Mona Berciu

Motivated by the success of the momentum average (MA) approximation for the single-polaron and bipolaron cases, in this dissertation we pursue the first generalization of this technique to the case of weakly-doped insulators beyond the Migdal limit. In search of a suitable starting point, we investigate the ground states of a variety of model Hamiltonians, inspired by real materials such as rare-earth nickelates, polyacetylene, barium bismuthates, and buckminsterfullerene. Our study of the ground state of the 2D rare-earth nickelate sheet shows that it exhibits cuprate-like orbital polarization unimpeded by weak to moderate electron-phonon coupling. This lends further support to the proposal for cuprate-like superconductivity in nickelate thin films and heterostructures. Investigating polyacetylene leads us to develop a semi-analytical technique that can be used to account for zero-point lattice energy in determining the equilibrium geometry of any crystal. The study of a simplified, one-dimensional model of buckminsterfullerene results in the first generalization of MA to a system with two polaron clouds, and a phonon-driven mechanism for exciton dissociation that could be responsible for photocurrent generation in these materials. Because both the nickelates and polyacetylene turn out to be too complex as starting points for MA generalization, we focus on a simplified, s-p chain model of the barium bismuthates. We ultimately succeed in generalizing MA for this band insulator – the first application of its kind. We compare the results to those from determinant quantum Monte Carlo to establish the accuracy of the MA results (with the MA results obtained at lower computational cost). Finally, we share in the development of the first open-sourced Python package implementing automatic MA equation generation via the GGCE method, which allows MA-style methods to be easily used by non-experts – and is expected to expand in the coming years, providing access to many more MA-style methods in the future.

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In this thesis we study the effects of the Peierls electron-phonon coupling in multi-band metal oxides systems. In contrast to the more commonly employed Holstein coupling, which is used in single-band models and is momentum-independent, the momentum-dependent Peierls coupling can explicitly treat coupling to multiple bands. We demonstrate the importance of using the Peierls coupling in modelling complex systems by looking at two examples. First, we investigate single polaron physics on a perovskite lattice inspired by BaBiO₃. We find that with Peierls coupling, the ground state momentum of the polaron jumps between high-symmetry points in the Brillouin zone as the coupling strength is increased. Because such sharp transitions are not possible in the Holstein model, it follows that it is not always feasible to map the more complex Peierls model onto the simpler Holstein model. Then, we add the Peierls coupling to Emery's three-band model for cuprate layers and study its effect on the polaron effective mass. We show that although the hole-phonon coupling strength is moderate to strong, it only causes a negligible increase in the effective mass, indicating that the effective coupling to the magnon-dressed quasiparticle is much reduced by the dressing. We explain the reason for this and describe how to treat the difference between lattice coupling to bare holes versus to correlations-dressed quasiparticles. Our results prove that it is essential to choose the proper coupling when describing multi-orbital systems, in order to deduce their properties appropriately. We propose to investigate more systematically the generic nontrivial behaviour brought by the momentum-dependent Peierls coupling by studying a diatomic 1D chain, to reinforce our conclusion.

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In this thesis, I investigate the behavior of particles dressed by quantum field excitations and random interactions.First I consider two-carrier states in the Peierls model describing the modulation of the particle hopping due to lattice distortions. I compute the spectral response using the Momentum Average approximation. Combining accurate numerical techniques and analytical arguments, I provide a complete picture of the Peierls bipolarons. It is found that polarons bind into strongly bound yet light bipolarons in the singlet sector, even at large values of the electron-phonon coupling strength. At finite electron fillings, these bipolarons may condense into a high-Tc superconductor. On the other hand, phonons mediate a repulsive interaction in the triplet sector, or equivalently (in one dimension), between two hard-core particles, in which case the ground-state dimers bound by sufficiently attractive bare interactions exhibit two sharp transitions, one of which is the first known example of a self-trapping transition at the two-carrier level. In both situations, phonons mediate pair-hopping effective interactions between the carriers. I further study some aspects of the excited spectrum for the two hard-core particles, a situation relevant to ultracold quantum simulators. It is found that the repulsive phonon-mediated interaction binds a repulsive bipolaron embedded in the excited spectrum.I then turn to the study of quenched randomness in an ultracold molecular plasma. I argue that the quenched ultracold plasma presents an experimental platform for studying quantum many-body physics of disordered systems in the long-time and finite energy-density limits. I analyze an experiment that quenches a plasma of nitric oxide to an ultracold system of Rydberg molecules, ions and electrons that exhibits a long-lived state of arrested relaxation. The qualitative features of this state fail to conform with classical models. I develop a microscopic quantum description for the arrested phase based on an effective many-body spin Hamiltonian that includes both dipole-dipole and van der Waals interactions. This effective model appears to offer a way to envision the essential quantum disordered non-equilibrium physics of this system.This thesis thus examines the quantum many-body response in interacting systems coupled to bosonic fields or in disordered environments.

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In this thesis we study the effects of finite temperature (T ) on the single-electron spectral function of doped magnetic insulators. First, we derive the low-temperature correction to the self-energy of a charge carrier injected with parallel spin into a ferromagnetic background which is modeled with both Heisenberg and Ising Hamiltonians so that differences due to gapless versus gapped magnons can be understood. Beside the expected thermal broadening of the T = 0 quasiparticle peak which becomes a resonance inside a continuum, we find that spectral weight istransferred to regions lying outside this continuum, because the carrier and a thermal magnon can bind into a spin-polaron. This work is valid in dimensions d ≥ 2, because it does not include the role of magnetic domains which are important in 1d. We then consider the role of these magnetic domains in 1d systems, for models where spin-polaron formation is impossible. We present Monte Carlo simulations for the spectral function of three related models of a charge carrier that is injected into an Ising chain. Both ferromagnetic and antiferromagnetic coupling between the Ising spins are considered. The interaction between the carrier and the Ising spins is also of Ising type. In two of the models the charge carrier is hosted by a different band, while in the third model it is hosted by the same band as the Ising spins. We find that thecarrier’s spectral function exhibits a distinctive fine structure due to its temporary entrapping inside small magnetic domains, and use these results to construct an accurate (quasi)analytic approximation for low and medium T . While at T = 0, for ferromagnetic order all three models have identical, low-energy quasiparticles, at finite T the low-energy behavior of the first two models remains equivalent, but that of the third model is controlled by rare events due to thermal fluctuations, which transfer spectral weight below the T = 0 quasiparticle peak, generating a pseudogaplike phenomenology. Taken together, our results show that the temperature evolution of the spectral weight of weakly doped magnetic insulators can be very complex.

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In this thesis, we investigated a set of theoretical models frequently used in the field of solid state physics. These models describe coupling of charge carriers to bosonic modes such as phonons or magnons. In particular, the Holstein model describes coupling of charge carriers to dispersionless phonons, whereas the Emery model describes coupling of charge carriers to magnons in hole-doped antiferromagnets.For the Holstein-like models, we studied how extending the model of the coupling beyond terms that are merely linear in the lattice distortion affects the ground state properties. Using appropriate extensions of the momentum average approximation, we could show that even small nonlinearities have a dramatic effect on the resulting quasi-particle's properties. We further investigated a particular type of nonlinear coupling, the double-well coupling model. After studying the properties of a single quasi-particle, we also showed that this system allows the formation of bound states between two charge carriers and a phonon cloud, the so-called bi-polaron. In contrast to the linear variation of the Holstein model, the resulting bi-polaron can be strongly bound yet lightweight. For the Emery model, we consider an experimentally relevant extension. The original model describes a single layer of CuO₂, relevant for the hole-doped cuprate superconductors. We consider recently synthesized layers of CuO, which can be viewed as two intercalated layers of CuO₂. The resulting system is similar to CuO₂, but different in important aspects. We use a variational method similar in spirit to MA but applicable to magnons instead of phonons to obtain the system's dispersion and compare it to that of the original CuO₂ layer. We observe a discrepancy between these dispersions that cannot be accounted for with a single-band model that is commonly used to model the CuO₂ dispersion. However, it has been a long-standing question whether or not this and other single-band models are appropriate for the description of cuprate physics. With our study of CuO, we demonstrated how a careful experimental analysis of this system can resolve that question.

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In this thesis, we investigated the spectral properties of polaronic quasiparticles resulting from the coupling of a charge carrier to the bosonic excitations of ordered environments. Holstein polarons, describing an electron locally coupled to dispersionless phonons, and spin polarons in hole-doped antiferromagnets are considered.The Green's function of the Holstein polaron in a lattice with various kinds of impurities is calculated using the momentum average approximation. The main finding is that the scenario where the mere effect of the coupling to phonons is to enhance the electron's effective mass is incomplete as the phonons are found to also renormalize the impurity potential the polaron interacts with. This formalism is applied first to the case of a single impurity and the range of parameters where the polaron's ground state is localized is identified. The lifetime of the polaron due to scattering from weak but extended disorder is then studied and it is shown that the renormalization of the disorder potential leads to deviation of the strong coupling results from Fermi's golden rule's predictions. Motivated by the hole-doped cuprate superconductors, the motion of a hole in a two-dimensional Ising antiferromagnet and its binding to an attractive impurity is studied next, based on a variational scheme that allows for configurations with a certain maximum number of spin flips. The role of the magnetic sublattices in determining the symmetry of the resulting bound states is discussed. Next, a more realistic model describing a hole in a CuO₂ layer which retains the O explicitly, is considered. By neglecting the fluctuations of the Cu spins and using a variational principle similar to that of the previous chapter, a semi-analytic solution for the Green's function of the hole in an infinite 2D lattice is constructed. The resulting quasiparticle dispersion shows the proper ground state and other features observed in experiments. The lack of importance of the background spin fluctuations is justified based on the hole's ability to move on the O sublattice without disturbing the Cu spins. Finally, the model is generalized to gauge the importance of other relevant O orbitals.

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Many condensed matter problems involve a particle coupled to its environment. The polaron, originally introduced to describe electrons in a polarizable medium, describes a particle coupled to a bosonic field. The Holstein polaron model, although simple, including only optical Einstein phonons and an interaction that couples them to the electron density, captures almost all of the standard polaronic properties. We herein investigate polarons that differ significantly from this behaviour. We study a model with phonon-modulated hopping, and find a radically different behaviour at strong couplings. We report a sharp transition, not a crossover, with a diverging effective mass at the critical coupling. We also look at a model with acoustic phonons, away from the perturbative limit, and again discover unusual polaron properties. Our work relies on the Bold Diagrammatic Monte Carlo (BDMC) method, which samples Feynman diagrammatic expansions efficiently, even those with weak sign problems. Proposed by Prokof'ev and Svistunov, it is extended to lattice polarons for the first time here. We also use the Momentum Average (MA) approximation, an analytical method proposed by Berciu, and find an excellent agreement with the BDMC results. A novel MA approximation able to treat dispersive phonons is also presented, along with a new exact solution for finite systems, inspired by the same formalism.

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To study transport properties, one needs to investigate the system of interest when coupled to biased external baths. This requires solving a master equation for this open quantum system. Obtaining this solution is very challenging, especially for large systems. This limits applications of the theories of open quantum systems, especially insofar as studies of transport in large quantum systems, of interest in condensed matter, is concerned.In this thesis, I propose three efficient methods to solve the Redfield equation --- an example of such a master equation. The first is an open-system Kubo formula, valid in the limit of weak bias. The second is a solution in terms of Green's functions, based on a BBGKY (Bogoliubov--Born--Green--Kirkwood--Yvon)-like hierarchy. In the third, the Redfield equation is mapped to a generalized Fokker-Planck equation using the coherent-state representation. All three methods, but especially the latter two, have much better efficiency than direct methods such as numerical integration of the Redfield equation via the Runge-Kutta method. For a central system with a d-dimensional Hilbert space, the direct methods have complexity of d³, while that of the latter two methods is on the order of order of polynomials of log d. The first method, besides converting the task of solving the Redfield equation to solving the much easier Schrödinger's equation, also provides an even more important conceptual lesson: the standard Kubo formula is not applicable to open systems.Besides these general methodologies, I also investigate transport properties of spin systems using the framework of the Redfield equation and with direct methods. Normal energy and spin transport is found for integrable but interacting systems. This conflicts with the well-known conjecture linking anomalous conductivity to integrability, and it also contradicts the relationship, suggested by some, between gapped systems (Jz > Jxy) and normal spin conductivity. I propose a new conjecture, linking anomalous transport to the existence of a mapping of the problem to one for non-interacting particles.

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In this work we present a highly efficient and accurate analytical approximation for the Green's function of a polaron: the momentum average (MA) approximation. It is obtained by summing all of the self-energy diagrams, but with each diagram averaged over the momenta of its free propagators. The result becomes exact for both zero bandwidth and for zero electron-phonon coupling, and is accurate everywhere in the parameter space. The approximation is first used to investigate the Holstein model. A detailed analysis of its accuracy is provided through diagrammatics and spectral weight sum rules. It is shown that the resulting Green's function satisfies exactly the first six spectral weight sum rules, and all higher order sum rules are satisfied with great accuracy. Comparison with numerical data also confirms this accuracy. We then show how to improve the MA approximation by systematically improving the accuracy of the self-energy diagrams in such a way that they can still all be summed efficiently. This allows us to fix some of the problems of the zeroth-order MA approximation. The quantitative agreement with numerical data is also improved.Next, we generalize the MA approximation to study the properties of models with momentum-dependent electron-phonon coupling, and then show that further improvements can be obtained based on variational considerations, using the 1D breathing-mode Hamiltonian as a specific example. For example, by using this variational MA, we obtain ground state energies within at most 0.3% error of the numerical data.Finally, we study the effects of a nearby surface on the spectral weight of a Holstein polaron. The broken translational symmetry is accounted for without any additional approximations, and the resulting inhomogeneous MA approximation continues to be accurate for all coupling strengths. We show that the surface changes properties significantly, with bulk values being recovered only very far away from it. We find that the electron-phonon coupling gives rise to an additional surface potential which is responsible for binding surface states even when they are not normally expected. These results demonstrate that interpretation in terms of bulk properties of spectroscopic data sensitive only to a few surface layers is not straightforward.

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