`Interaction annealing' to determine effective quantized valence and orbital structure: an illustration with ferro-orbital order in WTe$_2$

TL;DR

This work introduces an 'interaction annealing' framework to reveal fully quantized, low-energy electronic objects in strongly correlated materials by adiabatically suppressing charge fluctuations. It establishes an exact two-site Hubbard model as the theoretical backbone and demonstrates practical applicability with DFT (LDA+$U$) calculations on La${_2}$CuO${_4}$ and WTe${_2}$, uncovering a ferro-orbital-ordered, spin-0 OP2 state in the 5d system. The approach identifies dominant dressed structures and competing fixed points, explaining experimental observations such as lattice distortions and diamagnetism in WTe${_2}$. It provides a robust, computable route to analyze and predict local electronic structures in functional materials, with broad relevance to correlated electron phenomena.

Abstract

Correlated materials are known to display qualitatively distinct emergent behaviors at low energy. Conveniently, upon absorbing rapid quantum fluctuations, these rich low-energy behaviors can always be effectively described by dressed particles with fully quantized charge, spin, and orbital structure. Such a powerful and simple description is, however, difficult to access through bare particles used in most many-body computations, especially when fluctuations are strong such as in $4d$ and $5d$ compounds. To decipher the dominant quantized structure, we propose an easy-to-implement `interaction annealing' approach that utilizes suppressed charge fluctuation through enhancing ionic charging energy. We establish its theoretical foundation using an exactly treated two-site Hubbard model as a generic example. We then demonstrate its applications with more affordable density functional calculations to a representative $3d$ Mott insulator La${_2}$CuO${_4}$ and a highly fluctuating $5d$ semi-metal WTe${_2}$. In the latter, it reveals an emergent local electronic structure that makes possible an unprecedented explanation of several experimental observations. Finally, we demonstrate the effectiveness of this approach in studying competing local electronic structures in functional materials.

Figures (7)

• Figure 1: Illustration on obtaining effective quantized description through the interaction annealing procedure. Realistic Hamiltonian $H$ for bare particle $c_i^\dagger$ typically contains strong fluctuation that masks the dominant physics. Equivalently, the low-energy dynamics can always be described by fully quantized effective $\tilde{H}$ upon absorbing rapid fluctuations into dressed particles $\tilde{c}_i^\dagger$. The desired effective description can be obtained through the bare description, $H^\mathrm{(a)}[\{d_i^\dagger\}]$, of a fictitious 'interaction annealed' system with suppressed fluctuation, given $d_i^\dagger$'s resemblance to the dressed particles $\tilde{d}_i^\dagger$ and the "adiabatic connection" between $\tilde{d}_i^\dagger$ and $\tilde{c}_i^\dagger$ (see text).
• Figure 2: Lattice and orbital structures of WTe$_2$. (a) Top view of the $T_d$ lattice structure of WTe$_2$, represented via distortion (black arrows) of the higher symmetry $1T$ lattice structure, together with the local coordinate axis of W orbitals. (b) Color scheme for labeling the phase factor of orbitals. (c) Symmetry related (degenerate) $e'_g$ orbitals of W spherharm_web, and their superposition, $|e"_{g1}\rangle = \frac{1}{\sqrt{2}}(|e'_{g1}\rangle+|e'_{g2}\rangle)$ and $|e"_{g2}\rangle = \frac{1}{\sqrt{2}}(|e'_{g1}\rangle- |e'_{g2}\rangle)$ emerged in orbital polarized states that spontaneously break the symmetry. (d) The stable local electronic structures found using LDA+$U$, including low-spin (LS), orbital-polarized (OP), and high-spin (HS) configurations of W ions.
• Figure 3: Competing electronic structures via interaction annealing. Illustration of competing structures under the symmetric $1T$ structure of WTe$_2$, through a smooth evolution of (a) total energy (for 4 chemical formula units) upon reduction of $U$ and (b) local minimum in the energy contour against density matrices. The inset highlights the destabilization of some of the structures, corresponding to the vanishing of local energy minimum in (b). Due to the large energy scale of intra-atomic physics, configurations of distinct local structures separate into four groups, OP2+LS2 (red), HS3 (green), OP3 (blue), and LS4 (yellow) [c.f. Fig. \ref{['WTe2_structure']}(d)], but less sensitive to long-range ferro-orbital (FO), anti-ferro-orbital (AFO), ferromagnetic (FM), and antiferromagetic (AFM) long-range orders.
• Figure 4: Illustration of the emergent effective description with quantized valence, orbital, and spin structures, using the first-quantized representation of the two-site Hubbard model in its correlated regime ($t\ll U$). Left panel represents the Hamiltonian in the basis of product states of bare particles $c^\dagger_i$, while the right panel in the dressed particles $\tilde{c}^\dagger_i$.
• Figure 5: Same as Fig. \ref{['figs1']} but for the uncorrelated ($t\gg U$) regime, where the emergent objects resides in the effective bonding $\tilde{b}^\dagger_\mu$ and anti-bonding $\tilde{a}^\dagger_\mu$ orbitals.