  The eigenstate thermalization hypothesis describes how isolated many-body
quantum systems reach thermal equilibrium. However, quantum many-body scars and
Hilbert space fragmentation violate this hypothesis and cause nonthermal
behavior. We demonstrate that Hilbert space fragmentation may arise from
lattice geometry in a spin-1/2 model that conserves the number of domain walls.
We generalize a known, one-dimensional, scarred model to larger dimensions and
show that this model displays Hilbert space fragmentation on the Vicsek fractal
lattice and the two-dimensional lattice. Using Monte Carlo methods, the model
is characterized as strongly fragmented on the Vicsek fractal lattice when the
number of domain walls is either small or close to the maximal value. On the
two-dimensional lattice, the model is strongly fragmented when the density of
domain walls is low and weakly fragmented when the density of domain walls is
high. Furthermore, we show that the fragmentation persists at a finite density
of domain walls in the thermodynamic limit for the Vicsek fractal lattice and
the two-dimensional lattice. We also show that the model displays signatures
similar to Hilbert space fragmentation on a section of the second-generation
hexaflake fractal lattice and a modified two-dimensional lattice. We study the
autocorrelation function of local observables and demonstrate that the model
displays nonthermal dynamics.
