Quantum theory of spin waves in finite chiral spin chains

We calculate the effect of spin waves on the properties of finite-size spin chains with a chiral spin ground state observed on biatomic Fe chains deposited on iridium(001). The system is described with a Heisenberg model supplemented with a Dzyaloshinskii-Moriya coupling and a uniaxial single ion anisotropy that presents a chiral spin ground state. Spin waves are studied using the Holstein-Primakoff boson representation of spin operators. Both the renormalized ground state and the elementary excitations are found by means of Bogoliubov transformation, as a function of the two variables that can be controlled experimentally, the applied magnetic field and the chain length. Three main results are found. First, because of the noncollinear nature of the classical ground state, there is a significant zero-point reduction of the ground-state magnetization of the spin spiral. Second, there is a critical external field from which the ground state changes from chiral spin ground state to collinear ferromagnetic order. The character of the two lowest-energy spin waves changes from edge modes to confined bulk modes over this critical field. Third, in the spin-spiral state, the spin-wave spectrum exhibits oscillatory behavior as function of the chain length with the same period of the spin helix.

Figure 1

Schematic classical ground state for the mono strand (top) and the diatomic (bottom) chains. The parameters used in the calculation are given in the text. In both cases a magnetic field of 2T along the $(1,0,0)$ direction is applied.

Figure 2

Projections of the classical ground state ${\vec{\Omega }}_i$ over the $\widehat{x}$ (dashed blue) and the $\widehat{z}$ (solid red) directions for the monostrand chain with first nearest-neighbor interactions. In this case we take $B=1T$, $J=D=1$ meV.

Figure 3

Magnonic excitation analysis for the monostrand nearest-neighbor interaction chain as a function of magnetic fields along the $x$ axis. Energies for the first five excited states (a), dependence of the net spin along the $x$ axis for the classical ground state (b), and magnonic occupation on the ground state (c) and the first excited state (d). In these plots, black stands for null $\langle a_i^{†}{a}_i\rangle$ and white for maximal fluctuation.

Figure 4

Magnonic excitation properties for a 30-site biatomic Fe chain on Ir(001) as a function of a magnetic field along the $x$ axis. Energies for the first five excited states (a), dependence of the net spin along the $x$ axis for the classical ground (b), and magnonic occupation on the ground state (c) and the first excited state (d). In these plots, black denotes null $\langle a_i^{†}{a}_i\rangle$ and white denotes maximal fluctuation. Quantum fluctuations increase with the field in frustrated spin sites.

Figure 5

Six lowest-lying excitation energies as function of the chain's length for the spin spiral (left panel) and the ferromagnetic ground state (right panel). The calculation is 2T and 32T respectively.