Competing magnetic orders and multipolar Weyl fermions in 227 pyrochlore iridates

Letter

Competing magnetic orders and multipolar Weyl fermions in 227 pyrochlore iridates

Konstantinos Ladovrechis, Tobias Meng, and Bitan Roy

Phys. Rev. B 103, L241116 – Published 25 June 2021

Abstract

Owing to comparably strong spin-orbit coupling and Hubbard interaction, iridium based 227 pyrochlore oxides harbor a rich confluence of competing magnetic orders and emergent multipolar Weyl quasiparticles. Here we show that this family of materials is predominantly susceptible toward the nucleation of electronic noncoplanar all-in all-out (AIAO) and three-in one-out (3I1O) orders, respectively transforming under the singlet $A_{2 u}$ and triplet $T_{1 u}$ representations, supporting octupolar and dipolar Weyl fermions, and favored by strong on-site Hubbard and nearest-neighbor ferromagnetic interaction. Furthermore, a coplanar magnetic order generically appears as an intermediate phase between them. This order transforms under the triplet $T_{2 u}$ representation and also hosts octupolar Weyl fermions. With the AIAO and 3I1O phases possibly being realized in ${({Nd}_{1 - x} \Pr_{x})}_{2} {Ir}_{2} O_{7}$ when $x = 0$ and 1, respectively, the intervening $T_{2 u}$ order can in principle be found at an intermediate doping $(0 < x < 1)$ or in pressured (hydrostatic) ${Nd}_{2} {Ir}_{2} O_{7}$ .

Received 22 January 2021
Revised 22 April 2021
Accepted 17 June 2021

DOI:https://doi.org/10.1103/PhysRevB.103.L241116

Physics Subject Headings (PhySH)

Magnetic order Phase diagrams

Iridates Strongly correlated systems Weyl semimetal

Effective field theory Electron-correlation calculations Mean field theory Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Konstantinos Ladovrechis ¹, Tobias Meng ¹, and Bitan Roy ^2,3

¹Institute for Theoretical Physics and Würzburg-Dresden Cluster of Excellence ct.qmat, Technische Universität Dresden, 01069 Dresden, Germany
²Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Strasse 38, 01187 Dresden, Germany
³Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015, USA

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 103, Iss. 24 — 15 June 2021

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Noncoplanar (a) AIAO, (b) 2I2O, and (c) 3I1O arrangements of electronic spins on Ir tetrahedron. The AIAO and 2I2O orders transform under the singlet $A_{2 u}$ and triplet $T_{1 u}$ representation of cubic ( $O_{h}$ ) point group, respectively; their coupling with the low-energy Luttinger fermions are shown in Eqs. (3) and (4). The 3I1O order also transforms under $T_{1 u}$ representation. Bottom: Three possible coplanar arrangements of electronic spins following the irreducible $T_{2 u}$ representation. Their couplings with Luttinger fermions are shown in Eq. (5), with $T_{2 u}^{x} \equiv T_{2 u}^{(1)}, T_{2 u}^{y} \equiv T_{2 u}^{(2)}$ , and $T_{2 u}^{z} \equiv T_{2 u}^{(3)}$ .
Reuse & Permissions
Figure 2
Mean field phase diagram of an interacting Luttinger semimetal (LSM) in the presence of on-site Hubbard $(U)$ and nearest-neighbor ferromagnetic $(J)$ interactions for a fixed $J / U = 1 / 4$ [Eqs. (8, 9, 10)] at zero temperature, which should also be qualitatively applicable at sufficiently low temperatures close to the MIT. The noncoplanar $A_{2 u}$ and $T_{1 u}$ phases are separated by an intervening coplanar $T_{2 u}$ magnetic order (Fig. 1). Interaction couplings are dimensionless, obtained by taking $X Λ {(2 m_{p})}^{3 / 2} / (32 π^{3}) \to X$ for $X = U$ and $J$ , where $Λ$ is the ultraviolet momentum cutoff, $m_{p} = m_{1} m_{2} / \sqrt{m_{1}^{2} + m_{2}^{2}}$ and $α = {tan}^{- 1} (m_{1} / m_{2})$ [Eq. (1)]. A possible coexistence between adjacent phases can be realized above the dashed lines, which we estimate from mean field susceptibilities, that, however, can be renormalized due the presence of the primary dominant order [30]. Experimentally such a phase diagram can be constructed by changing the ionic radius of the Ln element [5] or applying hydrostatic or chemical pressure [16, 18]. Inset: Internal algebra among the components of three magnetic orders. Six vertices (center) of the hexagon are (is) occupied by the components of $T_{1 u}$ and $T_{2 u}$ ( $A_{2 u}$ ) magnetic orders [Eqs. (3, 4, 5)]. Mutually anticommuting matrix operators associated with them are connected by solid lines. When two order parameters mutually anticommute (even partially), coexistence between them gets energetically favored.
Reuse & Permissions
Figure 3
Distribution of the Abelian Berry curvature for octupolar (a) $A_{2 u}$ and (b) $T_{2 u}^{z}$ magnetic orders. Weyl nodes acting as source (sink) of the Berry curvature are represented by outward (inward) arrows of the corresponding Berry flux. For both magnetic orders, eight Weyl nodes are distributed in octupolar fashions. Four of them act as source and the remaining ones as sink of the Berry curvature. The Weyl nodes and the distributions of the Berry flux in (a) and (b) are related to each other by a rotation about the $k_{z}$ axis by $π / 4$ . Similarly, rotation by $π / 4$ about the $k_{x}$ ( $k_{y}$ ) axis causes rotation between the $A_{2 u}$ and $T_{2 u}^{x}$ ( $T_{2 u}^{y}$ ) orders.
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics