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arXiv:2401.12750 · Physical review. B./Physical review. B

Microscopic theory of field-tuned topological transitions in the Kitaev honeycomb model

Jagannath Das, Aman Kumar, Vikram Tripathi

Published Jan 21, 2025• 1 citation
arXivHTMLPDF

Abstract

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 $c=1/2,$ 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 $c=1/2$ 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.

DOI
10.1103/physrevb.111.l041111

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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.
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.
We study quantum phase transitions in the honeycomb Kitaev model under a magnetic field, focusing on the topological nature of Majorana fermion excitations. We find a gapless phase between the low-field gapless quantum spin liquid and the high-field gapped forced-ferromagnetic state for the antiferromagnetic Kitaev model in the [001] field by using the Majorana mean-field theory, in conjunction with the exact diagonalization and the spin-wave theory supporting the validity of this approach. The transition between the two gapless phases is driven by a topological change of the Majorana spectrum---line node formation interconnecting two Majorana cones. The peculiar change of the Majorana band topology is rationalized by a sign change of the effective Kitaev coupling by the magnetic field, which does not occur in the ferromagnetic Kitaev case. Upon tilting the magnetic field away from [001], the two gapless phases become gapped and topologically nontrivial, characterized by nonzero Chern numbers with different signs. The sign change of the Chern number leads to a reversal of the thermal edge current in the half-quantized thermal Hall effect.
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 show that a conical magnetic field H=(1,1,1)H can be used to tune the topological order and hence, anyon excitations of the Z_{2} quantum spin liquid in the isotropic antiferromagnetic Kitaev model. A novel topological order, featured with Chern number C=4 and Abelian anyon excitations, is induced in a narrow range of intermediate fields H_{c1}≤H≤H_{c2}. On the other hand, the C=1 Ising-topological order with non-Abelian anyon excitations, as previously known to be present at small fields, is found here to survive up to H_{c1}. The results are obtained by developing and applying a Z_{2} mean field theory that works at finite fields and is asymptotically exact in the zero field limit and the associated variational quantum Monte Carlo.
The competition between fractionalized spin-liquid states and magnetically ordered phases is an important paradigm in frustrated magnetism. Spin-orbit coupled Mott insulators with Ising-like magnetic anisotropies, such as Kitaev materials, are a particularly rich playground to explore this competition. In this work, we use effective field theory methods to show that a direct quantum phase transition can occur in two-dimensional (2D) Ising spin systems between a topologically ordered chiral spin liquid and a phase with magnetic long-range order. Such a transition can be protected by lattice symmetries and is described by a theory of massless Majorana fields coupled to non-Abelian SO(N)\text{SO}(N)SO(N) gauge fields with a Chern-Simons term. We further show that Euclidean Majorana zero modes bound to Z2{\mathbb{Z}}_{2}Z2​ monopole-instantons in the emergent non-Abelian gauge field are key to understanding spontaneous symmetry breaking in the ordered phase.
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.