Topological Lithography via External Field: Creating Topological States Anywhere Beyond the Edge
Topological insulators (TIs), recognized for their robust boundary states and unconventional phase transitions, have emerged as one of the most impactful discoveries in recent decades, attracting considerable interest across diverse fields. However, conventional TIs require topological contrasts between adjacent bulk regions, typically achieved by distinct symmetries, which limits their flexibility and broader applicability. In this work, we introduce a "lithography" approach to inducing topological states by applying external fields in time-reversal-symmetric systems. These states transcend the conventional bulk topology design paradigm and offer exceptional tunability: they can exist at geometric edges, within the bulk, or even be induced remotely. Because the external field is highly controllable, the induced states are programmable, reconfigurable, and can be easily tailored into desirable patterns. Theoretically, we demonstrate that the external field modifies the Jackiw-Rebbi mechanism, inducing a real-space topological transition characterized by a local topological marker (LTM). We further establish an extended valley bulk-edge correspondence, which explains both the conventional scenario and our findings. Our results, validated across mechanical, electronic, and acoustic platforms, highlight the broad applicability of this approach to various systems. This work not only advances the theory of topology but also enhances the diversity, tunability, and practical implementation of topological states and materials.