Novel Chiral Quantum Spin Liquids in Kitaev Magnets

Arnaud Ralko and Jaime Merino

Phys. Rev. Lett. 124, 217203 – Published 26 May 2020

Abstract

Quantum magnets with pure Kitaev spin exchange interactions can host a gapped quantum spin liquid with a single Majorana edge mode propagating in the counterclockwise direction when a small positive magnetic field is applied. Here, we show how under a sufficiently strong positive magnetic field a topological transition into a gapped quantum spin liquid with two Majorana edge modes propagating in the clockwise direction occurs. The Dzyaloshinskii-Moriya interaction is found to turn the nonchiral Kitaev’s gapless quantum spin liquid into a chiral one with equal Berry phases at the two Dirac points. Thermal Hall conductance experiments can provide evidence of the novel topologically gapped quantum spin liquid states predicted.

Received 24 October 2019
Accepted 13 May 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.217203

Physics Subject Headings (PhySH)

Exotic phases of matter Magnetic phase transitions Majorana fermions Topological phase transition

Chern insulators

Kitaev model Kitaev-Heisenberg model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Arnaud Ralko

Institut Néel, UPR2940, Université Grenoble Alpes et CNRS, Grenoble 38042, France

Jaime Merino

Departamento de Física Teórica de la Materia Condensada, Condensed Matter Physics Center (IFIMAC) and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, Madrid 28049, Spain

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Vol. 124, Iss. 21 — 29 May 2020

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Images

Figure 1
Phase diagram of Kitaev model with DM interaction under magnetic field. The full phase diagram of the Kitaev model in the presence of a magnetic field pointing in the [1,1,1] direction ( $t = 1$ ) is shown. Empty regions indicate gapless phases and the size of the hexagonal symbols indicate the size of the gap. Gray areas correspond to the gapped fully polarized (FP) phase with Chern number $ν = 0$ , the red to a gapped QSL with $ν = + 1$ denoted as ${GQSL}_{+ 1}$ . A gapped QSL with an unconventional Chern number of $ν = - 2$ , termed ${GQSL}_{- 2}$ (dark blue) occurs between the FP and the ${GQSL}_{+ 1}$ . Away the KQSL at $K = 1$ (open circle), an ungapped QSL, UQSL (thick red line) with equal Berry phases at the two Dirac points occurs for $B = 0$ and $K = 1$ at a nonzero DM, $D < 0.5$ , which becomes the ${GQSL}_{+ 1}$ around $D \sim 0.5 - 0.65$ , to finally become gapless and nontopological at a larger $D$ (dark gray). In this region, CR refers to classical regimes, beyond the accessibility of the present theory. The left inset shows the Majorana decomposition of the model considered while the right inset shows the first Brillouin zone and symmetry points of the honeycomb lattice.
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Figure 2
Chiral amplitudes in the Kitaev model from the Majorana mean-field theory. In (a) we show the NNN chiral Majorana amplitudes, $⟨ c_{i} c_{j} ⟩$ , induced in the ground state MMFT wave function either by a magnetic field perpendicular to the plane, $t = 1$ , or a DM vector oriented perpendicular to the plane ( $d = 1$ ). These are responsible for the chiral QSL with Chern number $ν = + 1$ ( ${GQSL}_{+ 1}$ ) discussed in the text. (b) Anisotropic amplitudes (the thickness corresponds to the different strength) arising from finite DM $d \neq 1$ (with no applied magnetic field) responsible for the opening of the gap for $0.5 < D < 0.65$ (with $K = 1$ and $d = 0$ ) in the ${GQSL}_{+ 1}$ . The snapshots show one of the two possible chiralities of the ground state ( $ν = + 1$ ). Note that (a) and (b) amplitudes are smoothly connected as shown in the phase diagram of Fig. 1.
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Figure 3
Majorana dispersions of the Kitaev model under a tilted external magnetic field and zero DM interaction. The evolution of the Majorana dispersions under a tilted $t = 1$ magnetic field comparing the case (a) with no applied magnetic field, $B = 0$ , consisting of gapless Majorana dispersions and flat flux bands (threefold degenerate) describing the $Z_{2}$ fluxes, (b) with $B ≃ 0.6$ , where a gap has already opened and the flux Majorana bands remain flat. This gapped QSL with Chern number $ν = + 1$ , ${GQSL}_{+ 1}$ , persists up to $B_{c} = 1.43$ is adiabatically connected with the KQSL, (c) with $B = 1.2$ , the Dirac cones have been washed out and the gapped fluxes are strongly distorted becoming dispersive, (d) with $B = 1.43$ , a gapped QSL with $ν = - 2$ , ${GQSL}_{- 2}$ , arises between $B_{c}$ and $B ≃ 1.64$ indicating a different topological state from the KQSL. $K \equiv 1$ in this plot.
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Figure 4
Dependence of the gap (top), the magnetic moment (middle) and the Chern number (bottom) with an applied magnetic field and zero DM interaction. The gap obtained from the MMFT changes from $Δ \propto B^{3}$ , expected from perturbation theory, to $Δ \propto B$ around $B^{*} \sim 0.6$ under the magnetic field $B = B (1, 1, 1) / \sqrt{3}$ . A topological transition from a gapped QSL with Chern number, $ν = + 1$ , ${GQSL}_{+ 1}$ , to a QSL with $ν = - 2$ , ${GQSL}_{- 2}$ , occurs at $B_{c} \sim 1.43$ . For $B ≳ 1.64$ a topologically trivial polarized insulator is stabilized. $K = 1$ in this plot.
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