Phantom helix states are a special set of degenerate eigenstates of the XXZ Heisenberg model which lie in the energy levels around zero energy and are bidirectionally equal. In this work, we study the helix state in the XXZ Heisenberg model with the Dzyaloshinskii-Moriya interaction (DMI). We show exactly that only the helix states in one direction remain unchanged in the presence of resonant DMI. Based on the Holstein-Primakoff transformation, the quantum spin model is mapped to a boson model, which allows us to understand the underlying mechanism. Furthermore, it also indicates that such phantom states can be separated from the spectrum by the strong DMI to enhance the robustness of the states. We demonstrate the dynamic formation processes of unidirectional phantom helix states by numerical simulations. The results indicate that the DMI, as expected, acts as a filter with high efficiency.
Phantom helix states are a special set of degenerate eigenstates of the XXZ Heisenberg model which lie in the energy levels around zero energy and are bidirectionally equal. In this work, we study the helix state in the XXZ Heisenberg model with the Dzyaloshinskii-Moriya interaction (DMI). We show exactly that only the helix states in one direction remain unchanged in the presence of resonant DMI. Based on the Holstein-Primakoff transformation, the quantum spin model is mapped to a boson model, which allows us to understand the underlying mechanism. Furthermore, it also indicates that such phantom states can be separated from the spectrum by the strong DMI to enhance the robustness of the states. We demonstrate the dynamic formation processes of unidirectional phantom helix states by numerical simulations. The results indicate that the DMI, as expected, acts as a filter with high efficiency.
In general, the summation of a set of sub-Hamiltonians cannot share a common eigenstate of each one, only if it is an unentangled product state, such as a phantom helix state in quantum spin system. Here we present a method, referred to as the building block method (BBM), for constructing possible spin-1/2 XXZ Heisenberg lattice systems possessing phantom helix states. We focus on two types of XXZ dimers as basic elements, with a non-Hermitian parity-time (PT ) field and Hermitian Dzyaloshinskii-Moriya interaction (DMI), which share the same degenerate eigenstates. Based on these two building blocks, one can construct a variety of Heisenberg quantum spin systems, which support helix states with zero energy. The underlying mechanism is the existence of a set of degenerate eigenstates. Furthermore, we show that such systems act as quantum spin graphs since they obey the analogs of Kirchhoff's laws for sets of spin helix states when the non-Hermitian PT fields cancel each other out. In addition, the dynamic response of the helix states for three types of perturbations is also investigated analytically and numerically. Our findings provide a way to study quantum spin systems with irregular geometries beyond the Bethe ansatz approach.
We systematically investigate possible helix states in the XXZ Heisenberg model with the Dzyaloshinskii-Moriya (DM) interaction. Exact solutions show that a set of precession helix states can be constructed by deliberate superposition of degenerate eigenstates of the Hamiltonian under the resonant condition. When a non-Hermitian balance boundary term is imposed as a quenching action, the quench dynamics shows that a steady helix state emerges from some easily prepared initial states, including saturated and maximally mixed ferromagnetic states, according to the analysis of the perturbation method. The corresponding dynamics for near resonant cases is also investigated numerically, indicating the robustness of the scheme. Our findings highlight the cooperation of non-Hermiticity and the DM interaction in a quantum spin system, suggesting a way for preparing a steady helix state in a non-Hermitian quantum spin system.
Abstract In general, the summation of a set of sub-Hamiltonians cannot share a common eigenstate of each one, only if it is an unentangled product state, such as a spin helix state in quantum spin system. Here we present a method, referred to as the building block method (BBM), for constructing possible spin-1/2 XXZ Heisenberg lattice systems possessing spin helix states. We focus on two types of XXZ dimers as basic elements, with a non-Hermitian parity-time ( P T ) magnetic field and Hermitian Dzyaloshinskii-Moriya interaction (DMI), which share the same degenerate eigenstates. Based on these two building blocks, one can construct a variety of Heisenberg quantum spin systems, which support helix states with zero energy. Furthermore, we show that such systems act as quantum spin graphs since they obey the analogs of Kirchhoff’s laws for sets of spin helix states when the non-Hermitian P T terms cancel each other out. In addition, the dynamic response of the helix states for three types of perturbations is also investigated analytically and numerically. Our findings provide a way to study quantum spin systems on irregular lattices.
Exact solutions for quantum many-body systems are rare and provide valuable insight to universal phenomena. Here we show experimentally in anisotropic Heisenberg chains that special helical spin patterns can have very long lifetimes. This finding confirms the recent prediction of phantom Bethe states, exact many-body eigenstates carrying finite momenta yet no energy. We theoretically find analogous stable spin helices in higher dimensions and in other non-integrable systems, where they imply non-thermalizing dynamics associated with quantum many-body scars. We use phantom spin helices to directly measure the interaction anisotropy which has a major contribution from short-range off-site interactions that have not been observed before. Phantom helix states open new opportunities for quantum simulations of spin physics and studies of many-body dynamics.
We demonstrate the existence of a special chiral ``phantom'' mode with some analogy to a Goldstone mode in the anisotropic quantum XXZ Heisenberg spin chain. The phantom excitations contribute zero energy to the eigenstate, but a finite fixed quantum of momentum k0. The mode exists not due to symmetry principles, but results from nontrivial scattering properties of magnons with momentum k0 given by the anisotropy via $cos{k}_{0}=\mathrm{\ensuremath{\Delta}}$. Different occupations of the phantom mode lead to energetical degeneracies between different magnetization sectors in the periodic case. This mode originates from special string-type solutions of the Bethe ansatz equations with unbounded rapidities, the phantom Bethe roots (PBRs). We derive criteria under which the spectrum contains eigenstates with PBRs, both in open and periodically closed integrable systems, for spin-1/2 and higher spins, and discuss the respective chiral eigenstates. The simplest of such eigenstates, the spin helix state, which is a periodically modulated state of chiral nature, is built up from the phantom excitations exclusively. Implications of our results for experiments are discussed.
We investigate special solutions to the Bethe Ansatz equations (BAE) for open integrable XXZ Heisenberg spin chains containing phantom (infinite) Bethe roots. The phantom Bethe roots do not contribute to the energy of the Bethe state, so the energy is determined exclusively by the remaining regular excitations. We derive the phantom Bethe roots criterion and focus on BAE solutions for mixtures of phantom roots and regular (finite) Bethe roots. We prove that in the presence of phantom Bethe roots, all eigenstates are split between two invariant subspaces, spanned by chiral shock states. Bethe eigenstates are described by two complementary sets of Bethe ansatz equations for regular roots, one for each invariant subspace. The respective ``semiphantom'' Bethe vectors are states of chiral nature, with chirality properties getting less pronounced when more regular Bethe roots are added. For the easy plane case, ``semiphantom'' Bethe states carry nonzero magnetic current and are characterized by quasiperiodic modulation of the magnetization profile, the most prominent example being the spin helix states (SHS). We illustrate our results investigating ``semiphantom'' Bethe states generated by one regular Bethe root (the other Bethe roots being phantom), with a simple structure of the invariant subspace, in all details. We obtain the explicit expressions for Bethe vectors and calculate the simplest correlation functions, including the spin-current for all the states in the single-particle multiplet.