Interorbital Hopping Research Articles

Interorbital antisymmetric hopping generated flat bands on kagome and pyrochlore Lattices

Flat bands are intriguing platforms for correlated and topological physics. Various methods have been developed to create flat bands utilizing lattice geometry, but the investigation of orbital symmetry in multiorbital materials is a new area of focus. Here, we introduce a site symmetry-based approach to emerging multiorbital 2D and 3D flat bands on the kagome and pyrochlore lattices. As a conceptual advance, the one-orbital flat bands are shown to originate as mutual eigenstates of isolated molecular motifs. Further developing the mutual eigenstate method for multiple orbitals transforming differently under the site symmetries, we derive interorbital hopping generated flat bands from the antisymmetric interorbital Hamiltonian and introduce group-theoretic descriptions of the flat band wavefunctions. Realizations of multiorbital flat bands in realistic materials are shown to be possible in the Slater-Koster formalism. Our findings provide new directions for exploring flat-band electronic structures for novel correlated and topological quantum states.

Parsing skin effect in a non-Hermitian spinless BHZ-like model

This work comprehensively investigates the non-Hermitian skin effect (NHSE) in a spinless Bernevig-Hughes-Zhang -like model in one dimension. It is generally believed that a system with non-reciprocal hopping amplitudes demonstrates NHSE. However, we show that there are exceptions, and more in-depth analyses are required to decode the presence of NHSE or its variants in a system. The fascinating aspects of our findings, depending on the inclusion of non-reciprocity in the inter-orbital hopping terms, concede the existence of conventional NHSE or NHSE at both edges and even a surprising absence of NHSE. The topological properties and the (bi-orthogonal) bulk-boundary correspondence, enumerated via computation of the (complex) Berry phase and spatial localization of the edge modes, highlight the topological phase transitions occurring therein. Further, to facilitate a structured discussion of the non-Hermitian model, we split the results into PT symmetric and non- PT symmetric cases with a view to comparing the two.

Giant Tunability of Rashba Splitting at Cation-Exchanged Polar Oxide Interfaces by Selective Orbital Hybridization.

The 2D electron gas (2DEG) at oxide interfaces exhibits extraordinary properties, such as 2D superconductivity and ferromagnetism, coupled to strongly correlated electrons in narrow d-bands. In particular, 2DEGs in KTaO3 (KTO) with 5d t2g orbitals exhibit larger atomic spin-orbit coupling and crystal-facet-dependent superconductivity absent for 3d 2DEGs in SrTiO3 (STO). Herein, by tracing the interfacial chemistry, weak anti-localization magneto-transport behavior, and electronic structures of (001), (110), and (111) KTO 2DEGs, unambiguously cation exchange across KTO interfaces is discovered. Therefore, the origin of the 2DEGs at KTO-based interfaces is dramatically different from the electronicreconstruction observed at STO interfaces. More importantly, as the interface polarization grows with the higher order planes in the KTO case, the Rashba spin splitting becomes maximal for the superconducting (111) interfaces approximately twice that of the (001) interface. The larger Rashba spin splitting couples strongly to the asymmetric chiral texture of the orbital angular moment, and results mainly from the enhanced inter-orbital hopping of the t2g bands and more localized wave functions. This finding has profound implications for the search for topological superconductors, as well as the realization of efficient spin-charge interconversion for low-power spin-orbitronics based on (110) and (111) KTO interfaces.

Stability of the interorbital-hopping mechanism for ferromagnetism in multi-orbital Hubbard models

The emergence of insulating ferromagnetic phase in iron oxychalcogenide chain system has been recently argued to be originated by interorbital hopping mechanism. However, the practical conditions for the stability of such mechanism still prevents the observation of ferromagnetic in many materials. Here, we study the stability range of such ferromagnetic phase under modifications in the crystal fields and electronic correlation strength, constructing a theoretical phase diagram. We find a rich emergence of phases, including a ferromagnetic Mott insulator, a ferromagnetic orbital-selective Mott phase, together with antiferromagnetic and ferromagnetic metallic states. We characterize the stability of the ferromagnetic regime in both the Mott insulator and the ferromagnetic orbital-selective Mott phase forms. We identify a large stability range in the phase diagram at both intermediate and strong electronic correlations, demonstrating the capability of the interorbital hopping mechanism in stabilizing ferromagnetic insulators. Our results may enable additional design strategies to expand the relatively small family of known ferromagnetic insulators.

Inter-orbital hopping effects on the superconducting state properties of a two-band BCS model

Inter-orbital hopping effects on the superconducting state properties of a two-band BCS model

Anisotropic Rashba coupling to polar modes in KTaO3

Motivated by the discovery of superconductivity in KTaO3-based heterostructures, we study a pairing mechanism based on spin-orbit assisted coupling between the conduction electrons and the ferroelectric (FE) modes present in the material. We use ab initio frozen-phonon computations to show a linear-in-momentum Rashba-like coupling with a strong angular dependence in momentum for the lower manifold, deviating from the conventional isotropic Rashba model. This implies the Rashba-like interaction with the polar modes has substantial L = 3 cubic harmonic corrections, which we quantify for each electronic band. The strong anisotropy of the Rashba interaction is captured by a microscopic toy model for the electrons. We find its origin to be the angular dependence in electronic momentum imposed by the kinetic term on the degenerate manifold. A comparison between the toy model and ab initio results indicates that additional symmetry allowed terms beyond odd-parity spin-conserving inter-orbital hopping processes are needed to describe the Rashba-like polar interaction between the electrons and the soft FE mode.

Proximity coupling induced significantly enhanced Curie temperature in van der Waals CrSBr/MoTe2 heterostructure

CrSBr monolayer is a promising ferromagnetic (FM) semiconductor with stable magnetic ground state, large bandgap, and high carrier density. However, its Curie temperature (TC) of about 146 K is still below room temperature. Herein, electronic and magnetic properties of a CrSBr/MoTe2 heterostructure are explored to boost the TC via the proximity effect using first-principles calculations. The long-range FM ordering in the CrSBr layer is enhanced both through an extra spin superexchange channel (Cr–Te–Cr) afforded by the MoTe2 substrate and the reduced degeneracy t2g orbitals of Cr atoms. Directly owing to the enhanced FM coupling, TC increases to 225 K. The strong p–d hybridizations increase the interorbital hopping between the t2g states through the Cr–Te–Cr superexchange channel. The hopping is related to the off diagonal matrix element of the velocity operator, which suggests that is necessary for not only the increased TC but also the nonzero Berry curvature. Additionally, the Fermi level (EF) is pushed into higher energy levels with the electron-doped CrSBr layer due to the charge transfer. The synergic effect of the EF shift from the transferred electron filling and the interorbital hopping from the Cr–Te interaction results in nonzero anomalous Hall conductivity in the heterostructure. Our work reveals that an interface is an uplifting way for engineering the magnetic and transport properties of 2D magnets, providing opportunities for fantastic spintronic devices.

Bardeen's tunneling theory applied to intraorbital and interorbital hopping integrals between dopants in silicon

We utilize Bardeen's tunneling theory to calculate intra- and interorbital hopping integrals between phosphorus donors in silicon using known orbital wave functions. While the two-donor problem can be solved directly, the knowledge of hoppings for various pairs of orbitals is essential for constructing multi-orbital Hubbard models for chains and arrays of donors. To assure applicability to long-range potentials, we rederive Bardeen's formula for the matrix element without assuming non-overlapping potentials. Moreover, we find a correction to the original expression allowing us to use it at short distances. We also show that accurate calculation of the lowest donor-pair eigenstates is possible based on these tunnel couplings, and we characterize the obtained states. The results are in satisfactory quantitative agreement with those obtained with the standard H\"uckel tight-binding method. The calculation relies solely on the wave functions in the barrier region and does not explicitly involve donor or lattice potentials, which has practical advantages. We find that neglecting the central correction potential in the standard method may lead to qualitatively incorrect results, while its explicit inclusion raises severe numerical problems, as it is contained in a tiny volume. In contrast, using wave functions obtained with this correction in the proposed method does not raise such issues. Nominally, the computational cost of the method is to calculate a double integral along the plane that separates donors. For donor separation in directions where valley interference leads to oscillatory behavior, additional averaging over the position of the integration plane is needed. Despite this, the presented approach offers a competitive computational cost as compared to the standard one. This work may be regarded as a benchmark of a promising method for calculating hopping integrals in lattice models.

Is the Orbital-Selective Mott Phase Stable against Interorbital Hopping?

The localization-delocalization transition is at the heart of strong correlation physics. Recently, there is great interest in multiorbital systems where this transition can be restricted to certain orbitals, leading to an orbital-selective Mott phase (OSMP). Theoretically, the OSMP is widely studied for kinetically decoupled orbitals, but the effect of interorbital hopping remains unclear. Here, we show how nonlocal interorbital hopping leads to local hybridization in single-site dynamical mean-field theory (DMFT). Under fairly general circumstances, this implies that, at zero temperature, the OSMP, involving the Mott-insulating state of one orbital, is unstable against interorbital hopping to a different, metallic orbital. We further show that the coherence scale below which all electrons are itinerant is very small and gets exponentially suppressed even if the interorbital hopping is not overly small. Within this framework, the OSMP with interorbital hopping may thus reach down to extremely low temperatures T, but not to T=0. Accordingly, it is part of a coherence-incoherence crossover and not a quantum critical point. We present analytical arguments supported by numerical results using the numerical renormalization group as a DMFT impurity solver. We also compare our findings with previous slave-spin studies.

Dynamical mean-field theory of two-orbital Hubbard model

The effect of interorbital hopping on orbital selective Mottness in a two-band correlation system is investigated by using the dynamical mean-field theory with the Lanczos method as impurity solver. The phase diagrams of the two-orbital Hubbard model with non-local interorbital hopping (<i>t</i><sub>12</sub>) , where the orbital selective Mott phases (OSMP) region is expanded by the increasing of the interorbital hopping. We compare the results obtained by self-consistent relations of Bethe lattice and squate lattice based on DMFT procedure, and the general OSMP physics of Bethe lattice is consistent with that of the square lattice while the critical points of two kinds of lattices are different. We extend the method to the study of half-filled Ba<sub>2</sub>CuO<sub>4–<i>δ</i></sub> materials. By comparing with the band structure obtained from the density functional theory (DFT), it is found that the isotropic inter-orbital hopping has a great influence on the bandstructure. The DFT bandstructure in DMFT is considered, and the phase diagram of the material is obtained. The half-filled Ba<sub>2</sub>CuO<sub>4–<i>δ</i></sub> should be orbital-selective Mott compound according to our results.

Orbital frustration and topological flat bands

We expand the concept of frustration in Mott insulators and quantum spin liquids to metals with flat bands. We show that when inter-orbital hopping $t_2$ dominates over intra-orbital hopping $t_1$, in a multiband system with strong spin-orbit coupling $\lambda$, electronic states with a narrow bandwidth $W\sim t_2^2/\lambda$ are formed compared to a bandwidth of order $t_1$ for intra-orbital hopping. We demonstrate the evolution of the electronic structure, Berry phase distributions for time-reversal and inversion breaking cases, and their imprint on the optical absorption, in a tight binding model of $d$-orbital hopping on a honeycomb lattice. Going beyond quantum Hall effect and twisted bilayer graphene, we provide an alternative mechanism and a richer materials platform for achieving flat bands poised at the brink of instabilities toward novel correlated and fractionalized metallic phases.

Octupolar order and Ising quantum criticality tuned by strain and dimensionality: Application to d -orbital Mott insulators

Recent experiments have discovered multipolar orders in a variety of $d$-orbital Mott insulators. Motivated by uncovering the exchange interactions which underlie octupolar order proposed in the osmate double perovskites, we study a two-site model using exact diagonalization on a five-orbital Hamiltonian, incorporating spin-orbit coupling (SOC) and interactions, and including both intra-orbital and inter-orbital hopping. Using an exact Schrieffer-Wolff transformation, we then extract an effective pseudospin Hamiltonian for the non-Kramers doublets, uncovering dominant ferrooctupolar coupling driven by the interplay of two distinct intra-orbital hopping terms. Using classical Monte Carlo simulations on the face-centered cubic lattice, we obtain a ferrooctupolar transition temperature which is in good agreement with experiments on the osmate double perovskites. We also explore the impact of uniaxial strain and dimensional tuning via ultrathin films, which are shown to induce a transverse field on the Ising octupolar order. This suppresses $T_c$ and potentially allows one to access octupolar Ising quantum critical points. We discuss possible implications of our results for a broader class of materials which may host such non-Kramers doublet ions.

Impact of spin-orbit interaction on the phase diagram and anisotropy of the in-plane critical magnetic field at the superconducting LaAlO3/SrTiO3 interface

The two-dimensional electron gas at the interface between ${\mathrm{LaAlO}}_{3}$ and ${\mathrm{SrTiO}}_{3}$ (LAO/STO) exhibits gate-tunable superconductivity with a characteristic domelike shape of the critical temperature (${T}_{c}$) in the phase diagram. As shown recently [Phys. Rev. B 102, 085420 (2020)], such an effect can be explained as a consequence of the extended $s\text{\ensuremath{-}}\mathrm{wave}$ symmetry of the gap within an intersite real space pairing scenario, leading to a good agreement between the experiment and theory. In this work, we turn to a detailed analysis of the influence of spin-orbit coupling on the LAO/STO phase diagram by considering separately the atomic component as well as the interorbital hopping induced by the broken inversion symmetry at the interface. In particular, we analyze the optimal carrier concentration for which the maximal ${T}_{c}$ is reached relative to the Lifshitz transition point. We find that the misalignment between the two can be significantly enhanced by the spin-orbit splitting of the bands, combined with the fact that superconductivity sets in when the Fermi level passes the anticrossing induced by the spin-orbital hybridization. In the presence of the external in-plane magnetic field, our calculations show fourfold anisotropy with the paramagnetic limit largely exceeded for ${B}_{||}$ directed along the high symmetry points [01] and [01]. The obtained electron concentration dependence of ${B}_{c||}$ reproduces the characteristic domelike shape reported in experiments and the estimated value of ${B}_{c||}$ corresponds to that measured experimentally.

Effect of strain-induced orbital splitting on the magnetic excitations in undoped cuprates

We investigate the magnetic excitations in view of the recent reports suggesting that the spin-wave energy may exhibit a significant dependence on the in-plane strain of a thin film of ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$. The nature of dependence, as we find, can be explained naturally within a two-orbital model based on the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and ${d}_{3{z}^{2}\ensuremath{-}{r}^{2}}$ orbitals. In particular, as the orbital-splitting energy between the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and ${d}_{3{z}^{2}\ensuremath{-}{r}^{2}}$ orbitals increases with compressive strain, the zone-boundary spin-wave energy hardens. However, the hardening persists only until the orbital splitting reaches $\ensuremath{\sim}2$ eV, beyond which there is no significant change. The behavior of zone-boundary spin-wave energy is explained in terms of the extent of hybridization between one of the exchange-split ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ bands which is nearly half-filled and the ${d}_{3{z}^{2}\ensuremath{-}{r}^{2}}$ band. The role of second order antiferromagnetic superexchange process involving the interorbital hopping is also discussed.

Emergence of pure spin current in doped excitonic magnets

An excitonic magnet hosts a condensate of spin-triplet excitons composed of conduction-band electrons and valence-band holes, and may be described by the two-orbital Hubbard model. When the Hamiltonian has the nearest-neighbor interorbital hopping integrals with $d$-wave symmetry and the number of electrons is slightly away from half filling, the $\boldsymbol{k}$-space spin texture appears in the excitonic phase with a broken time-reversal symmetry. We then show that, applying electric field to this doped excitonic magnet along a particular direction, a pure spin current emerges along its orthogonal direction. We discuss possible experimental realization of this type of the pure spin current in actual materials.

Orbital Mott transition in two dimensional pyrochlore lattice

We study orbital Mott transition in two dimensional pyrochlore lattice, using a two orbital Hubbard model with only inter-orbital electronic hopping. We use a real space Monte Carlo based approach to study the model at finite temperature, and establish temperature-interaction phase diagram that highlights the Mott transition, orbital ordering, and spectral trends, and possible window of pseudogap. Due to only inter-orbital hopping, the Mott insulator ‘generates’ ferro exchange resulting in ferro-orbital ordering, with Tcorr/t peaked at ≈0.2 around U/t ≈ 6. The optical conductivity shows unusual two peak feature due to two dimensional pyrochlore lattice.

Orbital angular momentum analysis for giant spin splitting in solids and nanostructures

Giant spin splitting (GSS) of electronic bands, which is several orders of magnitude greater than the standard Rashba effect has been observed in various systems including noble-metal surfaces and thin films of transition-metal dichalcogenides. Previous studies reported that orbital angular momentum (OAM) is not quenched in some GSS materials and that the atomic spin-orbit interaction (SOI) generates spin splitting in some solid states via the interorbital hopping. Although the unquenched OAM may be closely related to the interorbital hopping, their relationship is hardly studied in the aspect of using the unquenched OAM as a control parameter of GSS. Here, we analyze OAM in GSS materials by using the interorbital-hopping mechanism and first-principles calculations. We report that the interatomic hopping between different-parity orbitals, which is generated by specific broken mirror symmetry, produces k-dependent OAM, resulting in valley-dependent GSS in WSe2 monolayer, Rashba-type GSS in Au (111) surface, and Dresselhaus-type GSS in bulk HgTe. We also demonstrate systematic control of OAM by pressure, external fields, and substrates, thereby controlling the spin splitting, and discuss the temperature dependence of OAM. Our results provide a simplified picture for systematic design and control of GSS materials.

Microscopic theory of Dzyaloshinsky-Moriya interaction in pyrochlore oxides with spin-orbit coupling

Pyrochlore oxides show several fascinating phenomena, such as the formation of heavy fermions and the thermal Hall effect. Although a key to understanding some phenomena may be the Dzyaloshinsky-Moriya (DM) interaction, its microscopic origin is unclear. To clarify the microscopic origin, we constructed a $t_{2g}$-orbital model with the kinetic energy, the trigonal-distortion potential, the multiorbital Hubbard interactions, and the $LS$ coupling, and derived the low-energy effective Hamiltonian for a $d^{1}$ Mott insulator with the weak $LS$ coupling. We first show that lack of the inversion center of each nearest-neighbor V-V bond causes the odd-mirror interorbital hopping integrals. Those are qualitatively different from the even-mirror hopping integrals, existing even with the inversion center. We next show that the second-order perturbation using the kinetic terms leads to the ferromagnetic and the antiferromagnetic super exchange interactions. Then, we show the most important result: the third-order perturbation terms using the combination of the even-mirror hopping integral, the odd-mirror hopping integral, and the $LS$ coupling causes the DM interaction due to the mirror-mixing effect, where those hopping integrals are necessary to obtain the antisymmetric kinetic exchange and the $LS$ coupling is necessary to excite the orbital angular momentum at one of two sites. We also show that the magnitude and sign of the DM interaction can be controlled by changing the positions of the O ions and the strength of the Hubbard interactions. We discuss the advantages in comparison with the phenomenological theory and Moriya's microscopic theory, applicability of our mechanism, and the similarities and differences between our case and the strong-$LS$-coupling case.

Multiorbital kinetic effects on charge ordering of frustrated electrons on the triangular lattice

The role of the multiorbital effects on the emergence of frustrated electronic orders on the triangular lattice at half filling is investigated through an extended spinless fermion Hubbard model. By using two complementary approaches, unrestricted Hartree-Fock and exact diagonalizations, we unravel a very rich phase diagram controlled by the strength of both local and off-site Coulomb interactions and by the interorbital hopping anisotropy ratio $t'/t$. Three robust unconventional electronic phases, a pinball liquid, an inverse pinball liquid, and a large-unit-cell $\sqrt{12} \times \sqrt{12}$ droplet phase, are found to be generic in the triangular geometry, being controlled by the band structure parameters. The latter are also stabilized in the isotropic limit of our microscopic model, which recovers the standard SU(2) spinful extended single-band Hubbard model.

Spontaneous parity breaking in spin-orbital coupled systems

Effects of spontaneous parity breaking by charge, spin, and orbital orders are investigated in a two-band Hubbard model on a honeycomb lattice. This is a minimal model in which the inter-orbital hopping, atomic spin-orbit coupling, and strong electron correlation give rise to fascinating properties, such as the magnetoelectric effects, quantum spin Hall effect, and spin or valley splitting in the band structure. We perform the symmetry analysis of possible broken-parity states and the mean-field analysis of their competition. We find that the model at 1/4 filling exhibits a spin-orbital composite ordered state and a charge ordered state, in addition to a paramagnetic quantum spin-Hall insulator. We show that the composite ordered phase exhibits two types of magnetoelectric responses. The charge ordered state shows spin splitting in the band structure, while the topological nature varies depending on electron correlations.