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This book discusses the essential properties of carbon nanotubes and 2D graphene systems. The book focuses on the fundamental excitation properties of a large range of graphene-related materials, presenting a new theoretical framework that couples electronic properties and e-e Coulomb interactions together in order to thoroughly explore Coulomb excitations and decay rates in carbon-nanotube-related systems.

Non-Abelian topological order is a coveted state of matter with remarkable properties, including quasiparticles that can remember the sequence in which they are exchanged 1 – 4 . These anyonic excitations are promising building blocks of fault-tolerant quantum computers 5 , 6 . However, despite extensive efforts, non-Abelian topological order and its excitations have remained elusive, unlike the simpler quasiparticles or defects in Abelian topological order. Here we present the realization of non-Abelian topological order in the wavefunction prepared in a quantum processor and demonstrate control of its anyons. Using an adaptive circuit on Quantinuum’s H2 trapped-ion quantum processor, we create the ground-state wavefunction of D 4 topological order on a kagome lattice of 27 qubits, with fidelity per site exceeding 98.4 per cent. By creating and moving anyons along Borromean rings in spacetime, anyon interferometry detects an intrinsically non-Abelian braiding process. Furthermore, tunnelling non-Abelions around a torus creates all 22 ground states, as well as an excited state with a single anyon—a peculiar feature of non-Abelian topological order. This work illustrates the counterintuitive nature of non-Abelions and enables their study in quantum devices. A trapped-ion quantum processor is used to create ground-states and excitations of non-Abelian topological order on a kagome lattice of 27 qubits with high fidelity.

Quasi-one-dimensional chains of atoms can be effectively described by one-dimensional Dirac-type equation. Crystal structure of the chain is reflected by pseudo-spin of the quasi-particles. In the article, we present a simple framework where supersymmetric transformation is utilized to generate an interaction between two, initially non-interacting systems described by pseudo-spin-one Dirac-type equation. In the presented example, the transformation converts two asymptotically non-interacting atomic chains into a saw chain locally. The model possesses a flat band whose energy can be fine-tuned deliberately.

Materials with interacting quantum spins that nevertheless do not order magnetically down to the lowest temperatures are candidates for a materials class called quantum spin liquids (QSLs). QSLs are characterized by long-range quantum entanglement and are tricky to study theoretically; an even more difficult task is to experimentally prove that a material is a QSL. Broholm et al. take a broad view of the state of the field and comment on the upcoming challenges. Science , this issue p. eaay0668 Spin liquids are quantum phases of matter with a variety of unusual features arising from their topological character, including “fractionalization”—elementary excitations that behave as fractions of an electron. Although there is not yet universally accepted experimental evidence that establishes that any single material has a spin liquid ground state, in the past few years a number of materials have been shown to exhibit distinctive properties that are expected of a quantum spin liquid. Here, we review theoretical and experimental progress in this area.

Skyrmions are topologically stable quasiparticles that have been predicted and demonstrated in quantum fields, solid-state physics and magnetic materials, but only recently observed in electromagnetic fields. Here we review the recent advances in optical skyrmions within a unified topological framework. Starting from fundamental theories and classification of skyrmionic states, we describe generation and topological control of different kinds of skyrmions in evanescent, structured and spatiotemporal optical fields. We further highlight generalized classes of optical topological quasiparticles beyond skyrmions and outline the emerging applications, future trends and open challenges. A complex vectorial field structure of optical quasiparticles with versatile topological characteristics emerges as an important feature in modern spin optics, imaging, metrology, optical forces, structured light, and topological and quantum technologies. Advances in the understanding of optical skyrmions, within a unified topological framework, are reviewed. The field structure of such optical quasiparticles, and their topological characteristics, may be useful for fields ranging from imaging to quantum technologies.

In two-dimensional honeycomb ferromagnets, bosonic magnon quasiparticles (spin waves) may either behave as massless Dirac fermions or form topologically protected edge states. The key ingredient defining their nature is the next-nearest-neighbor Dzyaloshinskii-Moriya interaction that breaks the inversion symmetry of the lattice and discriminates chirality of the associated spin-wave excitations. Using inelastic neutron scattering, we find that spin waves of the insulating honeycomb ferromagnetCrI3(TC=61K) have two distinctive bands of ferromagnetic excitations separated by a∼4meVgap at the Dirac points. These results can only be understood by considering a Heisenberg Hamiltonian with Dzyaloshinskii-Moriya interaction, thus providing experimental evidence that spin waves inCrI3can have robust topological properties potentially useful for dissipationless spintronic applications.

The transition-metal-based kagome metals provide a versatile platform for correlated topological phases hosting various electronic instabilities. While superconductivity is rare in layered kagome compounds, its interplay with nontrivial topology could offer an engaging space to realize exotic excitations of quasiparticles. Here, we use scanning tunneling microscopy to study a newly discoveredZ2topological kagome metalCsV3Sb5with a superconducting ground state. We observe charge modulation associated with the opening of an energy gap near the Fermi level. When across single-unit-cell surface step edges, the intensity of this charge modulation exhibits aπ-phase shift, suggesting a three-dimensional2×2×2charge density wave ordering. Interestingly, while conventional Caroli–de Gennes–Matricon bound states are observed inside the superconducting vortex on the Sb surfaces, a robust zero-bias conductance peak emerges that does not split in a large distance when moving away from the vortex center on the Cs2×2surfaces, resembling the Majorana bound states arising from the superconducting Dirac surface states inBi2Te3/NbSe2heterostructures. Our findings establishCsV3Sb5as a promising candidate for realizing exotic excitations at the confluence of nontrivial lattice geometry, topology and multiple electronic orders.

Anyons are quasiparticles that, unlike fermions and bosons, show fractional statistics when two of them are exchanged. Here, we report the experimental observation of anyonic braiding statistics for the ν = 1/3 fractional quantum Hall state by using an electronic Fabry–Perot interferometer. Strong Aharonov–Bohm interference of the edge mode is punctuated by discrete phase slips that indicate an anyonic phase θ anyon = 2π/3. Our results are consistent with a recent theory that describes an interferometer operated in a regime in which device charging energy is small compared to the energy of formation of charged quasiparticles, which indicates that we have observed anyonic braiding. An interferometer device is used to detect the quantum-mechanical phase that is gained when two anyons are braided around each other. The fractional value of the phase proves that these quasiparticles are neither bosons nor fermions.

Majorana zero modes (MZMs) are spatially localized, zero-energy fractional quasiparticles with non-Abelian braiding statistics that hold promise for topological quantum computing. Owing to the particle-antiparticle equivalence, MZMs exhibit quantized conductance at low temperature. By using variable-tunnel–coupled scanning tunneling spectroscopy, we studied tunneling conductance of vortex bound states on FeTe0.55Se0.45 superconductors. We report observations of conductance plateaus as a function of tunnel coupling for zero-energy vortex bound states with values close to or even reaching the 2e²/h quantum conductance (where e is the electron charge and h is Planck’s constant). By contrast, no plateaus were observed on either finite energy vortex bound states or in the continuum of electronic states outside the superconducting gap. This behavior of the zero-mode conductance supports the existence of MZMs in FeTe0.55Se0.45.

In a joint experimental and theoretical effort, we report on the formation of a macrodroplet state in an ultracold bosonic gas of erbium atoms with strong dipolar interactions. By precise tuning of the s -wave scattering length below the so-called dipolar length, we observe a smooth crossover of the ground state from a dilute Bose-Einstein condensate to a dense macrodroplet state of more than 2×104atoms . Based on the study of collective excitations and loss features, we prove that quantum fluctuations stabilize the ultracold gas far beyond the instability threshold imposed by mean-field interactions. Finally, we perform expansion measurements, showing that although self-bound solutions are prevented by losses, the interplay between quantum stabilization and losses results in a minimal time-of-flight expansion velocity at a finite scattering length.