Abstract Aligning with the ongoing search for quantum spin liquids (QSLs), identifying QSLs in Kitaev magnets has attracted significant research interest during the past decade. Nevertheless, it remains a significant challenge. One of the major difficulties is that Kitaev QSL is typically fragile to competing interactions such as off-diagonal exchanges, which are ubiquitous in real materials owing to spin-orbit coupling and crystal-field effects. This, in turn, generates many intriguing field-induced novel phases and the thermal Hall effect. In this review, we focus on the interplay between the Kitaev interaction and off-diagonal Γ and Γ′ exchange from a numerical perspective. The review discusses certain representative exotic phases such as Γ spin liquid, nematic ferromagnet, spin-flop phase, and distinct chiral spin states with spontaneous time-reversal symmetry breaking. It also presents quantum phase diagrams of anisotropic Kitaev-Γ chains that exhibit kaleidoscopy for both ordered and disordered phases.
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
[111] 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
[111]-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.
We study the quantum phase diagram of the spin-
1/2 Heisenberg model on the kagome lattice with first-, second-, and third-neighbor interactions
J1,J2, and
J3 by means of density matrix renormalization group. For small
J2 and
J3, this model sustains a time-reversal invariant quantum spin-liquid phase. With increasing
J2 and
J3, we find in addition a
q=(0,0) N\'eel phase, a chiral spin-liquid phase, an apparent valence-bond crystal phase, and a complex noncoplanar magnetically ordered state with spins forming the vertices of a cuboctahedron known as a cuboc1 phase. Both the chiral spin liquid and cuboc1 phase break time-reversal symmetry in the sense of spontaneous scalar spin chirality. We show that the chiralities in the chiral spin liquid and cuboc1 are distinct, and that these two states are separated by a strong first-order phase transition. The transitions from the chiral spin liquid to both the
q=(0,0) phase and to time-reversal symmetric spin liquid, however, are consistent with continuous quantum phase transitions.
The phase diagram of the
XXZ spin-1 quantum magnet on the kagome lattice is studied for all cases where the
z-component nearest-neighbor spin interaction
Jz is antiferromagnetic. Besides
Jz and the nearest-neighbor in-plane spin interaction
Jxy, the system is also parametrized by an on-site anisotropic term
D(Sz)2. In the zero magnetic field case, the six previously introduced phases, found using various methods, are the nondegenerate gapped photon phase, which breaks no space symmetry or spin symmetry; the sixfold degenerate phase with plaquette order, which breaks both time-reversal symmetry and translational symmetry; the ``superfluid'' (ferromagnetic) phase with an in-plane global U(1) symmetry broken when
Jxy<0; the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order when
Jxy>0; the nematic phase when
D<0 and large; and a phase with resonating dimers on each hexagon. We obtain all of these phases and partial information about their quantum phase transitions in a single framework by studying condensation of defects in the sixfold plaquette phases. The transition between nematic phase and the sixfold degenerate plaquette phase is potentially an unconventional second-order critical point. In the case of a nonzero magnetic field along $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$, another ordered phase with translation symmetry broken is opened up in the nematic phase. Due to the breaking of time-reversal symmetry by the field, a supersolid phase emerges between the sixfold plaquette order and the superfluid phase. This phase diagram might be accessible in nickel compounds, organic compound $m\text{\ensuremath{-}}\mathrm{MPYNN}∙\mathrm{B}{\mathrm{F}}_{4}$, or optical lattices of atoms with three degenerate states on every site.
We study the spin-1/2 Kitaev honeycomb gapless spin liquid in the presence of Stone-Wales-type local lattice defects with odd-sided plaquettes. While the clean Kitaev model has no finite-temperature phase transitions, we find that introducing a finite defect density n_{d}≈10^{-4}-10^{-2} produces a true phase transition with a sizeable T_{c}≈2n_{d} in units of the Kitaev exchange. The resulting non-Abelian chiral quantum spin liquid exhibits scalar spin chirality and electron orbital magnetization which peak near lattice defects. This disorder-driven instability relies on an emergent long range ferromagnetic interaction r^{-γ} (γ≈2.7) between defect chiralities, mediated by the nearly gapless fermions, with implications for topology generation in Dirac cones with fluctuating mass terms.
Motivated by the search for unconventional orders in frustrated quantum magnets, we present a multi-method investigation into the nature of the quantum phase diagram of the spin-
1/2 Heisenberg model on the maple-leaf lattice with three symmetry-inequivalent nearest-neighbor interactions. It has been argued that the parameter regime with antiferromagnetic couplings on hexagons
Jh and ferromagnetic couplings on triangles
Jt and dimer
Jd bonds, is potentially host to a cornucopia of emergent phases with unconventional orders. Our analysis indeed identifies an extended region where any conventional dipolar magnetic order is absent. A hexagonal singlet state is found in the region around
Jd=Jt=0, while a dimerized hexagonal singlet order of a lattice nematic character appears proximate to the phase boundary with the c
120∘ antiferromagnetic order. Interestingly, upon traversing the bulk of the paramagnetic (PM) region, we find a variety of distinct correlation profiles, which are qualitatively different from those of the hexagonal singlet and dimerized hexagonal singlet orders but feature no appreciable spin-nematic response, while the boundary with the ferromagnetic phase shows evidence of spin-nematic order. This PM region is thus likely host to an ensemble of nonmagnetic phases which could putatively include quantum spin liquids. Our phase diagram is built from a complementary application of state-of-the-art implementations of the cluster mean-field and pseudo-fermion functional renormalization group approaches, together with an unconstrained Luttinger-Tisza treatment of the model providing insights from the semi-classical limit.