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arXiv:1901.05578 · Physical review. B./Physical review. B

Block excitonic condensate at n=3.5 in a spin-orbit coupled t2g multiorbital Hubbard model

Nitin Kaushal, Alberto Nocera, Gonzalo Álvarez, Adriana Moreo, Elbio R. Dagotto

Published Apr 8, 2019• 6 citations
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Abstract

Theoretical studies recently predicted the condensation of spin-orbit excitons at momentum $q=\ensuremath{\pi}$ in ${t}_{2g}^{4}$ spin-orbit coupled three-orbital Hubbard models at electronic density $n=4$. In parallel, experiments involving iridates with noninteger 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 an 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 $\ensuremath{\lambda}$ couplings, we found an excitonic condensate at the unexpected momentum $q=\ensuremath{\pi}/2$ involving ${j}_{\mathrm{eff}}=3/2,m=\ifmmode\pm\else\textpm\fi{}1/2$, and ${j}_{\mathrm{eff}}=1/2,m=\ifmmode\pm\else\textpm\fi{}1/2$ bands in the triplet channel, coexisting with an also unexpected block magnetic order. We also present the entire $\ensuremath{\lambda}$ vs $U$ phase diagram, at a fixed and robust Hund coupling. Interestingly, this new ``block excitonic phase'' is present even at large values of $\ensuremath{\lambda}$, 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.

DOI
10.1103/physrevb.99.155115

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Theoretical studies recently predicted the condensation of spin-orbit excitons at momentum q=\ensuremathπq=\ensuremath{\pi}q=\ensuremathπ in t2g4{t}_{2g}^{4}t2g4​ spin-orbit coupled three-orbital Hubbard models at electronic density n=4n=4n=4. In parallel, experiments involving iridates with noninteger 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 an excitonic condensate at n=3.5n=3.5n=3.5 in a one-dimensional Hubbard model with a degenerate t2g{t}_{2g}t2g​ sector, when in the presence of spin-orbit coupling. At intermediate Hubbard UUU and spin-orbit \ensuremathλ\ensuremath{\lambda}\ensuremathλ couplings, we found an excitonic condensate at the unexpected momentum q=\ensuremathπ/2q=\ensuremath{\pi}/2q=\ensuremathπ/2 involving jeff=3/2,m=\ifmmode±\else\textpm\fi1/2{j}_{\mathrm{eff}}=3/2,m=\ifmmode\pm\else\textpm\fi{}1/2jeff​=3/2,m=\ifmmode±\else\textpm\fi1/2, and jeff=1/2,m=\ifmmode±\else\textpm\fi1/2{j}_{\mathrm{eff}}=1/2,m=\ifmmode\pm\else\textpm\fi{}1/2jeff​=1/2,m=\ifmmode±\else\textpm\fi1/2 bands in the triplet channel, coexisting with an also unexpected block magnetic order. We also present the entire \ensuremathλ\ensuremath{\lambda}\ensuremathλ vs UUU phase diagram, at a fixed and robust Hund coupling. Interestingly, this new ``block excitonic phase'' is present even at large values of \ensuremathλ\ensuremath{\lambda}\ensuremathλ, unlike the n=4n=4n=4 excitonic phase discussed before. Our computational study helps to understand and predict the possible magnetic phases of materials with d3.5{d}^{3.5}d3.5 valence and robust spin-orbit coupling.
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The condensation of spin-orbit-induced excitons in t2g4{t}_{2g}^{4}t2g4​ electronic systems is attracting considerable attention. At large Hubbard UUU, antiferromagnetism was proposed to emerge from the Bose-Einstein Condensation (BEC) of triplons (Jeff=1{J}_{\text{eff}}=1Jeff​=1). Here, we show that even at intermediate UUU 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 UUU, and this BEC strong coupling regime is smoothly connected to the intermediate UUU excitonic insulator region. We solved the degenerate three-orbital Hubbard model with spin-orbit coupling ($\ensuremath{\lambda}$) 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 $\ensuremath{\lambda}$ versus UUU phase diagrams for both one- and two-dimensional lattices. Our main result is that at intermediate Hubbard UUU, increasing $\ensuremath{\lambda}$ at fixed UUU the system transitions from an incommensurate spin-density-wave metal to a Bardeen-Cooper-Schrieffer (BCS) excitonic insulator, with coherence length rcoh{r}_{\mathrm{coh}}rcoh​ of O(a)\mathcal{O}(a)O(a) and O(10a)\mathcal{O}(10a)O(10a) in 1d1d1d and 2d2d2d, respectively, with aaa being the lattice spacing. Further increasing $\ensuremath{\lambda}$, the system eventually crosses over to the BEC limit (with ${r}_{\mathrm{coh}}\ensuremath{\ll}a$).
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Excitonic phases described by the quantum condensation of electron-hole pairs (or excitons) are studied in the two-orbital Hubbard model with the Hund's rule coupling J . Using the variational cluster approximation, we calculate the ground-state energy and order parameters, and show that the Hund's rule coupling always stabilizes the excitonic spin-density- wave state and destabilizes the excitonic charge-density-wave state. The pair coherence lengths calculated in the spin-singlet and spin-triplet states indicate that only the spin-triplet excitons are paired more tightly with increasing J . The single-particle spectrum and density of states are also calculated to clarify the characters of these excitonic density-wave states.
We present a comprehensive study of the spin excitations---as measured by the dynamical spin structure factor $S(q,\ensuremath{\omega})$---of the so-called block-magnetic state of low-dimensional orbital-selective Mott insulators. We realize this state via both a multi-orbital Hubbard model and a generalized Kondo-Heisenberg Hamiltonian. Due to various competing energy scales present in the models, the system develops periodic ferromagnetic islands of various shapes and sizes, which are antiferromagnetically coupled. The $2\ifmmode\times\else\texttimes\fi{}2$ particular case was already found experimentally in the ladder material BaFe2Se3\mathrm{Ba}{\mathrm{Fe}}_{2}{\mathrm{Se}}_{3}BaFe2​Se3​ that becomes superconducting under pressure. Here we discuss the electronic density as well as Hubbard and Hund coupling dependence of $S(q,\ensuremath{\omega})$ using density matrix renormalization group method. Several interesting features were identified: (1) An acoustic (dispersive spin-wave) mode develops. (2) The spin-wave bandwidth establishes a new energy scale that is strongly dependent on the size of the magnetic island and becomes abnormally small for large clusters. (3) Optical (dispersionless spin excitation) modes are present for all block states studied here. In addition, a variety of phenomenological spin Hamiltonians have been investigated but none matches entirely our results that were obtained primarily at intermediate Hubbard UUU strengths. Our comprehensive analysis provides theoretical guidance and motivation to crystal growers to search for appropriate candidate materials to realize the block states, and to neutron scattering experimentalists to confirm the exotic dynamical magnetic properties unveiled here, with a rich mixture of acoustic and optical features.