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arXiv:2212.13271 · Physical review. B./Physical review. B

Proximate Dirac spin liquid in the honeycomb lattice J1−J3 XXZ model: Numerical study and application to cobaltates

Anjishnu Bose, Manodip Routh, Sreekar Voleti, Sudip Kumar Saha, Manoranjan Kumar, Tanusri Saha‐Dasgupta, Arun Paramekanti

Published Nov 16, 2023• 17 citations
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Abstract

Recent theoretical and experimental work suggests that the honeycomb cobaltates, initially proposed as candidate Kitaev quantum magnets, are in fact described by a pseudospin-$1/2$ easy-plane spin Hamiltonian with nearest-neighbor ferromagnetic (FM) exchange ${J}_{1}$ being frustrated by antiferromagnetic third-neighbor exchange ${J}_{3}$ and weaker compass anisotropies. Using exact diagonalization and density-matrix renormalization group (DMRG) calculations, we show that this model exhibits FM order at small ${J}_{3}/{J}_{1}$ and zigzag (ZZ) order at large ${J}_{3}/{J}_{1}$, separated by an intermediate phase, which we label as $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}$. This $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}$ phase is shown to exhibit spin-liquid-like correlations in DMRG, although we cannot preclude weak broken symmetries, e.g., weak Ising type N\'eel order, given the limits on our explored system sizes. Using a modified parton mean field theory and variational Monte Carlo on Gutzwiller projected wave functions, we show that the optimal FM and ZZ orders as well as the intermediate $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}$ state are proximate to a ``parent'' Dirac spin liquid (SL). This Dirac SL is shown to capture the broad continuum in the temperature and magnetic field dependent terahertz spectroscopy of ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$, and the reported low-temperature metallic thermal conductivity in ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$ and ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$ upon incorporating disorder-induced broadening.

DOI
10.1103/physrevb.108.174422

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Can we realize Kitaev quantum spin liquids in the $J_\text{eff}=1/2$ honeycomb cobaltates?

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Frustrated magnetism provides a rich platform for realizing exotic quantum phases in condensed matter systems. Among these, quantum spin liquids (QSLs) are a particularly intriguing class of states that exhibit no magnetic ordering and preserve all symmetries down to zero temperature. Just as conventional magnets can be thought of as spin crystals,QSLs represent their liquid analogs. This thesis explores two broad types of QSLs in two-dimensional magnets, with a focus on their dynamic properties and the nature of transitions from conventional magnetic states into QSL phases. QSLs are topological phases that host fractionalized excitations known as spinons. In these states, mobile spinons are coupled to an emergent gauge field, forming a highly entangled phase of matter. The first part of this thesis focuses on Dirac spin liquids (DSLs), a class of QSLs where spinons exhibit a gapless Dirac dispersion. To study the dynamicsof these spin liquids beyond leading-order mean-field theory, we develop a novel large-N fermionic limit of gapless spinons. In this limit, all gauge excitations are suppressed, and the leading-order corrections instead arise from order parameter fluctuations, which may condense and drive transitions into symmetry-broken phases from the parent DSL. Incorporating random phase approximation (RPA) corrections to the spin susceptibility, we find that sharp, spin-wave–like modes are recovered in the excitation spectrum. This methodology successfully captures characteristic features of the dynamics near phase transitions and in weakly symmetry-broken phases proximate to a DSL. We apply this approach to microscopic models, including the J1-J2 XXZ model on the triangular lattice and the J1-J3 XXZ model on the honeycomb lattice. These models closely represent a wide range of frustrated materials, such as triangular and honeycomb cobaltates and triangular ytterbium compounds. Our theoretical predictions show excellent agreement with experimentally measured inelastic neutron scattering spectra from these materials. Gapless spinons can also sometimes become gapped through chiral symmetry breaking, leading to the formation of a chiral spin liquid (CSL). The CSL is an archetypal bosonic realization of a fractional quantum Hall (FQH) state at half-filling, and has been proposed as a time-reversal–breaking ground state in a variety of frustrated spin models. The second part of this thesis addresses the intriguing possibility of realizing a CSL through the "melting" of a chiral spin crystal—a conventional magnet with non-coplanar ordering. Such a transition offers a physically appealing analogue to the melting of a Wigner crystal into a fermionic FQH liquid. Focusing on the kagome lattice, where chiral interactions are knownto stabilize a CSL ground state, we study the effects of additional long-range Heisenberg interactions and demonstrate a phase transition into a non-coplanar octahedral magnetic order. To describe the universal class of transition between a CSL and a non-coplanar anti-ferromagnetic order, we then develop a novel Chern-Simons–matter field theory framework, constrained by symmetry principles and symmetry fractionalization patterns. This thesis thus provides new insights into the dynamical signatures and phase transitions of quantum spin liquids, offering a bridge between theoretical models and experimental observations in frustrated magnetic systems.

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Unveiling magnetic interactions of ruthenium trichloride via constraining direction of orbital moments: Potential routes to realize a quantum spin liquid

Yusheng Hou, Hongjun Xiang, X. G. Gong

Recent experiments reveal that the honeycomb ruthenium trichloride $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$ is a prime candidate of the Kitaev quantum spin liquid (QSL). However, there is no theoretical model which can properly describe its experimental dynamical response due to the lack of a full understanding of its magnetic interactions. Here, we propose a general scheme to calculate the magnetic interactions in systems (e.g., $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$) with nonnegligible orbital moments by constraining the directions of orbital moments. With this scheme, we put forward a minimal ${J}_{1}\ensuremath{-}{K}_{1}\ensuremath{-}{\mathrm{\ensuremath{\Gamma}}}_{1}\ensuremath{-}{J}_{3}\ensuremath{-}{K}_{3}$ model for $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$ and find that: (I) The third nearest neighbor (NN) antiferromagnetic Heisenberg interaction ${J}_{3}$ stabilizes the zigzag antiferromagnetic order; (II) The NN symmetric off-diagonal exchange ${\mathrm{\ensuremath{\Gamma}}}_{1}$ plays a pivotal role in determining the preferred direction of magnetic moments and generating the spin wave gap. An exact diagonalization study on this model shows that the Kitaev QSL can be realized by suppressing the NN symmetric off-diagonal exchange ${\mathrm{\ensuremath{\Gamma}}}_{1}$ and the third NN Heisenberg interaction ${J}_{3}$. Thus, we not only propose a powerful general scheme for investigating the intriguing magnetism of ${J}_{\mathrm{eff}}=1/2$ magnets, but also point out future directions for realizing the Kitaev QSL in the honeycomb ruthenium trichloride $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$.

arXiv:1612.00761•Physical review. B./Physical review. B•Aug 7, 2017•74 citations
Recent theoretical and experimental work suggests that the honeycomb cobaltates, initially proposed as candidate Kitaev quantum magnets, are in fact described by a pseudospin-1/21/21/2 easy-plane spin Hamiltonian with nearest-neighbor ferromagnetic (FM) exchange J1{J}_{1}J1​ being frustrated by antiferromagnetic third-neighbor exchange J3{J}_{3}J3​ and weaker compass anisotropies. Using exact diagonalization and density-matrix renormalization group (DMRG) calculations, we show that this model exhibits FM order at small J3/J1{J}_{3}/{J}_{1}J3​/J1​ and zigzag (ZZ) order at large J3/J1{J}_{3}/{J}_{1}J3​/J1​, separated by an intermediate phase, which we label as SL\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}SL\ifmmode~\else˜\fi. This SL\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}SL\ifmmode~\else˜\fi phase is shown to exhibit spin-liquid-like correlations in DMRG, although we cannot preclude weak broken symmetries, e.g., weak Ising type N'eel order, given the limits on our explored system sizes. Using a modified parton mean field theory and variational Monte Carlo on Gutzwiller projected wave functions, we show that the optimal FM and ZZ orders as well as the intermediate SL\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}SL\ifmmode~\else˜\fi state are proximate to a ``parent'' Dirac spin liquid (SL). This Dirac SL is shown to capture the broad continuum in the temperature and magnetic field dependent terahertz spectroscopy of BaCo2(AsO4)2{\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}BaCo2​(AsO4​)2​, and the reported low-temperature metallic thermal conductivity in Na2Co2TeO6{\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}Na2​Co2​TeO6​ and BaCo2(AsO4)2{\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}BaCo2​(AsO4​)2​ upon incorporating disorder-induced broadening.
Can we realize Kitaev quantum spin liquids in the Jeff=1/2J_\text{eff}=1/2Jeff​=1/2 honeycomb cobaltates?
The quest for Kitaev quantum spin liquids has led to great interest in honeycomb quantum magnets with strong spin-orbit coupling. It has been recently proposed that even Mott insulators with 3d3d3d transition metal ions, having nominally weak spin-orbit coupling, can realize such exotic physics. Motivated by this, we study the rhombohedral honeycomb cobaltates CoTiO3_33​, BaCo2_22​(PO4_44​)2_22​, and BaCo2_22​(AsO4_44​)2_22​, using ab initio\textit{ab initio}ab initio density functional theory, which takes into account realistic crystal field distortions and chemical information, in conjunction with exact diagonalization numerics. We show that these Co2+^{2+}2+ magnets host jeff=1/2j_\text{eff}=1/2jeff​=1/2 local moments with highly anisotropic ggg-factors, and we extract their full spin Hamiltonians including longer-range and anisotropic exchange couplings. For CoTiO3_33​, we find a nearest-neighbor easy-plane ferromagnetic XXZXXZXXZ model with additional bond-dependent anisotropies and interlayer exchange, which supports three-dimensional (3D) Dirac nodal line magnons. In contrast, for BaCo2_22​(PO4_44​)2_22​ and BaCo2_22​(AsO4_44​)2_22​, we find a strongly suppressed interlayer coupling, and significant frustration from additional third-neighbor antiferromagnetic exchange mediated by P/As. Such bond-anisotropic J1J_1J1​-J3J_3J3​ spin models can support collinear zig-zag or coplanar spiral ground states; we discuss their dynamical spin correlations which reveal a gapped Goldstone mode, and argue that the effective parameters of these pseudospin-1/21/21/2 models may be strongly renormalized by coupling to a low energy spin-exciton. Our results call for re-examining proposals for realizing Kitaev spin liquids in the honeycomb cobaltates.
Frustrated magnetism provides a rich platform for realizing exotic quantum phases in condensed matter systems. Among these, quantum spin liquids (QSLs) are a particularly intriguing class of states that exhibit no magnetic ordering and preserve all symmetries down to zero temperature. Just as conventional magnets can be thought of as spin crystals,QSLs represent their liquid analogs. This thesis explores two broad types of QSLs in two-dimensional magnets, with a focus on their dynamic properties and the nature of transitions from conventional magnetic states into QSL phases. QSLs are topological phases that host fractionalized excitations known as spinons. In these states, mobile spinons are coupled to an emergent gauge field, forming a highly entangled phase of matter. The first part of this thesis focuses on Dirac spin liquids (DSLs), a class of QSLs where spinons exhibit a gapless Dirac dispersion. To study the dynamicsof these spin liquids beyond leading-order mean-field theory, we develop a novel large-N fermionic limit of gapless spinons. In this limit, all gauge excitations are suppressed, and the leading-order corrections instead arise from order parameter fluctuations, which may condense and drive transitions into symmetry-broken phases from the parent DSL. Incorporating random phase approximation (RPA) corrections to the spin susceptibility, we find that sharp, spin-wave–like modes are recovered in the excitation spectrum. This methodology successfully captures characteristic features of the dynamics near phase transitions and in weakly symmetry-broken phases proximate to a DSL. We apply this approach to microscopic models, including the J1-J2 XXZ model on the triangular lattice and the J1-J3 XXZ model on the honeycomb lattice. These models closely represent a wide range of frustrated materials, such as triangular and honeycomb cobaltates and triangular ytterbium compounds. Our theoretical predictions show excellent agreement with experimentally measured inelastic neutron scattering spectra from these materials. Gapless spinons can also sometimes become gapped through chiral symmetry breaking, leading to the formation of a chiral spin liquid (CSL). The CSL is an archetypal bosonic realization of a fractional quantum Hall (FQH) state at half-filling, and has been proposed as a time-reversal–breaking ground state in a variety of frustrated spin models. The second part of this thesis addresses the intriguing possibility of realizing a CSL through the "melting" of a chiral spin crystal—a conventional magnet with non-coplanar ordering. Such a transition offers a physically appealing analogue to the melting of a Wigner crystal into a fermionic FQH liquid. Focusing on the kagome lattice, where chiral interactions are knownto stabilize a CSL ground state, we study the effects of additional long-range Heisenberg interactions and demonstrate a phase transition into a non-coplanar octahedral magnetic order. To describe the universal class of transition between a CSL and a non-coplanar anti-ferromagnetic order, we then develop a novel Chern-Simons–matter field theory framework, constrained by symmetry principles and symmetry fractionalization patterns. This thesis thus provides new insights into the dynamical signatures and phase transitions of quantum spin liquids, offering a bridge between theoretical models and experimental observations in frustrated magnetic systems.
Recent work has shown that the triangular lattice spin-$\frac{1}{2} {J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg and XXZ antiferromagnets may exhibit coplanar or supersolid orders proximate to a gapless Dirac spin-liquid phase. We explore a distinct $\text{SU}(2N)\ifmmode\times\else\texttimes\fi{}\text{SU}(M)$ fermionic parton approach, complemented by variational Monte Carlo calculations for the spin-12\frac{1}{2}21​ model, to study the phase diagram of these models. We also calculate their dynamical spin response including parton interactions within a random-phase approximation, and discuss implications for neutron scattering on triangular lattice cobaltates Ba3CoSb2O9,Na2BaCo(PO4)2,K2Co(SeO3)2,Rb2Co(SeO3)2{\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}, {\mathrm{Na}}_{2}\mathrm{BaCo}{({\mathrm{PO}}_{4})}_{2}, {\mathrm{K}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}, {\mathrm{Rb}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}Ba3​CoSb2​O9​,Na2​BaCo(PO4​)2​,K2​Co(SeO3​)2​,Rb2​Co(SeO3​)2​, and Yb-based magnet KYbSe2{\mathrm{KYbSe}}_{2}KYbSe2​.
We study the ground-state (GS) properties of the frustrated spin-12\frac{1}{2}21​ J1{J}_{1}J1​--J2{J}_{2}J2​--J3{J}_{3}J3​ Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (${J}_{1}=\ensuremath{-}1$) exchange and frustrating antiferromagnetic next-nearest-neighbor (J2>0{J}_{2}>0J2​>0) and next-next-nearest-neighbor (J3>0{J}_{3}>0J3​>0) exchanges, for the case J3=J2{J}_{3}={J}_{2}J3​=J2​. We use the coupled-cluster method implemented to high orders of approximation, complemented by the Lanczos exact diagonalization of a large finite lattice with 32 sites, in order to calculate the GS energy, magnetic order parameter, and spin-spin correlation functions. In one scenario we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic (``striped'') collinear order at ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at J2cl=17{J}_{2}^{\mathrm{cl}}=\frac{1}{7}J2cl​=71​ ($\ensuremath{\approx}0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region 110<J2<15\frac{1}{10}<{J}_{2}<\frac{1}{5}101​<J2​<51​. Hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model. We compare results for the present ferromagnetic case (with ${J}_{1}=\ensuremath{-}1$) to previous results for the corresponding antiferromagnetic case (with J1=+1{J}_{1}=+1J1​=+1). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter J2{J}_{2}J2​. However, there are considerable differences in the behavior of the order parameters for the two models for ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with J1=+1{J}_{1}=+1J1​=+1) breaks down at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.60$, whereas the ``equivalent'' point for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) occurs at ${J}_{2}^{c}\ensuremath{\approx}0.11$. Unlike in the antiferromagnetic model (with J1=+1{J}_{1}=+1J1​=+1), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N\'eel and striped order) for J2c1<J2<J2c2{J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}J2c1​​<J2​<J2c2​​, with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order. Instead the evidence for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) points to one of two scenarios: either there is a direct first-order transition between the two magnetically ordered phases, as mentioned above; or there exists an intervening phase between them in the very narrow range $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range 110<J2<15\frac{1}{10}<{J}_{2}<\frac{1}{5}101​<J2​<51​.
We explore the phase diagram of the Kitaev-Heisenberg model with nearest neighbor interactions on the honeycomb lattice using the exact diagonalization of finite systems combined with the cluster mean field approximation, and supplemented by the insights from analytic approaches: the linear spin-wave and second-order perturbation theories. This study confirms that by varying the balance between the Heisenberg and Kitaev term, frustrated exchange interactions stabilize in this model either one of four phases with magnetic long range order: N\'eel phase, ferromagnetic phase, and two other phases with coexisting antiferromagnetic and ferromagnetic bonds, zigzag and stripy phase, or one of two distinct spin-liquid phases. Out of these latter disordered phases, the one with ferromagnetic Kitaev interactions has a substantially broader range of stability as the neighboring competing ordered phases, ferromagnetic and stripy, have very weak quantum fluctuations. Focusing on the quantum spin-liquid phases, we study spatial spin correlations and dynamic spin structure factor of the model by the exact diagonalization technique, and discuss the evolution of gapped low-energy spin response across the quantum phase transitions between the disordered spin liquid and phases with long range magnetic order.
Recent experiments reveal that the honeycomb ruthenium trichloride $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$ is a prime candidate of the Kitaev quantum spin liquid (QSL). However, there is no theoretical model which can properly describe its experimental dynamical response due to the lack of a full understanding of its magnetic interactions. Here, we propose a general scheme to calculate the magnetic interactions in systems (e.g., $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$) with nonnegligible orbital moments by constraining the directions of orbital moments. With this scheme, we put forward a minimal ${J}_{1}\ensuremath{-}{K}_{1}\ensuremath{-}{\mathrm{\ensuremath{\Gamma}}}_{1}\ensuremath{-}{J}_{3}\ensuremath{-}{K}_{3}$ model for $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$ and find that: (I) The third nearest neighbor (NN) antiferromagnetic Heisenberg interaction J3{J}_{3}J3​ stabilizes the zigzag antiferromagnetic order; (II) The NN symmetric off-diagonal exchange ${\mathrm{\ensuremath{\Gamma}}}_{1}$ plays a pivotal role in determining the preferred direction of magnetic moments and generating the spin wave gap. An exact diagonalization study on this model shows that the Kitaev QSL can be realized by suppressing the NN symmetric off-diagonal exchange ${\mathrm{\ensuremath{\Gamma}}}_{1}$ and the third NN Heisenberg interaction J3{J}_{3}J3​. Thus, we not only propose a powerful general scheme for investigating the intriguing magnetism of Jeff=1/2{J}_{\mathrm{eff}}=1/2Jeff​=1/2 magnets, but also point out future directions for realizing the Kitaev QSL in the honeycomb ruthenium trichloride $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$.