Thermally Activated Peierls Dimerization in Ferromagnetic Spin Chains

We demonstrate that a Peierls dimerization can occur in ferromagnetic spin chains activated by thermal fluctuations. The dimer order parameter and entanglement measures are studied as functions of the modulation of the magnetic exchange interaction and temperature, using a spin-wave theory and the density-matrix renormalization group. We discuss the case where a periodic modulation is caused by spin-phonon coupling and the case where electronic states effectively induce such a modulation. The importance of the latter for a number of transition metal oxides is highlighted.

Figure 1

Phase diagrams for the dimerized AFM and FM $S=1/2$ Heisenberg chains. The dotted-dashed line depicts the second-order SP transition from the uniform ($U$) to the dimerized ($D$) phase for the AFM chain. For the FM chain, the $D$ phase exists only at finite $T$ (in units of $k_B=1$) and only if $K<K_c\simeq 0.118|J|$—the transition is either second or first order, as shown by solid and dashed lines, respectively, and changes its character at the TCP. The insets show the order parameter $\delta (T)$ for (a) the FM chain with $K/|J|=0.1$ and (b) the AFM chain with $K/J=2$ (the lines are guides to the eye).

Figure 2

Order parameter ${\Delta }_{SS}^{-}$ (3) as a function of $\delta$ for the $S=1$ dimerized FM. The lines (symbols) denote the MSWT (TMRG) results. The inset shows the phase diagrams obtained by TMRG for the Hamiltonian (1) with $S=1/2$ (solid line) and $S=1$ (dashed line) with both axes scaled appropriately. The phase transition is first (second) order for $T<T_{\mathrm{TCP}}$ ($T>T_{\mathrm{TCP}}$). The TCP is marked by a dot (square) for $S=1/2$ ($S=1$).

Figure 3

TMRG results for $S=1/2$: (a) $B_{s(w)}$ for the strong (solid line) and the weak bond (dashed line), (b) ${\Delta }_{SS}^{-}$ for the same values of $\delta$ as in (a), (c) the entanglement entropy $S_{s(w)}^{\mathrm{ent}}$ for different $\delta$ as a function of temperature, and (d) $S_{s(w)}^{\mathrm{ent}}$ as a function of $\delta$ for two temperatures.

Figure 4

Phase diagram and dimerization parameters ${\delta }_S$ and ${\delta }_{\tau }$ for the spin-orbital model (5) with ${\gamma }_H=0.1$ in MF decoupling. The dashed lines denote the phase boundaries between the uniform ($U$) and dimerized ($D$) phases.