2025/01/17 by Shu Shen, Shen, Shu, Jianqing Yu +1 · 2 citations
Physics and Astronomy · Computer Science · Materials Science · #Quantum and electron transport phenomena #Quantum Computing Algorithms and Architecture #Graphene research and applications
paper · pdf · doi:10.48550/arxiv.2501.10530
The purpose of this article is to show a geometric version of Zabrodin-Wiegmann conjecture for an integer Quantum Hall state. Given an effective reduced divisor on a compact connected Riemann surface, using the canonical holomorphic section of the associated canonical line bundle as well as certain initial data and local normalisation data, we construct a canonical non-zero element in the determinant line of the cohomology of the p-tensor power of the line bundle. When endowed with proper metric data, the square of the L2 -norm of our canonical element is the partition function associated to an integer Quantum Hall state. We establish an asymptotic expansion for the logarithm of the partition function when p→ +∞. The constant term of this expansion includes the holomorphic analytic torsion and matches a geometric version of Zabrodin-Wiegmann's prediction. Our proof relies on Bismut-Lebeau's embedding formula for the Quillen metrics, Bismut-Vasserot and Finski's asymptotic expansion for the analytic torsion associated to the higher tensor product of a positive Hermitian holomorphic line bundle.