Presented in this thesis are a set of projects which lie at the intersection between strong correlations and topological phases of matter. The first of these projects is a treatment of an infinite dimensional generalization of the SSH model with local Coulomb interactions which is solved exactly using DMFT-NRG. Observed in the solution is power-law augmentation of the non-interacting density of states, as well as a Mott transition. This calculation represents an exact solution to an interacting topological insulator in the strongly correlated regime at zero temperature. The second set of projects involves the development of methods for formulating non-interacting auxiliary models for strongly correlated systems. These auxiliary models are able to capture the full dynamics of the original strongly correlated model, but with only completely non-interacting degrees of freedom, defined in an enlarged Hilbert space. We motivate the discussion by performing the mapping analytically for simple interacting systems using non-linear canonical transformations via a Majorana decomposition. For the nontrivial class of interacting quantum impurity models, the auxiliary mapping is established numerically exactly for finite-size systems using exact diagonalization, and for impurity models in the thermodynamic limit using NRG, both at zero and finite temperature. We find that the auxiliary systems take the form of generalized SSH models, which inherit the topological characteristics of those models. These generalized SSH models are also formalized and investigated in their own right as novel systems. Finally, we apply this methodology to study the Mott transition in the Hubbard model. In terms of the auxiliary system, we find that the Mott transition can be understood as a topological phase transition, which manifests as the formation and dissociation of topological domain walls.
Presented in this thesis are a set of projects which lie at the intersection between strong correlations and topological phases of matter. The first of these projects is a treatment of an infinite dimensional generalization of the SSH model with local Coulomb interactions which is solved exactly using DMFT-NRG. Observed in the solution is power-law augmentation of the non-interacting density of states, as well as a Mott transition. This calculation represents an exact solution to an interacting topological insulator in the strongly correlated regime at zero temperature. The second set of projects involves the development of methods for formulating non-interacting auxiliary models for strongly correlated systems. These auxiliary models are able to capture the full dynamics of the original strongly correlated model, but with only completely non-interacting degrees of freedom, defined in an enlarged Hilbert space. We motivate the discussion by performing the mapping analytically for simple interacting systems using non-linear canonical transformations via a Majorana decomposition. For the nontrivial class of interacting quantum impurity models, the auxiliary mapping is established numerically exactly for finite-size systems using exact diagonalization, and for impurity models in the thermodynamic limit using NRG, both at zero and finite temperature. We find that the auxiliary systems take the form of generalized SSH models, which inherit the topological characteristics of those models. These generalized SSH models are also formalized and investigated in their own right as novel systems. Finally, we apply this methodology to study the Mott transition in the Hubbard model. In terms of the auxiliary system, we find that the Mott transition can be understood as a topological phase transition, which manifests as the formation and dissociation of topological domain walls.
Over the past 30 years experimental observations have demonstrated the existence of a variety of quantum phases of matter not admissible to a classification in terms of the Landau theory of symmetry breaking. Examples include, but are not limited to the fractional quantum
Hall states and frustrated quantum magnets. Theoretical evidence supports the idea that such phases can exist in a large class of zero temperature strongly correlated condensed matter systems.
In this thesis we study a particular case of such systems called topological phases of matter. Such phases are characterised by the presence of non-local correlations which are manifest in properties such as degenerate groundstates that depend on the global topology of the
system and the emergence of topological excitations. Remarkably the classification of such materials is profoundly tied to the mathematical construction of topological quantum field theories (TQFT).
In this thesis we utilise this connection to explore possible candidate Hamiltonian models for topological phases of matter. Our methodology is that of reverse engineering effective local Hamiltonians from a class of discrete TQFT’s called state sums.
In chapter 5 we develop a construction to canonically associate to any state-sum TQFT a corresponding local Hilbert space and Hamiltonian defining a candidate model for a topological phase of matter.
In chapter 9 we develop a candidate model of topological phases using ideas from higher gauge theory and higher category theory. In particular we define a Hamiltonian realisation of the Yetter Homotopy 2-type TQFT which describes a topological gauge theory, where the
gauge symmetry is given by a finite 2-group and relate a class of such models to the construction of Walker-Wang.
Building on the Hamiltonian construction for state-sum TQFT’s, in Part III of this thesis we develop an algebraic approach to understanding the topological excitations of such theories, we call tube-algebras. In chapter 10 we develop a general construction for defining tube-algebras for any unitary state-sum TQFT and describe the general features. In chapter 12 we apply this construction to the
Dijkgraaf-Witten TQFT in 1+1, 2+1 and 3+1D. In chapter 13 we
apply this construction to topological higher lattice gauge theories and compare the results the Dijkgraaf-Witten TQFT.
In this thesis some new ideas to perform the analysis of Strongly Correlated Electronic Systems (SCES) are developed. In particular the use of non-linear canonical transformations is considered thoroughly. Using such transformations it is possible, in some circumstances, to simplify the quantum problem redefining the fermionic degrees of freedom used to describe the system. To understand and use effectively these non-linear transformations it is convenient to work in the Majorana fermion representation, i.e., to represent the quantum mechanical operators in terms of Majorana fermions. These objects can be imagined as algebraic constituents of the fermionic degrees of freedom. In a fermionic system, different equivalent sets of (emergent) Majorana fermions can be used to build the fermionic operators that characterize the system. The non-linear transformations can be seen as a way to mix these equivalent sets. Thanks to this insight, it becomes possible to determine the full structure of the group of canonical transformations and to identify an advantageous framework, which allows their use in the study of a generic SCES system. To test these statements the Hubbard and the Kondo lattice models were intensively studied making use of non-linear canonical transformations, obtaining interesting results in both cases. For example, in the Hubbard model a free fermion mean-field description of the paramagnetic Mott insulator was identified, while in the Kondo lattice it was possible to describe already at mean-field level the spin-selective Kondo insulating phase, consistently (from a quantitative and qualitative point of view) with the known numerical results. Moreover the method elaborated for the study of the Hubbard model is suitable for a systematic generalization to other situations and shows great room for improvement. These results prove that, thanks to the redefinition of the degrees of freedom used in the analysis of the system, it becomes possible to obtain quite non-trivial results already at mean-field level, or to consider very involved (but meaningful) correlated quantum states via simple variational trial states. This will potentially permit a more judicious and profitable choice of the fundamental degrees of freedom, allowing for an improvement of the efficiency of the analytical and numerical techniques used in the analysis of many SCES systems.
Materials with strongly correlated electrons show a multitude of unconventional phenomena, such as superconductivity or magnetism, that cannot be explained in a single-particle picture and necessitate to take many-body effects into account. A route to explain the origin of these phenomena consists in constructing effective models for these materials that are then solved, often using numerical techniques. Paradigmatic models for two different classes of strongly correlated electron systems are investigated in this thesis: The Hubbard model is a paradigmatic model to describe the Mott metal-insulator transition (MIT) and the Kondo lattice model describes the low-energy physics in heavy fermion compounds. Both models are analyzed numerically using different variations of a quantum cluster technique, the variational cluster approximation (VCA). The first part focuses on the MIT, as it is investigated experimentally in quasi-two-dimensional charge-transfer salts. Its nature and associated universality class are still heavily discussed and both experiments and theoretical approaches come to different conclusions. In this thesis, the Mott transition is investigated on an anisotropic two-dimensional Hubbard model at half-filling. By using a control parameter, which induces this anisotropy, strong evidence for Mott quantum criticality is found in weakly coupled Hubbard chains. The results at zero temperature show that the second-order critical end point Tc of the interaction-driven metal-insulator transition can be tuned down to zero at strong anisotropy. Further results for the antiferromagnetic phase suggest a similar picture and motivate adding the anisotropy as a new axis in the phase diagram to account for a low-temperature critical end point of the Mott transition. The second part of this thesis focuses on unconventional phases that emerge in a paradigmatic model for heavy fermion systems, the Kondo lattice model. Using the VCA, the two-dimensional Kondo lattice model is investigated in the paramagnetic phase as well as in phases with broken symmetry. An antiferromagnetic phase at weak coupling and a phase with Kondo-singlet formation at strong coupling are analyzed. Within the antiferromagnetically ordered region, two phases with different Fermi-surface topology, which are separated by a discontinuous transition, are identified. The model is also tested for s-wave superconductivity as found recently using dynamical mean-field theory, but no indications for robust local pairing are seen within VCA. Instead, nodal d-wave superconductivity is found and analyzed in a large range of couplings and electron fillings, and its interplay with antiferromagnetic order at weak coupling is discussed. This motivates further studies on extended models that might allow for a close comparison with experiments on heavy fermion systems. Understanding the formation of unconventional phases in strongly correlated electron systems is not only of fundamental interest, but it is also essential for designing new functional materials. Although the results of approximate cluster techniques have to be handled with care, the new aspects presented in this thesis pave the way for further investigations in this direction.
In recent years, topology has become increasingly prevalent in condensed matter physics. It has allowed us to understand, and even predict, a variety of striking and remarkable physical phenomena. The study of strongly interacting systems has similarly lavished us with a diverse range of exotic phases and unconventional transitions, many of which are still poorly understood. In this thesis we will explore the interplay between topology and interactions in an effort to uncover new and novel phases. First we study how interactions impact the quantum phase transition between a topologically non-trivial phase and a trivial phase. The combination of interactions and the low-energy degrees of freedom associated with the transition leads to the emergence of a dome of lattice-symmetry breaking nematic order. Such behaviour is reminiscent of a number of strongly correlated electronic systems. We move on to study the strongly interacting limit of one of the earliest and best-known non-interacting topological phases, Haldane's model of a Chern insulator. Recently realized with ultracold atoms in a shaken optical lattice, the model has a non-trivial topological invariant associated with its band structure. In the strongly interacting limit the spin degrees of freedom are all that survive and we find a rich phase diagram of magnetically ordered phases, using a combination of both classical and quantum techniques. Supplementing the model with an additional term we can `quantum-melt' one of these ordered states to produce a disordered, liquid state that we positively identify as a chiral spin liquid, a highly entangled state of matter with fractionalised excitations. We generalise this mechanism to other two dimensional lattices, uncovering a possible unifying framework with which to understand the emergence of chiral spin liquids in lattice spin models. Finally, motivated by groundbreaking experiments in the ultracold atoms community, we investigate a model of two-component bosons with an artificial spin-orbit coupling. The interplay between the lattice, interactions and spin-orbit coupling produces a variety of unusual superfluid phases. Using a novel Monte Carlo technique we reveal the finite temperature phase diagram that appears close to the Mott transition.
In this thesis, we describe methods for solving quantum impurity models out of equilibrium, and consider several applications of these methods. Quantum impurity models describe systems in which a small subsystem (the impurity) is coupled to a large external environment (the bath). These models are useful in describing a wide variety of physical systems, including quantum dots, absorbed adatoms, and molecular junctions. They are also important for the study of lattice models, such as the Hubbard model, where they appear as auxiliary problems within the framework of dynamical mean field theory (DMFT). Many interesting phenomena in these systems, such as transport through a quantum dot, or optical pumping of a Mott insulator, occur away from equilibrium. Theoretical description of these phenomena requires methods that are capable of handling both real time evolution and strong correlation effects presenting a formidable challenge. In this thesis, we describe several methods for the solution of non-equilibrium quantum impurity models which meet this challenge. These methods are based on an expansion in the coupling between the impurity and the bath known as the hybridization expansion. Chapter 1 gives an introduction to the problems addressed in this thesis. Chapter 2 provides a brief overview of Keldysh formalism for nonequilibrium problems. Chapter 3 describes the bare hybridization expansion for the Anderson impurity model as well as its bold extensions. In particular we derive the non-crossing approximation (NCA), the one-crossing approximation (OCA), the inchworm expansion, and a bold expansion for the impurity Green's function. Chapter 4 describes the implementation of a quantum Monte Carlo algorithm based on the inchworm expansion. The rest of the thesis describes applications of these algorithms to problems of physical interest. Chapters 5 and 6 apply the methods previously described to understanding the Kondo effect in quantum dot systems. The first chapter examines the behavior of the Kondo effect under an applied voltage bias, a classic non-equilibrium strongly-correlated problem. The second chapter examines different ways of measuring the cloud of electrons which screen the impurity spin in the Kondo regime. Chapters 7 and 8 apply the same methods to the Hubbard model using the Dynamical mean-field theory (DMFT) mapping. The first chapter studies the Hubbard model in equilibrium using the numerically exact inchworm quantum Monte Carlo method. The second chapter studies the possibility of dynamic switching between the metal and Mott-insulator states of the Hubbard model using the one-crossing approximation (OCA). Finally chapter 9 gives some overall conclusions and suggests future directions for the development of hybridization expansion methods.
This dissertation examines the impact of long-range interactions in lattice models of low-dimensional strongly correlated systems with a focus on their relevance to anomalous electronic transport. Traditionally, the Hubbard model forms the basis of theoretical and numerical investigations of strongly correlated quantum matter. This effective model constitutes the simplest model of electronic correlation and relies on an approximation that truncates electron-electron interactions to a local, on-site potential. While this approximation has proved valid in several instances, certain unexplained phenomena in correlated materials challenge us to develop new perspectives and approaches.In the first part of this thesis, we discuss the historical motivation for our current work from both a theoretical and an experimental viewpoint. In particular, we cover the discovery of bad metallic transport and the associated breakdown of Boltzmann theory of electron transport. We further review examples of strongly correlated systems that exhibit bad metallic transport and other exotic phenomena that suggest the possibility of poorly screened electronic interactions acting as a common microscopic mechanism. This general feature justifies the development of a theoretical framework that is capable of examining long-range interactions in a controlled, systematic manner.Motivated by this, we analyze a long-range interacting lattice model for spinless electrons in two dimensions via exact diagonalization on a finite cluster in real space. As this method is susceptible to finite-size errors, we discuss at length its implementation and technical details designed specifically to reduce these errors, namely the use of twisted boundary conditions and the Ewald summation. We determine the phase diagram at zero temperature and demonstrate that the long-range interactions act as a source of self-generated disorder that drives the development of a strongly correlated pseudogap phase, in a manner reminiscent of the classical Efros-Shklovskii Coulomb gap. Furthermore, we establish that this phenomenon is independent of lattice geometry, a fact that represents the broad potential of long-range interactions in giving rise to a novel form of strongly correlated behavior.Following our zero-temperature study, we address the question of whether this self-generated disorder can survive in the presence of thermal fluctuations and local Coulomb repulsion (Hubbard terms) relevant in Mott systems, a situation closer to those in experimental systems. By means of classical Monte Carlo calculations, we show that not only does the self-generated disorder survive, but that it also acts as a microscopic mechanism that generates bad metallic transport. Upon further investigation, we determine that the long-range interactions, in combination with quantum fluctuations, cause a transient localization of the charge carriers, thereby suppressing the conductivity values below those predicted in conventional Boltzmann theory, as observed experimentally.In conclusion, we have established that long-range interactions in electronic models act as a microscopic mechanism that is capable of causing exotic phenomena in numerous materials. We conclude that a proper treatment of both quantum fluctuations and long-range interactions should next be studied with the finite temperature Lanczos method, which will act as a bridge between our exact diagonalization and Monte Carlo studies. Overall, this dissertation emphasizes the capability of long-range interactions to produce novel phenomena and elicits further thought-provoking questions concerning the interplay with Mott-Hubbard physics and external disorder, crucial in metallic systems.