BCS-BEC crossover in a ${\left(t_{2g}\right)}^4$ excitonic magnet

The condensation of spin-orbit-induced excitons in $t_{2g}^4$ electronic systems is attracting considerable attention. At large Hubbard $U$, antiferromagnetism was proposed to emerge from the Bose-Einstein Condensation (BEC) of triplons ($J_{\text{eff}}=1$). Here, we show that even at intermediate $U$ regimes, the spin-orbit exciton condensation is possible leading also to staggered magnetic order. The canonical electron-hole excitations (excitons) transform into local triplon excitations at large $U$, and this BEC strong coupling regime is smoothly connected to the intermediate $U$ excitonic insulator region. We solved the degenerate three-orbital Hubbard model with spin-orbit coupling ($\lambda$) in one dimension using the density matrix renormalization group, while in two dimensions we use the Hartree-Fock approximation (HFA). Employing these techniques, we provide the full $\lambda$ versus $U$ phase diagrams for both one- and two-dimensional lattices. Our main result is that at intermediate Hubbard $U$, increasing $\lambda$ at fixed $U$ the system transitions from an incommensurate spin-density-wave metal to a Bardeen-Cooper-Schrieffer (BCS) excitonic insulator, with coherence length $r_{\mathrm{coh}}$ of $\mathcal{O}\left(a\right)$ and $\mathcal{O}\left(10a\right)$ in $1d$ and $2d$, respectively, with $a$ being the lattice spacing. Further increasing $\lambda$, the system eventually crosses over to the BEC limit (with $r_{\mathrm{coh}}\ll a$).

Figure 1

Visual representation of our main results, all supported by actual DMRG and Hartree Fock calculations. In (a), the single-particle excitations of the $j_{\text{eff}}=3/2$ and ${\mathrm{j}}_{\text{eff}}=1/2$ bands are shown at intermediate $U$, where near the chemical potential a gap opens due to the formation of excitons (note electron and hole have the same $m$). The $j_{\text{eff}}=3/2$ and $j_{\text{eff}}=1/2$ bands open gaps near momentum $q\approx 0$ and $q\approx \pi$, respectively. In (b), the real-space perspective of the excitonic state (at intermediate $U$) is shown, where the exciton's mean radius (characterized by the coherence length) decreases by increasing $\lambda$. In (c), the excitonic condensation in strong coupling is depicted. The local exciton creation operator leads to the creation of both a triplon and a quintuplon when acting on $|J_{\text{eff}}=0\rangle$. Including kinetic energy, i.e., the tight-binding term, the triplon and quintuplon excitations gain bandwidths, and decreasing $\lambda$ leads to the BEC of the triplon component.

Figure 2

DMRG results at $U/W=2$. (a) and (b) show the real-space spin-spin correlations at $\lambda /W=0$ and $\lambda /W=0.29$, repectively. In (c) and (d), the real-space orbital-orbital correlations are shown at $\lambda /W=0.0$ and $\lambda /W=0.29$, respectively. The spin structure factor $S\left(q\right)$ and orbital structure factors $L\left(q\right)$ are in (e) and (f), respectively, for various $\lambda \mathrm{s}$. In (a)–(d), the system size was $L=32$ and one site is fixed at the center, i.e., $i^{\prime }=15$. A system size $L=16$ is used for (e,f).

Figure 3

Results calculated using DMRG for a one-dimensional chain of $L=16$ sites. In (a) and (b), the orbital structure factor $L\left(q\right)$ and spin structure factor $S\left(q\right)$ are shown, respectively, at $U/W=10$ and various $\lambda /W$'s. (c) shows the measure of staggering in excitonic correlations ${\mathrm{\Delta }}_{1/2}$ for various values of $\lambda$ and $U$.

Figure 4

(a) and (b) show the momentum distribution function of excitons $\mathrm{\Delta }\left(q\right)$ for system sizes $L=16,24,\text{and}32$ at two representative points of the BCS and BEC regions, respectively. (c) and (d) show the coherence length for two paths inside the excitonic condensate regime, shown in the phase diagram Fig. 6. The pair-pair susceptibilities $\mathrm{\Delta }(q,\omega )$ are in (e) at $U/W=10$ and $\lambda /W=0.2$ i.e. in the nonmagnetic insulator (NMI) region, and in (f) at $U/W=10$ and $\lambda /W=0.11,$ i.e., in the BEC region. All the calculations used DMRG.

Figure 5

The single-particle spectral function calculated using DMRG. The noninteracting band structure is shown in (a). (b) contains $A_{jm}(q,\omega )$ for $\lambda =0$ and $U/W=2.0$, in the IC-SDW metallic phase. In (d) and (e), $A_{jm}(q,\omega )$ at $\lambda /W=0.29$ and $U/W=2.0$ is shown inside the excitonic insulator phase ($r_{\mathrm{coh}}=1.019$). (c) and (f) show the single-particle DOS $[{\rho }_{jm}\left(\omega \right)]$ at $\lambda =0.0$ and 0.29, respectively. The scaling of the charge gap is in inset (g) for ($U/W=2,\lambda /W=0$) and ($U/W=2,\lambda /W=0.29$).

Figure 6

$\lambda /W$ vs $U/W$ phase diagram calculated using DMRG for a one-dimensional system. The two red arrows corresponds to the paths used in (c) and (d) of Fig. 4. The vertical red arrow is chosen at $U/W=1.75$, and depicts Path-1 of Fig. 4. The diagonal red arrow corresponds to the Path-2 used in Fig. 4. The notation RBI, IC-SDW, FM, IOO, EC, AFM, and NMI stands for relativistic band insulator, incommensurate spin-density wave, ferromagnetic, excitonic condensate, antiferromagnetic, and nonmagnetic insulator, respectively.

Figure 7

(a) shows the $\lambda$ vs $U$ phase diagram for the square lattice, calculated using the Hartree-Fock approximation. In (b) and (c), the momentum distribution function of excitons $\mathrm{\Delta }\left(\mathbf{q}\right)$ is shown at $\lambda =0.53$ and $\lambda /W=0.592$, respectively, for fixed $U/W=0.5$ and a $24\times 24$ system. The coherence length for various values of $\lambda$, at fixed $U/W=0.5$, for system sizes $8\times 8$, $16\times 16$, and $24\times 24$ are in (d). The $N(\pi ,\pi )$, $S(\pi ,\pi )$, and $L(\pi ,\pi )$ are in (e) for various values of $\lambda$, at fixed $U/W=0.5$ and using a $24\times 24$ cluster.