Multi higher-order Dirac and nodal line semimetals

In recent years, there has been a surge of interest in exploring higher-order topology and their semi-metallic counterparts, particularly in the context of Dirac, Weyl, and nodal line semimetals, termed as higher-order Dirac semimetal (HODSM), higher-order Weyl semimetal, and higher-order nodal line semimetal (HONLSM). The HODSM phase exhibits hinge Fermi arcs (FAs) with a quantized higher-order topological invariant. Conversely, the HONLSM phase is a hybrid-order topological phase manifesting both drumhead-like surface states and hinge FAs as a signature of first- and second-order topology, and also possesses both first- and second-order topological invariants. In this work, we investigate a tight binding model for multi-HODSM (mHODSM) hosting multiple hinge FAs having a quantized quadrupolar winding number (QWN) greater than one. Furthermore, we obtain a multi-HONLSM (mHONLSM) phase from the mHODSM by applying an external magnetic field, which breaks the $PT$-symmetry. The mHONLSM phase possesses both the dipolar winding number, non-vanishing only inside the nodal loops, being the representative invariant for first-order topology, and the QWN, featuring both drumhead-like surface states and multiple hinge FAs. We study the spectral properties of the mHODSM and mHONLSM in different geometries. We also investigate the hinge FA-mediated transport in HONLSM employing a two-terminal setup.