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.
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.
Quantum phases of matter can be classified into two kinds: gapped or gapless, depending on whether the equilibrium ground state is separated from excited states by a finite energy window or not in the thermodynamic limit. Topology has played an important role in the understanding of both gapped and gapless phases of matter. This thesis uncovers several robust physical properties of both kinds of systems which are largely unexplored before. \\ For gapped systems, we study topological orders, which are characterized by the existence of anyonic quasiparticles with fractional braiding statistics. In the first part, we introduce a family of Abelian quantum Hall insulators, at filling factors
ν=p/2q for bosonic systems and
ν=p/(p+2q) for fermionic systems, with
p and
q being two coprime integers. These states are termed the \textit{nondiagonal} quantum Hall states, which are constructed in a coupled wire model that exhibits an intimate relation to the nondiagonal conformal field theory and has a constrained pattern of motion for anyons. We unveil a nontrivial interplay between charge symmetry and translation symmetry in this system, and establish the nondiagonal quantum Hall states as symmetry-enriched topological orders. In the second part, we introduce a related model of interacting electrons, but in the absence of a strong quantizing magnetic field, known as the \textit{toric code insulator}. It is constructed from a two-dimensional array of strongly coupled one-dimensional topological superconductors, which feature electron fractionalization and possesses anyonic excitations described by the
Z2 topological order. The motion of anyons is again constrained, and different types of anyons can be related by the translation symmetry. Nondiagonal quantum Hall states and the toric code insulator are convincing examples that itinerant electronic systems can realize interesting symmetry-enriched topological orders.\\ For gapless quantum matters we focus on a particularly familiar class, the metals, but we uncover their topological aspects which were much less appreciated. Every metal is characterized by a Fermi surface of gapless excitations, and the Fermi surface surrounds a manifold in the momentum space, known as the Fermi sea. The ground state of a non-interacting metal, or a Fermi gas, consists of all single-electron states within the Fermi sea. The shape of the Fermi sea has a natural topological characterization in terms of the Euler characteristic
χF, and a fundamental question that we address in this thesis is: \textbf{does
χF imply any robust physical properties in metals, and how can they be measured?} In the third part of the thesis,
χF is connected to physically measurable quantities, such as equal-time density-density correlation functions, and multipartite entanglement measures, such as the mutual information. Particularly, for three dimensions, we establish that the 4-partite mutual information is a robust probe of the Fermi sea topology even in the presence of Fermi-liquid interactions. In the last part, we focus on two dimensions, and propose an experimentally accessible platform to probe
χF for any two-dimensional electron gas (2DEG) that respects time-reversal symmetry. The proposed experimental setup is a Josephson
π-junction, and we predict that the electrical transport in the direction \textit{along} the junction exhibits a current-rectification effect that is controlled by the underlying Fermi sea topology. Due to the relation to Andreev state transport, we term this effect \textit{topological Andreev rectification}. The size of rectification is quantified by a \textit{nonlocal rectified conductance}, which is established to attain an integer quantized value that equals
χF (in unit of
e2/h). Extensive numerical simulations are performed to support our theoretical proposal, and material platforms such as InAs heterostructures and graphene are identified as promising candidates to demonstrate this effect.
Quantum phase transitions of metals involve changes in the Fermi surface, and can be divided into three categories. The first two categories involve symmetry breaking, and lead to a deformation or reconstruction of the Fermi surface. The third category involves a change in the volume enclosed by the Fermi surface without any symmetry breaking: one phase is a Fermi liquid (FL) with the conventional Luttinger volume, while the other phase is a `fractionalized Fermi liquid' (FL*), which has a non-Luttinger volume Fermi surface accompanied by a spin liquid with fractionalized excitations. It is a relatively simple matter to obtain a FL*-FL transition in Kondo lattice models. However, the FL*-FL transition can also be present in single-band Hubbard-like models: this is efficiently described by the `ancilla' method, which shows that the transition is a `flipped' Kondo lattice transition. This single-band FL*-FL transition is argued to apply to the metallic states of the hole-doped cuprates. In the clean limit, the critical properties of the quantum transitions in the three categories are distinct, but all lead to perfect metal transport in the quantum-critical regime. Impurity-induced `Harris disorder', with spatial fluctuations in the local position of the quantum critical point, is a relevant perturbation to the clean critical points. In the presence of Harris disorder, all three categories exhibit strange metal behavior, which can be described by a universal two-dimensional Yukawa-Sachdev-Ye-Kitaev model.
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.