Magnetic excitations in one-dimensional spin-orbital models

We study the dynamics and thermodynamics of one-dimensional spin-orbital models relevant for transition-metal oxides. We show that collective spin, orbital, and combined spin-orbital excitations with infinite lifetime can exist, if the ground state of both sectors is ferromagnetic. Our main focus is the case of effectively ferromagnetic (antiferromagnetic) exchange for the spin (orbital) sector, and we investigate the renormalization of spin excitations via spin-orbital fluctuations using a boson-fermion representation. We contrast a mean-field decoupling approach with results obtained by treating the spin-orbital coupling perturbatively. Within the latter self-consistent approach we find a significant increase of the linewidth and additional structures in the dynamical spin structure factor as well as Kohn anomalies in the spin-wave dispersion caused by the scattering of spin excitations from orbital fluctuations. Finally, we analyze the specific heat $c(T)$ by comparing a numerical solution of the model obtained by the density-matrix renormalization group with perturbative results. At low temperatures $T$ we find numerically $c(T)~T$ pointing to a low-energy effective theory with dynamical critical exponent $z=1$.

Figure 1

(a) Ground state of the FM spin-orbital model, Eq. (1.1), (b) a spin excitation, and (c) a coupled spin-orbital excitation. The two orbitals per site are assumed to be degenerate (the splitting is only for clarity of presentation).

Figure 2

Nearest-neighbor spin and orbital correlation functions for the spin-orbital model (2.1) with $\Gamma =1$ as a function of temperature $T$ in units of $J$ (we set $k_B=1$). In both panels $y=0.4$, 0.3, 0.2, 0.15, 0.1, 0.05, 0.0, $-0.1$, $-0.2$, $-0.3$ in the arrow direction. The spin correlations switch from AF to FM at $y=0.1$ (dashed lines). The dotted lines in the upper (lower) panel correspond to the limiting values $1$ and $-1.4015$ ($-1/\pi$ and $1/\pi$), respectively.

Figure 3

(a) Phase diagram of the Hamiltonian (3.1) with $\Gamma =1$ and $x=1$ in mean-field decoupling. The shaded area represents the dimerized phase. The phase transition at $T_2$ is first order whereas the transition at $T_1$ is of second order. The two transition lines merge at the tricritical point $y_{\mathrm{tp}}$. (b) Dimerization parameters ${\delta }_S$ and ${\delta }_{\tau }$ for $x=1$ and $y=0.14$. The lines are guides to the eye. The shaded area marks the temperature range where the dimerization is nonzero.

Figure 4

(a) Dynamical spin structure factor $S(q,\omega )$ as obtained for $T/\left|J_S\right|=0.1$ and $0⩽q⩽\pi$. The dashed line indicates the upper boundary of the two-magnon continuum. The dots are projections of the peak positions onto the $(q,\omega )$ plane. They are connected by the dotted line which is a guide to the eye. (b) Dynamical spin structure factor $S(q,\omega )$ for the same parameters at $q=4\pi /5$ (solid line) and the corresponding density of states (dashed line).

Figure 5

Magnon dispersion ${\omega }_B(q)$ (dashed line) for $y=-1$ and fermionic two-particle continuum (shaded area).

Figure 6

Self-energy diagrams which contribute to a renormalization of the magnon Green's function (5.2) in a perturbation theory. (a), (b) Self-energy diagrams with momentum exchange between the magnon and the fermions, and (c) self-energy diagram without momentum exchange. All self-energy diagrams are second order. Bare fermionic propagators are shown by solid lines, whereas bare bosonic propagators ${\mathcal{G}}_B^0$ are shown as dashed lines.

Figure 7

Second-order diagram for a system of interacting fermions with momentum exchange.

Figure 8

Perturbative results for the BF model at zero temperature with $x=1$. In the left (right) panel $y=-1$ ($y=-2$). (a), (c) $S(q,\omega )$ with the inset showing the region for which $\mathrm{Im}\hspace{0.5em}{\Sigma }_{\mathrm{BF}}^{+}(q,\omega )$ is nonzero as a shaded area and the renormalized spin-wave dispersion as a dotted line. While $S(q,\omega )$ is sharply peaked at low momenta, a significant broadening occurs at higher $q$. Moreover we find additional structures which, as explained in the text, are due to coupled spin-orbital excitations. (b), (d) $-\mathrm{Im}\hspace{0.5em}{\Sigma }_{\mathrm{BF}}^{+}(q,\omega )$ as given in Eq. (5.5) for the corresponding values of $q$ shown in (a) and (c), respectively. The coupled spin-orbital excitations show up as peaks and edges in $\mathrm{Im}\hspace{0.5em}{\Sigma }_{\mathrm{BF}}^{+}(q,\omega )$ (notice the logarithmic scale).

Figure 9

Renormalized magnon dispersions ${\omega }_q$ for $x=1$ and selected values of $y$. Inset: The most pronounced Kohn anomalies occur at $q$ near $\pi /2$.

Figure 10

Magnon linewidth ${\Gamma }_q$ [FWHM of $S(q,\omega )$] at zero temperature (data points), as obtained for $x=1$ and the representative values of $y$ indicated in the plot. The lines are guides to the eye.

Figure 11

(a) $-\mathrm{Im}\hspace{0.5em}\Sigma (q,\omega )$ for $y=-1$ as in Fig. 8b but on a linear scale. The lines are a guide to the eye connecting the clearly visible features of the continuum of spin-orbital excitations (see text). (b) Collective spin wave ${\omega }_q$ for $y=-1$ (solid line) together with the dispersions of the spin-orbital excitation features (dashed and dashed-dotted lines) extracted from panel (a). All lie within the Bose-Fermi continuum (shaded area).

Figure 12

Dynamical spin structure factor $S(q,\omega )$ calculated perturbatively for the spin-orbital chain at temperature $T/J=0.1$ with $y=-1$. The structure factor is peaked at the renormalized magnon frequency ${\omega }_q$.

Figure 13

Magnon linewidth ${\Gamma }_q$ [FWHM of $S(q,\omega )$] as a function of $q$, as obtained for $y=-1$ and temperatures $T/J=0$, 0.1, and 0.2. The lines are guides to the eye.

Figure 14

Diagrammatic representations of the second-order contributions to the free energy as given by Eq. (6.6).

Figure 15

Specific heat per site $c/(\mathit{Jx}){}^2$ as a function of $T/(\mathit{Jx})$ for (a) $x=10$ and (b) $x=2$. The circles denote the numerical data from TMRG with the solid line obtained by a low-temperature fit of the TMRG data for the inner energy. The dashed lines correspond to the MF decoupling solution while the dashed-dotted lines are the results obtained by perturbation theory. The perturbative results are expected to be valid for $1/x^2\ll T/(\mathit{Jx})\ll 1$. Insets: Specific heat, $c_{\mathrm{MF}}$, within the MF solution with the contributions from the spin, $c_{\mathrm{MF}}^S$, and the orbital, $c_{\mathrm{MF}}^{\tau }$, sectors shown separately.

Figure 16

Perturbative contribution to the specific heat per site $c_2/(\mathit{Jx}){}^2$, Eq. (6.9), as a function of $T/(\mathit{Jx})$ for $x=10$ (solid line) and $x=2$ (dashed line).