Antiferromagnetic Heisenberg spin-1 chain: Magnetic susceptibility of the Haldane chain described using scaling

The phenomenological expression $\chi T∕(N{g}^2{\mu }_B^2∕k)=C_{1n}\exp (-W_{1n}∕T)+C_{2n}\exp (-W_{2n}∕T)$ describes very accurately the temperature dependence of the magnetic susceptibility computed for antiferromagnetic rings of Heisenberg spins $S=1$, whose size $n$ is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For $n$ large, the coefficients of the expression converge towards $C_1=0.125$, $W_1=0.451J$, $C_2=0.564$, $W_2=1.793J$ ($J$ is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie–Weiss temperature, the correlation length at $T=0$ and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of ${\mathrm{Y}}_2\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_5$ and of $\mathrm{Ni}{\left({\mathrm{C}}_2{\mathrm{H}}_8{\mathrm{N}}_2\right)}_2\mathrm{N}{\mathrm{O}}_2\mathrm{Cl}{\mathrm{O}}_4$ (NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.

Figure 1

(Color online) $\partial \log \left(T\right)∕\partial \log \left(\chi T\right)$ deduced from exact calculations of the susceptibility of AF Heisenberg rings of $n$ spins $S=1$ is shown vs $T∕J$ for $n=6$, 16, and 20. The linear regime at high temperature, which extrapolates to the origin of the axes, points an exponential solution of type (6) for $T$ larger than $T_M$, where $\partial \log \left(T\right)∕\partial \log \left(\chi T\right)=1$. A distinct exponential regime is observed at temperatures smaller than $T_m$ where $\partial \log \left(T\right)∕\partial \log \left(\chi T\right)=0.5$. The low temperature gap saturates to the Haldane gap value $\Delta$ when $n$ increases.

Figure 2

(Color online) Results of exact calculations of the susceptibility $\chi \left(T\right)$ of AF rings of $n$ Heisenberg spins $S=1$, for $n$ equal to 6, 8, 10, 16, and 20. For $T>T_M$, there is a high temperature Arrhenius regime, whereby the $\chi T$ data first depart from the theoretical Curie constant for $S=1$ $(C_H=\frac{2}{3})$. Below $T_m$, there is a distinct low temperature Arrhenius regime whose Curie constant is $C_{Ln}=2∕n$ and whose activation energy $W_{Ln}$ is the Haldane gap. The different parameters $C$ and $W$ for all $n$ are listed in Table .

Figure 3

(Color online) The same susceptibility data as in Fig. 2 are shown here versus $\log \left(T\right)$, with the best values of the coefficients $C_{1n}$,$W_{1n}$ and $C_{2n}$,$W_{2n}$, characterizing the low and the high temperature regimes in a single fit.

Figure 4

(Color online) Variation of the parameters $C_{Ln}$, $W_{Ln}$, $C_{Hn}$, and $W_{Hn}$ of the exponentials characterizing the asympotic low and high temperature Arrhenius regimes of $\chi T∕(N{g}^2{\mu }_B^2∕k)$.

Figure 5

(Color online) Best values of the parameters $C_{1n}$, $W_{1n}$, $C_{2n}$, and $W_{2n}$, characterizing the overall temperature dependence of the susceptibility of AF rings of $n$ spins $S=1$ from a two exponential fit.

Figure 6

(Color online) Fit of the magnetic susceptibility data of $\mathrm{Ni}{\left({\mathrm{C}}_2{\mathrm{H}}_8{\mathrm{N}}_2\right)}_2\mathrm{N}{\mathrm{O}}_2\mathrm{Cl}{\mathrm{O}}_4$ (NENP) along two crystallographic axes, giving from the $W_2$ value, $J=43\mathrm{K}$ along all axes. The anisotropy at lower temperature is considered by permitting the Haldane gap to vary from $0.53J$ along the $b$ axis to $0.4J$ along the $a$ axis instead of $0.447J$ expected for the isotropic case. An additional constant ${\chi }_0$ has been introduced to account for a residual contribution to the susceptibility that is expected to depend on axial symmetry as observed.