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Raúl Sánchez Galán

Publications and source records attributed to Raúl Sánchez Galán.

9 recordsLinked to original sources
arXiv2026

Exact admissibility radii for the lattice topological charge of quantum spin textures

How much can a magnetic field change before a quantum spin texture loses its skyrmion charge? For a triangulated spin-$s$ texture on $N$ quantum sites, we determine the smallest change in its local moments that makes the reconstructed topological charge ill-defined, allowing unequal error bounds and fixed boundary moments. Combined with spectral perturbation theory, this geometric threshold yields explicit magnetic-field intervals that preserve the ground-state charge, scaling as $1/N$ when the gap and geometric margin have positive limits. We apply the bounds to finite spin-$1/2$ systems with exchange and Dzyaloshinskii-Moriya interactions. In a rigorously verified seven-spin example the charge changes at a single certified field, while the excitation gap and all local spin polarizations remain bounded away from zero; there the geometric margin vanishes linearly.

math-ph↗
arXiv2026

Asymptotic Rounding Laws for Optimal Thresholds in Secretary Problems

We study second-order asymptotics for optimal thresholds in secretary-type problems. Under a unimodality assumption and a local affine expansion of the discrete payoff increments, we establish a general principle showing that sufficiently accurate rational approximations to the leading threshold proportion yield exact optimal integer thresholds. Continued-fraction convergents provide an immediate application. We also derive an asymptotic rounding law and apply the results to several variants of the secretary problem with fixed and random horizons.

math.PR↗
arXiv2026

Quantum-classical crossover in finite spin-1/2 rings with Dzyaloshinsky-Moriya interaction

We study finite spin-$\tfrac12$ rings with nearest-neighbor Heisenberg exchange, Dzyaloshinsky--Moriya interaction, and an external magnetic field. We introduce an interpolation parameter between the fully quantum Hamiltonian and a state-dependent mean-field description. Using dissipative Gisin--Schrödinger dynamics, we analyze the resulting quantum--classical crossover through local magnetization, connected spin correlations, single-site entropy, and the saturation field of the fully polarized state.

cond-mat.str-el↗
arXiv2026

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗
arXiv2026

Vortex Harmonic Spinors on the Nappi-Witten Space

We establish a correspondence between vortex equations on flat Riemann surfaces and harmonic spinors on the Nappi--Witten space, the group manifold of a central extension of the Euclidean group $SE(2)$. Vortex configurations lift naturally to this setting, producing explicit solutions of a twisted Dirac equation. Using the conformal flatness of the Nappi--Witten metric, these solutions induce harmonic spinors on four-dimensional Minkowski space. This yields a geometric construction of Abelian magnetic zero-modes on flat Minkowski spacetime from vortex data.

hep-th↗
arXiv2026

Bridging Quantum and Classical Descriptions of Spin Dynamics in a Dzyaloshinsky-Moriya Trimer

The spin dynamics of a trimer with Dzyaloshinsky-Moriya (DM) interaction are investigated within a unified Hamiltonian framework that connects quantum-mechanical and semiclassical descriptions. The interpolation between the two regimes is realised by solving the modified Gisin-Schrödinger equation, in which the relative weight of a quantum coherence and local mean-field contributions is continuously tuned. The resulting dynamical behaviour is analysed and summarised in a ground state diagram that illustrates how the character of the spin motion evolves from fully quantum to semiclassical as the DM interaction is treated at different levels of approximation. In the last part of the publication, the chiral spin dynamics proposed by Da-Wei Wang et al. is examined theoretically, taking into account its behaviour at the boundary between quantum and classical physics.

cond-mat.mes-hall↗
arXiv2025

Lower bounds for the reach and applications

The reach of a submanifold of $\mathbb{R}^N$ is defined as the largest radius of a tubular neighbourhood around the submanifold that avoids self-intersections. While essential in geometric and topological applications, computing the reach explicitly is notoriously difficult. In this paper, we introduce a rigorous and practical method to compute a guaranteed lower bound for the reach of a submanifold described as the common zero-set of finitely many smooth functions, not necessarily polynomials. Our algorithm uses techniques from numerically verified proofs and is particularly suitable for high-performance parallel implementations. We illustrate the utility of this method through several applications. Of special note is a novel algorithm for computing the homology groups of planar curves, achieved by constructing a cubical complex that deformation retracts onto the curve--an approach potentially extendable to higher-dimensional manifolds. Additional applications include an improved comparison inequality between intrinsic and extrinsic distances for submanifolds of $\mathbb{R}^N$, lower bounds for the first eigenvalue of the Laplacian on algebraic varieties and explicit bounds on how much smooth varieties can be deformed without changing their diffeomorphism type.

math.NA↗
arXiv2022

Harmonic forms on asymptotically AdS metrics

In this paper we study the rotationally invariant harmonic cohomology of a 2-parameter family of Einstein metrics $g$ which admits a cohomogeneity one action of $SU (2) \times U (1) $ and has AdS asymptotics. Depending on the values of the parameters, $g$ is either of NUT type, if the fixed-point locus of the $U (1) $ action is 0-dimensional, or of bolt type, if it is 2-dimensional. We find that if $g$ is of NUT type then the space of $SU (2) $-invariant harmonic 2-forms is 3-dimensional and consists entirely of self-dual forms; if $g$ is of bolt type it is 4-dimensional. In both cases we explicitly determine a basis. The pair $(g,F)$ for $F$ a self-dual harmonic 2-form is also a solution of the bosonic sector of $4D $ supergravity. We determine for which choices it is a supersymmetric solution and the amount of preserved supersymmetry.

hep-th↗