Spiral order in the honeycomb iridate ${\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$

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Spiral order in the honeycomb iridate ${Li}_{2} {IrO}_{3}$

Johannes Reuther, Ronny Thomale, and Stephan Rachel

Phys. Rev. B 90, 100405(R) – Published 12 September 2014

Abstract

The honeycomb iridates $A_{2} {IrO}_{3}$ (A = Na, Li) constitute promising candidate materials to realize the Heisenberg-Kitaev model (HKM) in nature, hosting unconventional magnetic as well as spin-liquid phases. Recent experiments suggest, however, that ${{Li}_{2} IrO}_{3}$ exhibits a magnetically ordered state of incommensurate spiral type which has not been identified in the HKM. We show that these findings can be understood in the context of an extended Heisenberg-Kitaev scenario satisfying all tentative experimental evidence: (i) the maximum of the magnetic susceptibility is located inside the first Brillouin zone, (ii) the Curie-Weiss temperature is negative relating to dominant antiferromagnetic fluctuations, and (iii) significant second-neighbor spin exchange is involved.

Received 30 April 2014
Revised 20 August 2014

DOI:https://doi.org/10.1103/PhysRevB.90.100405

Authors & Affiliations

Johannes Reuther¹, Ronny Thomale², and Stephan Rachel³

¹Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
²Institute for Theoretical Physics, University of Würzburg, Am Hubland, D-97074 Würzburg, Germany
³Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany

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Vol. 90, Iss. 10 — 1 September 2014

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Figure 1
(a) Different colors of the nearest (full lines) and next nearest (dashed lines) neighbor bonds on the honeycomb lattice represent Kitaev interactions of $S_{i}^{x} S_{j}^{x}$ type (blue), $S_{i}^{y} S_{j}^{y}$ type (red), and $S_{i}^{z} S_{j}^{z}$ type (green). (b) Extended Brillouin zone scheme (inner hexagon is the first Brillouin zone) of the honeycomb lattice. Ferromagnetic (FM) order manifests as peaks in the center of the first Brillouin zone, while antiferromagnetic (AFM) order resides at the corner of the extended zone scheme. The spiral order found in experiments corresponds to an ordering wave vector on the red ring well inside the first Brillouin zone. $k_{0}$ denotes the distance from the $Γ$ point to the first Brillouin zone boundary.
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Figure 2
Susceptibility profiles for the $J_{1} − J_{2}$ Heisenberg model Eq. (1) and $J_{1} = - 1$ . Thin black lines mark the boundary of the first Brillouin zone part within the extended Brillouin zone. For small $J_{2} > 0$ we first detect FM order. Above $J_{2} \approx 0.12$ the peaks split resulting in incommensurate spiral peaks; see main text for explanations. Bottom right: Peak position $k = | k |$ and Curie-Weiss temperature $Θ$ as a function of $J_{2}$ . $k_{0} = 2 π / 3$ is defined in Fig. 1. The gray shaded region is the parameter regime with spiral peaks inside the first Brillouin zone and negative Curie-Weiss temperature.
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Figure 3
Phase diagram of the extended HKM in Eq. (2), $g = 0.8$ . We find FM order, AFM order, incommensurate spiral order with wave vectors outside the first Brillouin zone (SP1), and incommensurate spiral order with wave vectors inside the first Brillouin zone (SP2). The green area indicates enhanced quantum fluctuations, possibly signaling a narrow nonmagnetic phase. The dashed line separates parameter regimes with positive from negative Curie-Weiss temperature.
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Figure 4
(a) Susceptibility profiles for the spiral phases of Fig. 3, along a cut with $ϕ_{2} = 0.8$ and Brillouin zone notation as in Fig. 2. For larger $ϕ_{1}$ , new ordering peaks emerge in the first Brillouin zone (SP2 order). Residual SP1 signatures persist, manifesting via shoulders marked by arrows. All plots display the $x x$ component of susceptibility. The corresponding $y y$ and $z z$ components result from $2 π / 3$ rotations in $k$ space. (b) Detailed migration profile of the ordering peaks (blue, red).
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Figure 5
(a) Absolute value $k$ of the wave vector at the ordering peak and the Curie-Weiss temperature in the SP1 phase (blue) and in the SP2 phase (red) as a function of $ϕ_{1}$ ( $ϕ_{2} = 0.8$ ). The jump in the peak position for $k$ is clearly observed. The gray shaded region marks the joint appearance of spiral peaks inside the first Brillouin zone and negative Curie-Weiss temperatures. (b) Cut through the susceptibility at $k_{x} = 0$ (blue) and $k_{x} = 2$ (green) as a function of $k_{y}$ . The Bragg-peak maximum is at $k = (0, 1.66) / a_{Ir - Ir}$ . (c) The spin pattern related to ${Li}_{2} {IrO}_{3}$ forms a nonplanar spiral.
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Physical Review B

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Spiral order in the honeycomb iridate ${Li}_{2} {IrO}_{3}$

Johannes Reuther, Ronny Thomale, and Stephan Rachel

Phys. Rev. B 90, 100405(R) – Published 12 September 2014

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Physical Review B

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Spiral order in the honeycomb iridate Li2IrO3

Johannes Reuther, Ronny Thomale, and Stephan Rachel

Phys. Rev. B 90, 100405(R) – Published 12 September 2014

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Spiral order in the honeycomb iridate ${Li}_{2} {IrO}_{3}$