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Journal of the Physical Society of Japan

Formulation of a New Theory for Optical Response and Exciton Effects in Insulating States Coming from Strong Electron-Phonon and Inter-Electron Interactions

Keiichirō Nasu

Published Jul 15, 1996• 10 citations
Publisher

Abstract

We formulate a new theory for optical response of many-electron systems in insulating states coming from strong electron-phonon (e-ph) and electron-electron (e-e) interactions. The e-ph coupling and the short range part of e-e repulsion are reduced into a time-dependent one-body Hamiltonian by the path-integral. While, the long range part of e-e repulsion is taken into account straightly, as a perturbation to this time-dependent Hamiltonian. To evaluate higher order terms of this perturbation, we have extended the Bloch-De Dominicis theorem, so that it can be used even in the cases where an unperturbud one-body Hamiltonian is time-dependent, as well as in the conventional time-independent cases. Using this theorem, we have formulated the optical response, wherein the long range part results in exciton effects or particle-hole attractions as a final-state interaction of the optical transition. Another path-integral is also formulated, in which the short and long range parts are treated on an equal footing.

DOI
10.1143/jpsj.65.2285

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A new approach to the problem of X-ray edge singularities and peaks of many-body origin observed in the optical spectra is presented. We first establish the analogy between a system of one hole interacting with many electrons and the polaron, where one electron interacts with many phonons. We approach both cases as an eigenvalue problem for the time independent Schrodinger equation, obtain expressions for the many-body eigenstate& in a convenient basis and use them to understand the behavior of the system. In the case of a localized hole with infinite mass, we diagonalize the Hamiltonian exactly using first a wavefunction and then a noncanonical transformation. technique. We -use the expressions for the eigenstates to obtain the exact analytic formulas for the optical spectra, accurate over the whole frequency range. We then apply our techniques to the case of a finite mass valence hole, which is realized in modulation doped quantum wells or wires. In this case we approximately diagonalize the Hamiltonian by treating exactly the most important terms leading to nonperturbative behavior. The terms neglected are identified and can be treated in perturbation theory. Our method provides a clear physical picture and valuable insight into the accurate treatment of many-body Hamiltonians.

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Electron-electron and electron-phonon interactions in strongly correlated systems

Gerardo Sica

In this work we investigate some aspects of the physics of strongly correlated systems by taking into account both electron-electron and electron-phonon interactions as basic mechanisms for reproducing electronic correlations in real materials. The relevance of the electron-electron interactions is discussed in the first part of this thesis in the framework of a self-consistent theoretical approach, named Composite Operator Method (COM), which accounts for the relevant quasi-particle excitations in terms of a set of composite operators that appear as a result of the modification imposed by the interactions on the canonical electronic fields. We show that the COM allows the calculation of all the relevant Green’s and correlation functions in terms of a number of unknown internal parameters to be determined self-consistently. Therefore, depending on the balance between unknown parameters and self-consistent equations, exact and approximate solutions can be obtained. By way of example, we discuss the application of the COM to the extended t-U- J-h model in the atomic limit, and to the two-dimensional single-band Hubbard model. In the former case, we show that the COM provides the exact solution of the model in one dimension. We study the effects of electronic correlations as responsible for the formation of a plethora of different charge and/or spin orderings. We report the phase diagram of the model, as well as a detailed analysis of both zero and finite temperature single-particle and thermodynamic properties. As far as the single-band Hubbard model is concerned, we illustrate an approximated selfconsistent scheme based on the choice of a two-field basis. We report a detailed analysis of many unconventional features that arise in single-particle properties, thermodynamics and system’s response functions. We emphasize that the accuracy of the COM in describing the effects of electronic correlations strongly relies on the choice of the basis, paving the way for possible multi-pole extensions to the twofield theory. To this purpose, we also study a three-field approach to the single-band Hubbard model, showing a significant step forward in the agreements with numerical data with respect to the two-pole results. The role of the electron-phonon interaction in the physics of strongly correlated systems is discussed in the second part of this thesis. We show that in highly polarizable lattices the competition between unscreened Coulomb and Fröhlich interactions results in a short-range polaronic exchange term Jp that favours the formation of local and light pairs of bosonic nature, named bipolarons, which condense with a critical temperature well in excess of hundred kelvins. These findings, discussed in the framework of the so-called polaronic t-Jp model, are further investigated in the presence of a finite on-site potential ~U , coming from the competition between on-site Coulomb and Fröhlich interactions. We discuss the role of ~U as the driving parameter for a small-to-large bipolaron transition, providing a possible explanation of the BEC-BCS crossover in terms of the properties of the bipolaronic ground state. Finally, we show that a hard-core bipolarons gas, studied as a charged Bose-Fermi mixture, allows for the description of many non Fermi liquid behaviours, allowing also for a microscopic explanation of pseudogap features in terms of a thermal-induced recombination of polarons and bipolarons, without any assumption on preexisting order or broken symmetries. [edited by Author]

•Apr 29, 2013•0 citations
We formulate a new theory for optical response of many-electron systems in insulating states coming from strong electron-phonon (e-ph) and electron-electron (e-e) interactions. The e-ph coupling and the short range part of e-e repulsion are reduced into a time-dependent one-body Hamiltonian by the path-integral. While, the long range part of e-e repulsion is taken into account straightly, as a perturbation to this time-dependent Hamiltonian. To evaluate higher order terms of this perturbation, we have extended the Bloch-De Dominicis theorem, so that it can be used even in the cases where an unperturbud one-body Hamiltonian is time-dependent, as well as in the conventional time-independent cases. Using this theorem, we have formulated the optical response, wherein the long range part results in exciton effects or particle-hole attractions as a final-state interaction of the optical transition. Another path-integral is also formulated, in which the short and long range parts are treated on an equal footing.
We develop a path-integral formulation which is free from Grassmann algebras to calculate absorption spectra of strongly correlated electron systems. The Coulomb repulsions, including not only the short-range part but also a long-range part, are reduced into a time-dependent one-body Hamiltonian with only two kinds of Ising spin variables per site. The sums over these spin configurations are performed according to the quantum Monte Carlo techniques. By this method we disclose the electron correlation effects on absorption spectra. The method is applied to the one-dimensional half-filled electron systems, where the ground state is an insulating state due to the strong long-range Coulomb repulsion and contains large quantum fluctuations by low-energy collective spin excitations. The absorption spectrum shows an infrared divergencelike structure, which seems to arise from the simultaneous excitation of the charge-transfer exciton and collective spin excitations. The spectrum also contains new peaks on the high energy side which correspond to the multiple excitation of electron-hole pairs. Such a multiple excitation is caused by the electron correlation effect, through which the photogenerated electron and hole can excite other electron-hole pairs. Our result clearly shows that charge and spin of the electron are inseparable in a many-body system in an insulating state caused by the strong long-range Coulomb repulsion.
Optical responses in an excitonic insulating (EI) system with strong electron correlation are studied. We adopt the two-orbital Hubbard model with a finite energy difference between the two orbitals where the spin state degree of freedom exists. This model is analyzed by the variational cluster approach. In order to include the local electron correlation effect, the vertex correction is taken into account in the formulation of the optical conductivity spectra. We calculate a finite-temperature phase diagram in which an EI phase appears between a low-spin band insulating state and a high-spin Mott insulating state. Characteristic components of the optical conductivity spectra consisting of a sharp peak and continuum appear in the EI phase. Integrated intensity almost follows the order parameter of the EI state, suggesting that this component is available to identify the EI phases and transitions.
We study the optical properties of an electron–hole two-band Hubbard model with interactions of electron–electron (hole–hole) repulsion U and electron–hole attraction U '. In the case of two types of correlated insulators at half filling, viz. , the Mott–Hubbard type and the biexciton type, interband linear absorption/gain spectra including excitonic effects are calculated analytically using two-site dynamical mean-field theory and ladder approximation. We show that an exciton-like peak structure emerges because of an “exciton” in the sea of correlated fermions when U ' is larger than a certain critical value. The binding energy of the photoinduced excitonic bound state and the spectral intensity are evaluated as a function of U '. The emergence and absence of the exciton-like peak suggests that there are two different insulating mechanisms in the Mott–Hubbard insulating state of the electron–hole system: one is attributable to only U and the other is attributable to both U and U '; the latter is called the excitonic Mott–Hubbard insulating state. The intrinsic difference of the mechanisms cannot be captured from only single-particle properties such as the density of states, but it appears clearly in optical properties reflecting two-particle correlations.
A new approach to the problem of X-ray edge singularities and peaks of many-body origin observed in the optical spectra is presented. We first establish the analogy between a system of one hole interacting with many electrons and the polaron, where one electron interacts with many phonons. We approach both cases as an eigenvalue problem for the time independent Schrodinger equation, obtain expressions for the many-body eigenstate& in a convenient basis and use them to understand the behavior of the system. In the case of a localized hole with infinite mass, we diagonalize the Hamiltonian exactly using first a wavefunction and then a noncanonical transformation. technique. We -use the expressions for the eigenstates to obtain the exact analytic formulas for the optical spectra, accurate over the whole frequency range. We then apply our techniques to the case of a finite mass valence hole, which is realized in modulation doped quantum wells or wires. In this case we approximately diagonalize the Hamiltonian by treating exactly the most important terms leading to nonperturbative behavior. The terms neglected are identified and can be treated in perturbation theory. Our method provides a clear physical picture and valuable insight into the accurate treatment of many-body Hamiltonians.
We study the problem of dielectric response in the strong-coupling regime of a charge-transfer insulator. The frequency and wave-number dependence of the dielectric function $\ensuremath{\varepsilon}(\mathbf{q},\ensuremath{\omega})$ and its inverse ${\ensuremath{\varepsilon}}^{\ensuremath{-}1}(\mathbf{q},\ensuremath{\omega})$ is the main object of consideration. We show that the problem, in general, cannot be reduced to a calculation within the Hubbard model, which takes into account only a restricted number of electronic states near the Fermi energy. The contribution of the rest of the system to the longitudinal response [i.e., to ${\ensuremath{\varepsilon}}^{\ensuremath{-}1}(\mathbf{q},\ensuremath{\omega})]$ is essential for the whole frequency range. With the use of the spectral representation of the two-particle Green's function we show that the problem may be divided into two parts: into the contributions of the weakly correlated subsystem and the Hubbard subsystem. For the latter we propose an approach that starts from the correlated paramagnetic ground state with strong antiferromagnetic fluctuations. We obtain a set of coupled equations of motion for the two-particle Green's function that may be solved by means of the projection technique. The solution is expressed by a two-particle basis that includes the excitonic states with electron and hole separated at various distances. We apply our method to the multiband Hubbard (Emery) model that describes layered cuprates. We show that strongly dispersive branches exist in the excitonic spectrum of the ``minimal'' Emery model (1/Ud=Up=tpp=0){(1/U}_{d}{=U}_{p}{=t}_{\mathrm{pp}}=0)(1/Ud​=Up​=tpp​=0) and consider the dependence of the spectrum on finite oxygen hopping tpp{t}_{\mathrm{pp}}tpp​ and on-site repulsion Up.{U}_{p}.Up​. The relationship of our calculations to electron-energy-loss spectroscopy is discussed.
In this work we investigate some aspects of the physics of strongly correlated systems by taking into account both electron-electron and electron-phonon interactions as basic mechanisms for reproducing electronic correlations in real materials. The relevance of the electron-electron interactions is discussed in the first part of this thesis in the framework of a self-consistent theoretical approach, named Composite Operator Method (COM), which accounts for the relevant quasi-particle excitations in terms of a set of composite operators that appear as a result of the modification imposed by the interactions on the canonical electronic fields. We show that the COM allows the calculation of all the relevant Green’s and correlation functions in terms of a number of unknown internal parameters to be determined self-consistently. Therefore, depending on the balance between unknown parameters and self-consistent equations, exact and approximate solutions can be obtained. By way of example, we discuss the application of the COM to the extended t-U- J-h model in the atomic limit, and to the two-dimensional single-band Hubbard model. In the former case, we show that the COM provides the exact solution of the model in one dimension. We study the effects of electronic correlations as responsible for the formation of a plethora of different charge and/or spin orderings. We report the phase diagram of the model, as well as a detailed analysis of both zero and finite temperature single-particle and thermodynamic properties. As far as the single-band Hubbard model is concerned, we illustrate an approximated selfconsistent scheme based on the choice of a two-field basis. We report a detailed analysis of many unconventional features that arise in single-particle properties, thermodynamics and system’s response functions. We emphasize that the accuracy of the COM in describing the effects of electronic correlations strongly relies on the choice of the basis, paving the way for possible multi-pole extensions to the twofield theory. To this purpose, we also study a three-field approach to the single-band Hubbard model, showing a significant step forward in the agreements with numerical data with respect to the two-pole results. The role of the electron-phonon interaction in the physics of strongly correlated systems is discussed in the second part of this thesis. We show that in highly polarizable lattices the competition between unscreened Coulomb and Fröhlich interactions results in a short-range polaronic exchange term Jp that favours the formation of local and light pairs of bosonic nature, named bipolarons, which condense with a critical temperature well in excess of hundred kelvins. These findings, discussed in the framework of the so-called polaronic t-Jp model, are further investigated in the presence of a finite on-site potential ~U , coming from the competition between on-site Coulomb and Fröhlich interactions. We discuss the role of ~U as the driving parameter for a small-to-large bipolaron transition, providing a possible explanation of the BEC-BCS crossover in terms of the properties of the bipolaronic ground state. Finally, we show that a hard-core bipolarons gas, studied as a charged Bose-Fermi mixture, allows for the description of many non Fermi liquid behaviours, allowing also for a microscopic explanation of pseudogap features in terms of a thermal-induced recombination of polarons and bipolarons, without any assumption on preexisting order or broken symmetries. [edited by Author]