  Identifying novel topological properties of topological quantum states of
matter, such as exemplified by the quantized Hall conductance, is a valuable
step towards realizing materials with attractive topological attributes that
guarantee their imperviousness to realistic imperfections, disorder and
environmental disturbances. Is the gravitational coupling coefficient of
topological quantum states of matter a promising candidate? Substantially
building on well established results for quantum Hall states, using
disclinations as tools for controlled creation of pristine spatial curvature
free of undesirable artifacts such as would interfere with the electronic
motion of interest, herein we report that a large class of lattice topological
states of matter exhibit gravitational response, i.e., charge response to
intrinsic spatial curvature. This phenomenon is characterized by a
topologically quantized coupling constant. Remarkably, the charge-gravity
relationship remains linear in the curvature, up to the maximum curvature
achievable on the lattice, demonstrating absence of higher order nonlinear
response. Our findings facilitate articulating the physical principles
underlying the topological quantization of the gravitational coupling constant,
in analogy with the insights offered by the Chern number description of the
quantized Hall conductance.
