Continuous quantum phase transitions in interacting many-body systems exhibit universal phenomena that are largely independent of microscopic details. An organizing principle distilled from canonical examples such as the magnetic transition in Ising models is that the low-energy physics of transitions between gapped phases is governed by scale-invariant quantum field theories with gapless degrees of freedom at isolated points in momentum space. In this paper, we show that this principle can fail in a radical way: an entire gapless Fermi surface can emerge at a quantum critical point separating two fully gapped insulators. These exotic phase transitions are dubbed \textit{metallogenic quantum critical points} (MGQCP). We construct a controlled family of effective field theories for MGQCPs between distinct fractional Chern insulators and propose a periodically modulated quantum Hall bilayer as a minimal microscopic setting. The critical theory exhibits an unusual mixture of physical signatures: while transport and compressibility follow scaling laws compatible with a conformal field theory, the electron spectral function develops a power-law non-analyticity at the Fermi momentum, characteristic of non-Fermi-liquid metals. We discuss how the concept of MGQCP may shed new light on critical phenomena in moire materials with tunable Chern bands. More generally, our results point to a broader class of quantum critical phenomena in which extended manifolds of gapless excitations emerge in momentum space despite being absent in both neighboring phases.
Continuous quantum phase transitions in interacting many-body systems exhibit universal phenomena that are largely independent of microscopic details. An organizing principle distilled from canonical examples such as the magnetic transition in Ising models is that the low-energy physics of transitions between gapped phases is governed by scale-invariant quantum field theories with gapless degrees of freedom at isolated points in momentum space. In this paper, we show that this principle can fail in a radical way: an entire gapless Fermi surface can emerge at a quantum critical point separating two fully gapped insulators. These exotic phase transitions are dubbed \textit{metallogenic quantum critical points} (MGQCP). We construct a controlled family of effective field theories for MGQCPs between distinct fractional Chern insulators and propose a periodically modulated quantum Hall bilayer as a minimal microscopic setting. The critical theory exhibits an unusual mixture of physical signatures: while transport and compressibility follow scaling laws compatible with a conformal field theory, the electron spectral function develops a power-law non-analyticity at the Fermi momentum, characteristic of non-Fermi-liquid metals. We discuss how the concept of MGQCP may shed new light on critical phenomena in moire materials with tunable Chern bands. More generally, our results point to a broader class of quantum critical phenomena in which extended manifolds of gapless excitations emerge in momentum space despite being absent in both neighboring phases.
We propose that quantum Hall bilayers in the presence of a periodic potential at the scale of the magnetic length can host examples of a deconfined metal-insulator transition (DMIT), where a Fermi-liquid (FL) metal with a generic electronic Fermi surface evolves into a gapped insulator (or an insulator with Goldstone modes) through a continuous quantum phase transition.The transition can be accessed by tuning a single parameter, and its universal critical properties can be understood using a controlled framework.At the transition, the two layers are effectively decoupled, where each layer undergoes a continuous transition from a FL to a generalized composite Fermi liquid (gCFL).The thermodynamic and transport properties of the gCFL are similar to the usual CFL, while its spectral properties are qualitatively different.The FL-gCFL quantum critical point hosts a sharply defined Fermi surface without long-lived electronic quasiparticles.Immediately across the transition, the two layers of gCFL are unstable to forming an insulating phase.We discuss the topological properties of the insulator and various observable signatures associated with the DMIT.
Symmetry-protected topological (SPT) phases host gapless surface states that are robust against weak, symmetry-preserving interactions. However, strong interactions can drive spontaneous breaking of the protecting symmetries, leading to gapped ordered surfaces. The phase boundary separating the gapless and ordered phases can host exotic conformal field theory states/universality classes typically not observed in conventional quantum phase transitions. We refer to these exotic quantum critical states as surface topological quantum critical points (sTQCPs). In this dissertation, we study attractive interaction-driven quantum criticality on the surface of three-dimensional topological insulators, a class of SPT phases hosting Dirac fermion surface states. When multiple Dirac cones are present, the interaction parameter space becomes intrinsically multi-dimensional, and the phase boundary separating the gapless and ordered phases forms a continuous manifold. Using renormalization group (RG) analysis, we show that in the limit of suppressed quantum fluctuations, the universality of this phase boundary is governed by conformal manifolds: continuous families of interacting conformal field theories characterized by exactly marginal operators. These manifolds are featureless in the sense that they host a single universality class. Their emergence is traced to continuous symmetries of the RG equations that are otherwise absent in the interaction Hamiltonian. We further demonstrate that higher-order quantum fluctuations break these emergent symmetries, fragmenting the conformal manifolds into isolated fixed points of varying infrared stabilities. Remarkably, we find that along the RG flow within the manifold, an EPR-like entanglement entropy in fermion flavor space always increases. Infrared stable conformal field theories correspond to maximally entangled interaction operators, while weakly entangled fixed points are unstable. These results establish the central role of entangled conformal operators and their entropy in shaping the universality classes of sTQCPs. Finally, we propose an experimentally accessible route to realizing surface quantum criticality by coupling a topological insulator surface to a metallic thin film. Strong electron–phonon interactions in the metal mediate effective attractive interactions between surface Dirac fermions, enhanced via a quantum-well resonance mechanism, providing a concrete pathway for accessing sTQCP physics in real materials.
Symmetry-protected topological (SPT) phases host gapless surface states that are robust against weak, symmetry-preserving interactions. However, strong interactions can drive spontaneous breaking of the protecting symmetries, leading to gapped ordered surfaces. The phase boundary separating the gapless and ordered phases can host exotic conformal field theory states/universality classes typically not observed in conventional quantum phase transitions. We refer to these exotic quantum critical states as surface topological quantum critical points (sTQCPs). In this dissertation, we study attractive interaction-driven quantum criticality on the surface of three-dimensional topological insulators, a class of SPT phases hosting Dirac fermion surface states. When multiple Dirac cones are present, the interaction parameter space becomes intrinsically multi-dimensional, and the phase boundary separating the gapless and ordered phases forms a continuous manifold. Using renormalization group (RG) analysis, we show that in the limit of suppressed quantum fluctuations, the universality of this phase boundary is governed by conformal manifolds: continuous families of interacting conformal field theories characterized by exactly marginal operators. These manifolds are featureless in the sense that they host a single universality class. Their emergence is traced to continuous symmetries of the RG equations that are otherwise absent in the interaction Hamiltonian. We further demonstrate that higher-order quantum fluctuations break these emergent symmetries, fragmenting the conformal manifolds into isolated fixed points of varying infrared stabilities. Remarkably, we find that along the RG flow within the manifold, an EPR-like entanglement entropy in fermion flavor space always increases. Infrared stable conformal field theories correspond to maximally entangled interaction operators, while weakly entangled fixed points are unstable. These results establish the central role of entangled conformal operators and their entropy in shaping the universality classes of sTQCPs. Finally, we propose an experimentally accessible route to realizing surface quantum criticality by coupling a topological insulator surface to a metallic thin film. Strong electron–phonon interactions in the metal mediate effective attractive interactions between surface Dirac fermions, enhanced via a quantum-well resonance mechanism, providing a concrete pathway for accessing sTQCP physics in real materials.
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.
Topology of the Fermi surface and universality of the metal-metal and metal-insulator transitions: d-dimensional Hatsugai-Kohmoto model as an example
The earlier theory [1] of the quantum phase transitions related to the change of the Fermi Surface Topology (FST) is advanced. For such transitions the Fermi surface as a quantum critical manifold determined by the Lee-Yang zeros, the order parameter P as the d-volume of the Fermi sea, and the special FST universality class were introduced in [1]. The exactly solvable Hatsugai-Kohmoto (HK) d-dimensional (d=1,2,3) model of interacting fermions is analyzed. We explore the relation between the Lee-Yang zeros, the Luttinger and the plateau (Oshikawa) theorems. The validity of the Luttinger theorem in the HK model is confirmed. It is shown that the order parameter P and the FST universality class describe the transitions between metal and band/Mott insulators, as well as the Lifshitz and van Hove gapless-to-gapless transitions. The gapless phases are established to be the Landau Fermi liquids (metals). In addition to the conventional paradigm with a continuous order parameter, we apply the homology theory to analyze the FST transitions. They are critical points of the Morse function. To quantify FST we use the Euler characteristic, which is calculated for each phase of the HK model. We claim that the FST universality class is robust with respect to interactions and other model details, under the condition that the critical points are non-degenerate.
The emergence of quasiparticles at low temperature provides a powerful organizing principle for many quantum phases of matter, ranging from conventional magnets and superconductors to exotic insulators with topological order. In this thesis, I describe my research in gapless quantum phases in which the framework of quasiparticles breaks down. The main characters are two categories of gapless phases that feature the interplay between strong interactions and two additional ingredients – Fermi surfaces and fractional statistics. Chapter 2 through Chapter 5 focus on strongly interacting metals with Fermi surfaces. The most salient examples are a class of Hertz-Millis models describing the onset of spontaneous symmetry breaking in a metallic environment. At the quantum critical point, gapless order parameter fluctuations destroy quasiparticles living on the Fermi surface, giving rise to a strongly coupled non-Fermi liquid metal. A key result of these chapters is the identification of an infinite-dimensional symmetry that survives in these non-Fermi liquid metals despite the death of quasiparticles. This infinite-dimensional symmetry and its quantum anomaly lead to a series of non-perturbative results on thermodynamics and transport, which are confirmed by perturbative diagrammatic calculations in special examples. Chapter 6 through Chapter 8 explore quantum phases in which anyonic quasiparticles with fractional statistics play an essential role. When parameters in the system are tuned to close the anyon energy gaps, the original anyons lose their coherence and a variety of novel phases emerge. A highlight in this direction is a new mechanism for topological superconductivity in itinerant abelian and non-abelian anyon fluids, which could make contact with experiments on doped fractional quantum anomalous Hall states in the near future.