Universal contributions to charge fluctuations in spin chains at finite temperature

At finite temperature, conserved charges undergo thermal fluctuations in a quantum many-body system in the grand canonical ensemble. The full structure of the fluctuations of the total $\mathrm{U}(1)$ charge $Q$ can be succinctly captured by the generating function $G(\ensuremath{\theta})=\ensuremath{\langle}\phantom{\rule{0.16em}{0ex}}{\mathrm{e}}^{\mathrm{i}\ensuremath{\theta}Q}\ensuremath{\rangle}$. For a 1D translation-invariant spin chain in the thermodynamic limit, the magnitude $|G(\ensuremath{\theta})|$ scales with the system size $L$ as $ln|G(\ensuremath{\theta})|=\ensuremath{-}\ensuremath{\alpha}(\ensuremath{\theta})L+\ensuremath{\gamma}(\ensuremath{\theta})$, where $\ensuremath{\gamma}(\ensuremath{\theta})$ is the scale-invariant contribution and may encode universal information about the underlying system. In this work, we investigate the behavior and physical meaning of $\ensuremath{\gamma}(\ensuremath{\theta})$ when the system is periodic. We find that $\ensuremath{\gamma}(\ensuremath{\theta})$ only takes nonzero values at isolated points of $\ensuremath{\theta}$, which is $\ensuremath{\theta}=\ensuremath{\pi}$ for all our examples. In two exemplary lattice systems, we show that $\ensuremath{\gamma}(\ensuremath{\pi})$ takes quantized values when the $\mathrm{U}(1)$ symmetry exhibits a specific type of 't Hooft anomaly with other symmetries. In other cases, we investigate how $\ensuremath{\gamma}(\ensuremath{\theta})$ depends on microscopic conditions (such as the filling factor) in field theory and exactly solvable lattice models.