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arXiv:1710.07595 · Physical review. B./Physical review. B

Robust non-Abelian spin liquid and a possible intermediate phase in the antiferromagnetic Kitaev model with magnetic field

Zheng Zhu, Itamar Kimchi, D. N. Sheng, Liang Fu

Published Jun 15, 2018• 126 citations
arXivHTMLPDF

Abstract

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 ${H}_{*}^{\text{AFM}}<H<{H}_{**}^{\text{AFM}}$, 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 ${\mathrm{RuCl}}_{3}$ under magnetic fields.

DOI
10.1103/physrevb.97.241110

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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∗FM\ensuremath≈0.02{H}_{*}^{\text{FM}}\ensuremath{\approx}0.02H∗FM​\ensuremath≈0.02, the antiferromagnetic (AFM) spin liquid remains robust up to a magnetic field that is an order of magnitude larger, H∗AFM\ensuremath≈0.2{H}_{*}^{\text{AFM}}\ensuremath{\approx}0.2H∗AFM​\ensuremath≈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.
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.
This paper studies the phase diagram for the SSS=1 Kitaev model using a density renormalization group method. The authors find a gapped Kitaev spin liquid with topological nature for both ferromagnetic and antiferromagnetic Kitaev couplings. Further, the the antiferromagnetic Kitaev coupling hosts a more robust topological spin liquid with an additional nonmagnetic intermediate phase
We microscopically construct an Abelian mutual Chern-Simons lattice gauge theory for magnetic field-tuned topological transitions in the Kitaev model and obtain a complete characterization of the phases, including their quasiparticles. At low fields for both ferro (FM) and antiferromagnetic (AFM) Kitaev interactions, we demonstrate non-Abelian Ising topological order (ITO), and explicitly construct the Majorana anyon as an intrinsic excitation---a twist defect in our Abelian gauge theory. For the AFM case, an Abelian chiral phase appears at intermediate fields with trivial topological order and fermionic bulk excitations. Remarkably, both the ITO phase and the intermediate phase have the same chiral central charge c=1/2,c=1/2,c=1/2, implying no change in the quantized thermal Hall response across the transition. For the FM case, there is a direct transition from ITO to a partially polarized nontopological phase. Our paper completes the proof of Kitaev's original proposal of the low-field ITO, with c=1/2c=1/2c=1/2 going beyond his mean-field arguments by including the crucial effect of the gauge fluctuations, and provides a resolution of the debate surrounding the intermediate field phase in the AFM case.
Topological spin liquids in two spatial dimensions are stable phases in the presence of a small magnetic field, but may give way to field-induced phenomena at intermediate field strengths. Sandwiched between the low-field spin liquid physics and the high-field spin-polarized phase, the exploration of magnetic phenomena in this intermediate regime, however, often remains elusive to controlled analytical approaches. Here we numerically study such intermediate-field magnetic phenomena for two representative Kitaev models (on the square-octagon and decorated honeycomb lattice) that exhibit either Abelian or non-Abelian topological order in the low-field limit. Using a combination of exact diagonalization and density matrix renormalization group techniques, as well as linear spin-wave theory, we establish the generic features of Kitaev spin liquids in an external magnetic field. While ferromagnetic models typically exhibit a direct transition to the polarized state at a relatively low field strength, antiferromagnetic couplings not only substantially stabilizes the topological spin liquid phase, but generically lead to the emergence of a distinct field-induced intermediate regime, separated by a crossover from the high-field polarized regime. Our results suggest that, for most lattice geometries, this regime generically exhibits significant spin canting, antiferromagnetic spin-spin correlations, and an extended proximate spin liquid regime at finite temperatures. Notably, we identify a symmetry obstruction in the original honeycomb Kitaev model that prevents, at least for certain field directions, the formation of such canted magnetism without breaking symmetries---consistent with the recent numerical observation of an extended gapless spin liquid in this case.
Non-Abelian anyons in quantum spin liquids (QSLs) provide a promising route to fault-tolerant topological quantum computation. In the exactly solvable Kitaev honeycomb model, such anyons of the QSL state can be bound to nonmagnetic spin vacancies and endowed with non-Abelian statistics by an infinitesimal magnetic field. Here, we investigate how this approach for stabilizing non-Abelian anyons extends to a finite magnetic field represented by a proper Zeeman term. Through large-scale density-matrix renormalization group simulations, we compute the vacancy-anyon binding energy as a function of magnetic field for both the ferromagnetic and antiferromagnetic Kitaev models. We find that anyon binding remains robust within the entire QSL phase for the ferromagnetic Kitaev model but breaks down already inside this phase for the antiferromagnetic Kitaev model. To compute a binding energy several orders of magnitude below the magnetic energy scale, we introduce both a refined definition and an extrapolation scheme based on carefully tailored perturbations.