Recent experimental observations have revealed the emergence of a topological state withina quantum critical regime, in a system where the description in terms of electronic quasiparticlesis no longer valid. Such a result is often presented as paradoxical from the viewpointof standard theoretical frameworks of condensed matter physics. In this work, we demonstraterigorously that this phenomenon is fully compatible with a five-dimensional geometricframework, in which topological properties are global invariants independent of any effectivefour-dimensional particle-based reduction. Our analysis relies exclusively on general structuralarguments and does not require any ad hoc assumption or material-specific adjustment.
Recent experimental observations have revealed the emergence of a topological state withina quantum critical regime, in a system where the description in terms of electronic quasiparticlesis no longer valid. Such a result is often presented as paradoxical from the viewpointof standard theoretical frameworks of condensed matter physics. In this work, we demonstraterigorously that this phenomenon is fully compatible with a five-dimensional geometricframework, in which topological properties are global invariants independent of any effectivefour-dimensional particle-based reduction. Our analysis relies exclusively on general structuralarguments and does not require any ad hoc assumption or material-specific adjustment.
The history of modern condensed matter physics may be regarded as the competition and reconciliation between Stoner’s and Anderson’s physical pictures, where the former is based on momentum–space descriptions focusing on long wave-length fluctuations while the latter is based on real-space physics emphasizing emergent localized excitations. In particular, these two view points compete with each other in various nonperturbative phenomena, which range from the problem of high [Formula: see text] superconductivity, quantum spin liquids in organic materials and frustrated spin systems, heavy-fermion quantum criticality, metal-insulator transitions in correlated electron systems such as doped silicons and two-dimensional electron systems, the fractional quantum Hall effect, to the recently discussed Fe-based superconductors. An approach to reconcile these competing frameworks is to introduce topologically nontrivial excitations into the Stoner’s description, which appear to be localized in either space or time and sometimes both, where scattering between itinerant electrons and topological excitations such as skyrmions, vortices, various forms of instantons, emergent magnetic monopoles, and etc. may catch nonperturbative local physics beyond the Stoner’s paradigm. In this review paper, we discuss nonperturbative effects of topological excitations on dynamics of correlated electrons. First, we focus on the problem of scattering between itinerant fermions and topological excitations in antiferromagnetic doped Mott insulators, expected to be relevant for the pseudogap phase of high [Formula: see text] cuprates. We propose that nonperturbative effects of topological excitations can be incorporated within the perturbative framework, where an enhanced global symmetry with a topological term plays an essential role. In the second part, we go on to discuss the subject of symmetry protected topological states in a largely similar light. While we do not introduce itinerant fermions here, the nonperturbative dynamics of topological excitations is again seen to be crucial in classifying topologically nontrivial gapped systems. We point to some hidden links between several effective field theories with topological terms, starting with one-dimensional physics, and subsequently finding natural generalizations to higher dimensions.
This thesis focuses on the effects of strong electron-electron interactions on topological phase transitions.In very general terms, topological phase transitions are typically associated with the formation of gapless states at the boundary between two gapped phases.The opening of a gap is usually associated with a spontaneous symmetry breaking that can obliterate the topological nature of the system.In this work we highlight a completely different destiny for a topological transition in the presence of interaction and we develop a theoretical framework to address this physics identifying the physical origin for the unconventional scenario.In particular, we show that strong electronic correlation can lead to a gap opening at the topological transition without any symmetry breaking.This turns the topological transition discontinuous, thereby giving rise to a quantum critical point connecting the continuous region with the discontinuous one.This unconventional scenario establishes a surprising connection between a topological transition and quantum criticality.By solving a Bernevig-Hughes-Zhang model supplemented with a local symmetric interaction, we demonstrate that the crucial role of orbital polarization naturally emerges, already at the mean-field level.This local parameter effectively captures the essence of the topological transition, complementing the established evaluation of topological invariants and enabling a Landau-like description.Within this framework, we analyze the interplay between band topology and the dynamical effects of electronic correlations, achieved by developing a non-perturbative method for calculating the effective free energy, the Variational Gaussian Approximation, which is compared with both a simple static mean-field, which is not able to capture any critical behavior for the topological transition and with the accurate Dynamical Mean-Field Theory which provides an important benchmark for the accuracy of the approach.We show indeed that dynamical quantum fluctuations can lead to gap opening without any symmetry breaking.The growth of these quantum effects as the interaction is increased turns indeed the topological transition from continuous to discontinuous.This change of behavior is associated with the presence of massive Dirac fermion at the transition point showing a Gross-Neveu critical behaviour near the quantum critical endpoint.We identify the gap opening as a condensed matter analog of the Coleman-Weinberg mechanism of mass generation.The theory presented offers a broad scope for extension in various directions, ranging from changing the topological model and modifying the interactions, to incorporating non-local effects, or even applying it to entirely trivial strongly correlated models.
Topological state characterization in interacting quasi-periodic systems remains challenging because conventional band-topology approaches become invalid in the presence of strong many-body correlations and broken translational symmetry. This study develops a generalized many-body Zak phase framework for investigating topological phase transitions in Bose–Einstein condensates confined within quasi-periodic potential wells. By introducing a quasimomentum manifold derived from quasi-periodic phase symmetry and constructing gauge-continuous ground-state families, topological invariants are directly extracted from many-body wavefunctions without relying on Bloch bands, Brillouin zones, or bulk energy gaps. The proposed approach is integrated with density matrix renormalization group calculations to establish a unified characterization of superfluid, localized, and critical regimes. Results show that a stable topologically nontrivial phase emerges at U/J = 3.2 and ∆/J = 2.8, where the momentum distribution exhibits symmetric peaks at k = ±π/2 with an amplitude of 1.87. Open-boundary analysis reveals strongly localized edge modes with approximately 90% density concentration near system boundaries, while the entanglement spectrum displays a symmetry-protected doubly degenerate structure. The results demonstrate that interaction and quasi-periodicity jointly stabilize nontrivial topological states and associated wave-localization phenomena. The proposed framework provides a rigorous methodology for analyzing topological phase evolution, geometric-phase transitions, and wave propagation characteristics in strongly correlated quasi-periodic systems.
Understanding correlation effects in topological phases and their transitions is a cutting-edge area of research in recent condensed matter physics. We study topological quantum phase transitions (TQPTs) between double-Weyl semimetals (DWSMs) and insulators, and argue that a novel class of quantum criticality appears at the TQPT characterized by emergent anisotropic non-Fermi-liquid behaviors, in which the interplay between the Coulomb interaction and electronic critical modes induces not only anisotropic renormalization of the Coulomb interaction but also strongly correlated electronic excitation in three spatial dimensions. Using the standard renormalization group methods, large N_{f} theory, and the ε=4-d method with a fermion flavor number N_{f} and spatial dimension d, we obtain the anomalous dimensions of electrons (η_{f}=0.366/N_{f}) in large N_{f} theory and the associated anisotropic scaling relations of various physical observables. Our results may be observed in candidate materials for DWSMs such as HgCr_{2}Se_{4} or SrSi_{2} when the system undergoes a TQPT.
This paper presents a systematic extension of the Global Fractional Quantization (GFQ) framework from particle physics to condensed matter systems. We establish a rigorous theoretical foundation for the quantization of collective observables in many-body quantum systems, demonstrating that sums of specific physical quantities over entire bands, ensembles, or collective states exhibit discrete, rational relationships. The framework is built upon four fundamental postulates—Gauge Constraint, Gauge Invariance, Gauge Periodicity, and Global State Symmetry—applied to appropriately defined U(N) collective multiplets in condensed matter contexts. We provide detailed mathematical derivations with complete proofs and demonstrate the framework’s applicability to several key condensed matter systems: topological insulators, superconductors, magnetic systems,and correlated electron materials. The theory predicts novel quantization conditions for sums of Berry curvatures, total polarization, collective spin moments, and superconducting phase differences,which we validate through both analytical models and numerical calculations with comprehensive error analysis. Our results reveal a hierarchical structure of quantization laws that correlate with the system’s dimensionality, symmetry, and fundamental nature (topological vs. trivial, gapped vs. gapless). The GFQ framework offers a new unifying perspective on emergent quantization phenomena across diverse condensed matter platforms and provides testable predictions for future experiments.
Opportunities for topological phase transitions in strongly correlated Fermi systems near a quantum critical point are explored as an alternative to collective scenarios for experimentally observed departures from standard Fermi-liquid behavior. Attention is focused on a quantum critical point at which the effective mass is divergent due to vanishing of the quasiparticle group velocity at the Fermi surface. Working within the original Landau quasiparticle theory, it is demonstrated that the quasiparticle picture can remain meaningful beyond the quantum critical point through rearrangements of the unstable normal Fermi surface and quasiparticle spectrum. Two possibilities emerge at zero temperature, depending on whether the quasiparticle interaction is regular or singular at zero momentum transfer. In the regular case, one type of topological phase transformation leads to a state with a multiconnected Fermi surface. In the singular case, another type of topological phase transition leads to an exceptional state containing a fermion condensate – the Fermi surface swells into a volume in momentum space, within which partial occupation prevails and quasiparticle energies are pinned to the chemical potential. As the temperature increases from zero to a characteristic value Tm, a crossover can occur from the state with multiple Fermi surfaces to that containing a fermion condensate.