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arXiv:2311.03334 · Physical review. B./Physical review. B

Emergent magnetic order in the antiferromagnetic Kitaev model in a [111] field

Will Holdhusen, Daniel Huerga, Gerardo Ortíz

Published May 6, 2024• 5 citations
arXivHTMLPDF

Abstract

The Kitaev spin liquid, stabilized as the ground state of the Kitaev honeycomb model, is a paradigmatic example of a topological ${\mathbb{Z}}_{2}$ quantum spin liquid. The fate of the Kitaev spin liquid in presence of an external magnetic field is at present, a subject of intense scrutiny due to recent experimental signatures pointing to a ${\mathbb{Z}}_{2}$ topological phase in certain materials, as well as theoretical studies predicting the emergence of a quantum spin liquid phase of debated nature. In this work, we employ hierarchical mean-field theory, a method based on the use of clusters preserving relevant symmetries and short-range quantum correlations, to investigate the quantum phase diagram of the antiferromagnetic Kitaev's model in a [111] magnetic field. By using clusters of 24 sites, we predict that the Kitaev spin liquid transits through two intermediate phases, characterized by emerging stripe and chiral orders, respectively, before entering the trivial partially polarized phase, differing from previous studies. We assess our results by performing exact diagonalization and computing the scaling of different observables, including the many-body Chern number and other topological quantities, thus establishing hierarchical mean-field theory as a method to study models of frustrated quantum magnetism potentially hosting topological quantum spin liquids.

DOI
10.1103/physrevb.109.174411

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The Kitaev spin liquid, stabilized as the ground state of the Kitaev honeycomb model, is a paradigmatic example of a topological Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquid. The fate of the Kitaev spin liquid in presence of an external magnetic field is at present, a subject of intense scrutiny due to recent experimental signatures pointing to a Z2{\mathbb{Z}}_{2}Z2​ topological phase in certain materials, as well as theoretical studies predicting the emergence of a quantum spin liquid phase of debated nature. In this work, we employ hierarchical mean-field theory, a method based on the use of clusters preserving relevant symmetries and short-range quantum correlations, to investigate the quantum phase diagram of the antiferromagnetic Kitaev's model in a [111] magnetic field. By using clusters of 24 sites, we predict that the Kitaev spin liquid transits through two intermediate phases, characterized by emerging stripe and chiral orders, respectively, before entering the trivial partially polarized phase, differing from previous studies. We assess our results by performing exact diagonalization and computing the scaling of different observables, including the many-body Chern number and other topological quantities, thus establishing hierarchical mean-field theory as a method to study models of frustrated quantum magnetism potentially hosting topological quantum spin liquids.
Kitaev magnets have emerged as pivotal systems for investigating frustrated magnetism, providing a unique platform to explore quantum phases governed by the interplay between bond-dependent anisotropy and external magnetic fields. However, the quantum phase diagrams, particularly near the dominant antiferromagnetic Kitaev regime, remain puzzling despite extensive studies. In this work, we perform unbiased exact diagonalization calculations of the Kitaev-ΓΓΓ model in a [111] magnetic field on a C6C_{6}C6​-symmetric 24-site cluster. By calculating the Z2\mathbb{Z}_2Z2​ flux density and the topological entanglement entropy, we reveal multiple phase transitions and identify signatures of both scalar and vector chiral orders in the intermediate-field regime between the Kitaev spin liquid and the polarized phase. As the negative ΓΓΓ interaction increases, we discover a proximate quantum spin liquid featured by a three-peak specific heat and a spin-flop phase at a moderate magnetic field. Our findings provide insight into the field-induced intermediate phases in the antiferromagnetic Kitaev model and pave the way for the hunt for emergent phases in real materials.
We investigate the non-Abelian topological chiral spin-liquid phase in the two-dimensional Kitaev honeycomb model subject to a magnetic field. By combining density matrix renormalization group and exact diagonalization we study the energy spectra, entanglement, topological degeneracy, and expectation values of Wilson loop operators, allowing for a robust characterization. While the ferromagnetic Kitaev spin liquid is already destroyed by a weak magnetic field with Zeeman energy ${H}_{*}^{\text{FM}}\ensuremath{\approx}0.02$, the antiferromagnetic (AFM) spin liquid remains robust up to a magnetic field that is an order of magnitude larger, ${H}_{*}^{\text{AFM}}\ensuremath{\approx}0.2$. Interestingly, for larger fields H∗AFM<H<H∗∗AFM{H}_{*}^{\text{AFM}}<H<{H}_{**}^{\text{AFM}}H∗AFM​<H<H∗∗AFM​, an intermediate gapless phase is observed, before a second transition to the high-field partially polarized paramagnet. We attribute this rich phase diagram, and the remarkable stability of the chiral topological phase in the AFM Kitaev model, to the interplay of strong spin-orbit coupling and frustration enhanced by the magnetic field. Our findings suggest relevance to recent experiments on RuCl3{\mathrm{RuCl}}_{3}RuCl3​ under magnetic fields.
We investigate the non-Abelian topological chiral spin liquid phase in the two-dimensional (2D) Kitaev honeycomb model subject to a magnetic field. By combining density matrix renormalization group (DMRG) and exact diagonalization (ED) we study the energy spectra, entanglement, topological degeneracy, and expectation values of Wilson loop operators, allowing for robust characterization. While the ferromagnetic (FM) Kitaev spin liquid is already destroyed by a weak magnetic field with Zeeman energy H∗FM≈0.02H_*^\text{FM} \approx 0.02H∗FM​≈0.02, the antiferromagnetic (AFM) spin liquid remains robust up to a magnetic field that is an order of magnitude larger, H∗AFM≈0.2H_*^\text{AFM} \approx 0.2H∗AFM​≈0.2. Interestingly, for larger fields H∗AFM<H<H∗∗AFMH_*^\text{AFM} < H < H_{**}^\text{AFM}H∗AFM​<H<H∗∗AFM​, an intermediate gapless phase is observed, before a second transition to the high-field partially-polarized paramagnet. We attribute this rich phase diagram, and the remarkable stability of the chiral topological phase in the AFM Kitaev model, to the interplay of strong spin-orbit coupling and frustration enhanced by the magnetic field. We discuss possible relevance of this result to RuCl3_33​ under various magnetic fields.
Abstract The S = 1/2 Kitaev honeycomb model has attracted significant attention as an exactly solvable example with a quantum spin liquid ground state. In an properly oriented external magnetic field, the system exhibits chiral Majorana edge modes with an associated quantized thermal Hall conductance, and a distinct spin-disordered phase emerges at intermediate field strengths, below the polarized phase. However, since material realizations of Kitaev magnetism invariably display competing exchange interactions, the stability of these exotic phases with respect to additional couplings is a key issue. Here, we report a 24-site exact diagonalization study of the Heisenberg-Kitaev model in a magnetic field applied in the [001] and [111] directions. By mapping the full phase diagram of the model and contrasting the results to recent nonlinear spin-wave calculations, we show that both methods agree well, thus establishing that quantum corrections substantially modify the classical phase diagram. Furthermore, we find that, in a [111] field, the intermediate-field spin-disordered phase is remarkably stable to Heisenberg interactions and may potentially end in a novel quantum tricritical point.
We have studied the anti-ferromagnetic Kitaev model on a honeycomb lattice under the Zeeman field, using an extensive Majorana mean-field analysis. When the magnetic field is along a specific Cartesian axis, we find that the emergent fields exhibit direction-dependent stabilization up to a certain critical strength of the external field. For a conical magnetic field, the characteristics of the emergent intermediate state are elusive. Our mean-field analysis reveals the existence of two distinct phases in the intermediate region. First, the system enters a disordered phase, where emergent-field densities converge to random values, and the Chern number is ill-defined. The magnitude of magnetization also fluctuates and remains less than unity, indicating a strong quantum effect. In the second phase, emergent-field densities attain vanishingly small values. In this phase, the magnetization components fluctuate heavily, but the magnitude of the magnetization vectors becomes unity, indicating highly degenerate classical ground states. We perform exact diagonalization calculations that qualitatively support some of the mean-field results. We extend our study to the anisotropic limit of the Kitaev coupling parameters. When the couplings are beyond the triangular inequality, the pure Kitaev model is known to host a topologically trivial gapped quantum spin liquid. We find that, for intermediate strengths of a conical magnetic field, topology shows a reentrant behavior.
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.