Spin-orbit coupling exists in materials in general. However, it entangles the spin and orbital degrees of freedom and complicates the model. Thus, theorists usually neglect the effects induced by spin-orbit coupling first and consider spin-orbit coupling as perturbation next. The non-perturbative effects brought up by spin-orbit coupling are thus often less studied or overlooked.\nOn the other hand, the majority in the study of interacting topological order focusing on the general structure of theories and made significant advances by leaving material details behind. It is thus important to find possible microscopic models that could realize the new phases in laboratories and benefits from the progress of theories to make experimental predictions.\nIn this thesis, we study the physical effects due to strong spin-orbit coupling from the perspective of searching new quantum orders and the non-trivial responses.\n(i) The first project, we propose the nontrivial dipolar-octupolar(DO) doublets on the pyrochlore lattice. By studying the most general symmetry allowed model at the localized and the itinerant limit for DO doublets, we found two 3D symmetry enriched topological orders and topological insulator correspondingly. (ii) In the second project, we analyze the 2D model descending from the localized limit of DO doublets on pyrochlore. The discrete onsite symmetry and space group symmetry could lead to a symmetry-enriched topological order with symmetry fractionalization pattern that cannot emerge from a spin model with continuous spin rotational symmetry. The non-trivial symmetry fractionalization pattern contributes to the striking numerical signal that can help identifying the topological order. (iii) In the third project, we develop a theory to understand the high-energy Raman signal in Sr2IrO4.
Spin-orbit coupling exists in materials in general. However, it entangles the spin and orbital degrees of freedom and complicates the model. Thus, theorists usually neglect the effects induced by spin-orbit coupling first and consider spin-orbit coupling as perturbation next. The non-perturbative effects brought up by spin-orbit coupling are thus often less studied or overlooked.\nOn the other hand, the majority in the study of interacting topological order focusing on the general structure of theories and made significant advances by leaving material details behind. It is thus important to find possible microscopic models that could realize the new phases in laboratories and benefits from the progress of theories to make experimental predictions.\nIn this thesis, we study the physical effects due to strong spin-orbit coupling from the perspective of searching new quantum orders and the non-trivial responses.\n(i) The first project, we propose the nontrivial dipolar-octupolar(DO) doublets on the pyrochlore lattice. By studying the most general symmetry allowed model at the localized and the itinerant limit for DO doublets, we found two 3D symmetry enriched topological orders and topological insulator correspondingly. (ii) In the second project, we analyze the 2D model descending from the localized limit of DO doublets on pyrochlore. The discrete onsite symmetry and space group symmetry could lead to a symmetry-enriched topological order with symmetry fractionalization pattern that cannot emerge from a spin model with continuous spin rotational symmetry. The non-trivial symmetry fractionalization pattern contributes to the striking numerical signal that can help identifying the topological order. (iii) In the third project, we develop a theory to understand the high-energy Raman signal in Sr2IrO4.
Motivated by experimental progress in the growth of heavy transition metal oxides, we theoretically study a class of lattice models of interacting fermions with strong spin-orbit coupling. Focusing on interactions of intermediate strength, we derive a low-energy effective field theory for a fully gapped, topologically ordered, fractionalized state with an eightfold ground-state degeneracy. This state is a fermionic symmetry-enriched topological phase with particle-number conservation and time-reversal symmetry. The topological terms in the effective field theory describe a quantized magnetoelectric response and nontrivial mutual braiding statistics of dynamical extended vortex loops with emergent fermions in the bulk. We explicitly compute the expected mutual statistics in a specific model on the pyrochlore lattice within a slave-particle mean-field theory. We argue that our model also provides a possible condensed-matter realization of oblique confinement.
In this thesis, we study the topological phases of quantum spin systems. One project is to investigate a class of anti-ferromagnetic SU(N) Heisenberg models, describing Mott insulators of fermionic ultra-cold alkaline earth atoms on the three-dimensional simple cubic lattice. Our large-N analysis maps a rich phase diagram. One particularly striking state we found spontaneously breaks lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state. In the other projects, we study the phenomenon of symmetry fractionalization on anyons as a tool to characterize two-dimensional symmetry enriched topological phases. In particular, we focus on how crystalline symmetries may fractionalize in gapped Z2 spin liquids. If the system has the symmetry of the square lattice, then there are 2080 symmetry fractionalization patterns possible. With exactly solvable models, we realize 487 of these in strictly two-dimensional systems. In addition, we succeed to understand why the remaining patterns cannot be found in the family of models we construct. Some can only appear on the surface of three-dimensional systems with non-trivial point group symmetry protected topological (pgSPT) order, whose boundary degrees of freedom transform non-locally under the symmetries. We construct a simple toy model to show this anomalous crystalline symmetry fractionalization phenomenon associated with a reflection. Moreover, our approach establishes the connection between the pgSPT phases and the topological phases with on-site symmetries in lower dimensions. This insight is very useful for classification of pgSPT orders in general.
A large part of modern condensed matter concerns non-local or global properties of systems under the broad term of topology. Closely intertwined with certain senses of topology are symmetries of the system, which physically constrain our description and offer non-perturbative insights into material properties. The range of possible phases of matter is greatly enriched by quantum mechanics, including certain types of non-local entanglement that cannot be understood in the usual symmetry-breaking paradigm of local order parameters. Beyond weakly interacting topology we find the stronger notion of \textit{topological order} in correlated systems, a prominent example being the S=21 Kitaev honeycomb model in a weak magnetic field.Through a remarkable combination of lattice geometry, strong spin-orbit coupling, and electron interactions, several material candidates have been put forward to realize Kitaev physics, including the 4d Mott insulator α-RuCl3 which offers tantalizing experimental hints of topological order. In this thesis we explore several aspects of Kitaev candidates and related systems under two broad themes. One involves symmetry-informed and constrained observables, including a proposed singularity in a thermodynamic quantity accompanying phase transitions between two Ising topological orders (ITOs). The ITO physics realized by the Kitaev model requires time-reversal symmetry to be broken, and much of the non-trivial physics is found at finite magnetic field. To this end, we study a microscopic spin model appropriate for α-RuCl3 and show that it exhibits an interesting field-induced paramagnetic state between the low- and high-field magnetic orders, consistent with recent experimental findings. We determine several properties of a lower-dimensional spin-S analogue of the Kitaev model, including a multi-critical point at finite field separating different topological phases. Finally, we derive quantum state purities of entangled photons emitted from a spin-triplet superconducting heterostructure and discuss its use in a new paradigm of entangled spectroscopy.
Topological order provides a new paradigm to view phases of matter. Unlike conventional symmetry breaking order, these states are not distinguished by different patterns of symmetry breaking, instead by their intricate mathematical structure, topology. By the bulk-boundary correspondence, the nontrivial topology of the bulk results in robust gapless excitations on symmetry preserving surfaces. We utilize both of these views to study topological phases together with the analysis of their quantized physical responses to perturbations.First we study the edge excitations of strongly interacting abelian fractional quantum Hall liquids on an infinite strip geometry. We use the infinite density matrix renormalization group method to numerically measure edge exponents in model systems, including subleading orders. Using analytic methods we derive a generalized Luttinger's theorem that relates momenta of edge excitations.Next we consider topological crystalline insulators protected by space group symmetry. After reviewing the general formalism, we present results about the quantization of the magnetoelectric response protected by orientation-reversing space group symmetries. We construct and analyze insulating and superconducting tight-binding models with glide symmetry in three dimensions to illustrate the general result. Following this, we derive constraints on weak indices of three dimensional topological insulators imposed by space group symmetries. We focus on spin-orbit coupled insulators with and without time reversal invariance and consider both symmorphic and nonsymmorphic symmetries.Finally, we calculate the response of metals and generalize the notion of the magnetoelectric effect to noninteracting gapless systems. We use semiclassical dynamics to study the magnetopiezoelectric effect, the current response to elastic strain in static external magnetic fields.
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.
Symmetry fractionalization describes the fascinating phenomena that excitations in a 2D topological system can transform under symmetry in a fractional way. For example in fractional quantum Hall systems, excitations can carry fractional charges while the electrons making up the system have charge one. An important question is to understand what symmetry fractionalization (SF) patterns are possible given different types of topological order and different global symmetries. A lot of progress has been made recently in classifying the SF patterns, providing deep insight into the strongly correlated experimental signatures of systems like spin liquids and topological insulators. We review recent developments on this topic. First, it was shown that the SF patterns need to satisfy some simple consistency conditions. More interestingly, it was realized that some seemingly consistent SF patterns are actually ‘anomalous’, i.e. they cannot be realized in strictly 2D systems. We review various methods that have been developed to detect such anomalies. Applying such an understanding to 2D spin liquid allows one to enumerate all potentially realizable SF patterns and propose numerical and experimental probing methods to distinguish them. On the other hand, the anomalous SF patterns were shown to exist on the surface of 3D systems and reflect the nontrivial order in the 3D bulk. We review examples of this kind where the bulk states are topological insulators, topological superconductors, or have other symmetry protected topological orders.