Extraction of topological information in Tomonaga-Luttinger liquids

We discuss expectation values of the twist operator $U$ appearing in the Lieb-Schultz-Mattis theorem (or the polarization operator for periodic systems) in excited states of the one-dimensional correlated systems $z_L^{(q,\pm)}\equiv\braket{\Psi_{q/2}^{\pm}|U^q|\Psi_{q/2}^{\pm}}$, where $\ket{\Psi_{p}^{\pm}}$ denotes the excited states given by linear combinations of momentum $2pk_{\rm F}$ with parity $\pm 1$. We found that $z_L^{(q,\pm)}$ gives universal values $\pm 1/2$ on the Tomonaga-Luttinger (TL) fixed point, and its signs identify the topology of the dominant phases. Therefore, this expectation value changes between $\pm 1/2$ discontinuously at a phase transition point with the U(1) or SU(2) symmetric Gaussian universality class. This means that $z_L^{(q,\pm)}$ extracts the topological information of TL liquids. We explain these results based on the free-fermion picture and the bosonization theory, and also demonstrate them in several physical systems.