Dimerized ferromagnetic Heisenberg chain

Ferromagnetic, in contrast to antiferromagnetic, Heisenberg chains can undergo a spin-Peierls dimerization only at finite temperatures. They show reentrant behavior as a function of temperature, which might play a role for systems with small effective elastic constants as, for example, monatomic chains on surfaces. We investigate the physical properties of the dimerized ferromagnetic Heisenberg chain using a modified spin-wave theory. We calculate the exponentially decaying spin- and dimer-correlation functions, analyze the temperature dependence of the corresponding coherence lengths, the susceptibility, as well as the static and dynamic spin-structure factors. By comparing with numerical data obtained by the density-matrix renormalization group applied to transfer matrices, we find that the modified spin-wave theory yields excellent results for all these quantities for a wide range of dimerizations and temperatures.

Figure 1

Magnon dispersions obtained from MSWT for various dimerizations. The lower branches correspond to ${\omega }_k^{-}$ while the upper branches are given by ${\omega }_k^{+}$.

Figure 2

Occupation numbers for the (a) lower and (b) upper branches for $S=1$ and $T=0.5J$ and various dimerizations as indicated.

Figure 3

Nearest neighbor spin-correlation functions on (a) the strong bonds and (b) the weak bonds as a function of dimerization for various temperatures and $S=1$. The solid lines show MSWT results. For comparison, TMRG data are shown as symbols. The inset shows data for low temperatures.

Figure 4

Spin-correlation functions (solid lines) and alternation correlation functions $\Delta \left(r\right)$ (dashed lines) for $S=1$ and $\delta =0.1$ and various temperatures as a functions of distance $r$ obtained by MSWT. For a given temperature, the exponential decay of both quantities is determined by the same correlation length.

Figure 5

Spin-correlation functions (solid lines) and alternation correlation functions (dashed lines) obtained by MSWT for $S=1$ and $T/J=0.1$ and various dimerizations as indicated.

Figure 6

Comparison of the functions $c(T,\delta )$ as given by Eq. (17) (lines) and TMRG data (symbols) for $S=1$ and various $\delta$ as indicated.

Figure 7

Comparison of $c(T,\delta )$ for $S=1/2$ and 1 as a function of temperature at $\delta =0.1$. The MSWT results (lines) are compared with TMRG data (circles and squares).

Figure 8

(a) Spin-correlation function $\langle {\mathit{S}}_j·{\mathit{S}}_{j+r}\rangle$ on a logarithmic scale for $S=1$, $T/J=0.1$, and $\delta =0.3$ obtained by TMRG compared with the analytical results from the asymptotic expansion, Eq. (14), and the short-range expression, Eq. (19), up to $r=4$. The solid line connects data obtained from a numerical solution of the MSWT, Eqs. (10) and (11), for integer $r$. (b) Same as above on a linear scale.

Figure 9

Difference between the spin correlations on weak and strong bonds [see Eq. (13)] for $S=1$, $T/J=0.1$, and two representative dimerization parameters. The filled symbols are data obtained from TMRG whereas the open symbols show results obtained by MSWT.

Figure 10

Inverse susceptibilities $1/\chi$ as a function of $T/J$ for $S=1$ and different values of $\delta$ as indicated. The lines show the results from MSWT, the symbols for $\delta <1$ TMRG data, and the symbols for $\delta =1$ the exact solution for decoupled dimers. An increasing dimerization suppresses $\chi$, and the decoupled dimers follow a Curie law $\chi \sim 1/T$.

Figure 11

Diagonal elements of the dynamic spin-structure factor $S(q,q,\omega )$ for a dimerized ferromagnetic chain and $S=1$, $T/J=0.1$: (a) $S(q,q,\omega )$ spectra for different momenta $q$ and $\delta =0.6$. The red dashed lines show the edge boundaries, and the symbols show the peak positions projected onto the $(q,\omega )$ plane. (b) Broadened spectra $I_{\Lambda }(q,q,\omega )$ for $q=4\pi /5$, for $\delta =0.4$, and for different $\Lambda =0,0.001,0.01,$ and $0.1J$. Vertical lines in the $\Lambda =0$ panel indicate the positions the of edge singularities.

Figure 12

Off-diagonal element of the dynamic spin-structure factor matrix $S(q,q+\pi ,\omega )$ for $q=\pi /5$, $S=1$, and $T/J=0.1$ with (a) $\delta =0.3$, and (b) $\delta =0.6$. The vertical lines indicate positions of edge singularities (see text).

Figure 13

Diagonal static spin-structure factor for $S=1$ and $T/J=0.1$. Symbols denote data from TMRG, lines results from MSWT. Main: Data for dimerizations $\delta =0$ (diamonds), $\delta =0.3$ (triangles), $\delta =0.6$ (squares), and $\delta =0.9$ (circles). Left inset: High-momentum peak for $\delta =0.9$ at $T/J=0.1$ (circles), $T/J=0.08$ (filled triangles), and $T/J=0.0625$ (open triangles). Right inset: Data for $\delta =0.9$ with contributions stemming from the lower and upper branches shown by dashed-dotted and dashed lines, respectively.

Figure 14

Off-diagonal elements of the static spin-structure factor matrix for $S=1$, $T/J=0.1$, and various dimerizations as a function of $q$. The symbols show numerical data obtained by TMRG, whereas the solid lines are results from MSWT; see Eq. (41).