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Low Temperature Physics

Magnetoelastic effects in the hyperhoneycomb Kitaev spin liquid

Aysel Shiralieva, A. Prokoshin, Natalia Borisovna Perkins

Published Sep 1, 2021• 3 citations
Publisher

Abstract

In recent years, several magnetic Mott insulators with strong spin-orbit coupling were suggested to be proximate to the Kitaev quantum spin liquid, whose one of the most exciting features is the fractionalization of spin excitations into itinerant Majorana fermions and static Z2 fluxes. Unfortunately, the ground states of these systems cannot be easily captured by experiment, remaining featureless to conventional local probes. Here, we propose to study the signatures of fractionalized excitations by exploiting their coupling to the lattice vibrations, dubbed magnetoelastic coupling, which arises from the fact that the interaction between spins depends on the relative distance between them. We argue that the magnetoelastic coupling can lead to the distinct modification of the phonon dynamics, which can be observed by measuring renormalized phonon spectrum, the sound attenuation, and the phonon Hall viscosity. This makes the phonon dynamics a promising tool for the characterization and identification of quantum spin liquid phases. In this work, we focus on the magnetoelastic effects in the three-dimensional Kitaev model realized on the hyperhoneycomb lattice. The hyperhoneycomb Kitaev spin liquid is particularly interesting since the strong Kitaev interaction was observed in the Kitaev magnet β-Li2IrO3, for which the spin-orbit entangled Jeff = 1/2 moments of iridium ions form precisely the hyperhoneycomb lattice.

DOI
10.1063/10.0005801

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In recent years, several magnetic Mott insulators with strong spin-orbit coupling were suggested to be proximate to the Kitaev quantum spin liquid, whose one of the most exciting features is the fractionalization of spin excitations into itinerant Majorana fermions and static Z2 fluxes. Unfortunately, the ground states of these systems cannot be easily captured by experiment, remaining featureless to conventional local probes. Here, we propose to study the signatures of fractionalized excitations by exploiting their coupling to the lattice vibrations, dubbed magnetoelastic coupling, which arises from the fact that the interaction between spins depends on the relative distance between them. We argue that the magnetoelastic coupling can lead to the distinct modification of the phonon dynamics, which can be observed by measuring renormalized phonon spectrum, the sound attenuation, and the phonon Hall viscosity. This makes the phonon dynamics a promising tool for the characterization and identification of quantum spin liquid phases. In this work, we focus on the magnetoelastic effects in the three-dimensional Kitaev model realized on the hyperhoneycomb lattice. The hyperhoneycomb Kitaev spin liquid is particularly interesting since the strong Kitaev interaction was observed in the Kitaev magnet β-Li2IrO3, for which the spin-orbit entangled Jeff = 1/2 moments of iridium ions form precisely the hyperhoneycomb lattice.
Candidate materials for the Kitaev spin liquid generically have residual interactions beyond the Kitaev coupling. It therefore becomes necessary to understand how signatures of the quantum spin liquid, e.g., fractionalization of the spin excitations, are affected by the presence of these interactions. Recently it was shown that phonon dynamics is an indirect but effective probe to study fractionalized excitations in the Kitaev spin liquid. Ultrasound experiments can measure sound attenuation, which should show characteristic temperature and angular dependence of the sound-attenuation coefficient if the scattering of phonons happens predominantly on Majorana fermions. So far the computation of the sound attenuation was only done in the pure spin-phonon coupled Kitaev model, without taking into account residual interactions. In order to understand experimental signatures, here we present a mean-field study of the sound attenuation in the generalized $J\text{\ensuremath{-}}K\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}$ model, which is relevant to candidate materials. Our findings show that as long as the system is in the spin liquid phase, characteristic features of the sound attenuation remain observable even in the presence of residual interactions.
The Kitaev honeycomb model provides a paradigmatic example of an exactly solvable quantum spin liquid (QSL), where spins fractionalize into itinerant Majorana fermions coupled to a static background of Z2{\mathbb{Z}}_{2}Z2​ gauge fluxes. This model has attracted significant interest due to its potential experimental realization in spin-orbit Mott insulators like $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. Among various experimental techniques, ultrasound measurements of sound attenuation have emerged as a promising approach to detect spin fractionalization in these materials. However, deviations from the ideal Kitaev model, often due to disorder, introduce localized modes that dominate the low-energy physics. To investigate these effects, we calculate the sound attenuation coefficient in the site-disordered Kitaev honeycomb model under an applied magnetic field that breaks time-reversal symmetry. Our analysis reveals that quasilocalized modes from quasivacancies impact the phonon self-energy, yet the sound attenuation coefficient maintains its sixfold symmetry and linear temperature dependence at low temperatures, even with disorder. This robustness underscores sound attenuation's reliability as a probe for spin fractionalization. Furthermore, we show that this linear behavior persists under an external field and in the random-flux sector, highlighting the persistent influence of fractionalized excitations, despite disorder.
The spin-orbit entangled nature of the magnetic moments in Kitaev materials is expected to give rise to a quantum spin liquid (QSL) state with fractional excitations. Using resonant inelastic x-ray scattering under a magnetic field, here the authors map the excitation spectra of $\ensuremath{\beta}$-Li2{}_{2}2​IrO3{}_{3}3​. They observe a dichotomy between a magnon-like response at low energies, originating from intertwined field-induced ground states, and a multi-spinon-like continuum at higher energies, originating from pairwise excitations of long-lived Majorana fermions of the proximate QSL.
Motivated by recent experiments on $\ensuremath{\beta}$-Li2{}_{2}2​IrO3{}_{3}3​, we study the phase diagram of the Heisenberg-Kitaev model on a three-dimensional lattice of tricoordinated Ir4+{}^{4+}4+, dubbed the hyperhoneycomb lattice. The lattice geometry of this material, along with Ir4+{}^{4+}4+ ions carrying Jeff=1/2{J}_{\mathrm{eff}}=1/2Jeff​=1/2 moments, suggests that the Heisenberg-Kitaev model may effectively capture the low-energy spin-physics of the system in the strong-coupling limit. Using a combination of semiclassical analysis, exact solution, and slave-fermion mean-field theory, we find, in addition to the spin liquid, four different magnetically ordered phases depending on the parameter regime. All four magnetic phases---the N\'eel, the polarized ferromagnet, the skew-stripy, and the skew-zig-zag---have collinear spin ordering. The three-dimensional Z2{Z}_{2}Z2​ spin liquid, which extends over an extended parameter regime around the exactly solvable Kitaev point, has a gapless Majorana mode with a deformed Fermi circle (codimensions, dc=2{d}_{c}=2dc​=2). We discuss the effect of the magnetic field and finite temperature on different phases that may be relevant for future experiments.
We study the self-energy of phonons, magnetoelastically coupled to the two-dimensional Kitaev spin model on the honeycomb lattice. Fractionalization of magnetic moments into mobile Majorana matter and a static Z2{\mathbb{Z}}_{2}Z2​ gauge field lead to a continuum of relaxation processes comprising two channels. Thermal flux excitations, which act as an emergent disorder, strongly affect the phonon renormalization. Above the flux proliferation temperature, the dispersion of a narrow quasiparticle-hole channel is suppressed in favor of broad and only weakly momentum-dependent features, covering large spectral ranges. Our analysis is based on complementary calculations in the low-temperature homogeneous gauge and a mean-field treatment of thermal gauge fluctuations, valid at intermediate and high temperatures.
The recently discovered three-dimensional hyperhoneycomb iridate, β -Li 2 IrO 3 , has raised hopes for the realization of the dominant Kitaev interaction between spin-orbit entangled local moments due to its near-ideal lattice structure. If true, this material may lie close to the sought-after quantum spin-liquid phase in three dimensions. Utilizing ab initio electronic structure calculations, we first show that the spin-orbit entangled basis, , correctly captures the low-energy electronic structure. The effective spin model derived in the strong-coupling limit supplemented by the ab initio results is shown to be dominated by the Kitaev interaction. We demonstrated that the possible range of parameters is consistent with a non-coplanar spiral magnetic order found in a recent experiment. All of these analyses suggest that β -Li 2 IrO 3 may be the closest among known materials to the Kitaev spin-liquid regime.