Metal-organic frameworks: Strain-tunable quantum phase transition and high-order topological insulator and emergent fermions

The exploration of topological quantum phases and their transitions has become a focus of intense research. In this paper, we introduce a family of two-dimensional (2D) metal-organic frameworks (MOFs) that host various emergent 2D fermions and second-order topological insulators (SOTIs) while also exhibiting strain-tunable quantum phase transitions. Using first-principles calculations, we demonstrate that these MOFs can exhibit either a narrow-band-gap semiconductor or a zero-band-gap semimetal state, with three key bands in the low-energy region. Importantly, we find that applying biaxial strain can induce a semiconductor-to-semimetal quantum phase transition or vice versa. We further reveal that the semiconductor phase may possess nontrivial properties, including corner states, indicative of a second-order topological phase. At the critical strain point, a 2D triple point emerges, which transitions into a double-Weyl-fermion state under additional strain. We also construct effective models to describe these emergent 2D fermions. Our findings highlight the potential of MOFs as a versatile platform for studying quantum phases, emergent fermions, and SOTIs, paving the way for future applications in quantum materials and nanoscale technologies.