Detecting the Emergent Continuous Symmetry of Criticality via a Subsystem's Entanglement Spectrum

The (emergent) symmetry of a critical point constitutes fundamental pieces of information for determining the universality class and effective field theory. However, the underlying symmetry thus far can be conjectured only indirectly from the dimension of the order parameters in symmetry-breaking phases, and its correctness requires further verification to avoid overlooking hidden order parameters, which by itself is also a difficult task. In this Letter, we introduce an unbiased numerical approach to identify the underlying (emergent) symmetry of a critical point in quantum many-body systems without prior knowledge about the corresponding low-energy effective field theory. By numerically calculating the reduced density matrix in a very small subsystem of the total system, the Anderson tower of states in the entanglement spectrum can be obtained, clearly reflecting the underlying (emergent) symmetry of criticality. This is attributed to the fact that the entanglement spectrum can reveal the broken symmetry of the ground state of entanglement Hamiltonian after cooling from the critical point along an extra entanglement-temperature axis.