Effect of intraband Coulomb repulsion on the excitonic spin-density wave

We present a study of the magnetic ground state of a two-band model with nested electron and hole Fermi surfaces and both interband and intraband Coulomb interactions. Our aim is to understand how the excitonic spin-density-wave (ESDW) state induced by the interband Coulomb repulsion is affected by the intraband interactions. We first determine the magnetic instabilities of our model in an unbiased way by employing the random-phase approximation (RPA) to calculate the static spin susceptibility in the paramagnetic state. From this, we construct the mean-field phase diagram, demonstrating the robustness of the ESDW against the intraband interaction. We then calculate the RPA transverse spin susceptibility in the ESDW state and show that the intraband Coulomb repulsion significantly renormalizes the paramagnon line shape and suppresses the spin-wave velocity. We conclude with a discussion of the relevance of this suppression for the commensurate ESDW state of Mn-doped Cr alloys.

Figure 1

(a) Band structure and (b) Fermi surface of the noninteracting model for $t_1=t_2=t$ and $E_G=3t$ at half-filling. The hole and electron pockets are nested by the vector ${\mathbf{Q}}_1=(\pi ,\pi )$. The Fermi surfaces for $E_G=0.05t$ and $E_G=1.5t$ are shown in (c) and (d), respectively, illustrating the weaker intraband nesting of parts of the electron (hole) Fermi pockets with the vectors ${\mathbf{Q}}_1=(\pi ,\pi )$, ${\mathbf{Q}}_2=(\pi ,0)$, and ${\mathbf{Q}}_3=(0,\pi )$.

Figure 2

Total static transverse spin susceptibility for three representative points of the parameter space for finite temperatures and $t_1=t_2=t$ and $E_G=3t$. Note the different logarithmic scales.

Figure 3

Ground-state MF phase diagrams for different parameters of the band structure at half-filling. Solid and dashed lines indicate first-order and second-order phase transitions, respectively. Note that the transition between the $U_{12}=0$ PM and the ESDW phases is of second order. In the shaded region, the MF ground state is the ESDW state, but the static spin susceptibility shows an IC AFM intraband instability above the critical temperature of the ESDW state. The red crosses show points where the critical temperatures of the ESDW and IC AFM states are equal.

Figure 4

(a) Logarithm of the imaginary part of the transverse spin susceptibility for $U_{11}=2U_{12}=3.6t$ and $\Delta =0.0213t$. The spin wave is visible as a dark feature for $\omega <2\Delta$ near $\mathbf{q}={\mathbf{Q}}_1$ and $\mathbf{q}=\mathbf{0}$. Note the logarithmic color scale. (b) Ratio of the total transverse spin susceptibility in panel (a) and the $U_{11}=0$ result presented in Ref. 17.

Figure 5

(a) Cuts through the excitation spectrum for $U_{11}=0$, $U_{12}=1.8t$, and $U_{11}=2U_{12}=3.6t$. Note the increase of weight of the susceptibility for $q_x<\pi /2$ for $U_{11}\ne 0$, while for $q_x>\pi /2$ the weight changes only close to $q_x=\pi$. (b) Comparison of the paramagnon line shape for the susceptibilities in (a). The lines are defined as in panel (a). (c) Paramagnon line shape for $\omega =0.1t$ and various $U_{11}$ along the line of constant $\Delta =0.0213t$ in the phase diagram [see Fig. 6b].

Figure 6

(a) Comparison of the numerical (open circles) and the approximate (solid and dash-dotted lines) results for the spin-wave velocity for $\Delta =0.0213t$ depending on the intraband Coulomb repulsion. (b) Line of $\Delta =0.0213t$ (dash-dotted) for which we determine the spin-wave velocity. We continue the constant-$\Delta$ line to the limit of metastability of the ESDW state within the $\mathrm{Hub}({\mathbf{Q}}_2+{\mathbf{Q}}_3)$ phase.