Geometric Origin Research Articles

A cluster of results on amplituhedron tiles

The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in N=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}=4$$\\end{document} super Yang–Mills theory. It generalizes cyclic polytopes and the positive Grassmannian and has a very rich combinatorics with connections to cluster algebras. In this article, we provide a series of results about tiles and tilings of the m=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=4$$\\end{document} amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for Gr4,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{ Gr}_{4,n}$$\\end{document}. Secondly, we exhibit a tiling of the m=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=4$$\\end{document} amplituhedron which involves a tile which does not come from the BCFW recurrence—the spurion tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for Gr4,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{ Gr}_{4,n}$$\\end{document}. This paper is a companion to our previous paper “Cluster algebras and tilings for the m=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=4$$\\end{document} amplituhedron.”

Geometrical origin of the Kodama vector

Geometrical origin of the Kodama vector

The cosmological switchback effect. Part II

Recent developments in static patch holography proposed that quantum gravity in de Sitter space admits a dual description in terms of a quantum mechanical theory living on a timelike surface near the cosmological horizon. In parallel, geometric observables associated with the Einstein-Rosen bridge of a black hole background were suggested to compute the computational complexity of the state dual to a gravitational theory. In this work, we pursue the study of the complexity=volume and complexity=action conjectures in a Schwarzschild-de Sitter geometry perturbed by the insertion of a shockwave at finite boundary times. This analysis extends previous studies that focused either on the complexity=volume 2.0 conjecture, or on the case of a shockwave inserted along the cosmological horizon. We show that the switchback effect, describing the delay in the evolution of complexity in reaction to a perturbation, is a universal feature of the complexity proposals in asymptotically de Sitter space. The geometric origin of this phenomenon is related to the causal connection between the static patches of de Sitter space when a positive pulse of null energy is inserted in the geometry.

T-duality and flavor symmetries in Little String Theories

We explore the T-duality web of 6D Heterotic Little String Theories, focusing on flavor algebra reducing deformations. A careful analysis of the full flavor algebra, including Abelian factors, shows that the flavor rank is preserved under T-duality. This suggests a new T-duality invariant in addition to the Coulomb branch dimension and the two-group structure constants. We also engineer Little String Theories with non-simply laced flavor algebras, whose appearance we attribute to certain discrete 3-form fluxes in M-theory. Geometrically, these theories are engineered in F-theory with non-Kähler favorable K3 fibers. This geometric origin leads us to propose that freezing fluxes are preserved across T-duality. Along the way, we discuss various exotic models, including two inequivalent Spin(32)/ℤ2 models that are dual to the same E8 × E8 theory, and a family of self-T-dual models.

Semi-symmetric metric gravity: From the Friedmann–Schouten geometry with torsion to dynamical dark energy models

In this paper, we develop a geometric generalization of General Relativity based on the semi-symmetric metric connection introduced by Friedmann and Schouten in 1924. Although the mathematical properties of this connection, which allows for torsion, have been extensively studied, its physical implications remain underexplored. We provide a detailed exposition of the differential geometric aspects of semi-symmetric connections and formulate the corresponding field equations induced by the specific form of torsion we are investigating. We consider the cosmological applications of the theory by deriving the generalized Friedmann equations in a flat, homogeneous and isotropic geometry. The Friedmann equations also include some supplementary terms as compared to their general relativistic counterparts, which can be interpreted as a geometric type dark energy. To evaluate the proposed theory, we consider three cosmological models — the first with constant effective density and pressure, the second with the dark energy satisfying a linear equation of state, and a third one with a polytropic equation of state. We also compare the predictions of the semi-symmetric metric gravitational theory with the observational data for the Hubble function, and with the predictions of the ΛCDM model. Our findings indicate that the semi-symmetric metric cosmological models give a good description of the observational data, and for certain values of the model parameters, they can reproduce almost exactly the predictions of the ΛCDM paradigm. Consequently, Friedmann’s initially proposed geometry emerges as a credible alternative to standard general relativity, in which dark energy has a purely geometric origin.

Angular Dependence of Hump‐Shape Hall Effects for Distinguishing between Karplus–Luttinger and Geometrical Origins

AbstractAmong the vast magnetic heterostructures explored in Condensed Matter Physics, two contrasting interpretations of the hump‐shaped Hall Effects remain ambiguous and debated, namely, the overlap of two opposite‐signed Karplus–Luttinger Hall loops associated with inhomogeneous collinear domains with perpendicular anisotropy, or the Geometrical/Topological Hall Effect emanated from hexagonal close‐packed lattice of Skyrmion ground state with smoothly varying non‐collinear moments. Their similarity in topology implies difficulty in discrimination via magnetic imaging. Here, this ambiguity is overcome and clarified by the divergence exponent of hump peak fields extracted from Hall measurements with magnetic field rotation on several heterostructures. Their difference in sensitivity to in‐plane fields reveals that the former mechanism involves higher uniaxial anisotropy than the latter, departing from the Skyrmion ground state regime by the Ginzburg–Landau framework of triple‐q spin‐wave superposition. Numerous material systems can be summarized into a single curve of divergence exponent versus the collinear quality factor, bridging the crossover of the two mentioned mechanisms.

Unraveling the Geometry-Driven C═C Epoxidation and C-H Hydroxylation Reactivity of Tetra-Coordinated Nonheme Iron(IV)-Oxo Complexes.

The electronic structure and reactivity of tetra-coordinated nonheme iron(IV)-oxo complexes have remained unexplored for years. The recent synthesis of a closed-shell iron(IV)-oxo complex [(quinisox)FeIV(O)]+ (1) has set up a platform to understand how such complexes compare with the celebrated open-shell iron-oxo chemistry. Herein, using density functional theory and ab initio calculations, we present an in-depth electronic structure investigation of the C═C epoxidation [oxygen atom transfer (OAT)] and C-H hydroxylation [hydrogen atom transfer (HAT)] reactivity of 1. Using a solvent-coordinated geometry of 1 (1') and other potential tetra-coordinated iron(IV)-oxo complexes bearing rigid ligands (2 and 3), we established the geometric origin of spin-state energetics and reactivity of 1. Complex 1 featuring a strong Fe-O bond exhibits OAT and HAT reactivity in its quintet state. The lowest quintet OAT pathway has a lower barrier by ∼4 kcal/mol than the quintet HAT pathway, corroborating the experimentally observed gas-phase OAT reactivity preference. A conventional HAT reactivity preference for 2 and a comparable OAT and HAT reactivity for 3 are observed. This further supports the geometry-driven reactivity preference for 1. Noncovalent interaction analyses reveal a pronounced π-π interaction between the substrate and ligand in the OAT transition state, rationalizing the origin of the observed reactivity preference for 1.

Inner walls or vortices? Crescent-shaped asymmetries in ALMA observations of protoplanetary discs

ABSTRACT Crescent-shaped asymmetries are common in millimetre observations of protoplanetary discs and are usually attributed to vortices or dust overdensities. However, they often appear on a single side of the major axis and roughly symmetric about the minor axis, suggesting a geometric origin. In this work, we interpret such asymmetries as emission from the exposed inner cavity walls of inclined discs and use them to characterize their vertical extent. Here we focus on the discs around CIDA 9 and RY Tau, first modelling their observations in visibility space with a simple geometric prescription for the walls, and then exploring more detailed radiative transfer models. Accounting for the wall emission yields significantly better residuals than purely axisymmetric models, and we estimate the dust scale height of these systems to be 0.4 au at 37 au for CIDA 9 and 0.2 au at 12 au for RY Tau. Finally, we identify crescent-shaped asymmetries in twelve discs, nine of which have constraints on their orientation – in all cases, the asymmetry appears on the far-side of the disc, lending support to the hypothesis that they are due to their inner rims. Modelling this effect in larger samples of discs will help to build a statistical view of their vertical structure.

Dynamical chiral Nernst effect in twisted Van der Waals few layers

The Nernst effect is a fundamental thermoelectric conversion phenomenon that was deemed to be possible only in systems with magnetic field or magnetization. In this work, we propose a novel dynamical chiral Nernst effect that can appear in two-dimensional van der Waals materials with chiral structural symmetry in the absence of any magnetic degree of freedom. This unconventional effect is triggered by time variation of an out-of-plane electric field, and has an intrinsic quantum geometric origin linked to not only the intralayer center-of-mass motion but also the interlayer coherence of electronic states. We demonstrate the effect in twisted homobilayer and homotrilayer transition metal dichalcogenides, where the strong twisted interlayer coupling leads to sizable intrinsic Nernst conductivities well within the experimental capacity. This work suggests a new route for electric control of thermoelectric conversion.

Large geometric polarization and magnetic behavior in the multiferroic quasi-2D SrNiF4 fluoride

In the last decades, multifunctional single-crystals that show multiferroic and magnetoelectric responses have attracted considerable attention due to the potential applications and physical phenomena involved in the entanglement of the ferroic orders. This paper explores the structural, ferroelectric, and magnetic properties of the unexplored layered SrNiF4 fluoride compound. We show that, in terms of the vibrational phonon modes, this fluoride compound shows a tangible ferroelectric behavior with an Ps = 14.8 μC cm−2 standing as the largest among its family. Besides, based on our findings such ferroelectric polarization presents a geometric origin that is strongly entangled with the octahedral rotations within the Γ2− phonon mode and enhanced by the structure’s Γ1+ mode. We also observed that the spontaneous polarization is coupled to the noncollinear antiferromagnetic ordering in the structure allowing a switching of the weak antiferromagnetic component when the polarization is reversed.

Spin flop transitions and complex magnetic behaviour in CoMoO4 powders and [formula omitted]-CoMoO4 thin films

Magnetic measurements were made on α-CoMoO4 powder and β-CoMoO4 thin film samples. α-CoMoO4 powder showed antiferromagnetic ordering below 13 K and two sharp spin flop transitions with complete magnetic saturation after the second transition (>5 T). A magnetic phase diagram for α-CoMoO4 was constructed, revealing three low-temperature magnetic phases: an antiferromagnetic phase at low field, an intermediate spin-flop phase, and a paramagnetic phase at high field. Two well-defined magnetisation plateaus were observed either side of the spin-flop phase: the first occurred with a moment per Co ion of 1/3 of spin saturation (where Co moments are completely aligned to the applied magnetic field), whereas the second occurred at a moment per Co ion matching calculations for spin saturation (3 μB per Co). No geometric origin for this factor of 1/3 could be found. Hysteresis was observed around the first spin-flop transition, indicating the presence of spin canting or magnetic domains under an applied field. The first successful fabrication of CoMoO4 thin films was achieved via RF magnetron sputtering, yielding β phase films which remain in the β phase upon cooling. The β-CoMoO4 thin films demonstrated antiferromagnetic ordering below 10 K. While two spin-flop transitions are seen in the β phase films (with a hysteresis around the first), spin saturation is not reached in fields up to 6 T, suggesting the existence of a third spin reorientation at higher field. A magnetic phase diagram was also constructed for the β-CoMoO4 film, which demonstrated the same three low-temperature magnetic phases, though with lower critical fields and transition temperatures. Comparisons with isostructural Ni and Mn molybdates showed that the key differences in the α and β phases arise from the different coordination of the Mo tetramers, indicating differences in magnetic properties in both phases arise from variations in the inter-tetramer exchange coupling.

The quantum geometric origin of capacitance in insulators

In band insulators, without a Fermi surface, adiabatic transport can exist due to the geometry of the ground state wavefunction. Here we show that for systems driven at a small but finite frequency ω, transport likewise depends sensitively on quantum geometry. We make this statement precise by expressing the Kubo formula for conductivity as the variation of the time-dependent polarization with respect to the applied field. We find that at linear order in frequency, the longitudinal conductivity results from an intrinsic capacitance determined by the ratio of the quantum metric and the spectral gap, establishing a fundamental link between the dielectric response and the quantum metric of insulators. We demonstrate that quantum geometry is responsible for the electronic contribution to the dielectric constant in a wide range of insulators, including the free electron gas in a quantizing magnetic field, for which we show the capacitance is quantized. We also study gapped bands of hBN-aligned twisted bilayer graphene and obstructed atomic insulators such as diamond. In the latter, we find its abnormally large refractive index to have a topological origin.

Lasing in Non-Hermitian Flat Bands: Quantum Geometry, Coherence, and the Fate of Kardar-Parisi-Zhang Physics.

We show that lasing in flat-band lattices can be stabilized by means of the geometrical properties of the Bloch states, in settings where the single-particle dispersion is flat in both its real and imaginary parts. We illustrate a general projection method and compute the collective excitations, which display a diffusive behavior ruled by quantum geometry through a peculiar coefficient involving gain, losses and interactions, and entailing resilience against modulational instabilities. Then, we derive an equation of motion for the phase dynamics and identify a Kardar-Parisi-Zhang term of geometric origin. This term is shown to exactly cancel whenever the real and imaginary parts of the laser nonlinearity are proportional to each other, or when the uniform-pairing condition is satisfied. We confirm our results through numerical studies of the π-flux diamond chain. This Letter highlights the key role of Bloch geometric effects in nonlinear dissipative systems and KPZ physics, with direct implications for the design of laser arrays with enhanced coherence.

Non-metricity with boundary terms: 𝖿(𝖰,𝖢) gravity and cosmology

We formulate f(Q,C) gravity and cosmology. Such a construction is based on the symmetric teleparallel geometry, but apart form the non-metricity scalar Q we incorporate in the Lagrangian the boundary term C of its difference from the standard Levi-Civita Ricci scalar R̊. We extract the general metric and affine connection field equations, we apply them at a cosmological framework, and adopting three different types of symmetric teleparallel affine connections we obtain the modified Friedmann equations. As we show, we acquire an effective dark-energy sector of geometrical origin, which can lead to interesting cosmological phenomenology. Additionally, we may obtain an effective interaction between matter and dark energy. Finally, examining a specific model, we show that we can obtain the usual thermal history of the universe, with the sequence of matter and dark-energy epochs, while the effective dark-energy equation-of-state parameter can be quintessence-like, phantom-like, or cross the phantom-divide during evolution.

Cosmological consequences of Brans–Dicke theory in 4D from 5D scalar-vacuum

We study 5D Brans–Dicke theory in the framework of (gravitational) baryogenesis and primordial light element formation. Such a model is able to explain the present cosmic accelerated expansion without recurring to matter fields in 5D or dark energy in 4D. In fact, the 5D to 4D reduction arises a space-matter tensor of geometrical origin that plays the role of a new ingredient on-shell. For an isotropic and homogeneous Universe, the cosmological equations admit a power-law solution of the scale factor, a(t)∼tα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a(t)\\sim t^\\alpha$$\\end{document}, we constrain the exponential factor α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} by using the present bounds on the matter–antimatter asymmetry in the Universe and Big Bang Nucleosynthesis. A possible connection with dark matter relic abundance is also discussed.

Geometric Origin of Non-Bloch PT Symmetry Breaking.

The parity-time (PT) symmetry of a non-Hermitian Hamiltonian leads to real (complex) energy spectrum when the non-Hermiticity is below (above) a threshold. Recently, it has been demonstrated that the non-Hermitian skin effect generates a new type of PT symmetry, dubbed the non-Bloch PT symmetry, featuring unique properties such as high sensitivity to the boundary condition. Despite its relevance to a wide range of non-Hermitian lattice systems, a general theory is still lacking for this generic phenomenon even in one spatial dimension. Here, we uncover the geometric mechanism of non-Bloch PT symmetry and its breaking. We find that non-Bloch PT symmetry breaking occurs by the formation of cusps in the generalized Brillouin zone (GBZ). Based on this geometric understanding, we propose an exact formula that efficiently determines the breaking threshold. Moreover, we predict a new type of spectral singularities associated with the symmetry breaking, dubbed non-Bloch van Hove singularity, whose physical mechanism fundamentally differs from their Hermitian counterparts. This singularity is experimentally observable in linear responses.

Berry-Curvature Engineering for Nonreciprocal Directional Dichroism in Two-Dimensional Antiferromagnets.

In two-dimensional antiferromagnets, we find that the mixed Berry curvature can be attributed as the geometrical origin of the nonreciprocal directional dichroism (NDD), which refers to the difference in light absorption between opposite propagation directions. This Berry curvature is closely related to the uniaxial strain in accordance with the symmetry constraint, leading to a highly tunable NDD, whose sign and strength can be tuned via strain direction. We choose the lattice model of MnBi_{2}Te_{4} as a concrete example. The coupling between mixed Berry curvature and strain also suggests the magnetic quadrupole of the Bloch wave packet as the macroscopic order parameter probed by the NDD in two dimensions, which is distinct from the multiferroic order P×M or the spin toroidal and quadrupole order within a unit cell in previous studies. Our work paves the way for the Berry-curvature engineering for optical nonreciprocity in two-dimensional antiferromagnets.

Evidence for a dynamic corona in the short-term time lags of black hole X-ray binary MAXI J1820+070

ABSTRACT In X-ray observations of hard state black hole X-ray binaries (BHXRBs), rapid variations in accretion disc and coronal power-law emission are correlated and show Fourier-frequency-dependent time lags. On short ($\sim$0.1 s) time-scales, these lags are thought to be due to reverberation and therefore may depend strongly on the geometry of the corona. Low-frequency quasi-periodic oscillations (QPOs) are variations in X-ray flux that have been suggested to arise because of geometric changes in the corona, possibly due to general relativistic Lense–Thirring precession. Therefore, one might expect the short-term time lags to vary on the QPO time-scale. We performed novel spectral-timing analyses on Neutron Star Interior Composition ExploreR observations of the BHXRB MAXI J1820+070 during the hard state of its outburst in 2018 to investigate how the short-term time lags between a disc-dominated and a coronal power-law-dominated energy band vary on different time-scales. Our method can distinguish between variability due to the QPO and broad-band noise, and we find a linear correlation between the power-law flux and lag amplitude that is strongest at the QPO frequency. We also introduce a new method to resolve the QPO signal and determine the QPO phase dependence of the flux and lag variations, finding that both are very similar. Our results are consistent with a geometric origin of QPOs, but also provide evidence for a dynamic corona with a geometry varying in a similar way over a broad range of time-scales, not just the QPO time-scale.

Conformal gravitational theories in Barthel–Kropina-type Finslerian geometry, and their cosmological implications

We consider dark energy models obtained from the general conformal transformation of the Kropina metric, representing an (alpha , beta )-type Finslerian geometry, constructed as the ratio of the square of a Riemannian metric alpha and the one-form beta . Conformal symmetries appear in many fields of physics, and they may play a fundamental role in our understanding of the Universe. We investigate the possibility of obtaining conformal theories of gravity in the osculating Barthel–Kropina geometric framework, where gravitation is described by an extended Finslerian-type model, with the metric tensor depending on both the base space coordinates and a vector field. We show that it is possible to formulate a family of conformal Barthel–Kropina theories in an osculating geometry with second-order field equations, depending on the properties of the conformal factor, whose presence leads to the appearance of an effective scalar field of geometric origin in the gravitational field equations. The cosmological implications of the theory are investigated in detail by assuming a specific relation between the component of the one-form of the Kropina metric and the conformal factor. The cosmological evolution is thus determined by the initial conditions of the scalar field and a free parameter of the model. We analyze in detail three cosmological models corresponding to different values of the theory parameters. Our results show that the conformal Barthel–Kropina model can provide an acceptable description of the observational data, and may represent a theoretically attractive alternative to the standard Lambda CDM cosmology.

Geometric local systems on very general curves and isomonodromy

We show that the minimum rank of a non-isotrivial local system of geometric origin on a suitably general n n -pointed curve of genus g g is at least 2 g + 1 2\sqrt {g+1} . We apply this result to resolve conjectures of Esnault-Kerz and Budur-Wang. The main input is an analysis of stability properties of flat vector bundles under isomonodromic deformations, which additionally answers questions of Biswas, Heu, and Hurtubise.