This thesis explores the interaction of symmetries and quantum anomalies on the low-energy dynamics and phase transitions of strongly correlated quantum matter. Beyond the traditional Landau-Ginzburg-Wilson paradigm, we investigate how the embedding of microscopic lattice symmetries into emergent continuum symmetries governs the stability of quantum critical points in (2+1) dimensions. Through the lens of 't Hooft anomalies and anomaly matching, we establish a theoretical framework for understanding the fate of various exotic quantum phase transitions. A central application of this framework is understanding the stability of emergent conformal field theories (CFTs) that describe gapless quantum critical points. The exact nature of these phase transitions is chiefly governed by how the microscopic lattice symmetries impose stringent conditions on the relevant operators allowed in the low-energy effective theory. When physical symmetries suppress relevant perturbations, an interacting conformal fixed point, such as (2+1)-dimensional quantum electrodynamics (QED₃), can stably emerge. Chapters 2 and 3 explore the consequences of this symmetry embedding through the phenomenon of "unnecessary'' quantum critical points, which uniquely describe continuous transitions occurring entirely within a single phase of matter. We demonstrate that the Dirac spin liquid (DSL) in lattice magnets can surprisingly describe a stable continuous phase transition within a single, conventional, symmetry breaking phase. In chapter 4, we apply these tools to a class of quantum phase transitions between featureless bosonic atomic insulators. Despite lacking topological order or protected edge modes, these insulators represent distinct phases because they can be "obstructed,'' meaning their Wannier centers are not pinned to the physical atomic sites. Surprisingly, we find that the transition between these topologically trivial states can host a conformally invariant state described by QED₃, suggesting that even transitions between trivial phases can harbor rich, fractionalized physics. Lastly, chapter 5 analyzes transitions from a composite Fermi liquid (CFL) to a chiral superconductor, employing more traditional field-theoretic techniques and making concrete connections with recent experiments in moire materials.
This thesis explores the interaction of symmetries and quantum anomalies on the low-energy dynamics and phase transitions of strongly correlated quantum matter. Beyond the traditional Landau-Ginzburg-Wilson paradigm, we investigate how the embedding of microscopic lattice symmetries into emergent continuum symmetries governs the stability of quantum critical points in (2+1) dimensions. Through the lens of 't Hooft anomalies and anomaly matching, we establish a theoretical framework for understanding the fate of various exotic quantum phase transitions. A central application of this framework is understanding the stability of emergent conformal field theories (CFTs) that describe gapless quantum critical points. The exact nature of these phase transitions is chiefly governed by how the microscopic lattice symmetries impose stringent conditions on the relevant operators allowed in the low-energy effective theory. When physical symmetries suppress relevant perturbations, an interacting conformal fixed point, such as (2+1)-dimensional quantum electrodynamics (QED₃), can stably emerge. Chapters 2 and 3 explore the consequences of this symmetry embedding through the phenomenon of "unnecessary'' quantum critical points, which uniquely describe continuous transitions occurring entirely within a single phase of matter. We demonstrate that the Dirac spin liquid (DSL) in lattice magnets can surprisingly describe a stable continuous phase transition within a single, conventional, symmetry breaking phase. In chapter 4, we apply these tools to a class of quantum phase transitions between featureless bosonic atomic insulators. Despite lacking topological order or protected edge modes, these insulators represent distinct phases because they can be "obstructed,'' meaning their Wannier centers are not pinned to the physical atomic sites. Surprisingly, we find that the transition between these topologically trivial states can host a conformally invariant state described by QED₃, suggesting that even transitions between trivial phases can harbor rich, fractionalized physics. Lastly, chapter 5 analyzes transitions from a composite Fermi liquid (CFL) to a chiral superconductor, employing more traditional field-theoretic techniques and making concrete connections with recent experiments in moire materials.
Symmetries play a central role in organizing many-body quantum phases of matter and in revealing universal low-energy behavior. Systems with gapless excitations are especially rich, as symmetries can act in intricate ways on low-energy degrees of freedom, while enhanced correlations and entanglement open possibilities not available in gapped settings. This thesis studies two settings in which symmetry considerations give rise to qualitatively new phenomena in gapless quantum matter and correlated electronic systems. In the first part of the thesis, I study how the framework developed for gapped topological phases can be generalized to gapless systems, especially those arising at topological phase transitions. I show that symmetry can refine a fixed conformal field theory universality class into distinct symmetry-enriched critical phases that cannot be smoothly connected without breaking symmetry or leaving the critical theory. In Chapter 1, I show that transitions between distinct symmetry-protected topological phases can exhibit exotic boundary critical behavior not seen in gapped systems. In Chapter 2, I show that distinct gapless symmetry-enriched phases can meet at an anomalous multicritical point. In Chapter 3, I show that spatial interfaces provide an operational indicator of these phases, and that distinct symmetry-enriched criticalities are separated by non-invertible conformal defects. Together, these results show that subtle distinctions in symmetry-enriched criticality have robust and physically detectable consequences. In the second part of the thesis, I study how symmetry and strong interactions govern charge carriers in correlated electron systems. In Chapter 4, I develop a microscopic theory of the Josephson diode effect in Josephson junctions, showing that nonreciprocal supercurrents arise universally from finite Cooper-pair momentum when time-reversal and inversion symmetry are simultaneously broken. In Chapter 5, I study spin-polaron formation in a flat-band ferromagnetic Hubbard model and show that binding occurs in a regime where the conventional mechanisms based on kinetic frustration or nontrivial band topology do not apply, but instead exotic composite charge carriers arise from quantum geometry.
States with gapless degrees of freedom are typically more complicated and less well- understood than systems possessing a gap. In this thesis, we study strongly-correlated systems described by gapless fermions. In most of the systems we consider, the fermionic excitations are emergent, i.e. they are not adiabatically connected to the electrons which constitute the fundamental building blocks of the system in question.\nWe start by proposing several novel states of matter which we claim are relevant to strongly-correlated systems. The rst state we present is a fractionalized Fermi liquid on the surface of a topological Kondo insulator. A Kondo insulator is a material which becomes insulating at low temperatures as a result of strong electron-spin interactions. When spin- orbit coupling is present, this insulator might be topological and, consequently, host robust gapless surface modes. Given the strong interactions and the decreased dimensionality of the surface, we propose that the spins and electrons there may decouple, resulting in the formation of a fractionalized Fermi liquid.\nWe next argue that quantum electrodynamics in 2+1 dimensions (QED3) with Nf = 4 fermion avours may describe a continuous, decon ned phase transition connecting the 120◦ coplanar Néel phase and the √12 × √12 valence bond solid phase of the triangular lattice antiferromagnet (AF). In addition to being a critical point, QED3 is also believed to describe a critical phase of matter called the Dirac spin liquid. Regardless of whether QED3 is manifest as a phase or a critical point, impurities and imperfections are always present in the real world, and it is therefore important to understand what e ects this may have. We show that when QED3 is perturbed by weak disorder, under certain circumstances, it ows to a new critical point in which both interactions and disorder are present.\nConclusively identifying a system described by QED3 has proven to be a di cult task, not only experimentally, but in numerical studies as well. Since QED3 is a critical theory, its spectrum on a torus is a universal quantity dependent only on low-energy degrees of freedom, such as the torus area. It follows that this data is accessible via exact diagonalization and can serve as an identifying signature of the theory. To this end, we calculate the QED3 torus spectrum using path integral methods.\nWe conclude this thesis with a comprehensive study of gapped Z2 spin liquids of the square lattice antiferrogmagnet, using a critical theory of Dirac fermions coupled to an SU(2) gauge eld as a starting point. There are many di erent Z2 spin liquid groundstates which preserve the same symmetry group. Even if one knows that a system has a Z2 spin liquid groundstate, it is by no means obvious which Z2 spin liquid is being realized. By starting from a gapless theory, the set of possibilities can be reduce considerably. Further, using certain recently proposed dualities, comparisons with theories formulated using bosons can be made.
We study a class of quantum phase transitions between featureless bosonic atomic insulators in (2+1) dimensions, where each phase exhibits neither topological order nor protected edge modes. Despite their lack of topology, these insulators may be ``obstructed'' in the sense that their Wannier centers are not pinned to the physical atomic sites. These insulators represent distinct phases, as no symmetry-preserving adiabatic path connects them. Surprisingly, we find that for certain lattices, the critical point between these insulators can host a conformally invariant state described by quantum electrodynamics in (2+1) dimensions (QED3). The emergent electrodynamics at the critical point can be stabilized if the embedding of the microscopic lattice symmetries suppresses the proliferation of monopoles, suggesting that even transitions between trivial phases can harbor rich and unexpected physics. We analyze the mechanism behind this phenomenon, discuss its stability against perturbations, and explore the embedding of lattice symmetries into the continuum through anomaly matching. In all the models we analyze, we confirm that the QED3 is indeed emergeable, in the sense that it is realizable from a local lattice Hamiltonian.
Emergence refers to the phenomena in many-body systems not obvious from microscopic details and fundamental laws. In this dissertation, I will present some of the works that study exotic emergent phe- nomena. In the first 2 chapters, one kind of gapless quantum spin liquid(QSL) called Dirac Spin Liquid(DSL) is examined, on its stability, experiment signatures and proximate symmetry breaking phases. Dirac fermions, photons and monopoles emerge in DSL, rendering it a strongly interacting phase, with an emergent symmetry group larger than that of the physical system. We establish DSL as likely a stable phase of matter on triangular/kagome lattices and connect DSL with symmetry-breaking magnetic orders on 2D lattices. Then the focus switches to more conventional insulators, and the issue of electric polarization. A topological response term to electromagnetic fields offers a modern definition for polarization. This also implies on the boundary of an insulator, Luttinger theorem is modified to include bulk polarization effects. Lastly, we study a route towards topological superconductivity from a chiral spin liquid(CSL) and construct a wave-function for superconductors without explicit electron pairing.
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].
This Thesis explores the formation of novel phases in Dirac systems that are driven by strong many-body interactions and collective quantum fluctuations. The central focus is the half-filled honeycomb lattice, which is the prototypical Dirac semimetal with low-energy excitations described by Dirac fermions. Of particular interest are exotic phases of matter that are not favoured at mean-field but are stabilised by fluctuations. Self-consistent path integral techniques are used to demonstrate that extended repulsive interactions favour charge order that breaks translational and other lattice symmetries. This charge order outcompetes interaction induced topological insulating phases, which are dominant at the mean-field level. The interacting quantum critical point was found to be near the critical point of a topological Lifshitz transition. Interacting Dirac fermions exhibit perhaps the simplest example of fermionic quantum criticality. Gross-Neveu-Yukawa field theories describe the universality class of semimetal-insulator transitions, which are driven by on-site and nearest neighbour repulsive interactions on the honeycomb lattice. The Dirac fermions couple to the dynamical order parameter fields and play a crucial role in determining the universal behaviour. Naturally, this variety of criticality is outside of the Ginzburg-Landau-Wilson paradigm. This idea is extended to include the translational symmetry breaking charge order. Here it is found that the breaking of crystal symmetries corresponds to Lorentz violating effective field theories. The criticality is analysed with the renormalisation group. The low energy excitations at the topological transition are gapless semi-Dirac fermionic quasiparticles that disperse linearly in one direction and quadratically in the other. The instabilities induced by generic short-range repulsions are investigated. The criticality of this anisotropic system is analysed with the renormalisation group in two spatial dimensions, and it is found to exhibit anisotropic order parameter correlations.