# 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.

## Key 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.