Lattice twist operators and vertex operators in sine-Gordon theory in one dimension

In one dimension, the exponential position operators introduced in a theory of polarization are identified with the twist operators appearing in the Lieb-Schultz-Mattis argument, and their finite-size expectation values $z_L^{\mathrm{}\left(q\right)\mathrm{}}$ measure the overlap between the q-fold degenerate ground state and an excited state. Insulators are characterized by $z_{\infty }\ne 0,$ and different states are distinguished by the sign of $z_L.$ We identify $z_L$ with ground-state expectation values of vertex operators in the sine-Gordon model. This allows an accurate detection of quantum phase transitions in the universality classes of the Gaussian and the $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_1$ Wess-Zumino-Novikov-Witten models. We apply this theory to the half-filled extended Hubbard model and obtain agreement with the level-crossing method.