We study the geometric phase for the ground state of a generalized one-dimensional non-Hermitian quantum
XY model, which has transverse-field-dependent intrinsic rotation-time reversal symmetry. Based on the exact solution, this model is shown to have a full real spectrum in multiple regions for the finite-size system. The result indicates that the phase diagram or exceptional boundary which separates the unbroken- and broken-symmetry regions corresponds to the divergence of the Berry curvature. The scaling behaviors of the ground-state energy and Berry curvature are obtained in an analytical manner for a concrete system.
In the traditional quantum theory, one-dimensional quantum spin models possess a factorization surface where the ground states are fully separable and have vanishing bipartite as well as multipartite entanglement. We report that in the non-Hermitian counterpart of these models, these factorization surfaces either can predict the exceptional points where the unbroken-to-broken transition occurs or can guarantee the spectrum to be real, thereby suggesting a procedure to reveal the unbroken phase. We first analytically demonstrate it for the nearest-neighbor rotation-time-
(RT) symmetric
XY model with uniform and alternating transverse magnetic fields, referred to as the
iATXY model. Exact diagonalization techniques are then employed to establish this fact for the
RT-symmetric
XYZ model with short- and long-range interactions as well as for the long-range
iATXY model. Moreover, we show that although the factorization surface prescribes the unbroken phase of the non-Hermitian model, the bipartite nearest-neighbor entanglement at the exceptional point is nonvanishing.
Non-Hermitian spin systems have attracted extensive interest due to their unconventional magnetic properties, rich entanglement resources, and unusual quantum criticality phenomena. In this paper, based on the exact solution of a one-dimensional non-Hermitian spin-1/2 XY model with rotation-time-reversal (
RT) symmetry [X. Z. Zhang et al., Phys. Rev. A 87, 012114 (2013)], we study the ground-state energy density, magnetization, correlation functions, quantum entanglement, and correlation in
RT-symmetric and
RT-broken phases as well as their characteristics at the exceptional point. We find that the energy density rises faster with the non-Hermitian parameter $|\ensuremath{\gamma}|$ in the symmetric region than in the broken one and is elevated rapidly at the exceptional point, and the decay of magnetization has similar results. We analyze the effect of $\ensuremath{\gamma}$ on the system and find that the energy density decreases linearly with the external magnetic field
h for $\ensuremath{\gamma}>0,$ while it is bifurcated when $\ensuremath{\gamma}=0$. In addition, the behaviors of the energy density indicate that the phase transition caused by symmetry breaking is second order, which is further demonstrated by the magnetization, quantum entanglement, etc. The numerical results of the correlation functions surprisingly indicate that the
RT-broken phase has quasi-long-range order, which is quite different from the Hermitian XY model. Especially, the crossover behavior of the ground-state entanglement shows that it increases with $\ensuremath{\gamma}$ in the symmetric region, which is opposite to the case of the broken one, and its maximum always appears at the exceptional point. The above behaviors at the phase boundary are actually ascribed to the fierce competition between $\ensuremath{\gamma}$ and
h, which results in a rapid decline of magnetization and the appearance of the maximum of entanglement.
The occurrence of parity-time reversal
(PT) symmetry breaking is discussed in a non-Hermitian spin chain. The Hermiticity of the model is broken by the presence of an alternating, imaginary, transverse magnetic field. A full real spectrum, which occurs if and only if all the eigenvectors are
PT symmetric, can appear only in presence of dimerization, i.e., only if the hopping amplitudes between nearest-neighbor spins assume alternate values along the chain. In order to make a connection between such system and the Hermitian world, we study the critical magnetic properties of the model and look for the conditions that would allow to observe the same phase diagram in the absence of the imaginary field. Such procedure amounts to renormalizing the spin-spin coupling amplitudes.
We investigate the phase diagram of a non-Hermitian XY chain with staggered Dzyaloshinskii--Moriya (DM) interactions using quantum-information methods. Through an alternating local-spin-rotation transformation, the model is mapped onto a standard DM-free XY chain, and the resulting transformed Hamiltonian exhibits rotation--time-reversal ($\RT$) symmetry. Combining correlation-function analysis with known phase results of the standard XY chain, we construct the phase diagram for the present system. We further adopt quantum-information-based quantities to systematically assess their performance in characterizing quantum phase transitions within this non-Hermitian system. Our results show that single-site entanglement can only detect the Luttinger-liquid (LL)--paramagnetic (PM) phase transition, which corresponds to the exceptional boundary across which $\RT$ symmetry is restored. In contrast, quantum discord (QD) and quantum coherence (QC) identify both phase boundaries: in addition to locating the exceptional boundary, their second-order derivatives resolve the ferromagnetic (FM)--LL transition inside the $\RT$-broken region. Moreover, measurements of QC along different directions capture the DM-induced relative rotation between the two sublattices, thereby distinguishing the staggered-DM chain from a zero-DM chain with identical effective parameters.
Abstract The XY spin chain is a paradigmatic example of a model solved by free fermions, in which the energy eigenspectrum is built from combinations of quasi-energies. In this article, we show that by extending the XY model’s anisotropy parameter
λ λ to complex values, it is possible for two of the quasi-energies to become degenerate. In the non-Hermitian XY model, these quasi-energy degeneracies give rise to exceptional points (EPs) where eigenvalues and their corresponding eigenvectors coalesce. The distinct
λ λ values at which EPs appear form concentric rings in the complex plane which are shown in the infinite system size limit to converge to the unit circle coinciding with the boundary between distinct topological phases. The non-Hermitian model is also seen to possess a line of broken
PT P T symmetry along the pure imaginary
λ λ -axis. For finite systems, there are four EP values on this broken
PT P T -symmetric line if the system size is a multiple of 4.