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arXiv:1908.10877 · Physical review. B./Physical review. B

Two-magnon bound states in the Kitaev model in a [111] field

Subhasree Pradhan, Niravkumar D. Patel, Nandini Trivedi

Published May 5, 2020• 16 citations
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Abstract

It is now well established that the Kitaev honeycomb model in a magnetic field along the [111] direction harbors an intermediate gapless quantum spin liquid (QSL) phase sandwiched between a gapped non-Abelian QSL at low fields $H<{H}_{c1}$ and a partially polarized phase at high fields $H>{H}_{c2}$. Here, we analyze the low-field and high-field phases and phase transitions in terms of single- and two-magnon excitations using exact diagonalization and density matrix renormalization group methods. We find that the energy to create a bound state of two-magnon ${\mathrm{\ensuremath{\Delta}}}_{p}$ becomes lower than the energy to create a single spin flip ${\mathrm{\ensuremath{\Delta}}}_{s}$ near ${H}_{c2}$. In the entire Kitaev spin liquid, ${\mathrm{\ensuremath{\Delta}}}_{p}<{\mathrm{\ensuremath{\Delta}}}_{s}$ and both gaps vanish at ${H}_{c2}$. We make testable predictions for magnon pairing that could be observable in Raman scattering measurements on Kitaev QSL candidate materials.

DOI
10.1103/physrevb.101.180401

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It is now well established that the Kitaev honeycomb model in a magnetic field along the [111] direction harbors an intermediate gapless quantum spin liquid (QSL) phase sandwiched between a gapped non-Abelian QSL at low fields H<Hc1H<{H}_{c1}H<Hc1​ and a partially polarized phase at high fields H>Hc2H>{H}_{c2}H>Hc2​. Here, we analyze the low-field and high-field phases and phase transitions in terms of single- and two-magnon excitations using exact diagonalization and density matrix renormalization group methods. We find that the energy to create a bound state of two-magnon \ensuremathΔp{\mathrm{\ensuremath{\Delta}}}_{p}\ensuremathΔp​ becomes lower than the energy to create a single spin flip \ensuremathΔs{\mathrm{\ensuremath{\Delta}}}_{s}\ensuremathΔs​ near Hc2{H}_{c2}Hc2​. In the entire Kitaev spin liquid, \ensuremathΔp<\ensuremathΔs{\mathrm{\ensuremath{\Delta}}}_{p}<{\mathrm{\ensuremath{\Delta}}}_{s}\ensuremathΔp​<\ensuremathΔs​ and both gaps vanish at Hc2{H}_{c2}Hc2​. We make testable predictions for magnon pairing that could be observable in Raman scattering measurements on Kitaev QSL candidate materials.
The Kitaev model with an applied magnetic field in the [Formula: see text] direction shows two transitions: from a nonabelian gapped quantum spin liquid (QSL) to a gapless QSL at [Formula: see text] and a second transition at a higher field [Formula: see text] to a gapped partially polarized phase, where K is the strength of the Kitaev exchange interaction. We identify the intermediate phase to be a gapless U(1) QSL and determine the spin structure function [Formula: see text] and the Fermi surface [Formula: see text] of the gapless spinons using the density matrix renormalization group (DMRG) method for large honeycomb clusters. Further calculations of static spin-spin correlations, magnetization, spin susceptibility, and finite temperature-specific heat and entropy corroborate the gapped and gapless nature of the different field-dependent phases. In the intermediate phase, the spin-spin correlations decay as a power law with distance, indicative of a gapless phase.
The main question we address is how to probe the fractionalized excitations of a quantum spin liquid (QSL), for example, in the Kitaev honeycomb model. By analyzing the energy spectrum and entanglement entropy, for antiferromagnetic couplings and a field along either [111] or [001], we find a gapless QSL phase sandwiched between the non-Abelian Kitaev QSL and polarized phases. Increasing the field strength towards the polarized limit destroys this intermediate QSL phase, resulting in a considerable reduction in the number of frequency modes and the emergence of a beating pattern in the local dynamical correlations, possibly observable in pump-probe experiments.
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.
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The pursuit of quantum spin liquid (QSL) in the Kitaev honeycomb magnets has drawn intensive attention recently. In particular, $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ has been widely recognized as a promising candidate for the Kitaev QSL. Although the compound exhibits an antiferromagnetic order under zero field, it is believed to be endowed with fractionalized excitations and can be driven to the QSL phase by magnetic fields. Here, based on a realistic $K\text{\ensuremath{-}}J\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ model for $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$, we exploit the exponential tensor renormalization group approach to explore the phase diagram of the compound under magnetic fields. We calculate the thermodynamic quantities, including the specific heat, Gr\"uneisen parameter, magnetic torque, magnetotropic susceptibility, and so on, under a magnetic field with a tilting angle $\ensuremath{\theta}$ to the c∗{c}^{*}c∗-axis perpendicular to the honeycomb plane. We find an extended QSL in the angle-field phase diagram determined with thermodynamic responses. The gapless nature of such field-induced QSL is identified from the specific heat and entropy data computed down to very low temperatures. The present study provides guidance for future high-field experiments for the QSL in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ and other candidate Kitaev magnets.