The exactly solvable quasi-Hermitian transverse Ising model

Abstract

A non-Hermitian deformation of the one-dimensional transverse Ising model is shown to have the property of quasi-hermiticity. The transverse Ising chain is obtained from the starting non-Hermitian Hamiltonian through a similarity transformation. Consequently, both the models have identical eigen spectra, although the eigenfunctions are different. The metric in the Hilbert space, which makes the non-Hermitian model unitary and ensures the completeness of states, has been constructed explicitly. Although the longitudinal correlation functions are identical for both the non-Hermitian and the Hermitian Ising models, the difference shows up in the transverse correlation functions, which have been calculated explicitly and are not always real. A proper set of Hermitian spin operators in the Hilbert space of the non-Hermitian Hamiltonian has been identified, in terms of which all the correlation functions of the non-Hermitian Hamiltonian become real and identical to that of the standard transverse Ising model. Comments on the quantum phase transitions in the non-Hermitian model have been made.