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Physical review. B./Physical review. B

Even-odd effect in multilayer Kitaev honeycomb magnets

Jaime Merino, Arnaud Ralko

Published Feb 27, 2025• 1 citation
Publisher

Abstract

Motivated by the three-dimensional structure of Kitaev materials we explore multilayer Kitaev models. The magnetic properties of a multilayer with an arbitrary number of Kitaev honeycomb layers stacked on top of each other coupled through a Heisenberg interaction $J$ is analyzed through Abrikosov fermion mean-field theory. We find quantum spin liquid (QSL) solutions which are gapped (ungapped) for even (odd) number of layers. The presence (absence) of the gap in even- (odd-) layered systems is associated with the spontaneous breaking (nonbreaking) of the inversion symmetry of the multilayer by the mean-field QSL solution. The good agreement between the Gutzwiller projected mean-field energy and the exact energy at large $J$ indicates that our QSL solution is adiabatically connected to the exact ground state. We also find that the Kitaev gapped chiral quantum spin liquid induced by external magnetic fields is stabilized by an antiferromagnetic interlayer coupling. Our results are relevant to the physics of $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ and ${\mathrm{H}}_{3}{\mathrm{LiIr}}_{2}{\mathrm{O}}_{6}$ which are examples of magnetically coupled multilayer Kitaev models.

DOI
10.1103/physrevb.111.085152

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Even-odd effect in multilayer Kitaev honeycomb magnets

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Motivated by the three-dimensional structure of Kitaev materials we explore multilayer Kitaev models. The magnetic properties of a multilayer with an arbitrary number of Kitaev honeycomb layers stacked on top of each other coupled through a Heisenberg interaction JJJ is analyzed through Abrikosov fermion mean-field theory. We find quantum spin liquid (QSL) solutions which are gapped (ungapped) for even (odd) number of layers. The presence (absence) of the gap in even- (odd-) layered systems is associated with the spontaneous breaking (nonbreaking) of the inversion symmetry of the multilayer by the mean-field QSL solution. The good agreement between the Gutzwiller projected mean-field energy and the exact energy at large JJJ indicates that our QSL solution is adiabatically connected to the exact ground state. We also find that the Kitaev gapped chiral quantum spin liquid induced by external magnetic fields is stabilized by an antiferromagnetic interlayer coupling. Our results are relevant to the physics of \ensuremathα\ensuremath-RuCl3\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}\ensuremathα\ensuremath-RuCl3​ and H3LiIr2O6{\mathrm{H}}_{3}{\mathrm{LiIr}}_{2}{\mathrm{O}}_{6}H3​LiIr2​O6​ which are examples of magnetically coupled multilayer Kitaev models.
Motivated by the three-dimensional structure of Kitaev materials we explore multilayer Kitaev models. The magnetic properties of a multilayer of an arbitrary number of Kitaev honeycomb layers stacked on top of each other coupled through a Heisenberg interaction, J, is analyzed through Abrikosov fermion mean-field theory. The system sustains quantum spin liquid (QSL) solutions which have different character depending on parity of the number of layers. While in even layered Kitaev models a gapped QSL emerges, odd-layered models host gapless QSLs. The projective symmetry group analysis of these solutions unravel a layer-to-layer inversion symmetry rather than an expected reflection. Although these QSLs retain features of the single layer Kitaev spin liquid (KSL), they should be regarded hybrid QSLs consisting on several KSLs. The good agreement at large-J between the energy of the Gutzwiller projected mean-field QSL and the exact energy indicates that such QSL Ansatz is adiabatically connected to the exact ground state. We also find that the Kitaev gapped chiral quantum spin liquid induced by external magnetic fields is stabilized by an antiferromagnetic interlayer coupling. Our results are relevant to the physics of alpha-RuCl3 and H3LiIr2O6 which are examples of magnetically coupled multilayer Kitaev models.
We investigate a bilayer Kitaev model in which two honeycomb layers are coupled by the Heisenberg interactions to discuss the effects of interlayer coupling on Kitaev quantum spin liquids (QSLs). In this model, there exists a local conserved quantity which results in no long-range spin correlations in the system. Using the exact diagonalization, bond-operator mean-field theory, and cluster expansion techniques, we study ground-state properties in the system. The obtained results suggest the existence of a first-order quantum phase transition between the Kitaev QSL and dimer-singlet states. We find that a one-triplet excitation from the dimer-singlet ground state is localized owing to the existence of the local conserved quantity. To examine finite-temperature properties, we make use of the thermal pure quantum state approach. We clarify that the double-peak structure in the specific heat inherent in the Kitaev QSL is maintained even above the quantum phase transition. The present results suggest that the Kitaev QSL is stable against the interlayer interference. Magnetic properties of multilayer Kitaev models are also addressed.
Kitaev's honeycomb-lattice spin-1/21/21/2 model has become a paradigmatic example for Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquids, both gapped and gapless. Here we study the fate of these spin-liquid phases in differently stacked bilayer versions of the Kitaev model. Increasing the ratio between the interlayer Heisenberg coupling ${J}_{\ensuremath{\perp}}$ and the intralayer Kitaev couplings Kx,y,z{K}^{x,y,z}Kx,y,z destroys the topological spin liquid in favor of a paramagnetic dimer phase. We study phase diagrams as a function of ${J}_{\ensuremath{\perp}}/K$ and Kitaev coupling anisotropies using Majorana-fermion mean-field theory, and we employ different expansion techniques in the limits of small and large ${J}_{\ensuremath{\perp}}/K$. For strongly anisotropic Kitaev couplings, we derive effective models for the different layer stackings that we use to discuss the quantum phase transition out of the Kitaev phase. We find that the phase diagrams depend sensitively on the nature of the stacking and anisotropy strength. While in some stackings and at strong anisotropies we find a single transition between the Kitaev and dimer phases, other stackings are more involved. Most importantly, we prove the existence of two novel macrospin phases, which can be understood in terms of Ising chains that can be either coupled ferromagnetically or remain degenerate, thus realizing a classical spin liquid. In addition, our results suggest the existence of a flux phase with spontaneous interlayer coherence. We discuss prospects for experimental realizations.
The ground state of the bilayer Kitaev model with the Heisenberg-type interlayer exchange interaction is investigated by means of the exact diagonalization. Calculating the ground-state energy, local quantity defined on each plaquette, and dynamical spin structure factor, we obtain results suggesting the existence of a quantum phase transition between the Kitaev quantum spin liquid (QSL) and dimer singlet states when the interlayer coupling is antiferromagnetic. On the other hand, increasing the ferromagnetic interlayer coupling, there exists no singularity in the physical quantities, suggesting that the S=1/2S=1/2S=1/2 Kitaev QSL state realized in each layer adiabatically connects to another QSL state realized in the S=1S=1S=1 Kitaev model. Thermodynamic properties are also studied by means of the thermal pure quantum state method.
We investigate the quantum spin liquid (QSL) ground state of anisotropic Kitaev model with antiferromagnetic (AFM) coupling under the [001][001][001] magnetic field with the finite-temperature Lanczos method (FTLM). In this anisotropic AFM Kitaev model with KX=KYK_{X}=K_{Y}KX​=KY​, KX+KY+KZ=−3KK_{X}+K_{Y}+K_{Z}=-3KKX​+KY​+KZ​=−3K, and KZ<−KK_{Z}<-KKZ​<−K, with magnetic field increasing, the gapped QSL experiences a transition to a gapless QSL at hc1=gμBHz1/Kh_{c1}=g\mu_{B}H_{z1}/Khc1​=gμB​Hz1​/K, to another gapless QSL with C6C_{6}C6​ rotational symmetry at hc2h_{c2}hc2​, and to a new U(1)U(1)U(1) gapless QSL between hc3h_{c3}hc3​ and hc4h_{c4}hc4​, respectively. These indicate that magnetic field could first turn the anisotropic gapped or gapless QSL back into the isotropic C6C_{6}C6​ gapless one and then make it to undergo the similar evolution as the isotropic case. Moreover, the critical magnetic fields hc1h_{c1}hc1​, hc2h_{c2}hc2​, hc3h_{c3}hc3​, and hc4h_{c4}hc4​ come up monotonically with the increasing Kitaev coupling; this suggests that the magnetic field can be applied to the modulation of the anisotropic Kitaev materials.
We investigate the phase diagram of a bilayer Kitaev honeycomb model with Ising interlayer interactions, deriving effective models via perturbation theory and performing Majorana mean-field theory calculations. We show that a diverse array of magnetic and topological phase transitions occur, depending on the direction of the interlayer Ising interaction and the relative sign of Kitaev interactions. When two layers have the same sign of the Kitaev interaction, a first-order transition from a Kitaev spin liquid to a magnetically ordered state takes place. The magnetic order points along the Ising axis and it is (anti)ferromagnetic for (anti)ferromagnetic Kitaev interactions. However, when two layers have opposite signs of the Kitaev interaction, we observe a notable weakening of magnetic ordering tendencies and the Kitaev spin liquid survives up to a remarkably larger interlayer exchange. Our mean-field analysis suggests the emergence of an intermediate gapped Z2{\mathbb{Z}}_{2}Z2​ spin-liquid state, which eventually becomes unstable upon vison condensation. The confined phase is described by a highly frustrated ${120}^{\ensuremath{\circ}}$ compass model. We furthermore use perturbation theory to study the model with the Ising axis pointing along the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ axis or lying in the xyxyxy plane. In both cases, our analysis reveals the formation of one-dimensional Ising chains, which remain decoupled in perturbation theory, resulting in a subextensive ground-state degeneracy. Our results highlight the interplay between topological order and magnetic ordering tendencies in bilayer quantum spin liquids.