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arXiv:1703.07202 · Physical review. B./Physical review. B

Ground-state phase diagram of an anisotropic spin- 12 model on the triangular lattice

Qiang Luo, Shijie Hu, Bin Xi, Jize Zhao, Xiaoqun Wang

Published Apr 7, 2017• 62 citations
arXivHTMLPDF

Abstract

Motivated by a recent experiment on the rare-earth material ${\mathrm{YbMgGaO}}_{4}$ [Y. Li et al., Phys. Rev. Lett. 115, 167203 (2015)], which found that the ground state of ${\mathrm{YbMgGaO}}_{4}$ is a quantum spin liquid, we study the ground-state phase diagram of an anisotropic spin-$1/2$ model that was proposed to describe ${\mathrm{YbMgGaO}}_{4}$. Using the density matrix renormalization-group method in combination with the exact-diagonalization method, we calculate a variety of physical quantities, including the ground-state energy, the fidelity, the entanglement entropy and spin-spin correlation functions. Our studies show that in the quantum phase diagram, there is a ${120}^{\ensuremath{\circ}}$ phase and two distinct stripe phases. The transitions from the two stripe phases to the ${120}^{\ensuremath{\circ}}$ phase are of the first order. However, the transition between the two stripe phases is not of the first order, which is different from its classical counterpart. Additionally, we find no evidence for a quantum spin liquid in this model. Our results suggest that additional terms may also be important to model the material ${\mathrm{YbMgGaO}}_{4}$. These findings will stimulate further experimental and theoretical works in understanding the quantum spin-liquid ground state in ${\mathrm{YbMgGaO}}_{4}$.

DOI
10.1103/physrevb.95.165110

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Motivated by a recent experiment on the rare-earth material YbMgGaO4{\mathrm{YbMgGaO}}_{4}YbMgGaO4​ [Y. Li et al., Phys. Rev. Lett. 115, 167203 (2015)], which found that the ground state of YbMgGaO4{\mathrm{YbMgGaO}}_{4}YbMgGaO4​ is a quantum spin liquid, we study the ground-state phase diagram of an anisotropic spin-1/21/21/2 model that was proposed to describe YbMgGaO4{\mathrm{YbMgGaO}}_{4}YbMgGaO4​. Using the density matrix renormalization-group method in combination with the exact-diagonalization method, we calculate a variety of physical quantities, including the ground-state energy, the fidelity, the entanglement entropy and spin-spin correlation functions. Our studies show that in the quantum phase diagram, there is a 120\ensuremath∘{120}^{\ensuremath{\circ}}120\ensuremath∘ phase and two distinct stripe phases. The transitions from the two stripe phases to the 120\ensuremath∘{120}^{\ensuremath{\circ}}120\ensuremath∘ phase are of the first order. However, the transition between the two stripe phases is not of the first order, which is different from its classical counterpart. Additionally, we find no evidence for a quantum spin liquid in this model. Our results suggest that additional terms may also be important to model the material YbMgGaO4{\mathrm{YbMgGaO}}_{4}YbMgGaO4​. These findings will stimulate further experimental and theoretical works in understanding the quantum spin-liquid ground state in YbMgGaO4{\mathrm{YbMgGaO}}_{4}YbMgGaO4​.
Based on exact diagonalization and density matrix renormalization group method, we show that an anisotropic triangular lattice Heisenberg spin model has three distinct quantum phases. In particular, a spin-liquid phase is present in the weak interchain coupling regime, which is characterized by an anisotropic spin structure factor with an exponential-decay spin correlator along the weaker coupling direction, consistent with the Cs2CuCl4{\mathrm{Cs}}_{2}\mathrm{Cu}{\mathrm{Cl}}_{4}Cs2​CuCl4​ compounds. In the obtained phase diagram, the spin-liquid phase is found to persist up to a relatively large critical anisotropic coupling ratio ${J}^{\ensuremath{'}}∕J=0.78$, which is stabilized by strong quantum fluctuations, with a parity symmetry distinct from two magnetic ordered states in the stronger coupling regime.
The anisotropic-exchange spin-1/2 model on the triangular lattice has been used to describe the rare-earth chalcogenides, which may have exotic ground states. We investigate the quantum phase diagram of the model by using the projected entangled pair state method, and compare it to the classical phase diagram. Besides two stripe-ordered phases, and the ${120}^{\ensuremath{\circ}}$ phase, there is also a multi-Q phase. We identify the multi-Q phase as a Z2{Z}_{2}Z2​ vortex state. No quantum spin liquid state is found in the phase diagram, contrary to the previous density matrix renormalization group calculations.
The celebrated Kitaev honeycomb model provides an analytically tractable example with an exact quantum spin liquid ground state. While in real materials, other types of interactions besides the Kitaev coupling (KKK) are present, such as Heisenberg (JJJ) and symmetric off-diagonal ($\mathrm{\ensuremath{\Gamma}}$) terms, these interactions can also be generalized to a triangular lattice. Here, we carry out a comprehensive study of the $J\ensuremath{-}K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model on a triangular lattice covering the whole parameter region, using a combination of exact diagonalization, classical Monte Carlo, and analytic methods, with an emphasis on the effects of the $\mathrm{\ensuremath{\Gamma}}$ term. In the HK limit ($\mathrm{\ensuremath{\Gamma}}=0$), we find five quantum phases which are quite similar to their classical counterparts. Among them, stripe-A and dual N\'eel phases are robust against the introduction of the $\mathrm{\ensuremath{\Gamma}}$ term, in particular, stripe A extends to the region connecting $K=\ensuremath{-}1$ and K=1K=1K=1 for $\mathrm{\ensuremath{\Gamma}}<0$. Though the ${120}^{\ensuremath{\circ}}$ N\'eel phase also extends to a finite $\mathrm{\ensuremath{\Gamma}}$, its region is largely reduced compared to the previous classical result. Interestingly, the ferromagnetic (dubbed FM-A) and stripe-B phases are unstable in response to an infinitesimal $\mathrm{\ensuremath{\Gamma}}$ interaction. Moreover, we find five additional phases for $\mathrm{\ensuremath{\Gamma}}\ensuremath{\ne}0$ which are elaborated by both the quantum and classical numerical methods. Part of the parameter space previously identified as the ${120}^{\ensuremath{\circ}}$ N\'eel phase in the classical study is found to give way to the modulated stripe phase. Depending on the sign of the $\mathrm{\ensuremath{\Gamma}}$ term, the FM-A phase transits into FM-B ($\mathrm{\ensuremath{\Gamma}}>0$) and FM-C ($\mathrm{\ensuremath{\Gamma}}<0$) phases with different spin orientations. Similarly, the stripe-B phase transits into stripe-C ($\mathrm{\ensuremath{\Gamma}}>0$) and stripe-A ($\mathrm{\ensuremath{\Gamma}}<0$) phases. Around the positive $\mathrm{\ensuremath{\Gamma}}$ point, due to the interplay of the Heisenberg, Kitaev, and $\mathrm{\ensuremath{\Gamma}}$ interactions, we find a possible quantum spin liquid in a noticeable region with a continuum in spin excitations.
Employing exact diagonalization, we systematically study the anisotropic Heisenberg model which is related to rare-earth triangular-lattice materials. We probe its full 3D phase diagram afresh and identify a large region of quantum spin liquid (QSL) phase which can extend to the QSL region of the J1–J2{J}_{1}\text{--}{J}_{2}J1​–J2​ triangular Heisenberg model. Furthermore, we explore the magnetization curves of different phases and reproduce the 1/3-magnetization plateau in the quantum spin liquid phase region. More importantly, to study the possible chemical disorders in real materials, we consider the randomness of exchange interactions and find no spin glass order. And there is a large region of random-singlet phase which contains strongly random spin networks, dominated by two-spin singlets, four-spin singlets and other singlet domains. Our comprehensive ED study can give detailed insightful understanding of the microscopic Hamiltonian related to the YbMgGaO4{\mathrm{YbMgGaO}}_{4}YbMgGaO4​ and some other related rare-earth triangular-lattice materials.
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Two-dimensional density-matrix renormalization group method is employed to examine the ground-state phase diagram of the Hubbard model on the triangular lattice at half-filling. The calculation reveals two discontinuities in the double occupancy with increasing the repulsive Hubbard interaction UUU at ${U}_{\mathrm{c}1}\ensuremath{\sim}7.8t$ and ${U}_{\mathrm{c}2}\ensuremath{\sim}9.9t$ (ttt being the hopping integral), indicating that there are three phases separated by first-order transitions. The absence of any singularity in physical quantities for $0\ensuremath{\le}U<{U}_{\mathrm{c}1}$ implies a metallic phase in this regime. For U>Uc2U>{U}_{\mathrm{c}2}U>Uc2​, the local spin density induced by an applied pinning magnetic field exhibits a three sublattice feature, which is compatible with the ${120}^{\ensuremath{\circ}}$ N\'eel-ordered state realized in the limit of $U\ensuremath{\rightarrow}\ensuremath{\infty}$. For Uc1<U<Uc2{U}_{\mathrm{c}1}<U<{U}_{\mathrm{c}2}Uc1​<U<Uc2​, a response to the applied pinning magnetic field is comparable to that in the metallic phase with a relatively large spin correlation length, but showing neither valence bond nor chiral magnetic order, which therefore resembles gapless spin liquid. However, the spin structure factor for the intermediate phase exhibits the maximum at the K\mathrm{K}K and ${\mathrm{K}}^{\ensuremath{'}}$ points in the momentum space, which is not compatible to spin liquid with a large spinon Fermi surface. The calculation also finds that the pairing correlation function monotonically decreases with increasing UUU and thus the superconductivity is unlikely in the intermediate phase.