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Dilemma in Strongly Correlated Materials: Hund's Metal Vs Relativistic Mott Insulator

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Dilemma in Strongly Correlated Materials: Hund's Metal Vs Relativistic Mott Insulator Dilemma in strongly correlated materials: Hund’s metal vs relativistic Mott insulator Gang Chen∗ Department of Physics and HKU-UCAS Joint Institute for Theoretical and Computational Physics at Hong Kong, The University of Hong Kong, Hong Kong, China We point out the generic competition between the Hund’s coupling and the spin-orbit coupling in correlated materials, and this competition leads to an electronic dilemma between the Hund’s metal and the relativistic insulators. Hund’s metals refer to the fate of the would-be insulators where the Hund’s coupling suppresses the correlation and drives the systems into correlated metals. Relativistic Mott insulators refer to the fate of the would-be metals where the relativistic spin-orbit coupling enhances the correlation and drives the systems into Mott insulators. These contradictory trends are naturally present in many correlated materials. We study the competition between Hund’s coupling and spin-orbit coupling in correlated materials and explore the interplay and the balance from these two contradictory trends. The system can become a spin-orbit-coupled Hund’s metal or a Hund’s assisted relativistic Mott insulator. Our observation could find a broad application and relevance to many correlated materials with multiple orbitals. Correlated quantum materials provide a rich platform to spin-orbit coupling, the system would be a correlated metal. explore different competing interactions. The simplest one It is the spin-orbit coupling that assists the electron correla- would be the competition between the electron kinetic energy tion and drives the Mott insulating behaviors. As the spin- and the Coulomb interactions between the electrons. This is orbit coupling is a relativistic effect, the spin-orbit-coupled captured by the well-known Hubbard model [1]. For the in- Mott insulator is often referred as the relativistic Mott insula- teger electron filling per site, a strong on-site electron inter- tor. Having explained the underlying ideas of Hund’s metal action would directly convert the system from a metal into and relativistic Mott insulator, one would immediately realize a Mott insulator with the formation of local moments [2– that, both of these two concepts are dealing with the correlated 4]. This “big” parent picture is decorated in many different materials with multiple orbitals, but they have rather opposite ways when extra interactions are included or emerge as the tendencies. Thus, the correlated materials would fall into a subleading effects. This includes, for example, the residual dilemma between Hund’s metal and relativistic Mott insula- interactions and the magnetic ground states of the Mott in- tor. The purpose of this work is to point out this dilemma sulators, the nature of the metallic states, the band structure through a specific example. The specific example in this work topology [5,6], the nature of Mott transition [7–9], the or- is simply adopted to explain the universal physics behind, and bital selectivity of Mott transition [10–12], etc. Two interest- should not be interpreted as a localized specifics without any ing decorations, Hund’s metal [13, 14] and relativistic Mott generalization. One should really extract the general message insulator [15, 16] that are discussed in this work, are from delivered by our specific example. We further design a calcu- the Hund’s coupling and from the spin-orbit coupling, respec- lation formalism to study the spin-orbit coupling, the Hund’s tively. They are two interesting ideas that emerge in the theory coupling and Mott physics in correlated materials, and hope of correlated electron materials over the past decade. to capture these competing effects at least in a crude manner. The Hund’s metal is concerned with how the correlated We start with an extended Hubbard model with multiple metallic regime is enhanced by the presence of the Hund’s coupling on multiply degenerate d-orbitals for many transi- tion metal compounds [13, 14]. Most often, it was argued that, the Hund’s coupling effectively reduces the electron correla- tion by harnessing the interaction energy gain in the spin sec- tor. Thus, the metallic regime is significantly expanded com- pared to the case without the Hund’s coupling. The relativis- tic Mott insulator is a concept about the role of strong spin- arXiv:2012.06752v1 [cond-mat.str-el] 12 Dec 2020 orbit coupling in correlated materials with multiple 4d/5d or- bitals or bands [15]. It can sometimes be relevant for systems with 3d electrons when the spin-orbit coupling becomes ac- tive. The strong spin-orbit coupling twists the electron mo- tions as it hops on the lattice and reduces the bandwidth of the electrons. Another description of this effect is that, the spin-orbit coupling breaks the whole bands into multiple spin- orbital-entangled subbands with much narrower bandwidths. FIG. 1. (a) The hopping between the t2g orbitals from neighboring As a result, the electron correlation is enhanced. The system sites on the square lattice. (b) The energy splitting of three-fold de- becomes a spin-orbit-coupled Mott insulator. If there is no generate t2g orbitals and the two-fold degenerate eg orbitals. 2 orbitals on a square lattice (see Fig.1). We consider the octa- bitals is very high so that we can safely neglect the presence of hedral crystal field environment for the transition metal ions. the upper eg orbitals. We restrict ourselves to the t2g orbitals. The two-fold degenerate eg orbitals are higher in the energy This is sufficient for revealing the physics that was previously than the three-fold degenerate t2g orbitals. We assume that, advocated. The extended Hubbard model on the t2g manifold the crystal field separation ∆ between the eg and the t2g or- is written as X X X X X λ X X H = tmncy c + Lµ σµ cy c + U cy cy c c i j imα jnα 2 mn αβ imα inβ imα imβ imβ imα hi ji m;n i m;n µ i m X hU0 X J X J0 X i + cy cy c c + cy cy c c + cy cy c c + ··· ; (1) 2 imα inβ inβ imα 2 imα inβ imβ inα 2 imα imβ inβ inα i m,n m,n m,n where m; n = 1; 2; 3 are the orbital indices corresponding to corporated into this formalism by including the J-interactions the yz; xz; xy orbitals for the t2g orbitals, α, β ="; # are the spin onto the spinon sector and properly decoupling the interaction indices, µ = x; y; z refers to the component for the spin and according to the orderings [29]. The latter approach attributes orbital angular momenta, and the spin indices are automati- the Mott localization to the U-interaction, and the magnetic y cally summed. The operator cimα (cimα) creates (annihilates) orders to the J-interactions, even though both interactions to- an electron on the m orbital with the spin quantum α. On gether give rise to the magnetism in many cases. Here we are the first line of Eq. (1), the first term describes the electron not interested in addressing the nature of the ground state for hopping between different orbitals and the neighboring sites the Mott regime, but to understand the variation of the Mott on the lattice, the second term describes the atomic spin-orbit transition. Thus, we simply assume the Mott side is a spin coupling, and the third term is the intra-orbital Coulomb in- liquid and explore the fate of Mott transition in the presence teraction. On the second line of Eq. (1), the first term is the of extra couplings. inter-orbital interaction, the second term is the Hund’s cou- For our purpose, we first take away the J-interactions, and pling, the third term describes the electron pair hopping, and decouple the extended Hubbard model into the spinon sector “··· ” refers to the extra interactions and effects that are not H f and the the charge sector Hθ with considered here. Owing to the lattice symmetries and the or- X X mn y X X X λ µ µ y bital orientations, only the nearest-neighbor intra-orbital hop- H f = t f f + Lmnσ f f i j imα jnα 2 αβ imα inβ ping is non-vanishing and is set to t (see Fig.1). The inter- hi ji m;n i m;n µ X X actions in Eq. (1) are the standard Kanamori interactions. We − h f y f ; (2) 0 0 i imα imα will take the atomic limit with J = J and U = U − 2J in our i m calculation, and the interactions can then be described as U- X iθ −iθ X U 2 H = χ e i j + h:c: + L + h L ; (3) interaction and J-interactions. This Hamiltonian is particu- θ i j 2 i i i larly relevant for e.g. V4+ ions with 3d1 electron configura- hi ji i 4+ 2 mn y tions in Sr VO [17–20], Mo ions with 4d electron config- iθi−iθ j mn P mn 2 4 where ti j = he it , χ = m;n t h f f i, hi is a La- 4+ 3 i j i j i j imα jnα urations in Sr2MoO4 [21], Re ions with 5d electron con- grangian multiplier for each site to enforce the Hilbert space figurations in Sr ReO , Ru4+ ions with 4d4 electron configu- P P y 2 4 constraint such that [ m α f f ] − n¯ = Li, and Li is an 4+ imα imα rations in Sr2RuO4 [22] and Ca2RuO4 [23–25], even Ir ion angular momentum variable conjugate to the U(1) phase θi.
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