Quantum phase transitions in non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric transverse-field Ising spin chains

Abstract

We present a theoretical study of quantum phases and quantum phase transitions occurring in non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric superconducting qubits chains described by a transverse-field Ising spin model. A non-Hermitian part of the Hamiltonian is implemented via imaginary staggered \textit{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 $\gamma$. A scaling procedure for the coherence length $\xi$ allows us to establish a complete quantum phase diagram of the system. We obtain two quantum phases for $J<0$, namely, $\mathcal{P}\mathcal{T}$-symmetry broken antiferromagnetic state and $\mathcal{P}\mathcal{T}$-symmetry preserved paramagnetic state, and the quantum phase transition line between them is the line of exception points. For $J>0$ the $\mathcal{P}\mathcal{T}$-symmetry of the ground state is retained in a whole region of parameter space of $J$ and $\gamma$, and a system shows \textit{two} intriguing quantum phase transitions between ferromagnetic and paramagnetic states for a fixed parameter $\gamma >1$. We also provide the qualitative quantum phase diagram $\gamma -J$ derived in the framework of the Bethe-Peierls approximation that is in a good accord with numerically obtained results.

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• Schrieffer-Wolff transformation for non-Hermitian systems: application for $\mathcal{P}\mathcal{T}$-symmetric circuit QED
• $\mathcal{P}\mathcal{T}$-Symmetry breaking in quantum spin chains with exceptional non-Hermiticities
• Stabilizing period-doubled density waves by spin-orbit coupling in Bose-Einstein condensates in optical lattices
• Quantum dynamics of frustrated Josephson junction arrays embedded in a transmission line: an effective XX spin chain with long-range interaction
• Many-Body Neural Network Wavefunction for a Non-Hermitian Ising Chain
• Parity-Time Symmetric Spin-1/2 Richardson-Gaudin Models