  The low-energy physics of Tomonaga-Luttinger liquids (TLLs) is controlled by
the Luttinger parameter. We demonstrate that this parameter can be extracted
from a single wave function for one-component TLLs with periodic boundary
condition. This method relies on the fact that TLLs are described by conformal
field theory in which crosscap states can be constructed. The overlaps between
the crosscap states and the ground state as well as some excited states are
proved to be universal numbers that directly reveal the Luttinger parameter. In
microscopic lattice models, crosscap states are formed by putting each pair of
antipodal sites into a maximally entangled state. Analytical and numerical
calculations are performed in a few representative models to substantiate the
conformal field theory prediction. The extracted Luttinger parameters are
generally quite accurate in finite-size systems with moderate lengths, so there
is no need to perform data fitting and/or finite-size scaling.
