Electron spin resonance shifts in $S=1$ antiferromagnetic chains

We discuss electron spin resonance (ESR) shifts in spin-1 Heisenberg antiferromagnetic chains with a weak single-ion anisotropy, based on several effective field theories: the O(3) nonlinear sigma model (NLSM) in the Haldane phase, free-fermion theories around the lower and the upper critical fields. In the O(3) NLSM, the single-ion anisotropy corresponds to a composite operator which creates two magnons at the same time and position. Therefore, even inside a parameter range where free magnon approximation is valid for thermodynamics, we have to take interactions among magnons into account in order to include the single-ion anisotropy as a perturbation. Although the O(3) NLSM is only valid in the Haldane phase, an appropriate translation of Faddeev-Zamolodchikov operators of the O(3) NLSM to fermion operators enables one to treat ESR shifts near the lower critical field in a similar manner to discussions in the Haldane phase. Our theory gives quantitative agreements with a numerical evaluation using quantum Monte Carlo simulation, and also with recent ESR experimental results on a spin-1 chain compound $\mathrm{Ni}{\left({\mathrm{C}}_5{\mathrm{H}}_{14}{\mathrm{N}}_2\right)}_2{\mathrm{N}}_3\left({\mathrm{PF}}_6\right)$.

Figure 1

Quantum Monte Carlo results of the normalized ESR shift (13) induced by the single-ion anisotropy (10) for temperatures $T/J=0.1$–$0.5$. The system size is $L=40$ sites. The lower critical field $H_{c1}=0.41J$ and the upper critical field $H_{c2}=4J$ are guided by the dotted and the dashed lines, respectively. There is an extremum around $H=H_{c1}+T$. This nonmonotonic behavior of the ESR shift is understood by the finite-temperature crossover.

Figure 2

Comparisons of QMC and analytic results at (a) $T/J=0.1$ and at (b) $T/J=0.2$. Open symbols (circles and triangles) represent QMC data. The solid curves denote the normalized shift in the dilute limit (36). The dashed curves correspond to (54).

Figure 3

The open circles denote the QMC data obtained for 40-site chains at $T/J=0.1$. The solid and dashed curves are derived from the free-fermion theory near $H_{c2}$. The former is $T>0$ data and the latter is $T=0$ data. The finite-temperature effect is irrelevant in the lower-field range $H\lesssim 3J$.

Figure 4

Comparison of the free-fermion theory (54) with the experimental data on NDMAP at a temperature $T=0.05J$ (Ref. 5). The solid curve is the free-fermion result (54) near $H_{c1}$ and the solid circles denote the experimental data. We used the parameters $J=30.0$ K, $D/J=0.25$, and $g_e=2.10$. The magnetic field is applied along the $c$ axis, which corresponds to $z_p=1$, $z_q=z_r=0$. The dashed curve is high-temperature paramagnetic resonance frequency $\omega =g_e{\mu }_{B}H$.

Figure 5

QMC data of the resonance frequency (64) at $T=0.1J$. The $g$ factor is set to $g_e{\mu }_B=1$ for simplicity. The dashed line $\omega =H$ corresponds to the paramagnetic resonance frequency. Several cases $D/J=0.1$, $0.2$, and $-0.1$ are shown. Note that there is a zero point $H_0\sim 3J$ where the shift (5) vanishes.

Figure 6

Quantum Monte Carlo results of the normalized ESR shift (A3) induced by the exchange anisotropy (66) for temperatures $T/J=0.1$–$0.5$. The system size is $L=40$ sites. The maximum around $H=H_{c1}+T$ is also found in this case. The field dependence of $Y_{J^{\prime }}(T,H)$ in a lower field region $H<J$ at low temperatures $T<0.3J$ looks similar to that of $Y_D(T,H)$ except the overall sign $Y_{J^{\prime }}(T,H)\propto -Y_D(T,H)$. In a relatively higher temperature $T>0.4J$, the nonmonotonic behavior of the normalized shift vanishes.

Figure 7

Comparisons of QMC and (A5) at $T/J=0.1$ and $0.2$. The open circles ($T/J=0.1$) and triangles ($T/J=0.2$) denote the QMC results. The solid (dashed) curve represents (A5) with $Z_2^{\prime }=0.41$ at $T/J=0.1$ ($T/J=0.2$).