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arXiv:1802.04273 · Physical review. B./Physical review. B

Classification and properties of quantum spin liquids on the hyperhoneycomb lattice

Biao Huang, Wonjune Choi, Yong Baek Kim, Yuan-Ming Lu

Published May 22, 2018• 14 citations
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Abstract

The family of ``Kitaev materials'' provides an ideal platform to study quantum spin liquids and their neighboring magnetic orders. Motivated by the possibility of a quantum spin liquid ground state in pressurized hyperhoneycomb iridate $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$, we systematically classify and study symmetric quantum spin liquids on the hyperhoneycomb lattice, using the Abrikosov-fermion representation. Among the 176 symmetric $U(1)$ spin liquids (and 160 ${Z}_{2}$ spin liquids), we identify eight ``root'' $U(1)$ spin liquids in proximity to the ground state of the solvable Kitave model on the hyperhonecyomb lattice. These eight states are promising candidates for possible $U(1)$ spin liquid ground states in pressurized $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$. We further discuss physical properties of these eight $U(1)$ spin liquid candidates, and show that they all support nodal-line-shaped spinon Fermi surfaces.

DOI
10.1103/physrevb.97.195141

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The family of Kitaev materials'' provides an ideal platform to study quantum spin liquids and their neighboring magnetic orders. Motivated by the possibility of a quantum spin liquid ground state in pressurized hyperhoneycomb iridate $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$, we systematically classify and study symmetric quantum spin liquids on the hyperhoneycomb lattice, using the Abrikosov-fermion representation. Among the 176 symmetric $U(1)$ spin liquids (and 160 ${Z}_{2}$ spin liquids), we identify eight root'' U(1)U(1)U(1) spin liquids in proximity to the ground state of the solvable Kitave model on the hyperhonecyomb lattice. These eight states are promising candidates for possible U(1)U(1)U(1) spin liquid ground states in pressurized \ensuremathβ\ensuremath-Li2IrO3\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}\ensuremathβ\ensuremath-Li2​IrO3​. We further discuss physical properties of these eight U(1)U(1)U(1) spin liquid candidates, and show that they all support nodal-line-shaped spinon Fermi surfaces.
We point out that the Kitaev materials may not necessarily support Kitaev spin liquid. It is well known that having a Kitaev term in the spin interaction is not the sufficient condition for the Kitaev spin-liquid ground state. Many other spin liquids may be stabilized by the competing spin interactions of the systems. We thus explore the possibilities of non-Kitaev spin liquids in the honeycomb Kitaev materials. We carry out a systematic classification of gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids using the Schwinger boson representation for the spin variables. The presence of strong spin-orbit coupling in the Kitaev materials brings new ingredients into the projective symmetry group classification of the non-Kitaev spin liquid. We predict the spectroscopic properties of these gapped non-Kitaev spin liquids. Moreover, among the gapped spin liquids that we discover, we identify the spin liquid whose spinon condensation leads to the zigzag magnetic order that was observed in Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ and $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. We further discuss the possibility of gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquid and the deconfined quantum criticality from the zigzag magnetic order to spin dimerization in pressurized $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$.
Motivated by recent numerical studies reporting putative quantum paramagnetic behavior in spin-1/21/21/2 Heisenberg models on the maple-leaf lattice, we classify Abrikosov fermion mean-field Ans\"atze of fully symmetric U(1) and Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquids within the framework of projective symmetry groups. We obtain a total of 17U(1)17\phantom{\rule{4pt}{0ex}}\text{U}(1)17U(1) and 12Z212\phantom{\rule{4pt}{0ex}}{\mathbb{Z}}_{2}12Z2​ algebraic PSGs, and, upon restricting their realization via mean-field Ans\"atze with nearest-neighbor amplitudes (relevant to the studied models), only 12 U(1) and 8 Z2{\mathbb{Z}}_{2}Z2​ distinct phases are obtained. We present both singlet and triplet fields for all Ans\"atze up to third nearest-neighbor bonds and discuss their spinon dispersions as well as their dynamical spin structure factors. We further assess the effects of Gutzwiller projection on the equal-time spin structure factors.
Motivated by recent experiments on $\ensuremath{\beta}$-Li2{}_{2}2​IrO3{}_{3}3​, we study the phase diagram of the Heisenberg-Kitaev model on a three-dimensional lattice of tricoordinated Ir4+{}^{4+}4+, dubbed the hyperhoneycomb lattice. The lattice geometry of this material, along with Ir4+{}^{4+}4+ ions carrying Jeff=1/2{J}_{\mathrm{eff}}=1/2Jeff​=1/2 moments, suggests that the Heisenberg-Kitaev model may effectively capture the low-energy spin-physics of the system in the strong-coupling limit. Using a combination of semiclassical analysis, exact solution, and slave-fermion mean-field theory, we find, in addition to the spin liquid, four different magnetically ordered phases depending on the parameter regime. All four magnetic phases---the N\'eel, the polarized ferromagnet, the skew-stripy, and the skew-zig-zag---have collinear spin ordering. The three-dimensional Z2{Z}_{2}Z2​ spin liquid, which extends over an extended parameter regime around the exactly solvable Kitaev point, has a gapless Majorana mode with a deformed Fermi circle (codimensions, dc=2{d}_{c}=2dc​=2). We discuss the effect of the magnetic field and finite temperature on different phases that may be relevant for future experiments.
Motivated by recent numerical discovery of a gapped spin liquid phase in spin-1/21/21/2 triangular-lattice ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model, we classify symmetric Z2{Z}_{2}Z2​ spin liquids on triangular lattice in the Abrikosov-fermion representation. We find 20 phases with distinct spinon symmetry quantum numbers, eight of which have their counterparts in the Schwinger-boson representation. Among them we identify two promising candidates (#1 and #20), which can realize a gapped Z2{Z}_{2}Z2​ spin liquid with up to next nearest neighbor mean-field amplitudes. We analyze their neighboring magnetic orders and valence bond solid patterns, and find one state (#20) that is connected to 120-degree Neel order by a continuous quantum phase transition. We also identify gapped nematic Z2{Z}_{2}Z2​ spin liquids in the neighborhood of the symmetric states and find three promising candidates (#1, #6, and #20).
The Kitaev model, which hosts a quantum spin liquid (QSL) in the ground state, was originally defined on a two-dimensional honeycomb lattice, but can be straightforwardly extended to any tricoordinate lattices in any spatial dimensions. In particular, the three-dimensional (3D) extensions are of interest as a realization of 3D QSLs, and some materials like βββ-Li2_{2}2​IrO3_{3}3​, γγγ-Li2_22​IrO3_33​, and βββ-ZnIrO3_{3}3​ were proposed for the candidates. However, the phase diagrams of the models for those candidates have not been fully elucidated, mainly due to the limitation of numerical methods for 3D frustrated quantum spin systems. Here we study the Kitaev-Heisenberg model defined on a 3D hyperhoneycomb lattice, by using the pseudofermion functional renormalization group method. We show that the ground-state phase diagram contains the QSL phases in the vicinities of the pristine ferromagnetic and antiferromagnetic Kitaev models, in addition to four magnetically ordered phases, similar to the two-dimensional honeycomb case. Our results respect the four-sublattice symmetry inherent in the model, which was violated in the previous study. Moreover, we also show how the phase diagram changes with the anisotropy in the interactions. The results provide a reference for the search of the hyperhoneycomb Kitaev materials.
In quantum magnetism, Kitaev's honeycomb model is well known for its gapped and gapless spin liquid ground states in which the elementary spin degrees of freedom fractionalize into Majorana fermions and a Z2{\mathbb{Z}}_{2}Z2​ gauge field. In this paper, the authors discuss the gapless spin liquids arising in three-dimensional Kitaev models and show that the gapless Majorana fermions form metallic states which, depending on the underlying lattice structure, exhibit Fermi surfaces, nodal lines, or Weyl points.