  We introduce a one-dimensional lattice model whose hopping amplitudes are
modulated for equally spaced sites. Such mosaic lattice exhibits many
interesting topological and localization phenomena that do not exist in the
regular off-diagonal lattices. When the mosaic modulation is commensurate with
the underlying lattice, topologically nontrivial phases with zero- and
nonzero-energy edge modes are observed as we tune the modulation, where the
nontrivial regimes are characterized by quantized Berry phases. If the mosaic
lattice becomes incommensurate, Anderson localization will be induced purely by
the quasiperiodic off-diagonal modulations. The localized eigenstate is found
to be centered on two neighboring sites connected by the quasiperiodic hopping
terms. Furthermore, both the commensurate and incommensurate off-diagonal
mosaic lattices can host Chern insulators in their two-dimensional
generalizations. Our work provides a platform for exploring topological phases
and Anderson localization in low-dimensional systems.
