美国普林斯顿大学Ali Yazdani团队研究了分形霍夫施塔特能谱的光谱学。这一研究成果于2025年2月26日发表在《自然》杂志上。

霍夫施塔特蝴蝶是磁场中二维晶格中非相互作用电子的预测能谱，是自然界中最显著的分形结构之一。在每个晶格晶胞的磁通量量子的合理比率下，该光谱显示了反映其递归构造的能级自相似分布。对于大多数材料，霍夫施塔特蝴蝶是在实验室规模的磁场无法实现的实验条件下预测的。最近，电输运研究为霍夫施塔特蝴蝶理论存在于人工设计的具有较大晶格常数的材料中提供了证据，例如具有莫尔超晶格的材料。然而，到目前为止，霍夫施塔特近50年前预测的分形能谱的直接光谱仍然遥不可及。

研究组使用高分辨率扫描隧道显微镜/光谱学（STM/STS）来研究扭曲双层石墨烯（TBG）中预测的第二魔角附近的平坦电子带，这是霍夫施塔特能谱光谱研究的理想设置。研究结果表明，平坦莫尔带被细分为离散的霍夫施塔特子带，并识别出该光谱自相似性的实验特征。此外，该测量揭示了一个随电子密度动态演变的光谱，由于强相关性、库仑相互作用和TBG中电子的量子简并的综合作用，显示了超出霍夫施塔特原始模型的现象。

附：英文原文

Title: Spectroscopy of the fractal Hofstadter energy spectrum

Author: Nuckolls, Kevin P., Scheer, Michael G., Wong, Dillon, Oh, Myungchul, Lee, Ryan L., Herzog-Arbeitman, Jonah, Watanabe, Kenji, Taniguchi, Takashi, Lian, Biao, Yazdani, Ali

Issue&Volume: 2025-02-26

Abstract: Hofstadter’s butterfly, the predicted energy spectrum for non-interacting electrons confined to a two-dimensional lattice in a magnetic field, is one of the most remarkable fractal structures in nature1. At rational ratios of magnetic flux quanta per lattice unit cell, this spectrum shows self-similar distributions of energy levels that reflect its recursive construction. For most materials, Hofstadter’s butterfly is predicted under experimental conditions that are unachievable using laboratory-scale magnetic fields1,2,3. More recently, electrical transport studies have provided evidence for Hofstadter’s butterfly in materials engineered to have artificially large lattice constants4,5,6, such as those with moiré superlattices7,8,9,10. Yet, so far, direct spectroscopy of the fractal energy spectrum predicted by Hofstadter nearly 50years ago has remained out of reach. Here we use high-resolution scanning tunnelling microscopy/spectroscopy (STM/STS) to investigate the flat electronic bands in twisted bilayer graphene (TBG) near the predicted second magic angle11,12, an ideal setting for spectroscopic studies of Hofstadter’s spectrum. Our study shows the fractionalization of flat moiré bands into discrete Hofstadter subbands and discerns experimental signatures of self-similarity of this spectrum. Moreover, our measurements uncover a spectrum that evolves dynamically with electron density, showing phenomena beyond that of Hofstadter’s original model owing to the combined effects of strong correlations, Coulomb interactions and the quantum degeneracy of electrons in TBG.

DOI: 10.1038/s41586-024-08550-2

Source: https://www.nature.com/articles/s41586-024-08550-2