Hybridization between bosonic excitations can produce topological bands, but their Chern numbers conceal how constituent modes generate the resulting Berry geometry. Here we derive an exact decomposition of Berry curvature in multicomponent bosonic Bogoliubov-de Gennes systems into weighted constituent-sector contributions and a term associated with single-quasiparticle entanglement. The latter couples momentum-space variations of the sector imbalance to a gauge-invariant relative Berry connection and has an equivalent gauge-covariant pseudospin representation. Applying this framework to magnon-phonon excitations in a ferromagnetic skyrmion crystal, we find that magnetoelastic coupling generates topological low-energy hybrid bands despite the trivial topology of the two lowest uncoupled magnon bands and degeneracies among folded phonons. The resulting Chern numbers are assembled through constructive, entanglement-assisted, and entanglement-compensated geometric mechanisms. In particular, bands with identical Chern numbers can exhibit opposite entanglement contributions. Our approach thus resolves nonquantized internal geometric contributions hidden within quantized hybrid-band invariants, without requiring an effective two-level Hamiltonian.
Hybridization between bosonic excitations can produce topological bands, but their Chern numbers conceal how constituent modes generate the resulting Berry geometry. Here we derive an exact decomposition of Berry curvature in multicomponent bosonic Bogoliubov-de Gennes systems into weighted constituent-sector contributions and a term associated with single-quasiparticle entanglement. The latter couples momentum-space variations of the sector imbalance to a gauge-invariant relative Berry connection and has an equivalent gauge-covariant pseudospin representation. Applying this framework to magnon-phonon excitations in a ferromagnetic skyrmion crystal, we find that magnetoelastic coupling generates topological low-energy hybrid bands despite the trivial topology of the two lowest uncoupled magnon bands and degeneracies among folded phonons. The resulting Chern numbers are assembled through constructive, entanglement-assisted, and entanglement-compensated geometric mechanisms. In particular, bands with identical Chern numbers can exhibit opposite entanglement contributions. Our approach thus resolves nonquantized internal geometric contributions hidden within quantized hybrid-band invariants, without requiring an effective two-level Hamiltonian.
Recent experiments in moiré quantum materials exhibit quantized Hall states without an external magnetic field, motivating continuum mechanisms based on smooth moiré-periodic pseudospin textures. We present a controlled theory of skyrmion Chern bands generated by such textures. An exact local SU(2) transformation reveals an emergent non-Abelian gauge field; for large branch splitting we perform an operator-level Schrieffer-Wolff expansion, yielding a single-branch Hamiltonian together with systematically dressed physical operators that define the projected interacting theory beyond strict adiabaticity. The leading dynamics is governed by a U(1) Berry connection whose flux is set by the skyrmion density, while controlled non-adiabatic corrections are fixed by the texture's real-space quantum geometric tensor. In a Landau-level representation built from the averaged emergent field, moiré-periodic modulations induce Umklapp-resolved deformations of Girvin-MacDonald-Platzman kinematics and microscopic sources of excess optical quantum weight above the topological lower bound. Assuming a gapped Hall phase, we further derive a skyrmion-crystal effective field theory with a universal Berry-phase term and a noncommutative magnetophonon. Our results provide experimentally accessible signatures for twisted transition-metal dichalcogenide homobilayers and rhombohedral graphene aligned with hexagonal boron nitride.
Bosonic Bogoliubov-de Gennes (BBdG) Hamiltonians describe the excitations of weakly interacting Bose condensates as well as photonic systems under parametric driving. Their topological features have been studied mainly by utilizing a generalized symplectic version of the Berry curvature and related Chern numbers. However, a full characterization of geometrical features in BBdG systems is still lacking. Here, we propose a symplectic quantum geometric tensor (SQGT), whose imaginary part leads to the previously studied symplectic Berry curvature, while the real part gives rise to a symplectic quantum metric, providing a natural distance measure in the space of bosonic Bogoliubov modes. The SQGT is directly related to observable properties of BBdG systems. We show how to measure all components of the SQGT by extracting excitation rates in response to periodic modulations of the system’s parameters. Moreover, we connect the symplectic Berry curvature to a generalized symplectic anomalous velocity term for Bogoliubov-Bloch wave packets. We test our results for a bosonic Bogoliubov-Haldane model.
We generalize the concept of topological invariants for mixed states based on the ensemble geometric phase (EGP) to two-dimensional band structures. In contrast to the geometric Uhlmann phase for density matrices, the EGP leads to a proper Chern number for Gaussian, finite-temperature, or nonequilibrium steady states. The Chern number can be expressed as an integral of the ground-state Berry curvature of a fictitious lattice Hamiltonian, constructed from single-particle correlations. For the Chern number to be nonzero the fictitious Hamiltonian has to break time-reversal symmetry.
A magnetic impurity embedded in a metal is collectively screened by Fermi-surface quasiparticles into a many-body spin-singlet ground state, forming a Kondo cloud of size ξK∼ℏvF/(kBTK). This kinematic picture collapses in flat bands, where vF=0 and the hierarchy of dispersive energy shells is absent. Here we show that the missing organizing principle is quantum geometry. A magnetic impurity coupled to an isolated flat band selects a single active bath mode: a coherent superposition of flat-band Bloch states weighted by the hybridization factor v(k), while all orthogonal flat-band modes remain dark. The resulting flat-band Kondo problem is a quantum geometric molecule, with an algebraic Kondo scale set by the total projected hybridization strength rather than a logarithmic-renormalization scale. In real space, the impurity-bath spin correlation defines a quantum geometric Kondo cloud. Its cloud-size tensor admits a gauge-invariant decomposition into a hybridization-weighted quantum metric, a dressed Berry-connection covariance, and a positive hybridization-gradient term, yielding the lower bound ξK2≥∑kρ(k)Trg(k), where ρ(k)=∣v(k)∣2/∑k∣v(k)∣2. Our result reveals that, in flat bands, Kondo screening is governed by the quantum geometry and interference structure of the impurity-selected Bloch wave packet, rather than Fermi-surface kinematics.
We study the ground-state topology and quasiparticle properties in bosonic Mott insulators with two-dimensional spin-orbit couplings in cold atomic optical lattices. We show that the many-body Chern and spin-Chern number can be expressed as an integral of the quasihole Berry curvatures over the Brillouin zone. Using a strong-coupling perturbation theory, for an experimentally feasible spin-orbit coupling, we compute the Berry curvature and the spin Chern number and find that these quantities can be generated purely by interactions. We also compute the quasiparticle dispersions, spectral weights, and the quasimomentum space distribution of particle and spin density, which can be accessed in cold-atom experiments and used to deduce the Berry curvature and Chern numbers.
The study of topological magnetic excitations have attracted widespread attention in the past few years.The wide variety of ground state phases realized in different two-dimensional magnets have emerged as versatile platforms for realizing magnetic analogues of topological phases uncovered in electronic systems over the past two decades.Dzyaloshinskii-Moriya(DM) interactions, that are ubiquitous in many quantum magnets, and has been demonstrated to induce a non-trivial topology in the magnetic excitations in many quantum magnets.In particular, signatures of DM-interaction induced topological features observed in the dispersion of magnetic excitations in the geometrically frustrated Shastry-Sutherland compound SrCu 2 (BO 3 ) 2 and honeycomb ferromagnet CrI 3 have provided the motivation behind the bulk of the research presented in this thesis.In the first chapter we introduce the background and motivation of our study.We showed that the vastly available materials with a possibility of nonzero symmetry allowed DM-interaction motivate us to study the minimal models of interacting spins related to the materials.In the second chapter we have described the methods that is used to study the spin systems.To describe the spin excitations in the long-range ordered system and dimerized system, we use Holstein-Primakoff transformation and bond-operator formalism respectively.The Schwinger Boson mean field theory is applicable for any generic magnetic ground state and useful to study the system at high temperatures compared with Holstein-Primakoff transformation.After transforming the Hamiltonian into a quadratic bosonic Hamiltonian, it is diagonalized by using Bogoliubov-Valatin transformation.Different observables like Berry-curvature, Chern-number, Dynamical Spin structure factor etc. are cal-vii culated by using the single particle wave-function obtained from Bogoliubov-Valatin transformation.A non-zero Berry curvature is a signature of topologically protected edge states in a stripe geometry of a lattice and a systematic way of calculating edge states is also explained in the second chapter.Moreover in the second chapter the usefulness of symmetry in obtaining band degeneracies and the allowed spin-spin interaction terms are also described.The past theoretical studies show the presence of topological magnons in honeycomb ferromagnetic models due to presence of DM-interaction on a next-nearest neighbour bonds.Recently this kind of DM-interaction is detected in the honeycomb ferromagnet CrI 3 .This type of DM-interaction induces chiral edge states of magnon in the system and so we named it as chiral DM-interaction.In the third chapter we show that breaking of inversion symmetry at the center of honeycomb structure gives rise to a extra DM-interaction which we named as anti-chiral DM-interaction.We studied the system by using Schwinger Boson mean field theory and Holstein-Primakoff transformation and show that when the antichiral DM-interaction dominates over the chiral DM-interaction, the direction of velocity of the edge states become in the same direction (antichiral edge states).Due to conservation of total number of spins in the system a bulk current flows in the opposite direction relative to the antichiral edge current.In this study we suggested the way to break the inversion symmetries in the materials CrSrTe 3 , CrGrTe 3 , AFe 2 (PO 4 ) 2 (A=Ba, Cs, K, La) to achieve antichiral DM-interaction.The presence of antichiral edge states can be detected by using inelastic neutron scattering by detecting the band tilting at K and K points of magnon bands.Moreover we showed that spin noise spectroscopy at the edges is useful measurements to detect the presence of antichiral edge modes.Magnetic force microscopy is also a promising tool to detect the antichiral magnons at edges.The underlying spin-Hamiltonian of the antiferromagnetic systems in the materials like rare-earth tetraborides(RB 4 , R=Er,Tm) and U 2 Pd 2 In can be described by using extended Shastry-Sutherland models.In previous theoretical studies it is shown that there is a non-collinear spin-state known as flux state exists in the Shastry-Sutherland model in presence of out of plane DM-interaction.Although the presence of perpendicular component of DM-interactions in the rare-earth materials are not known, it can be induced artificially by using circularly polarized light or using heavy-metal alloys.In the fourth chapter, we studied the magnetic excitations in the flux state on Shastry-Sutherland lattice in-First of all, I would like to thank my research supervisor Prof. Pinaki Sengupta for his continuous support and guidance throughout my Ph.D. study on and off the field.whenever I went to his office with a problem he always welcomed me and assisted me.He has encouraged and inspired me to become an independent researcher and developed my critical thinking.I am really grateful to him as a person who introduced me the field of research.Moreover my scientific writing and presentation skills also improved due to his suggestions.I am really very thankful for being so flexible and allowing me to choose the work and project which interests me most.