Noncollinear Magnetic Structure and Multipolar Order in Eu2Ir2O7

Abstract

The magnetic properties of the pyrochlore iridate material Eu 2 Ir 2 O 7 (5d 5 ) have been studied based on first principles calculations, where the crystal field splitting Δ, spin-orbit coupling (SOC) λ, and Coulomb interaction U within Ir 5d orbitals all play significant roles. The ground state phase diagram has been obtained with respect to the strength of SOC and Coulomb interaction U, where a stable antiferromagnetic ground state with all-in-all-out (AIAO) spin structure has been found. In addition, another antiferromagnetic state with energy close to AIAO has also been found to be stable. The calculated nonlinear magnetization of the two stable states both have the d-wave pattern but with a π=4 phase difference, which can perfectly explain the experimentally observed nonlinear magnetization pattern. Compared with the results of the nondistorted structure, it turns out that the trigonal lattice distortion is crucial for stabilizing the AIAO state in Eu 2 Ir 2 O 7 . Furthermore, besides large dipolar moments, we also find considerable octupolar moments in the magnetic states. DOI: 10.1103/PhysRevLett.119.187203 The ordering of electronic states is one of the fundamental problems in condensed matter physics. In 3d transition metal compounds, the ordered states can be described quite well by the product of orders in orbital and spin subspaces [1], because the spin-orbit coupling (SOC) here is weak and can be treated perturbatively. However, in rare-earth compounds [2], the SOC is strong enough to bind the orbital and spin degrees of freedom into rigid objects described by the total angular momentum, and the ordered states can then be well understood in terms of the moments with high angular momentum that splits into atomic multiplets under crystal field. Unlike the above two limits, the situation in 4d and 5d transition metal compounds is unique [3]. On one hand, the SOC is strong enough to combine the orbital and spin degrees of freedom to form some complex orders. While on the other hand, the SOC is still far away from the limit where the low energy physics can be entirely determined within the subspace with a fixed total angular momentum. In fact, the SOC strength in these compounds is comparable to that of the crystal field so that the magnetic orders there can involve multiple total angular momentum states.The pyrochlore iridates [4,5] are typical 5d transition metal compounds with many novel properties already being discussed extensively in the literature, including the noncollinear magnetic order [6][7][8][9][10][11][12], the metal-insulator transition [13][14][15][16][17][18][19][20], the anomalous Hall effect [21][22][23], the topological insulator and the Weyl semimetal phase [3,6,7,15,[24][25][26][27][28][29][30][31], and the chiral metallic states in the domain wall [32][33][34]. Among them, the magnetic order is the most fundamental one that determines most of the electronic properties. In Ref. [6], by using the density functional theory plus U (DFT þ U) method, Wan et al. obtained an all-in-al...

Stability of the Weyl-semimetal phase on the pyrochlore lattice

“…This can be seen by comparing equations (26)- (29) to the spins given in figure 1(b) in [37]. We conclude that the tight-binding model of [37] corresponds to small s | | t in our phase diagram. Our results differ from [18,30,37] in that the rotation angle always equals π/2 in these works.…”

“…We conclude that the tight-binding model of [37] corresponds to small s | | t in our phase diagram. Our results differ from [18,30,37] in that the rotation angle always equals π/2 in these works. Instead, we find the angle to vary, as shown in figure 4.…”

“…Since there are three ways to split four spins into two pairs and the free energy is invariant under inverting all spins, there are six degenerate equilibrium states. The rAIAO phase is the same one that Wang et al [37] find at large U for the ideal pyrochlore structure and denote by AF2. This can be seen by comparing equations (26)- (29) to the spins given in figure 1(b) in [37].…”

“…The rAIAO phase is the same one that Wang et al [37] find at large U for the ideal pyrochlore structure and denote by AF2. This can be seen by comparing equations (26)- (29) to the spins given in figure 1(b) in [37]. We conclude that the tight-binding model of [37] corresponds to small s | | t in our phase diagram.…”

“…The r2I2O order breaks the cubic symmetry. The axis ΓX in the plot is the one distinguished by the order, i.e., the (100) direction for the solution in equations (34)- (37). The horizontal red line marks the chemical potential.…”