Universal thermal charge fluctuations in 1D systems with quantum anomaly

In recent years, a universal quantity known as the topological disorder parameter (TDP), analogous to topological entanglement entropy, has been identified in 2D topological phases. This quantity emerges as a sub-leading term in the scaling of disorder operators. In all known cases, the extraction of TDP can be reduced to evaluating thermal expectation of a global symmetry transformation in the 1D edge theory, which is described by a conformal field theory (CFT). In this work, we establish the TDP as an intrinsic property of a 1D spin chain possessing U(1) symmetry. In two exemplary systems, we show that it appears as a universal contribution to thermal charge fluctuations when the U(1) symmetry exhibits type-III anomaly with other symmetries. We demonstrate this in exactly-solvable models and also provide arguments based on matrix product density operators. Our results extend the CFT-based understanding to any finite temperature.