Quantum phase transitions in non-Hermitian [formula omitted]-symmetric transverse-field Ising spin chains

Abstract

We present a theoretical study of quantum phases and quantum phase transitions occurring in non-Hermitian PT-symmetric superconducting qubits chains described by a transverse-field Ising spin model. A non-Hermitian part of the Hamiltonian is implemented via imaginary staggered longitudinal magnetic field, which corresponds to a local staggered gain and loss terms. By making use of a direct numerical diagonalization of the Hamiltonian for spin chains of a finite size N, we explore the dependencies of the energy spectrum, including the energy difference between the first excited and the ground states, the spatial correlation function of local polarization (z-component of local magnetization) on the adjacent spins interaction strength J and the local gain (loss) parameter γ. A scaling procedure for the correlation length ξ allows us to establish a complete quantum phase diagram of the system. We obtain two quantum phases for J<0, namely, PT-symmetry broken antiferromagnetic state and PT-symmetry preserved paramagnetic state, and the quantum phase transition line between them is the line of exception points. For J>0 the PT-symmetry of the ground state is retained in a whole region of parameter space of J and γ, and a system shows two intriguing quantum phase transitions between ferromagnetic and paramagnetic states for a fixed parameter γ>1. We also provide the qualitative quantum phase diagram γ−J derived in the framework of the Bethe–Peierls approximation that is in a good accord with numerically obtained results.