Continuous anomaly

January 26, 2021

In many cases, an anomaly corresponds to a SPT order or a topological order in one higher dimension.[1][2] This kind of anomalies are labelled by a discrete index. However, for continuous symmetries, Ref. [1] pointed out that there can be a special kind of anomalies, which are labelled by a continuous index, and will be referred to as continuous anomaly. Ref [1] gave an example of continuous anomaly: a 2+1D system with an U(1) symmetry and an unquantized Hall conductance. (This example may be closely related to a recent work on the anomaly of emergent loop-U(1) symmetry[3]). The mixed U(1)xRd anomaly discussed in Ref [4] is another example of continuous anomaly, that is labelled by the continuous particle density ρ.

Continuous anomalies also correspond to gapped states in one higher dimension with topological terms, but now the topological terms have a continuous coefficient.[1] For example, the 2+1D continuous U(1) anomaly of unquantized Hall conductance is characterized by a 3+1D gapped state with a topological term θ∫ F∧F. Had the coefficient been discrete, the bulk gapped state with the topological term would correspond to a SPT order or a topological order, and the anomaly would correspond to a SPT order or a topological order in one higher dimension.  However, for continuous anomalies (such as the 2+1D U(1) continuous anomaly), the coefficient of the topological term is continuous and can be smoothly tuned to zero. So the bulk gapped state with this kind of topological term does not correspond to non-trivial phase, and continuous anomalies does not correspond to a SPT phase in one higher dimension, but rather correspond to a bulk gapped state with a unquantized topological term, which can be viewed as a pseudo SPT state. 

We like to stress that a continuous anomaly for a symmetry is not robust against all symmetry preserving deformations of the Hamiltonian. For example, the mixed U(1)xRd anomaly is robust only against all symmetry preserving deformations that keep the particle density of the ground state unchanged (ie keep the U(1) representation of the ground state unchanged). This is a general feature of continuous anomaly.