Novel Block Excitonic Condensate at $n=3.5$ in a Spin-Orbit Coupled $t_{2g}$ Multiorbital Hubbard Model

TL;DR

This study predicts a novel excitonic condensate at electron density 3.5 in a one-dimensional spin-orbit coupled $t_{2g}$ Hubbard model, revealing new magnetic phases relevant for materials with $d^{3.5}$ valence.

Contribution

It introduces the first evidence of a block excitonic condensate at $n=3.5$ in a $t_{2g}$ Hubbard model with spin-orbit coupling, expanding understanding of magnetic phases in such systems.

Findings

Existence of a block excitonic phase at $n=3.5$ and $q=\pi/2$

Coexistence of excitonic condensate with block magnetic order

Phase diagram showing robustness of the phase at large spin-orbit coupling

Abstract

Theoretical studies recently predicted the condensation of spin-orbit excitons at momentum $q$=$\pi$ in $t_{2g}^4$ spin-orbit coupled three-orbital Hubbard models at electronic density $n=4$. In parallel, experiments involving iridates with non-integer valence states for the Ir ions are starting to attract considerable attention. In this publication, using the density matrix renormalization group technique we present evidence for the existence of a novel excitonic condensate at $n=3.5$ in a one-dimensional Hubbard model with a degenerate $t_{2g}$ sector, when in the presence of spin-orbit coupling. At intermediate Hubbard $U$ and spin-orbit $\lambda$ couplings, we found an excitonic condensate at the unexpected momentum $q$=$\pi/2$ involving $j_{\textrm{eff}}=3/2,m=\pm1/2$ and $j_{\textrm{eff}}=1/2,m=\pm1/2$ bands in the triplet channel, coexisting with an also unexpected block magnetic order. We also present the entire $\lambda$ vs $U$ phase diagram, at a fixed and robust Hund coupling. Interestingly, this new `block excitonic phase' is present even at large values of $\lambda$, unlike the $n=4$ excitonic phase discussed before. Our computational study helps to understand and predict the possible magnetic phases of materials with $d^{3.5}$ valence and robust spin-orbit coupling.

Figures

Equations

$$H_{int}=(U-2J_H)\frac{N(N-1)}{2}+J_H(\frac{5}{2}N-2S^2-\frac{L^2}{2})$$

$${\rho }_{jm}(\omega <\mu )=-\frac{1}{\pi L}\sum _{i}Im[⟨{\psi }_0|a_{i,jm}^{†}\frac{1}{\omega -H+E_g+i\eta }{a}_{i,jm}|{\psi }_0⟩],$$

$${\rho }_{jm}(\omega >\mu )=-\frac{1}{\pi L}\sum _{i}Im[⟨{\psi }_0|a_{i,jm}\frac{1}{\omega +H+E_g+i\eta }{a}_{i,jm}^{†}|{\psi }_0⟩],$$

$${\rho }_{jm}(\omega ,c)=-\frac{1}{\pi }Im[⟨{\psi }_0|a_{c,jm}^{†}\frac{1}{\omega \pm H+E_g+i\eta }{a}_{c,jm}|{\psi }_0⟩],$$

Full text

SUPPLEMENTARY INFORMATION for

**Novel Block Excitonic Condensate at n=3.5 in a

Spin-Orbit Coupled $t_{2g}$ Multiorbital Hubbard Model

** by N. Kaushal, A. Nocera, G. Alvarez, A. Moreo, and E. Dagotto

I Two sites problem in strong coupling ($\lambda=0$)

In this section, we discuss the 2-sites three-orbital Hubbard model in the strong coupling limit ($U/t>>1$) for $n=3.5$ i.e a total of 7 electrons. Before discussing further details, below we show the Coulomb interaction term of the Hamiltonian in the rotationally invariant form (we have used $U^{{}^{\prime}}=U-2J_{H}$) which we use to calculate the energy of each atom given the occupation, spin moment ($S$), and orbital moment ($L$):

[TABLE]

If we work at $t=0$, atomic limit, the ground state (GS) of the 2-sites system consists only of $d^{3}$ and $d^{4}$ sites which have $(S,L)$=$(3/2,0)$ and $(1,1)$, respectively. Using Eq.(1), the total energy $E=E_{1}+E_{2}$ can be calculated to be $7U/2$. The total spin moment $S_{total}=S_{1}+S_{2}$ of the GS can be $\{1/2,3/2,5/2\}$ and the total orbital moment $L_{total}$ is 1.

Now to understand the effect of a nonzero kinetic energy, we perform exact diagonalization of the 2-site clusters with finite hopping. As soon as we turn-on the hopping, we noticed the degeneracy in $S_{total}$ breaks, as shown in Fig. 1(b). Interestingly all $S_{total}$ manifolds splits in their high-energy anti-bonding (dashed lines) and low-energy bonding (solid lines) states. The GS for finite hopping has a saturated total spin moment i.e. $S_{total}=5/2$, thus the GS has ferromagnetic ordering. The GS state has a degeneracy $6\times 3=18$ fold, where “$6$” and “$3$” arise from $SU(2)$-symmetries in the spin sector of the $S=5/2$ and $L=1$ space. We also noticed that for $U/t>>1$, for all $S_{total}$ manifolds, the energies for the bonding(-) and anti-bonding(+) states change linearly with $t$ as $\frac{7U}{2}\pm\tilde{t}$, where the $\tilde{t}$ for $S_{total}=5/2$, $3/2$, $1/2$ manifolds are $t$, $\frac{2}{3}t$, and $\frac{1}{3}t$, respectively.

The above discussed bonding and anti-bonding states can be interpreted in terms of momentum $q=0$ and $q=\pi$ states, respectively, of the incipient bands which fully develops for the larger system and leads to band formation near the chemical potential. The presence of bands near the chemical potential is indeed confirmed by the density-of-state calculations performed on 4 and 16 sites systems using Lanczos and DMRG techniques, as shown in the main paper. We also noticed that the density-of-states of the 2 sites system can explain most of the features present in $\rho(\omega)$ of 4-site clusters in strong coupling limit. In Fig. 1(c) we show the $\rho(\omega-\mu)$ for the 2-site cluster in the strong coupling limit, and in Fig. 1(a) we show all the “N+1” and “N-1” particle states leading to excitations above and below the chemical potential, respectively. For 4-site clusters we observed that these peaks are present nearly at the same values $\omega^{\prime}s$ in terms of $U$, as shown in the main paper.

II Single particle density-of-states

As discussed in the main paper, we calculated the density-of-states $\rho_{jm}(\omega)$ using the DMRG correction-vector target method. The electron ($\omega<\mu$) and hole ($\omega>\mu$) components of $\rho_{jm}(\omega)$ are defined as:

[TABLE]

[TABLE]

where $i$ is the site index. In the approach above, the sum over all the sites has to be done, which means that for every $\omega$ an “$L$” no. of runs has to be performed for each site $i$ with different correction vectors $|x(\omega)\rangle=\frac{1}{\omega+i\eta+E_{g}+H}a_{i,jm}^{\dagger}|\psi_{0}\rangle$, and similarly for the electron part of $\rho(\omega)$. To reduce the cost by $``L"$ times we perform the simulations only with respect to the central site i.e. calculating the following:

[TABLE]

where $c$ is a central site. This becomes exact for a system with periodic boundary conditions i.e. $\rho_{jm}(\omega,c)=\rho_{jm}(\omega)$, but our DMRG simulations are done for $L=16$ sites with open boundary conditions, for which calculating $\rho_{jm}(\omega,c)$ is still a good approximation.

To reproduce the data shown in this publication, the open source DMRG++ program and input files are available at https://g1257.github.io/dmrgPlusPlus/ .