Thermodynamics of the one-dimensional frustrated Heisenberg ferromagnet with arbitrary spin

The thermodynamic quantities (spin-spin correlation functions $\langle {\mathbf{S}}_0{\mathbf{S}}_n\rangle$, correlation length $\xi$, spin susceptibility $\chi$, and specific heat $C_V$) of the frustrated one-dimensional $J_1$-$J_2$ Heisenberg ferromagnet with arbitrary spin quantum number $S$ below the quantum critical point, i.e., for $J_2<\left|J_1\right|/4$, are calculated using a rotation-invariant Green-function formalism and full diagonalization as well as a finite-temperature Lanczos technique for finite chains of up to $N=20$ sites. The low-temperature behavior of the susceptibility $\chi$ and the correlation length $\xi$ are well described by $\chi =\frac{2}{3}{S}^4\left(|J_1|-4J_2\right)T^{-2}+A{S}^{5/2}{\left(|J_1|-4J_2\right)}^{1/2}{T}^{-3/2}$ and $\xi =S^2\left(|J_1|-4J_2\right)T^{-1}+B{S}^{1/2}{\left(|J_1|-4J_2\right)}^{1/2}{T}^{-1/2}$ with $A\approx 1.1\cdots 1.2$ and $B\approx 0.84\cdots 0.89$. The vanishing of the factors in front of the temperature at $J_2=\left|J_1\right|/4$ indicates a change of the critical behavior of $\chi$ and $\xi$ at $T\to 0$. The specific heat may exhibit an additional frustration-induced low-temperature maximum when approaching the quantum critical point. Such a maximum appears for $S=1/2$ and $S=1$, but was not found for $S>1$.

Figure 1

Spin-spin correlation functions $\langle {\mathbf{S}}_0{\mathbf{S}}_1\rangle /S^2$ (NN), $\langle {\mathbf{S}}_0{\mathbf{S}}_2\rangle /S^2$ (NNN), and $\langle {\mathbf{S}}_0{\mathbf{S}}_{10}\rangle /S^2$ obtained by the RGM for spin quantum numbers $S=1$ (solid), $3/2$ (long-dashed), and $S=2$ (short-dashed) and frustration parameters $J_2=0$ (black), $0.1$ (red), and $0.2$ (blue).

Figure 2

Susceptibility $\chi$ for spin quantum number $S=1$ and frustration parameters $J_2=0$ (black), $0.1$ (red), $0.2$ (blue), and $0.22$ (violet) [solid lines: RGM for $N\to \infty$, dashed lines: RGM for $N=12$, symbols: ED for $N=12$]. Inset: Susceptibility $\chi$ for $J_2=0.1$ and $S=1/2$, 1, $3/2$, and 2 (from bottom to top). The data for $S=1/2$ are taken from Ref. 9.

Figure 3

Correlation length $\xi$ for spin quantum number $S=1$ and frustration parameters $J_2=0$ (black), $0.1$ (red), $0.2$ (blue), and $0.22$ (violet). Inset: Correlation length $\xi$ for $J_2=0.1$ and $S=1/2$, 1, $3/2$, and 2 (from bottom to top). The data for $S=1/2$ are taken from Ref. 9.

Figure 4

RGM results for $\frac{3}{2}\chi T^2/S^4$ (upper panel) and $\xi T/S^2$ (lower panel) versus $\sqrt{T}$ for spin quantum numbers $S=1$ (solid), $3/2$ (long-dashed), and $S=2$ (short-dashed) and frustration parameters $J_2=0$ (black), $0.1$ (red), and $0.2$ (blue).

Figure 5

Next-to-leading coefficients $y_1/S^{5/2}$ (upper panel) and $x_1/S^{1/2}$ (lower panel) in dependence on $J_2$ obtained by fitting the low-temperature data of $\chi$ and $\xi$ (see main text). The symbols represent the data points for $S=1/2$, 1, $3/2$, 2, and $S=5$, and the lines of the same color show the corresponding fit curves of these points using the fit function $f\left(J_2\right)=a\sqrt{1-b{J}_2}$. The thick black solid line shows the quadratic fit for $S=1/2$ without the data point at $J_2=0.22$ as used in Ref. 9. Insets: Leading coefficients $y_0$ (upper panel) and $x_0$ (lower panel) in dependence on $J_2$ obtained by fitting the low-temperature data of $\chi$ and $\xi$ (see main text). The values for $y_0/S^4$ as well as for $x_0/S^2$ practically coincide for $S=1/2$, 1, $3/2$, 2, and $S=5$ used to determine $y_0/S^4$ and $x_0/S^2$.

Figure 6

Specific heat for $S=1/2$ (green), $S=1$ (black), $S=3/2$ (red), and $S=2$ (blue) obtained by RGM for $J_2=0$ (solid), $J_2=0.2$ (long-dashed), and $J_2=0.22$ (short-dashed). Inset: ED results for $N=8$ and $S=1$ (black), $3/2$ (red), and 2 (blue) for $J_2=0.2$ (solid) and $J_2=0.22$ (dashed).

Figure 7

Exact diagonalization (ED) and finite-temperature Lanczos (FTL) results for the low-temperature maximum in the specific heat for $S=1$ and $J_2=0.2$ (upper panel) and $J_2=0.22$ (lower panel) for chain lengths $N=10,12,14,16,18$, and 20. For $N=14$ we present both the exact ED and the approximate FTL data to demonstrate the accruracy of the FTL technique down to very low temperatures.