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 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 investigate strongly correlated phenomena in quantum spin systems. In the first part of this work, we study geometrically frustrated antiferromagnets (AFMs). Generalizing the SU(2) Heisenberg Hamiltonian to Sp(N) symmetry, we obtain, in the large-N limit, the mean-field phase diagrams for the planar pyrochlore and cubic AFMs. We then use gauge theories to consider fluctuation effects about their respective mean-field configurations. We find, in addition to conventional Neel states, a plethora of novel magnetically disordered phases: two kinds of spin liquids, Z2 in 2+1D and U(1)in 3+1D, and several valence bond solids such as two and three-dimensional plaquette and columnar singlet states. We use the same approach to study the diamond lattice AFM which possesses extended classical ground state degeneracy. We demonstrate that quantum and entropic fluctuations lift this degeneracy in different ways. In the second part of the thesis, we study ultracold spinor atoms confined in optical lattices. We first demonstrate the feasibility of experimental realization of rotor models using ultracold spin-one Bose atoms in a spin-dependent and disordered optical lattice. We show that the ground state of such disordered rotor models with quadrupolar interactions can exhibit biaxial nematic ordering in the disorder-averaged sense, and suggest an imaging experiment to detect the biaxial nematicity in such systems. Finally, using variational wavefunction methods, we study the Mott phases and superfluid-insulator transition of spin-three bosons in an optical lattice with an anisotropic two dimensional optical trap. We chart out the phase diagrams for Mott states with n = 1 and n = 2 atoms per lattice site. We show that the long-range dipolar interaction stabilizes a state characterized by antiferromagnetic chains made of ferromagnetically aligned spins. We also obtain the mean-field phase boundary for the superfluid-insulator transition, and show that inside the superfluid phase and near the superfluid-insulator phase boundary, the system undergoes a first order antiferromagnetic-ferromagnetic spin ordering transition.
The physics of strongly interacting particles is often extraordinarily rich, leading to the \textit{emergence} of fundamentally new phases and phenomena that cannot exist in the absence of correlations. This thesis is devoted to the study of different classes of quantum systems with such strong correlations. Programmable quantum simulators based on Rydberg atom arrays have recently emerged as versatile platforms to probe quantum many-body phases and their dynamics. In Chapter 2, we start with theories of quantum phase transitions (QPTs) in one-dimensional chains of Rydberg atoms. We demonstrate the existence of a novel QPT in the universality class of the Z3 chiral clock model, calculate its critical exponents, and discuss their measurement with the quantum Kibble-Zurek mechanism. Proceeding to two dimensions, Chapter 3 describes the zero-temperature phase diagram of a square-lattice Rydberg atom array, revealing several experimentally realizable density-wave-ordered phases and QPTs. For a different geometry, namely, the kagome lattice studied in Chapter 4, we identify a regime that constitutes a likely candidate for hosting a phase with long-range quantum entanglement and topological order. Based on mappings to dimer models and gauge theories, we argue for the existence of a quantum spin liquid phase that has proved elusive in solid-state materials. In the next part of this thesis, we focus on similarly strongly correlated phases in the realm of quantum materials. Motivated by recent observations of an enhanced thermal Hall response in the high-temperature cuprate superconductors, in Chapter 5, we study the thermal Hall conductivity, κxy, of chiral spin-liquid ans\"{a}tze on the square lattice, focusing on the contribution from neutral (bosonic or fermionic) spinons. Thereafter, going beyond mean-field theory, we consider the corrections due to emergent gauge fields, which yield a response of opposite sign to that of the fermionic matter. We additionally consider the contribution of phonons to κxy and establish its relation to the spinon Hall viscosity. Thermal Hall measurements also leave intriguing fingerprints in Kitaev materials like α-RuCl3, and in this context, Chapter 6 investigates the unquantized κxy obtained in a gapless spinon Fermi surface state.
Ever since quantum mechanics and relativity appeared at the beginning of the 20th century, physics has been rocked to its foundations. Novel ideas and applications surged relentlessly from the minds of those brave enough to venture into the unknown new worlds revealed by the new sciences. This year marks the centenary of Heisenberg’s quantum mechanics, and the exploration has not waned; rather, the frontiers of our knowledge now extend beyond fundamental theories into the complex playground of symmetry, topology, and emergence. We are faced with the challenge of understanding and predicting the behavior of complex systems in which the interplay of all three is essential for explaining their unique phenomena. Amid such an insurmountable challenge, novel viewpoints and signatures can often guide us through this dense web of complexity. In this doctoral thesis, we explore distinct scenarios where the intertwining of symmetry and topology naturally leads to previously unexplored phenomena, and subtle signatures arise from the combination of old and new theoretical frameworks in the weak and strong coupling regimes. In part one, we study weakly interacting topological phases. In chapter 2, we introduce the idea of time-reversal invariant finite-size topology. We identify signatures of a higher-dimensional bulk topological phase within thin films and wires; we probe this via Wilson loop spectroscopy and electromagnetic responses. Chapter 3 deals with extending previously studied topological skyrmion phases to time-reversal invariant systems. A generalization of the time-reversal parity invariant corresponds to a bulk-boundary texture of the pseudo-spin degrees of freedom. Moreover, our skyrmion invariant is directly linked to a reduced entanglement spectrum with protected spectral flow. Part two of this thesis is dedicated to strongly interacting electrons, where the effective description is that of a network of spins displaying the unconventional physics of spin liquids. In chapter 4, we describe how fractionalization can be generalized in classical spin liquids to yield irrational magnetic moments of dilution clusters. These clusters act as gauge charges for the low-energy lattice gauge theory. Moreover, we show that for a higher-rank gauge theory, distinctive emergent non-decaying spin textures and interactions are hallmarks of the unusual topological magnet. Chapter 5 studies a fragile spin liquid in three dimensions, which displays fractionalization but no algebraically decaying correlations. We find that not only is the Heisenberg model disordered, but the Ising model is as well. Perturbing away, we find a special point where quantum fluctuations stabilize a Z_2 quantum spin liquid.
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.
In this thesis, we investigate several topics pertaining to emergent collective quantum phenomena in the domain of correlated fermions, using the quantum Monte Carlo method. They display exotic low temperature phases as well as phase transitions which are beyond the Landau–Ginzburg theory. The interplay between three key points is crucial for us: fermion statistics, many body effects and topology. We highlight the following several achievements: 1. Successful modeling of continuum field theories with lattice Hamiltonians, 2. their sign-problem-free Monte Carlo simulations of these models, 3. and numerical results beyond mean field descriptions. First, we consider a model of Dirac fermions with a spin rotational invariant inter- action term that dynamically generates a quantum spin Hall insulator. Surprisingly, an s-wave superconducting phase emerges due to the condensation of topological de- fects of the spin Hall order parameter. When particle-hole symmetry is present, the phase transition between the topological insulator and the superconducting phase is an example of a deconfined quantum critical point(DQCP). Although its low energy effec- tive field theory is purely bosonic, the exact conservation law of the skyrmion number operator rules out the possibility of realizing this critical point in lattice boson models. This work is published in Ref. [1]. Second, we dope the dynamically generated quantum spin Hall insulator mentioned above. Hence it is described by a field theory without Lorentz invariance due to the lack of particle-hole symmetry. This sheds light on the extremely hot topic of twisted bilayergraphene: Why is superconductivity generated when the repulsive Coulomb interaction is much stronger than the electron-phonon coupling energy scale? In our case, Cooper pairs come from the topological skyrmion defects of the spin current order parameter, which are charged. Remarkably, the nature of the phase transition is highly non-mean-field-like: one is not allowed to simply view pairs of electrons as single bosons in a superfluid-Mott insulator transition, since the spin-current order parameter can not be ignored. Again, due to the aforementioned skyrmions, the two order parameters are intertwined: One phase transition occurs between the two symmetry breaking states. This work is summarized in Ref. [2]. Third, we investigate the 2 + 1 dimensional O(5) nonlinear sigma model with a topological Wess-Zumino-Witten term. Remarkably, we are able to perform Monte Carlo calculations with a UV cutoff given by the Dirac Landau level quantization. It is a successful example of simulating a continuous field theory without lattice regularization which leads to an additional symmetry breaking. The Dirac background and the five anti-commuting Dirac mass terms naturally introduce the picture of a non-trivial Berry phase contribution in the parameter space of the five component order parameter. Using the finite size scaling method given by the flux quantization, we find a stable critical phase in the low stiffness region of the sigma model. This is a candidate ground state of DQCP when the O(5) symmetry breaking terms are irrelevant at the critical point. Again, it has a bosonic low energy field theory which is seemingly unable to be realized in pure boson Hamiltonians. This work is summarized in Ref. [3].
A rich variety of phases can exist in quantum systems. For example, the fractional quantum Hall states have persistent topological characteristics that derive from strong interaction. This thesis uses the exact diagonalization method to investigate quantum lattice models with strong interaction. Our research topics revolve around quantum phase transitions between novel phases. The goal is to find the best schemes for realizing these novel phases in experiments. We studied the fractional Chern insulator and its transition to uni-directional stripes of particles. In addition, we studied topological Mott insulators with spontaneous time-reversal symmetry breaking induced by interaction. We also studied emergent kinetics in one-dimensional lattices with spin-orbital coupling. The exact diagonalization method and its implementation for studying these systems can easily be applied to study other strongly correlated systems.