We present a first-principles investigation of the spin-ice state in Dy_2Ti_2O_7 using a magnetic source-free exchange and correlation functional, implemented in the Castep electronic-structure code. By comparing results from the conventional local spin-density approximation, we show that a spin-ice state in Dy_2Ti_2O_7 can be reliably obtained by removing the magnetic sources from the exchange and correlation contributions to the potential, and we contrast this against the computed ground states of other frustrated pyrochlore magnets.

The kagome lattice of spin-1/2 copper atoms in herbertsmithite is conjectured to sustain a quantum spin liquid state with spinon quasiparticles. Ideally, the kagome crystal planes are each separated by a plane of spinless zinc atoms. However, in real crystals, some spin-1/2 copper atoms substitute randomly onto these inter-kagome zinc sites. Here we reconceptualize such 'impurity' atoms as quantum witness spins whose dynamics is designed to probe the spin liquid state. We then introduce spin noise spectroscopy to measure the frequency and temperature dependence of witness spin dynamics, demonstrating that their phenomenology is consistent with extensive interactions between witness spins mediated by propagation of spinons through a quantum spin liquid. Ultimately, a sharp transition occurs at around 260 mK, below which the properties of both spin noise and magnetic susceptibility suggest that the witness spins form a spin glass phase. Among the theoretical models considered, we demonstrate that our observations are only consistent with spinon-mediated interactions between witness spins by either a Z2 or U(1) quantum spin liquid, with the former model more closely matching the data. Our work demonstrates that quantum mechanical witness spins may now conceivably be used as a widely applicable probe of quantum spin liquid physics.

Dynamical heterogeneity, in which transitory local fluctuations occur in the conformation and dynamics of constituent particles, is widely hypothesized to be essential to the evolution of supercooled liquids into the structural glass state. Yet its microscopic spatiotemporal phenomenology is challenging to detect directly in molecular glass forming liquids. Because recent theoretical advances predict that corresponding dynamical heterogeneity could occur in supercooled magnetic monopole fluids (Proc. Nat. Acad. Sci. 112, 8549 (2015)), we searched for such phenomena in Dy2Ti2O7. By measuring its microsecond-resolved spontaneous magnetization fluctuations [Formula: see text] we detected a sharp bifurcation in monopole noise characteristics below [Formula: see text], with the appearance of powerful spontaneous monopole current bursts. This intense dynamics emerges upon entering the supercooled monopole fluid regime, reaches maximum strength near [Formula: see text] and then collapses along with coincident loss of ergodicity approaching [Formula: see text]. Moreover, when the four-point dynamical susceptibility [Formula: see text] is determined directly from temperature dependence of correlations in [Formula: see text], it evolves as predicted when dynamical heterogeneity is present, revealing its simultaneously and rapidly escalating length and time scales, [Formula: see text] and [Formula: see text]. This overall phenomenology greatly expands our empirical knowledge of supercooled monopole fluids and, more generally, demonstrates techniques for detection of the time sequence, magnitude, statistics, and correlations of dynamical heterogeneity, access to which may greatly accelerate fundamental vitrification studies.

The development of Kondo lattice coherence in UTe2 leads to the formation of a heavy Fermi liquid state from which superconductivity emerges at lower temperature. In Kondo lattice systems, the nuclear magnetic resonance (NMR) and muon Knight shift have proven to be particularly sensitive to the properties of the developing heavy-electron fluid. Here we report muon Knight shift measurements on high-quality UTe2 single crystals grown by a molten salt-flux method. Together with previous data from a single crystal grown by a chemical-vapor transport method, our results show the contribution of the heavy-electron liquid to the muon Knight shift increases below a crossover temperature T* ~ 30 K in accord with a universal scaling function of T/T* for heavy-fermion materials. An observed departure from this universal scaling below a temperature T ~ 12 K at certain muon stopping sites signifies a reversal of the Kondo hybridization and a relocalization of U 5f moments with an antiferromagnetic coupling. The preservation of universal scaling at a different muon site demonstrates a coexistence of itinerant and localized 5f electron states preceding the superconducting phase transition.

Abstract In Chapter 50 we show the inevitability of a Big Bang from our geometrical viewpoint and its reliance on smooth spacetimes.

Abstract Having formulated a theory of gravity, we are now going to use it on the largest conceivable problem: the origin and fate of our Universe. In Chapter 15, we introduce the ideas behind the field of cosmology and in particular the cosmological principle, which says that the Universe is spatially homogeneous and isotropic. We also investigate two model universes (i) an empty universe (Universe 0) has flat spacetime described by a Minkowski metric field; (ii) the de Sitter model (Universe 1) is spatially flat, but has a spacetime with a constant curvature.

Abstract In Chapter 16 we investigate the possible geometries describing homogeneous, isotopic spacetimes with constant curvature. These Robertson-Walker spacetimes are all isotropic and homogeneous and are classified as closed (k = 1), flat (k = 0) and open (k=−1). They evolve as a function of time and we examine the effect they have on the measured redshift.

Abstract The ideas in the previous chapter are developed in Chapter 43 and we use differential geometry to understand the conservation of charge.

An emerging concept for identification of different types of spin liquids [C. Broholm et al. , Science 367 , eaay0668 (2020)] is through the use of spontaneous spin noise [S. Chatterjee, J. F. Rodriguez-Nieva, E. Demler, Phys. Rev. B 99 , 104425 (2019)]. Here, we develop spin noise spectroscopy for spin liquid studies by considering Ca 10 Cr 7 O 28 , a material hypothesized to be either a quantum or a spiral spin liquid (SSL). By enhancing techniques introduced for magnetic monopole noise studies [R. Dusad et al. , Nature 571 , 234–239 (2019)], we measure the time and temperature dependence of spontaneous flux Φ ( t , T ) and thus magnetization M ( t , T ) of Ca 10 Cr 7 O 28 samples. The resulting power spectral density of magnetization noise S M ω , T reveals intense spin fluctuations with S M ω , T ∝ ω - α ( T ) and 0.84 < α T < 1.04 . Both the variance σ M 2 T and the correlation function C M t , T of this spin noise undergo crossovers at a temperature T ∗ ≈ 450 mK . While predictions for quantum spin liquids are inconsistent with this phenomenology, those from Monte–Carlo simulations of a two-dimensional (2D) SSL state in Ca 10 Cr 7 O 28 yield overall quantitative correspondence with the measured frequency and temperature dependences of S M ω , T , C M t , T , and σ M 2 T , thus indicating that Ca 10 Cr 7 O 28 is an SSL.

Abstract We apply all of the ideas developed in the last few chapters to a number of different model universes. The Robertson-Walker spaces filled with a perfect fluid provide a range of models, many of which start with a Big Bang. The Einstein Universe is static, but unstable; the Einstein-de Sitter model expands without bound and the Lemaître model has a coasting period. The standard models of cosmology have zero cosmological constant Λ, and all begin with a Big-Bang singularity.

Centrosymmetric GdRu2Si2 exhibits a variety of multi-Q magnetic states as a function of temperature and applied magnetic field, including a square skyrmion-lattice phase. The material's behavior is strongly dependent on the direction of the applied field, with different phase diagrams resulting for fields applied parallel or perpendicular to the crystallographic c axis. Here, we present the results of muon-spin relaxation (μ+SR) measurements on single crystals of GdRu2Si2. Our analysis is based on the computation of muon stopping sites and consideration of quantum zero-point motion effects of muons, allowing direct comparison with the underlying spin textures in the material. The muon site is confirmed experimentally, using angle-dependent measurements of the muon Knight shift. Using transverse-field μ+SR with fields applied along either the [001] or [100] crystallographic directions, we distinguish between the magnetic phases in this system via their distinct muon response, providing additional evidence for the skyrmion and meron-lattice phases, while also suggesting the existence of RKKY-driven muon hyperfine coupling. Zero-field μ+SR provides clear evidence for a transition between two distinct magnetically ordered phases at 39 K. Published by the American Physical Society 2025

Abstract We describe a set of coordinates which are useful for studying black hole singularities. These Kruskal-Szekeres coordinates successfully remove the singularity at the Schwarzschild radius. They suggest that the coordinates can be extended to cover, apparently unphysical, regions of spacetime. Kruskal coordinates and their link to Schwarzschild coordinates can be understood by analogy to the relation between Minkowski coordinates and accelerating Minkowski coordinates. The conformal structure of the extended system reveals how the singularity can be thought of as living at the edge of spacetime.

Ruddlesden-Popper oxide phases in the LaxSr2-xCo0.5Ir0.5O4 (0 < x < 1) solid solution can be converted to the corresponding LaxSr2-xCo0.5Ir0.5O4-yHy oxyhydride phases, by topochemical reaction with LiH, in which the hydride ions are substituted exclusively onto the equatorial anion sites of the host framework. Analysis reveals that oxyhydride phases in the range 0.5 < x < 1 adopt LaxSr2-xCo0.5Ir0.5O2+xH2-x compositions which maintain a constant Co1+, Ir3+ oxidation-state combination (confirmed by Co K-edge XANES data), with the presence of low-spin d6 Ir3+ being consistent with the covalent stabilization of the metastable oxyhydride phases via strong Ir-H σ-bonds. Phases at the lanthanum-poor end of the solid solution (x < 0.5) adopt LaxSr2-xCo0.5Ir0.5O4-yHy compositions with lower hydride concentrations (y < 1.5). Magnetisation and μSR data indicate that all the LaxSr2-xCo0.5Ir0.5O4-yHy oxyhydride phases exhibit strong magnetic frustration, attributed to the large-scale cation and anion disorder, and resulting in glassy magnetic behaviour at low temperature.

Muon spectroscopy has become a leading tool for the investigation of local magnetic fields in condensed matter physics, finding applications in the study of superconductivity, magnetism, ionic diffusion in battery materials, and numerous other fields. Though the muon yields quantitative information about the material, this can only be fully interpreted if the nature of the muon site and its stability is fully understood. Electronic structure calculations are of paramount importance for providing this understanding, particularly through a group of techniques that has become known as DFT + μ , density functional theory including the presence of the implanted muon. We describe how these electronic structure calculations can be used to underpin muon spectroscopy, and some examples of the science that follows from this, as well as some of the available software tools that are currently being developed.

Abstract We discuss various theories that have been developed to unify general relativity and quantum mechanics in order to produce a theory of quantum gravity. These include string theory and loop quantum gravity.

Abstract In Chapter 14, we review some of the successes of general relativity that follow from the formalism described in this part of the book. In particular: Many of the topics described in this chapter will then be unpacked in more detail in the rest of the book.

Abstract In the previous chapter we have investigated the possible geometries describing homogeneous, isotopic spacetimes with constant curvature and in Chapter 17 we fill these with matter. Filling the Robertson-Walker universes with a perfect fluid gives us the Friedmann equations for the evolution of the scale factor a(t) with time. These can be given in terms of the dust, radiation and vacuum-energy content of the universe.

YbZn_{2}GaO_{5} is a promising candidate for realizing a quantum spin liquid (QSL) state, particularly owing to its lack of significant site disorder. Pulsed-field magnetometry at 0.5 K shows magnetization saturating near 15 T, with a corrected saturation moment of 2.1(1)μ_{B} after subtracting the van Vleck contribution. Our zero-field μSR measurements down to milliKelvin temperatures provide evidence for a dynamic ground state and the absence of magnetic order. To probe fluctuations in the local magnetic field at the muon site, we performed longitudinal field μSR experiments. These results provide evidence for spin dynamics with a field dependence that is consistent with a U1A01 Dirac quantum spin liquid as a plausible description of the ground state.