large, a new state appears in addition to extended and self-trapped polarons: an impurity-induced soliton. We interpret this as the break-down of linear response, reminiscent of what we observe in a polar liquid. Quantum LB model has been proposed to be realizable in ultracold Bose gases in cavities. We thus discuss the experimental feasibility, and propose a setup which is believed to exhibit the aforementioned polaronic and solitonic states. We eventually generalize the polaron formalism to the case of impurities that couple quadratically to a nearly-critical field; hence called the ''quadratic polaron''. The Hertz-Millis field theory and its generalization to the case of magnetic transition in helimagnets, is taken as a toy model. The phase diagram of the bare model contains both second-order and fluctuation-induced first-order quantum phase transitions. We propose a semi-classical scenario in which the impurity and the field couple quadratically. The polaron properties in the vicinity of these transitions are calculated in different dimensions. We observe that the quadratic coupling in three dimensions, even in the absence of the critical modes with finite wavelength, leads to a jump-like localization of the polaron. In lower dimensions, the transition behavior remains qualitatively similar to those in the case of linear coupling, namely the critical modes must have a finite wavelength to localize the particle.