Energy Structure of a Finite Haldane Chain in ${\mathrm{Y}}_2{\mathrm{BaNi}}_{0.96}{\mathrm{Mg}}_{0.04}{\mathrm{O}}_5$ Studied by High Field Electron Spin Resonance

This Letter presents the fine structure of energy levels for the edge states of a Haldane chain. In order to investigate the edge states, we have performed high field and multifrequency electron spin resonance (ESR) measurements of finite length $S=1$ antiferromagnetic chains in ${\mathrm{Y}}_2{\mathrm{BaNi}}_{0.96}{\mathrm{Mg}}_{0.04}{\mathrm{O}}_5$. Owing to the high spectral resolution by high fields and high frequencies, observed ESR signals can be separated into the contributions of the finite chains with various chain lengths. Our results clearly show that the edge spins actually interact with each other through the quantum spin chain and the interaction depends on the chain length $N$. This $N$ dependence has been obtained experimentally for the first time, and shows that the correlation length $\xi$ in the real system is somewhat larger than that calculated by a simple Heisenberg model.

Figure 1

ESR spectra for various frequencies at 4.2 K in the magnetic field $B$ parallel to the $c$ axis.

Figure 2

Frequency-field diagrams for the applied field $B$ along different directions (a) $B\parallel a$, (b) $B\parallel b$, and (c) $B\parallel c$. The solid and open circles represent the resonance points for the main peak and secondary peaks, respectively. The dashed lines indicate the resonance modes for free $S=1/2$ spins.

Figure 3

Energy levels for a $S=1$ spin with $D=-8.97$ and $E=-2.98\mathrm{K}$ in the applied field $B$ along different directions (a) $B\parallel a$, (b) $B\parallel b$, and (c) $B\parallel c$.

Figure 4

$N$ dependence of $\gamma (N)$. The solid circles and open triangles represent $\gamma (N)$ obtained by our experiment and calculated in Ref. 16, respectively.