Relating non-Hermitian and Hermitian quantum systems at criticality*

We demonstrate three different transformations that relate Hermitian and non-Hermitian quantum systems at criticality. For the case preserving both the physical spectrum (PS) and the entanglement spectrum (ES), the entanglement entropy properties are identical for both Hermitian and non-Hermitian systems. In terms of the conformal field theory (CFT), the corresponding central charges extracted from the logarithmic scaling of the entanglement entropy are identical for both Hermitian and non-Hermitian systems. The second transformation preserves the PS but not the ES. The entanglement entropy scalings are different and lead to different central charges. We demonstrate this transformation by the dilation method for the free fermion cases, where the central charge c = −2 system can be mapped to c = 1 system. The last one is the Galois conjugate transformation that does not preserve both PS and ES. We demonstrate the Galois conjugate transformation in the anyonic golden chain model. The tricritical Ising model/3-state Potts mode can relate to the Lee-Yang model with negative central charges.