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arXiv:1601.05512 · Physical review. B./Physical review. B

Clues and criteria for designing a Kitaev spin liquid revealed by thermal and spin excitations of the honeycomb iridate Na2IrO3

Youhei Yamaji, Takafumi J. Suzuki, Takuto Yamada, Sei-ichiro Suga, Naoki Kawashima, Masatoshi Imada

Published May 24, 2016• 106 citations
arXivHTMLPDF

Abstract

Contrary to the original expectation, ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ is not a Kitaev's quantum spin liquid (QSL) but shows a zigzag-type antiferromagnetic order in experiments. Here, we propose experimental clues and criteria to measure how a material in hand is close to the Kitaev's QSL state. For this purpose, we systematically study thermal and spin excitations of a generalized Kitaev-Heisenberg model studied by Chaloupka et al., Phys. Rev. Lett. 110, 097204 (2013) and an effective ab initio Hamiltonian for ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ proposed by Yamaji et al., Phys. Rev. Lett. 113, 107201 (2014), by employing a numerical diagonalization method. We reveal that closeness to the Kitaev's QSL is characterized by the following properties, besides trivial criteria such as reduction of magnetic ordered moments and N\'eel temperatures. (1) Two peaks in the temperature dependence of specific heat at ${T}_{\ensuremath{\ell}}$ and ${T}_{h}$ caused by the fractionalization of spin to two types of Majorana fermions. (2) In between the double peak, a prominent plateau or shoulder pinned at $\frac{R}{2}ln2$ in the temperature dependence of entropy, where $R$ is the gas constant. (3) Failure of the linear spin wave approximation at the low-lying excitations of dynamical structure factors. (4) Small ratio ${T}_{\ensuremath{\ell}}/{T}_{h}$ close to or less than 0.03. According to the proposed criteria, ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ is categorized to a compound close to the Kitaev's QSL, and is proven to be a promising candidate for the realization of the QSL if the relevant material parameters can further be tuned by making thin film of ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ on various substrates or applying axial pressure perpendicular to the honeycomb networks of iridium ions. Applications of these characterization to ${({\mathrm{Na}}_{1\ensuremath{-}x}{\mathrm{Li}}_{x})}_{2}{\mathrm{IrO}}_{3}$ and other related materials are also discussed.

DOI
10.1103/physrevb.93.174425

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Contrary to the original expectation, Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ is not a Kitaev's quantum spin liquid (QSL) but shows a zigzag-type antiferromagnetic order in experiments. Here, we propose experimental clues and criteria to measure how a material in hand is close to the Kitaev's QSL state. For this purpose, we systematically study thermal and spin excitations of a generalized Kitaev-Heisenberg model studied by Chaloupka et al., Phys. Rev. Lett. 110, 097204 (2013) and an effective ab initio Hamiltonian for Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ proposed by Yamaji et al., Phys. Rev. Lett. 113, 107201 (2014), by employing a numerical diagonalization method. We reveal that closeness to the Kitaev's QSL is characterized by the following properties, besides trivial criteria such as reduction of magnetic ordered moments and N'eel temperatures. (1) Two peaks in the temperature dependence of specific heat at T\ensuremathℓ{T}_{\ensuremath{\ell}}T\ensuremathℓ​ and Th{T}_{h}Th​ caused by the fractionalization of spin to two types of Majorana fermions. (2) In between the double peak, a prominent plateau or shoulder pinned at R2ln2\frac{R}{2}ln22R​ln2 in the temperature dependence of entropy, where RRR is the gas constant. (3) Failure of the linear spin wave approximation at the low-lying excitations of dynamical structure factors. (4) Small ratio T\ensuremathℓ/Th{T}_{\ensuremath{\ell}}/{T}_{h}T\ensuremathℓ​/Th​ close to or less than 0.03. According to the proposed criteria, Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ is categorized to a compound close to the Kitaev's QSL, and is proven to be a promising candidate for the realization of the QSL if the relevant material parameters can further be tuned by making thin film of Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ on various substrates or applying axial pressure perpendicular to the honeycomb networks of iridium ions. Applications of these characterization to (Na1\ensuremath−xLix)2IrO3{({\mathrm{Na}}_{1\ensuremath{-}x}{\mathrm{Li}}_{x})}_{2}{\mathrm{IrO}}_{3}(Na1\ensuremath−x​Lix​)2​IrO3​ and other related materials are also discussed.
The pursuit of quantum spin liquid (QSL) in the Kitaev honeycomb magnets has drawn intensive attention recently. In particular, $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ has been widely recognized as a promising candidate for the Kitaev QSL. Although the compound exhibits an antiferromagnetic order under zero field, it is believed to be endowed with fractionalized excitations and can be driven to the QSL phase by magnetic fields. Here, based on a realistic $K\text{\ensuremath{-}}J\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ model for $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$, we exploit the exponential tensor renormalization group approach to explore the phase diagram of the compound under magnetic fields. We calculate the thermodynamic quantities, including the specific heat, Gr\"uneisen parameter, magnetic torque, magnetotropic susceptibility, and so on, under a magnetic field with a tilting angle $\ensuremath{\theta}$ to the c∗{c}^{*}c∗-axis perpendicular to the honeycomb plane. We find an extended QSL in the angle-field phase diagram determined with thermodynamic responses. The gapless nature of such field-induced QSL is identified from the specific heat and entropy data computed down to very low temperatures. The present study provides guidance for future high-field experiments for the QSL in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ and other candidate Kitaev magnets.
We investigate the dynamical properties of Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​. For five effective models proposed for Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​, we numerically calculate dynamical structure factors (DSFs) with an exact diagonalization method. An effective model obtained from ab initio calculations explains inelastic neutron scattering experiments adequately. We further calculate excitation modes based on linearized spin-wave theory. The spin-wave excitation of the effective models obtained by ab initio calculations disagrees with the low-lying excitation of DSFs. We attribute this discrepancy to the location of Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ in a parameter space close to the phase boundary with the Kitaev spin-liquid phase.
An effective low-energy Hamiltonian of itinerant electrons for iridium oxide Na2IrO3 is derived by an ab initio downfolding scheme. The model is then reduced to an effective spin model on a honeycomb lattice by the strong coupling expansion. Here we show that the ab initio model contains spin-spin anisotropic exchange terms in addition to the extensively studied Kitaev and Heisenberg exchange interactions, and allows us to describe the experimentally observed zigzag magnetic order, interpreted as the state stabilized by the antiferromagnetic coupling of the ferromagnetic chains. We clarify possible routes to realize quantum spin liquids from existing Na2IrO3.
The magnetic-field response of the Mott-insulating honeycomb iridate Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ is investigated using torque magnetometry measurements in magnetic fields up to 60 T. A peak-dip structure is observed in the torque response at magnetic fields corresponding to an energy scale close to the zigzag ordering ($\ensuremath{\approx}15\phantom{\rule{0.28em}{0ex}}\mathrm{K}$) temperature. Using exact diagonalization calculations, we show that such a distinctive signature in the torque response constrains the effective spin models for these classes of Kitaev materials to ones with dominant ferromagnetic Kitaev interactions, while alternative models with dominant antiferromagnetic Kitaev interactions are excluded. We further show that, at high magnetic fields, long range spin correlation functions decay rapidly, pointing to a transition to a long-sought-after field-induced quantum spin liquid beyond the peak-dip structure, suggesting this to be a common feature of the family of Kitaev systems. Kitaev systems are thus revealed to be excellent candidates for field-induced quantum spin liquids, similar physics having been suggested in another Kitaev system $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$.
Motivated by the recent synthesis of two insulating Li2IrO3{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}Li2​IrO3​ polymorphs, where Ir4+{\mathrm{Ir}}^{4+}Ir4+ Seff=1/2{S}_{\mathrm{eff}}=1/2Seff​=1/2 moments form 3D (``harmonic'') honeycomb structures with threefold coordination, we study magnetic Hamiltonians on the resulting $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ hyperhoneycomb lattice and $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ stripyhoneycomb lattice. Experimentally measured magnetic susceptibilities suggest that Kitaev interactions, predicted for the ideal ${90}^{\ensuremath{\circ}}$ Ir-O-Ir bonds, are sizable in these materials. We first consider pure Kitaev interactions, which lead to an exactly soluble 3D quantum spin liquid (QSL) with emergent Majorana fermions and Z2{\mathrm{Z}}_{2}Z2​ flux loops. Unlike 2D QSLs, the 3D QSL is stable to finite temperature, with ${T}_{c}\ensuremath{\approx}|K|/100$. On including Heisenberg couplings, exact solubility is lost. However, by noting that the shortest closed loop $\ensuremath{\ell}$ is relatively large in these structures, we construct an $\ensuremath{\ell}\ensuremath{\rightarrow}\ensuremath{\infty}$ approximation by defining the model on the Bethe lattice. The phase diagram of the Kitaev-Heisenberg model on this lattice is obtained directly in the thermodynamic limit, using tensor network states and the infinite-system time-evolving-block-decimation (iTEBD) algorithm. Both magnetically ordered and gapped QSL phases are found, the latter being identified by an entanglement fingerprint.
We investigate the ground state properties of Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ based on numerical calculations of the recently proposed ab initio Hamiltonian represented by Kitaev and extended Heisenberg interactions. To overcome the limitation posed by small tractable system sizes in the exact diagonalization study employed in a previous study [Y. Yamaji et al., Phys. Rev. Lett. 113, 107201 (2014)], we apply a two-dimensional density matrix renormalization group and an infinite-size tensor-network method. By calculating at much larger system sizes, we critically test the validity of the exact diagonalization results. The results consistently indicate that the ground state of Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ is a magnetically ordered state with zigzag configuration in agreement with experimental observations and the previous diagonalization study. Applications of the two independent methods in addition to the exact diagonalization study further uncover a consistent and rich phase diagram near the zigzag phase beyond the accessibility of the exact diagonalization. For example, in the parameter space away from the ab initio value of Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ controlled by the trigonal distortion, we find three phases: (i) an ordered phase with the magnetic moment aligned mutually in 120 degrees orientation on every third hexagon, (ii) a magnetically ordered phase with a 16-site unit cell, and (iii) an ordered phase with presumably incommensurate periodicity of the moment. It suggests that potentially rich magnetic structures may appear in A2IrO3{A}_{2}{\mathrm{IrO}}_{3}A2​IrO3​ compounds for AAA other than Na. The present results also serve to establish the accuracy of the first-principles approach in reproducing the available experimental results thereby further contributing to finding a route to realize the Kitaev spin liquid.