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arXiv:1708.07856 · SciPost Physics

Spin Liquid versus Spin Orbit Coupling on the Triangular Lattice

Jason Iaconis, Chunxiao Liu, Gábor Halász, Leon Balents

Published Jan 19, 2018• 63 citations
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Abstract

In this paper, we explore the relationship between strong spin-orbit coupling and spin liquid physics. We study a very general model on the triangular lattice where spin-orbit coupling leads to the presence of highly anisotropic interactions. We use variational Monte Carlo to study both U(1) U ( 1 ) quantum spin liquid states and ordered ones, via the Gutzwiller projected fermion construction. We thereby obtain the ground state phase diagram in this phase space. We furthermore consider effects beyond the Gutzwiller wavefunctions for the spinon Fermi surface quantum spin liquid, which are of particular importance when spin-orbit coupling is present.

DOI
10.21468/scipostphys.4.1.003

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In this paper, we explore the relationship between strong spin-orbit coupling and spin liquid physics. We study a very general model on the triangular lattice where spin-orbit coupling leads to the presence of highly anisotropic interactions. We use variational Monte Carlo to study both U(1) U ( 1 ) quantum spin liquid states and ordered ones, via the Gutzwiller projected fermion construction. We thereby obtain the ground state phase diagram in this phase space. We furthermore consider effects beyond the Gutzwiller wavefunctions for the spinon Fermi surface quantum spin liquid, which are of particular importance when spin-orbit coupling is present.
Motivated by recent experiments on Ba3{}_{3}3​NiSb2{}_{2}2​O9{}_{9}9​, we investigate possible quantum spin liquid ground states for spin S=1S=1S=1 Heisenberg models on the triangular lattice. We use variational Monte Carlo techniques to calculate the energies of microscopic spin liquid wave functions where spin is represented by three flavors of fermionic spinon operators. These energies are compared with the energies of various competing three-sublattice ordered states. Our approach shows that the antiferromagnetic Heisenberg model with biquadratic term and single-ion anisotropy does not have a low-temperature spin liquid phase. However, for an SU(3)-invariant model with sufficiently strong ring-exchange terms, we find a paired chiral quantum spin liquid with a Fermi surface of deconfined spinons that is stable against all types of ordering patterns we considered. We discuss the physics of this exotic spin liquid state in relation to the recent experiment and suggest new ways to test this scenario.
Motivated by the recent progress in the spin-orbit-coupled triangular lattice spin liquid candidate YbMgGaO4{\mathrm{YbMgGaO}}_{4}YbMgGaO4​, we carry out a systematic projective symmetry group analysis and mean-field study of candidate U(1)U(1)U(1) spin-liquid ground states. Due to the spin-orbital entanglement of the Yb moments, the space-group symmetry operation transforms both the position and the orientation of the local moments, and hence it brings different features for the projective realization of the lattice symmetries from the cases with spin-only moments. Among the eight U(1)U(1)U(1) spin liquids that we find with the fermionic parton construction, only one spin-liquid state, which was proposed and analyzed by Yao Shen et al. [Nature (London) 540, 559 (2016)] and labeled as U1A00 in the present work, stands out and gives a large spinon Fermi surface and provides a consistent explanation for the spectroscopic results in YbMgGaO4{\mathrm{YbMgGaO}}_{4}YbMgGaO4​. Further connection of this spinon Fermi surface U(1)U(1)U(1) spin liquid with YbMgGaO4{\mathrm{YbMgGaO}}_{4}YbMgGaO4​ and the future directions are discussed. Finally, our results may apply to other spin-orbit-coupled triangular lattice spin-liquid candidates, and more broadly, our general approach can be well extended to spin-orbit-coupled spin-liquid candidate materials.
We show that the spin-liquid phase of the half-filled Hubbard model on the triangular lattice can be described by a pure spin model. This is based on a high-order strong coupling expansion (up to order 12) using perturbative continuous unitary transformations. The resulting spin model is consistent with a transition from three-sublattice long-range magnetic order to an insulating spin-liquid phase, and with a jump of the double occupancy at the transition. Exact diagonalizations of both models show that the effective spin model is quantitatively accurate well into the spin-liquid phase, and a comparison with the Gutzwiller projected Fermi sea suggests a gapless spectrum and a spinon Fermi surface.
Strongly correlated electron systems have the potential to host very exoticphases of matter. In order to have relevance to real materials, this exoticphysics often must emerge from relatively simple models. The quantumwavefunctions which describe such phases may bear little resemblance to theoriginal microscopic models. In these cases a variety of complex analytictools often must be supplemented with controlled numerical calculations tofully understand the essential behavior of these models. In this dissertation,we study such quantum phases of matter and their relationship to real materials.We focus on three main problems. First, we explore the relationship betweenstrong spin-orbit coupling and spin liquid physics by studying a very generalmodel on the triangular lattice where spin-orbit coupling leads to the presenceof highly anisotropic interactions. We use variational Monte Carlo to study bothU(1) quantum spin liquid states and ordered ones, via the Gutzwiller projectedfermion construction. We thereby obtain the ground state phase diagram in thisphase space. We furthermore consider effects beyond the Gutzwiller wavefunctionsfor the spinon Fermi surface quantum spin liquid, which are of particularimportance when spin-orbit coupling is present.Second we show that the interplay between a high density two-dimensional electrongas and localized electrons in a neighboring Mott insulator leads to kineticmagnetism unique to the Mott/band insulator interface. Our study is based upona bilayer Hubbard model at U=inftyU=\\inftyU=infty with a potential difference between thetwo layers. We combine analytic results with DMRG simulations to show thatmagnetism, and especially kinetic ferromagnetism, is greatly enhanced relative by theproximity of the two subsystems. Third we study the effect of interactions on the properties of a model 2Dtopological Kondo insulator phase. We introduce a model Hamiltonian which webelieve captures the essential physics of the different competing phases.Perhaps the most dramatic example of many-body physics in symmetry protectedtopological phases is the existence of exotic gapless edge states. We comment onthe potentially dramatic effects that interactions can have on such edge states.
By using Gutzwiller projected fermionic wave functions and variational Monte Carlo technique, we study the spin-1/21/21/2 Heisenberg model with the first-neighbor (J1{J}_{1}J1​), second-neighbor (J2{J}_{2}J2​), and additional scalar chiral interaction ${J}_{\ensuremath{\chi}}{\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}({\mathbf{S}}_{j}\ifmmode\times\else\texttimes\fi{}{\mathbf{S}}_{k})$ on the triangular lattice. In the nonmagnetic phase of the ${J}_{1}\ensuremath{-}{J}_{2}$ triangular model with $0.08\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.16$, recent density-matrix renormalization group (DMRG) studies [Zhu and White, Phys. Rev. B 92, 041105(R) (2015) and Hu, Gong, Zhu, and Sheng, Phys. Rev. B 92, 140403(R) (2015)] find a possible gapped spin liquid with the signal of a competition between a chiral and a Z2{Z}_{2}Z2​ spin liquid. Motivated by the DMRG results, we consider the chiral interaction ${J}_{\ensuremath{\chi}}{\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}({\mathbf{S}}_{j}\ifmmode\times\else\texttimes\fi{}{\mathbf{S}}_{k})$ as a perturbation for this nonmagnetic phase. We find that with growing ${J}_{\ensuremath{\chi}}$, the gapless U(1) Dirac spin liquid, which has the best variational energy for ${J}_{\ensuremath{\chi}}=0$, exhibits the energy instability towards a gapped spin liquid with nontrivial magnetic fluxes and nonzero chiral order. We calculate topological Chern number and ground-state degeneracy, both of which identify this flux state as the chiral spin liquid with fractionalized Chern number C=1/2C=1/2C=1/2 and twofold topological degeneracy. Our results indicate a positive direction to stabilize a chiral spin liquid near the nonmagnetic phase of the ${J}_{1}\ensuremath{-}{J}_{2}$ triangular model.
The U(1)U(1)U(1) Dirac spin liquid might realize an exotic phase of matter whose low-energy properties are described by quantum electrodynamics in 2+12+12+1 dimensions, where gapless modes exist but spinons and gauge fields are strongly coupled. Its existence was proposed in frustrated Heisenberg models in the presence of frustrating superexchange interactions by the (Abrikosov) fermionic representation of the spin operators [Wen, Phys. Rev. B 65, 165113 (2002)], supplemented by the Gutzwiller projection. Here, we construct charge-QQQ monopole excitations in the Heisenberg model on the triangular lattice with nearest-neighbor (J1{J}_{1}J1​) and next-nearest-neighbor (J2{J}_{2}J2​) couplings. In the highly frustrated regime, singlet and triplet monopoles with Q=1Q=1Q=1 become gapless in the thermodynamic limit; in addition, the energies for generic QQQ agree with field-theoretical predictions, obtained for a large number of gapless fermion modes. Finally, we consider localized gauge excitations, in which magnetic $\ensuremath{\pi}$ fluxes are concentrated in the triangular plaquettes (in analogy with Z2{\mathbb{Z}}_{2}Z2​ visons), showing that these kinds of states do not play a relevant role at low energies. All our findings lend support to a stable U(1)U(1)U(1) Dirac spin liquid in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the triangular lattice.