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arXiv:2304.00555

Nematic chiral spin liquid in a Kitaev magnet under external magnetic field

Shun-Yao Yu, Han Li, Qi-Rong Zhao, Yuan Gao, Xiaoyu Dong, Zheng-Xin Liu, Wei Li, Shou-Shu Gong

Published Apr 2, 2023• 0 citations
arXivHTMLPDF

Abstract

The possible existence of a quantum spin liquid (QSL) phase, an exotic state of matter with long-range quantum entanglement and fractionalized excitations, in $α$-RuCl$_3$ has sparked widespread interests in exploring QSLs in various Kitaev models under magnetic fields. Recently, a $K$-$J$-$Γ$-$Γ'$ model has been proposed to accurately describe the compound by fitting the measured thermodynamic data, where $J$ is Heisenberg interaction, and $Γ$, $Γ'$ are off-diagonal exchanges on top of the dominant Kitaev coupling $K$. Based on this effective model, an intermediate QSL phase in presence of an out-of-plane fields along the $[1 1 1]$ direction, between the low-field zigzag order and high-field polarized phase, has been predicted. By combining density matrix renormalization group (DMRG), exponential tensor renormalization group (XTRG), and variational Monte Carlo (VMC) calculations, we address the nature of this QSL phase in the honeycomb $K$-$J$-$Γ$-$Γ'$ model under the $[1 1 1]$-direction field. Our DMRG calculations find the algebraic-like decay of spin correlation function, the finite spin scalar chiral order, and lattice nematic order. Together with the XTRG results of power-law specific heat at low temperature, our findings naturally suggest a gapless nematic chiral spin liquid. On the other hand, our VMC study finds a gapped nematic chiral spin liquid with the variational energy very close to that obtained in DMRG. As VMC finds a very small gap that is beyond the current resolution of both ground-state DMRG and finite-temperature XTRG calculations on finite-width cylinders, we resort the full clarification for the nature of the intermediate QSL to future studies. Lastly, we discuss the implications of our results to the recent experiment on $α$-RuCl$_3$ and the QSL-like phase in a similar $K$-$Γ$-$Γ'$ model.

DOI
10.48550/arxiv.2304.00555

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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.
Motivated by recent experimental observations in α-RuCl_{3}, we study the K-Γ model on the honeycomb lattice in an external magnetic field. By a slave-particle representation and variational Monte Carlo calculations, we reproduce the phase transition from zigzag magnetic order to a field-induced disordered phase. The nature of this state depends crucially on the field orientation. For particular field directions in the honeycomb plane, we find a gapless Dirac spin liquid, in agreement with recent experiments on α-RuCl_{3}. For a range of out-of-plane fields, we predict the existence of a Kalmeyer-Laughlin-type chiral spin liquid, which would show an integer-quantized thermal Hall effect.
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.
Abstract Pursuing the exotic quantum spin liquid (QSL) state in the Kitaev material α-RuCl3 has intrigued great research interest recently. A fascinating question is on the possible existence of a field-induced QSL phase in this compound. Here we perform high-field magnetization measurements of α-RuCl3 up to 102 T employing the non-destructive and destructive pulsed magnets. Under the out-of-plane field along the c* axis (i.e., perpendicular to the honeycomb plane), two quantum phase transitions are uncovered at respectively 35 T and about 83 T, between which lies an intermediate phase as the predicted QSL. This is in sharp contrast to the case with in-plane fields, where a single transition is found at around 7 T and the intermediate QSL phase is absent instead. By measuring the magnetization data with fields tilted from the c* axis up to 90° (i.e., in-plane direction), we obtain the field-angle phase diagram that contains the zigzag, paramagnetic, and QSL phases. Based on the K-J-Γ- Γ′{{{\Gamma }}}^{{\prime} }Γ′ Γ ′ model for α-RuCl3 with a large Kitaev term we perform density matrix renormalization group simulations and reproduce the quantum phase diagram in excellent agreement with experiments.
We study the ground-state properties of the quantum spin liquid (QSL) phases of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the triangular lattice with nearest- (J1)({J}_{1})(J1​), next-nearest- (J2)({J}_{2})(J2​), and third-neighbor (J3)({J}_{3})(J3​) interactions by using the density-matrix renormalization group (DMRG) method. By combining parallel DMRG with SU(2)\mathrm{SU}(2)SU(2) spin rotational symmetry, we are able to obtain accurate results on large cylinders with length up to Lx=48{L}_{x}=48Lx​=48 and circumference Ly=6−−12{L}_{y}=6--12Ly​=6−−12. Our results suggest that the QSL phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model is gapped, which is characterized by the absence of a gapless mode and by short-range spin-spin and dimer-dimer correlations. In the presence of J3{J}_{3}J3​ interaction, we find that a critical QSL with a single gapless mode emerges. While both spin-spin and scalar chiral-chiral correlations are short ranged, dimer-dimer correlations are quasi long ranged and decay as a power law at long distances.
Recent experimental evidence for a field-induced quantum spin liquid (QSL) in ααα-RuCl3_33​ calls for an understanding for the ground state of honeycomb Kitaev model under a magnetic field. In this work we address the nature of an enigmatic gapless paramagnetic phase in the antiferromagnetic Kitave model, under an intermediate magnetic field perpendicular to the plane. Combining theoretical and numerical efforts, we identify this gapless phase as a U(1)U(1)U(1) QSL with spinon Fermi surfaces. We also reveal the nature of continuous quantum phase transitions involving this U(1)U(1)U(1) QSL, and obtain a phase diagram of the Kitaev model as a function of bond anisotropy and perpendicular magnetic field.