Collective Modes in Excitonic Magnets: Dynamical Mean-Field Study

We present a dynamical mean-field study of dynamical susceptibilities in the two-band Hubbard model. Varying the model parameters we analyze the two-particle excitations in the normal as well as in the ordered phase, an excitonic condensate. The two-particle dynamical mean-field theory spectra in the ordered phase reveal the gapless Goldstone modes arising from spontaneous breaking of continuous symmetries. We also observe the gapped Higgs mode, characterized by vanishing of the gap at the phase boundary. Qualitative changes observed in the spin susceptibility can be used as an experimental probe to identify the excitonic condensation.

Figure 1

The sketch of crystal field vs temperature ($\mathrm{\Delta }-T$) phase diagram (b) with marked cuts, along which the susceptibilities are calculated. The $1P$ spectral function at selected $(\mathrm{\Delta },T)$ points marked with stars: violet (3.55, $1/11$), green (3.55, $1/40$), and blue (3.8, $1/40$).

Figure 2

Evolution of the excitonic modes of dynamical susceptibility in the $U^2(1)$ model ($t_{ab}=0$) across $\mathrm{\Delta }$-driven transition ($T=1/40$). The columns correspond to $-\mathrm{Im}{\chi }_{\gamma \gamma }^{OO}(\mathbf{k},\omega )$ with $O^{\gamma }=I^x$, $I^y$, $R^x$, $R^y$ (left to right) along the high-symmetry lines in the two-dimensional Brillouin zone. The rows from top to bottom correspond to $\mathrm{\Delta }=3.9$, 3.8, 3.65, 3.55, 3.45 with ${\mathrm{\Delta }}_c\approx 3.75$. [The red line separates the normal state from the PEC phase.]

Figure 3

(a) The sound velocity $v_s$ of the GMs, the phase mode (${\chi }_{yy}^{RR}$, blue symbols), and the spin rotation mode (${\chi }_{xx}^{II}$, black symbols) in the $U^2(1)$ model as a function of the crystal field $\mathrm{\Delta }$. The dotted lines show the corresponding strong-coupling results. (b) The Higgs gap in the $U(1)$ model with $t_{ab}=0.02$ and 0.06 as a function of $\mathrm{\Delta }$. The line is a guide for eyes.

Figure 4

The same susceptibilities as in Fig. 2 ($T=1/40$) in the vicinity of the $\mathrm{\Gamma }$ point for the $U(1)$ model with cross-hopping $t_{ab}=0.06$. The rows from top to bottom correspond to $\mathrm{\Delta }=3.675$, 3.60, 3.5, with ${\mathrm{\Delta }}_c\approx 3.65$ [the red line separates the normal state from the PEC phase].

Figure 5

(a) Evolution of dynamical spin susceptibility $-\mathrm{Im}{\chi }_{zz}^{SS}(\mathbf{k},\omega )$ across the $\mathrm{\Delta }$-driven transition in the $U^2(1)$ model of Fig. 2 (the asterisk marks the normal phase). (b) The corresponding static susceptibilities $\mathrm{Re}{\chi }_{zz}^{SS}(\mathbf{k},0)$ throughout the Brillouin zone.

Figure 6

The same susceptibilities of $U^2(1)$ model as in Fig. 2 calculated across the thermally driven transition for $\mathrm{\Delta }=3.55$. The rows from top to bottom correspond to temperatures $T=1/11$, $1/16$, $1/30$, $1/40$ with $T_c\approx 1/13$.

Figure 7

(a) Evolution of dynamical spin susceptibility $-\mathrm{Im}{\chi }_{zz}^{SS}(\mathbf{k},\omega )$ across the thermally driven phase transition as in Fig. 6 for temperatures $T=1/11$, $1/20$, $1/30$ (the asterisk marks the normal phase). (b) The corresponding static susceptibilities $\mathrm{Re}{\chi }_{zz}^{SS}(\mathbf{k},0)$ throughout the Brillouin zone.