arXiv subjects

Kaiyuan Gu

Publications and source records attributed to Kaiyuan Gu.

7 recordsLinked to original sources

arXiv2026

Uniformity in rational torsion and small points on abelian varieties

In this paper, we propose a method to study the {\it Uniform Boundedness Conjecture} and the {\it Lang-Silverman Conjecture} for abelian varieties $A$ defined over a global field $K$; the latter is a uniform lower bound on the heights of non-torsion rational points. Our method is inspired by Vojta's proof of the Mordell Conjecture (Faltings's Theorem). Over function fields of characteristic $0$, a recent breakthrough of Looper-Yap (arXiv:2603.23396) proves both conjectures with inexplicit bounds. In our paper, we give a new proof of both conjectures with explicit bounds, which also depend polynomially on the field $K$ unless $A/K$ admits a factor of good reduction everywhere. We also prove explicit bounds for elliptic curves over function fields of characteristic $p>0$. Over number fields, we prove both conjectures under a suitable high-dimensional Szpiro conjecture (weaker than Hindry's version, Conjecture 3.4 of https://webusers.imj-prg.fr/~marc.hindry/MW-size.pdf) that we propose.

math.NT↗

arXiv2025

Ferrofluids under oscillatory magnetic fields

Ferrofluids exhibit two canonical interfacial instabilities, a static Rosensweig (normal-field) instability that produces a lattice of peaks and a dynamical Faraday instability that produces parametrically excited standing waves. Here we present a systematic phase diagram of ferrofluid surface states driven by a purely AC vertical magnetic field with zero mean. Scanning a broad range of frequencies and field amplitudes, we resolve two robust branches: a Faraday-wave regime that includes a stable square lattice and a Rosensweig-like peak--valley regime indistinguishable in morphology from Rosensweig peaks. The Faraday-onset boundary is well described by a power law close to $\sqrt{f}$, while the Rosensweig-like peak onset becomes essentially frequency independent at low viscosity. The wave vector of the square lattice grows linearly with frequency over our accessible band. We present a surface-wave theory that captures the full phenomenology, including the emergence of Rosensweig peaks under zero-mean AC driving, the near-$\sqrt{f}$ scaling of the phase boundaries, the linear growth of the selected wave vector with frequency, and the preference for square over hexagonal lattices.

cond-mat.soft↗

arXiv2024

On Special Subvarieties of the Universal Semi-abelian Scheme and Pink Conjectures

The universal Poincar\'e torsor, or more generally the universal semi-abelian scheme, can be viewed as a mixed Shimura variety. We give a classification of special subvarieties of the universal semi-abelian scheme of arbitrary toric rank. Given this classification, we show that the Zilber-Pink conjecture for mixed Shimura varieties implies the Zilber-Pink conjecture for semi-abelian varieties, correcting an error in an unpublished manuscript of Pink. Moreover, we give a more reasonable reformulation of the Relative Manin-Mumford Conjecture for semi-abelian schemes.

math.AG↗

arXiv2024

Inevitable First Order Phase Transitions in 3D Quantum Hall Systems

Recent experiments suggest that low carrier density three-dimensional (3D) metals ZrTe$_5$ and HfTe$_5$ exhibit the 3D quantum Hall (QH) effect with Hall resistivity plateaus and a metal-insulator transition in strong magnetic fields. The conventional 3D QH theory requires a fixed period charge density wave (CDW), which is however not observed experimentally. We investigate alternative non-CDW mechanisms by considering a 3D metal in strong magnetic fields with electrons coupled to a boson (e.g., phonon) field. We show that the model exhibits inevitable first order phase transitions at jumps of the number of occupied Landau level bands, which do not involve CDW. These transitions may drive the system into a phase separation state with percolation transitions. We further show this can lead to Hall resistivity quasi-plateaus similar to that observed experimentally, and can provide a natural explanation for the metal-insulator transition.

cond-mat.mes-hall↗

arXiv2023

Hermitian Bulk -- Non-Hermitian Boundary Correspondence

Non-Hermitian band theory distinguishes between line gaps and point gaps. While point gaps can give rise to intrinsic non-Hermitian band topology without Hermitian counterparts, line-gapped systems can always be adiabatically deformed to a Hermitian limit. Here we show that line-gap topology and point-gap topology can be intricately connected: topological line-gapped systems in $d$ dimensions induce nontrivial point-gap topology on their $(d-1)$-dimensional boundaries when suitable internal and spatial symmetries are present. Since line-gapped systems essentially realize Hermitian topological phases, this establishes a correspondence between Hermitian bulk topology and intrinsic non-Hermitian boundary topology. For the correspondence to hold, no non-Hermitian perturbations are required in the bulk itself, so that the bulk can be purely Hermitian. Concomitantly, the presence of non-Hermitian perturbations in the bulk does not affect any results as long as they do not close the bulk line gap. On the other hand, non-Hermitian perturbations are essential on the boundary to open a point gap. The non-Hermitian boundary topology then further leads to higher-order skin modes, as well as chiral and helical hinge modes, that are protected by point gaps and hence unique to non-Hermitian systems. We identify all the internal symmetry classes where bulk line-gap topology induces boundary point-gap topology as long as an additional spatial symmetry is present, and establish the correspondence between their topological invariants. There also exist some symmetry classes where the Hermitian edge states remain stable, in the sense that even a point gap cannot open on the boundary.

cond-mat.mes-hall↗

arXiv2022

Magic angles in twisted bilayer graphene near commensuration: Towards a hypermagic regime

The Bistritzer-MacDonald continuum model (BM model) describes the low-energy moiré bands for twisted bilayer graphene (TBG) at small twist angles. We derive a generalized continuum model for TBG near any commensurate twist angle, which is characterized by complex interlayer hoppings at commensurate $AA$ stackings (rather than the real hoppings in the BM model), a real interlayer hopping at commensurate $AB/BA$ stackings, and a global energy shift. The complex phases of the $AA$ stacking hoppings and the twist angle together define a single angle parameter $ϕ_0$. We compute the model parameters for the first six distinct commensurate TBG configurations, among which the $38.2^\circ$ configuration may be within experimentally observable energy scales. We identify the first magic angle for any $ϕ_0$ at a condition similar to that of the BM model. At this angle, the lowest two moiré bands at charge neutrality become flat except near the $\boldsymbolΓ_M$ point and retain fragile topology but lose particle-hole symmetry. We further identify a hypermagic parameter regime centered at $ϕ_0 = \pmπ/2$ where many moiré bands around charge neutrality (often $8$ or more) become flat simultaneously. Many of these flat bands resemble those in the kagome lattice and $p_x$, $p_y$ 2-orbital honeycomb lattice tight-binding models.

cond-mat.mes-hall↗

arXiv2020

Type-II Ising superconductivity and anomalous metallic state in macro-size ambient-stable ultrathin crystalline films

Recent emergence of two-dimensional (2D) crystalline superconductors has provided a promising platform to investigate novel quantum physics and potential applications. To reveal essential quantum phenomena therein, ultralow temperature transport investigation on high quality ultrathin superconducting films is critically required, although it has been quite challenging experimentally. Here we report a systematic transport study on the ultrathin crystalline PdTe2 films grown by molecular beam epitaxy (MBE). Interestingly, a new type of Ising superconductivity in 2D centrosymmetric materials is revealed by the detection of large in-plane critical field more than 7 times Pauli limit. Remarkably, in perpendicular magnetic field, we provide solid evidence of anomalous metallic state characterized by the resistance saturation at low temperatures with high quality filters. The robust superconductivity with intriguing quantum phenomena in the macro-size ambient-stable ultrathin PdTe2 films remains almost the same for 20 months, showing great potentials in electronic and spintronic applications.

cond-mat.supr-con↗