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arXiv:2304.13378 · Physical review. B./Physical review. B

Quantum liquids of the S=32 Kitaev honeycomb and related Kugel-Khomskii models

W. M. H. Natori, Hui-Ke Jin, Johannes Knolle

Published Aug 4, 2023• 18 citations
arXivHTMLPDF

Abstract

The $S=\frac{3}{2}$ Kitaev honeycomb model (KHM) is unique among the spin-$S$ Kitaev models due to a massive ground-state quasidegeneracy that hampered previous numerical and analytical studies. In a recent work [Jin et al., Nat. Commun. 13, 3813 (2022)], we showed how an SO(6) Majorana parton mean-field theory of the $S=\frac{3}{2}$ isotropic KHM explains the anomalous features of this Kitaev spin liquid (KSL) in terms of an emergent low-energy Majorana flat band. Away from the isotropic limit, the $S=\frac{3}{2}$ KSL generally displays a quadrupolar order with gapped or gapless Majorana excitations, features that were quantitatively confirmed by density-matrix renormalization group simulations. In this paper, we explore the connection between the $S=\frac{3}{2}$ KHM with Kugel-Khomskii models and discover exactly soluble examples for the latter. We perform a symmetry analysis for the variational parton mean-field Ans\"atze in the spin and orbital basis for different quantum liquid phases of the $S=\frac{3}{2}$ KHM. Finally, we investigate a proposed time-reversal symmetry-breaking spin liquid induced by [111] single-ion anisotropy and elucidate its topological properties as well as experimental signatures, e.g., an unquantized thermal Hall response.

DOI
10.1103/physrevb.108.075111

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The S=32S=\frac{3}{2}S=23​ Kitaev honeycomb model (KHM) is unique among the spin-SSS Kitaev models due to a massive ground-state quasidegeneracy that hampered previous numerical and analytical studies. In a recent work [Jin et al., Nat. Commun. 13, 3813 (2022)], we showed how an SO(6) Majorana parton mean-field theory of the S=32S=\frac{3}{2}S=23​ isotropic KHM explains the anomalous features of this Kitaev spin liquid (KSL) in terms of an emergent low-energy Majorana flat band. Away from the isotropic limit, the S=32S=\frac{3}{2}S=23​ KSL generally displays a quadrupolar order with gapped or gapless Majorana excitations, features that were quantitatively confirmed by density-matrix renormalization group simulations. In this paper, we explore the connection between the S=32S=\frac{3}{2}S=23​ KHM with Kugel-Khomskii models and discover exactly soluble examples for the latter. We perform a symmetry analysis for the variational parton mean-field Ans"atze in the spin and orbital basis for different quantum liquid phases of the S=32S=\frac{3}{2}S=23​ KHM. Finally, we investigate a proposed time-reversal symmetry-breaking spin liquid induced by [111] single-ion anisotropy and elucidate its topological properties as well as experimental signatures, e.g., an unquantized thermal Hall response.
Abstract The S=3/2 Kitaev honeycomb model (KHM) is a quantum spin liquid (QSL) state coupled to a static Z2 gauge field. Employing an SO(6) Majorana representation of spin3/2’s, we find an exact representation of the conserved plaquette fluxes in terms of static Z2 gauge fields akin to the S=1/2 KHM which enables us to treat the remaining interacting matter fermion sector in a parton mean-field theory. We uncover a ground-state phase diagram consisting of gapped and gapless QSLs. Our parton description is in quantitative agreement with numerical simulations, and is furthermore corroborated by the addition of a [001] single ion anisotropy (SIA) which continuously connects the gapless Dirac QSL of our model with that of the S=1/2 KHM. In the presence of a weak [111] SIA, we discuss an emergent chiral QSL within a perturbation theory.
The exactly solvable spin-1/2 Kitaev model on a honeycomb lattice has drawn significant interest, as it offers a pathway to realizing the long-sought-after quantum spin liquid. Building upon the Kitaev model, Yao and Lee introduced another exactly solvable model on an unusual star lattice featuring non-abelian spinons. The additional pseudospin degrees of freedom in this model could provide greater stability against perturbations, making this model appealing. However, a mechanism to realize such an interaction in a standard honeycomb lattice remains unknown. Here, we provide a microscopic theory to obtain the Yao-Lee model, on a honeycomb lattice by utilizing strong spin-orbit coupling of anions edge-shared between two e g ions in the exchange processes. This mechanism leads to the desired bond-dependent interaction among spins rather than orbitals, unique to our model, implying that the orbitals fractionalize into gapless Majorana fermions and fermionic octupolar excitations emerge. Since the conventional Kugel-Khomskii interaction also appears, we examine the phase diagram, including these interactions, using classical Monte Carlo simulations and exact diagonalization techniques. Our findings reveal a broad region of disordered states that break rotational symmetry in the bond energy, suggesting intriguing behavior reminiscent of a spin-orbital liquid.
The formation of quantum spin liquids in frustrated magnets represents an exciting possibility due to the rather exotic features they harbor, including fractionalized excitations and emergent gauge fields. \nUnfortunately, they are notoriously difficult to study as there are often no good analytical methods available and quantum Monte Carlo simulations are hindered by the negative sign problem. \nThe Kitaev honeycomb model is a notable exception of a frustrated quantum model which is exactly solvable and which hosts a number of distinct quantum spin liquid ground states. \nAs such, it allows for a rare opportunity to study the physics of spin liquids with full analytical control. \n \nIn this thesis, we study the fractionalization of spin-1/2 moments into Majorana fermions and an emergent Z2 gauge field in a generalization of the Kitaev honeycomb model to a number of three-dimensional lattices. \nWhile the excitations of the gauge field are always gapped, the fermionic quasiparticles may exhibit a gapless dispersion, forming fully two-dimensional Fermi surfaces, symmetry protected nodal lines, or topological Weyl nodes. \nWe show that one can deduce rather general constraints on the possible gapless excitations by making use of an object called the projective symmetry group. \nIn doing so we provide a scheme for classifying the various gapless Kitaev spin liquids. \nA thorough analysis is carried out for a number of these spin liquids, primarily investigating the stability of the gapless modes and the novel features resulting from their sometimes non-trivial topology, as well as their effects on certain equal-time correlation functions.
The Kitaev model on a honeycomb lattice with bond-dependent Ising interactions offers an exactly solvable model of a quantum spin liquid (QSL) with fractionalized excitations: gapped Z2{Z}_{2}Z2​ fluxes and gapless linearly dispersing Majorana fermions in the isotropic limit (Kx=Ky=Kz{K}_{x}={K}_{y}={K}_{z}Kx​=Ky​=Kz​). We explore the phase diagram along two axes, an external magnetic field hhh, applied out of plane of the honeycomb, and anisotropic interactions Kz{K}_{z}Kz​ larger than the other two (${K}_{x}={K}_{y}\ensuremath{\equiv}K$). For ${K}_{z}/K\ensuremath{\gg}2$ and h=0h=0h=0, the matter Majorana fermions have the largest gap, and the system is described by a gapped Z2{Z}_{2}Z2​ toric code. One of the central questions we address is whether the fractionalized excitations in the different phases have sharp signatures that can be detected in experiments. We show that while the response to single-spin excitations is broad, the spectral function corresponding to two-spin excitations across a bond has sharp signatures that can be attributed to specific anyons. In the toric code regime, the $\ensuremath{\epsilon}=e\ifmmode\times\else\texttimes\fi{}m$ fermion, formed from the bosonic Ising electric (eee) and magnetic (mmm) charges, disperses along a specific one-dimensional direction that provides a fingerprint of fractionalization. At lower Kz{K}_{z}Kz​ in the center of the Abelian phase, in a regime we dub the primordial fractionalized regime, the field generates a hybridization between the $\ensuremath{\epsilon}$ fermion and the Majorana matter fermion, resulting in a $\ensuremath{\psi}$ fermion which too has a distinct quasi-one-dimensional dispersion. All the other phases in the field-anisotropy plane are naturally obtained from this primordial soup. These highly constrained fractonlike dispersions can be observable by inelastic light and neutron scattering, thereby providing ``smoking gun'' signatures of fractionalization in the QSL phase. Our analysis is based on calculations of susceptibilities, topological entanglement entropy, and excitation dynamics, obtained using exact diagonalization and density matrix renormalization group, and supported by perturbation theory.
Theory of Dirac Spin-Orbital Liquids: monopoles, anomalies, and applications to SU(4)SU(4)SU(4) honeycomb models
Dirac spin liquids represent a class of highly-entangled quantum phases in two dimensional Mott insulators, featuring exotic properties such as critical correlation functions and absence of well-defined low energy quasi-particles. Existing numerical works suggest that the spin-orbital SU(4)SU(4)SU(4) symmetric Kugel-Khomskii model of Mott insulators on the honeycomb lattice realizes a Dirac spin-orbital liquid, described at low energy by (2+1)d(2+1)d(2+1)d quantum electrodynamics (QED3_33​) with Nf=8N_f=8Nf​=8 Dirac fermions. We generalize methods previously developed for SU(2)SU(2)SU(2) spin systems to analyze the symmetry properties and stability of the Dirac spin-orbital liquid. We conclude that the standard Dirac state in the SU(4)SU(4)SU(4) honeycomb system, based on a simple parton mean-field ansatz, is intrinsically unstable at low energy due to the existence of a monopole perturbation that is allowed by physical symmetries and relevant under renormalization group flow. We propose two plausible alternative scenarios compatible with existing numerics. In the first scenario, we propose an alternative U(1)U(1)U(1) Dirac spin-orbital liquid, which is similar to the standard one except for its monopole symmetry quantum numbers. This alternative U(1)U(1)U(1) state represents a stable gapless phase. In the second scenario, we start from the standard U(1)U(1)U(1) Dirac liquid and Higgs the U(1)U(1)U(1) gauge symmetry down to Z4\mathbb{Z}_4Z4​. The resulting Z4\mathbb{Z}_4Z4​ Dirac spin-orbital liquid is stable. We also discuss the continuous quantum phase transitions from the Z4\mathbb{Z}_4Z4​ Dirac liquids to conventional symmetry-breaking orders, described by the QED3_33​ theory with Nf=8N_f=8Nf​=8 supplemented with a critical charge-444 Higgs field. We discuss possible ways to distinguish these scenarios in numerics. We also extend previous calculations of the quantum anomalies of QED3_33​ and match with generalized lattice Lieb-Schultz-Mattis constraints.
We study the quadrupolar Kitaev model, an S=1S=1S=1 honeycomb-lattice model with frustrated bond-dependent quadrupolar interactions. Using complementary methods and expanding around controlled limits, we uncover several intertwined structures. First, a semiclassical variational analysis based on SU(3)\mathrm{SU}(3)SU(3) flavor theory reveals an extensively degenerate manifold of classical mean-field ground states, suggesting that quantum fluctuations may stabilize a quantum-disordered phase. Second, in the bond-anisotropic limit, perturbation theory is used to derive effective low-energy Hamiltonians, which crucially depend on the presence (or absence) of a residual symmetry M\mathcal{M}M of combined lattice reflection and discrete spin rotation. A Majorana parton construction uncovers an exact Z2\mathbb Z_2Z2​ gauge structure and motivates possible confined and deconfined phases driven by gauge-charge condensation, consistent with the effective theories obtained in anisotropic limit. Further, within the same parton formalism, different Majorana mean-field ansätze produce both gapless and gapped candidate quantum-disordered states, distinguished by linear versus projective implementations of M\mathcal MM. Our results highlight frustrated quadrupolar interactions as a route to quantum-disordered phases, relevant to S≥1S \geq 1S≥1 Kitaev materials and Rydberg-array quantum simulators.