  The interplay between topology and interaction always plays an important role
in condensed matter physics and induces many exotic quantum phases, while rare
transition metal layered material (TMLM) has been proved to possess both. Here
we report a TMLM Ta$_2$Pd$_3$Te$_5$ has the two-dimensional second-order
topology (also a quadrupole topological insulator) with correlated edge states
- Luttinger liquid. It is ascribed to the unconventional nature of the mismatch
between charge- and atomic- centers induced by a remarkable double-band
inversion. This one-dimensional protected edge state preserves the Luttinger
liquid behavior with robustness and universality in scale from micro- to macro-
size, leading to a significant anisotropic electrical transport through
two-dimensional sides of bulk materials. Moreover, the bulk gap can be
modulated by the thickness, resulting in an extensive-range phase diagram for
Luttinger liquid. These provide an attractive model to study the interaction
and quantum phases in correlated topological systems.
