Strongly correlated fermionic systems have been one of the most studied fields in condensed matter theory. The competition between Coulomb-repulsion and kinetic energy leads to a manifold of individual phases, especially in lower dimensions. Famous examples are Luttinger liquids in one dimensions, Mott-insulators or topological phases such as spin liquids. Recently a new class of non-trivial insulators has been introduced, the fragile Mott insulator (FMI). In this paper the authors extend the concept of an FMI to the case of ring-shaped molecules and study the interplay between symmetry, strong correlations as well as strong interactions which ultimately determine the electronic structure of the system.
Strongly correlated fermionic systems have been one of the most studied fields in condensed matter theory. The competition between Coulomb-repulsion and kinetic energy leads to a manifold of individual phases, especially in lower dimensions. Famous examples are Luttinger liquids in one dimensions, Mott-insulators or topological phases such as spin liquids. Recently a new class of non-trivial insulators has been introduced, the fragile Mott insulator (FMI). In this paper the authors extend the concept of an FMI to the case of ring-shaped molecules and study the interplay between symmetry, strong correlations as well as strong interactions which ultimately determine the electronic structure of the system.
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.
A fragile topological insulator (FTI) can be viewed as an almost-atomic insulator, with emergent negative charges localized at certain real-space points, even though the underlying lattice Hilbert space contains only positively charged states. Fragile topology in free-fermion systems has been fruitfully studied using modern topological band theory. However, the concept is well defined even for strongly correlated systems, and fragile states that cannot be realized within free-fermion band theory exist abundantly. In this work we propose a rather general parton construction for such correlation-enabled FTIs. In our parton construction the associated gauge symmetries are completely Higgsed, resulting in only short-range entangled states. The effective negative charges in the FTIs emerge naturally as the remnants of negatively charged partons. For spinful electrons with SU(2) spin-rotation symmetry, the fragile phases can be viewed as FTIs of charge-2, spin singlet bosonic Cooper pairs. We discuss a few examples of correlated FTIs for both spinless and spinful fermions, including some ``featureless Mott insulators'' on the honeycomb lattice previously discussed in the literature.
We study the effects of electronic correlations on fragile topology using dynamical mean-field theory. Fragile topological insulators (FTIs) offer obstruction to the formation of exponentially localized Wannier functions, but they can be trivialized by adding certain trivial degrees of freedom. For the same reason, FTIs do not host symmetry-protected flow of edge states between bulk bands in cylindrical boundary conditions but are expected to have a spectral flow between the fragile bands and other bands under certain twisted boundary conditions. We here analyze commonly observed effects of strong correlations, such as the Mott insulator transition and magnetism, on a known model hosting fragile topology. We show that in the nonmagnetic case, fragile topology, along with the twisted boundary states, is stable with interactions below a critical interaction strength. Above this interaction strength, a transition to the Mott insulating phase occurs, and the twisted boundary states disappear. Furthermore, by applying a homogeneous magnetic field, the fragile topology is destroyed. However, we show that a magnetic field can induce a topological phase transition which converts a fragile topological insulator to a Chern insulator. Finally, we study ferromagnetic solutions of the fragile topological model.
The interplay among symmetry, topology and condensed matter systems has deepened our understandings of matter and lead to tremendous recent progresses in finding new topological phases of matter such as topological insulators, superconductors and semi-metals. Most examples of the aforementioned topological materials are free fermion systems, in this thesis, however, we focus on their strongly correlated counterparts where electron-electron interactions play a major role. With interactions, exotic topological phases and quantum critical points with fractionalized quantum degrees of freedom emerge. In the first part of this thesis, we study the problem of resonant tunneling through a quantum dot in a spinful Luttinger liquid. It provides the simplest example of a (0+1)d system with symmetry-protected phase transitions. We show that the problem is equivalent to a two channel SU(3) Kondo problem and can be mapped to a quantum Brownian motion model on a Kagome lattice. Utilizing boundary conformal field theory, we find the universal peak conductance and compute the scaling behavior of the resonance line-shape.\nFor the second part, we present a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory is based on a SO(7)1=SO(7)1/(G2)1 x (G2)1 coset construction and implemented by a solvable two-dimensional network model. In addition, we predict a closely related ''anti-Fibonacci'' phase, whose topological order is characterized by the tricritical Ising model. Finally, we propose an interferometer that generalizes the Z2 Majorana interferometer and directly probes the Fibonacci non-Abelian statistics.\nFor the third part, we argue that a correlated fluid of electrons and holes can exhibit fractional quantum Hall effects at zero magnetic field. We first show that a Chern insulator can be realized as a free fermion model with p-wave(m=1) excitonic pairing. Its ground state wavefunction is then worked out and generalized to m>1. We give several pieces of evidence that this conjectured wavefunction correctly describes a topological phase, dubbed ''fractional excitonic insulator'', within the same universality class as the corresponding Laughlin state at filling 1/m. We present physical arguments that gapless states with higher angular momentum pairing between energy bands are conducive to forming the fractional excitonic insulator in the presence of repulsive interactions. Without interactions, these gapless states appear at topological phase transitions which separate the trivial insulator from a Chern insulator with higher Chern number. Since the nonvanishing density of states at these higher angular momentum band inversion transitions can give rise to interesting many-body effects, we introduce a series of minimal lattice models realizations in two dimensions. We also study the effect of rotational symmetry broken electron-hole exciton condensation in our lattice models using mean field theory.
We discuss the appearance of quantum orders in the Hubbard model for interacting electrons, at half-filling. Such phases do not have local order parameters and need to be characterized by the quantum mechanical properties of their ground state. On one hand, we study the Mott transition from a metal to a spin liquid insulator in two dimensions, of potential relevance to some layered organic compounds. The correlation-driven transition occurs at fixed filling and involves fractionalization of the electron: upon entering the insulator, a Fermi surface of neutral spinons coupled to an internal gauge field emerges. We focus on the transport properties near the quantum critical point and find that the emergent gauge fluctuations play a key role in determining the universal scaling. Second, motivated by a class of three-dimensional transition metal oxides, the pyrochlore iridates, we study the interplay of non-trivial band topology and correlations. Building on the strong spin orbit coupling in these compounds, we construct a general microscopic Hubbard model and determine its mean-field phase diagram, which contains topological insulators, Weyl semimetals, axion insulators and various antiferromagnets. We also discuss the effects many-body correlations on theses phases. We close by examining a fractionalized topological insulator that combines the two main themes of the thesis: fractionalization and non-trivial band topology. Specifically, we study how the two-dimensional protected surface states of a topological Mott insulator interact with a three-dimensional emergent gauge field. Various correlation effects on observables are identified.
Abstract: The metal–insulator transition (MIT) constitutes one of the most profound and persistent problems in condensed matter physics, sitting at the intersection of electronic structure theory, many-body physics, and emergent quantum phenomena. Conventional band theory, which successfully classifies materials as metals or insulators according to the filling of their electronic bands, fails dramatically when confronted with a class of compounds transition metal oxides chief among them that ought to be metallic by band-counting arguments yet are robustly insulating in the laboratory. The resolution of this paradox required abandoning the independent-electron picture entirely and recognizing that strong Coulomb repulsion between electrons can, by itself, drive localization. This insight, formalized by Mott and later distilled into the Hubbard model, established correlation-driven insulation as a phenomenon distinct from band insulation. The present paper offers a theoretical exploration of the MIT problem, tracing its conceptual development from the Mott criterion through the Hubbard Hamiltonian, the Brinkman–Rice scenario of quasiparticle mass divergence, and the watershed development of dynamical mean-field theory (DMFT), which rendered the Mott transition quantitatively tractable for the first time. We argue that the competition between the electronic bandwidth and the on-site interaction encoded in the ratio U/W provides the unifying organizing principle for the field. Building toward the modern frontier, we examine how the Mott transition connects to quantum phase transitions and quantum criticality, where zero-temperature instabilities governed by quantum rather than thermal fluctuations produce non-Fermi-liquid behavior and anomalous scaling. Case studies of vanadium oxides, rare-earth nickelates, and doped cuprates illustrate the diversity and subtlety of real correlated materials. We conclude that the MIT remains a central and unresolved problem whose continued study is increasingly entangled with the development of quantum materials and quantum technologies. Keywords: Metal–Insulator Transition; Mott Insulator; Strongly Correlated Electrons; Hubbard Model; DMFT; Quantum Criticality; Quantum Materials; Condensed Matter Physics