Recent experiments on triangular lattice organic Mott insulators have found evidence for a two-dimensional (2D) spin liquid in close proximity to the metal-insulator transition. A Gutzwiller wave function study of the triangular lattice Heisenberg model with a four-spin ring exchange term appropriate in this regime has found that the projected spinon Fermi sea state has a low variational energy. This wave function, together with a slave particle-gauge theory analysis, suggests that this putative spin liquid possesses spin correlations that are singular along surfaces in momentum space, i.e., ``Bose surfaces.'' Signatures of this state, which we will refer to as a ``spin Bose metal'' (SBM), are expected to manifest in quasi-one-dimensional (quasi-1D) ladder systems: the discrete transverse momenta cut through the 2D Bose surface leading to a distinct pattern of 1D gapless modes. Here, we search for a quasi-1D descendant of the triangular lattice SBM state by exploring the Heisenberg plus ring model on a two-leg triangular strip (zigzag chain). Using density matrix renormalization group (DMRG) supplemented by variational wave functions and a bosonization analysis, we map out the full phase diagram. In the absence of ring exchange the model is equivalent to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain, and we find the expected Bethe-chain and dimerized phases. Remarkably, moderate ring exchange reveals a new gapless phase over a large swath of the phase diagram. Spin and dimer correlations possess singular wave vectors at particular ``Bose points'' (remnants of the 2D Bose surface) and allow us to identify this phase as the hoped for quasi-1D descendant of the triangular lattice SBM state. We use bosonization to derive a low-energy effective theory for the zigzag spin Bose metal and find three gapless modes and one Luttinger parameter controlling all power law correlations. Potential instabilities out of the zigzag SBM give rise to other interesting phases such as a period-3 valence bond solid or a period-4 chirality order, which we discover in the DMRG. Another interesting instability is into a spin Bose-metal phase with partial ferromagnetism (spin polarization of one spinon band), which we also find numerically using the DMRG.
In our previous work [Phys. Rev. Lett. 121, 046401 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.046401], we found a quantum spin liquid phase with a spinon Fermi surface in the two dimensional spin-1/2 Heisenberg model with four-spin ring exchange on a triangular lattice. In this work we dope the spinon Fermi surface phase by studying the t-J model with four-spin ring exchange. We perform density matrix renormalization group calculations on four-leg cylinders of a triangular lattice and find that the dominant pair correlation function is that of a pair density wave; i.e., it is oscillatory while decaying with distance with a power law. The doping dependence of the period is studied. This is the first example where a pair density wave is the dominant pairing in a generic strongly interacting system where the pair density wave cannot be explained as a composite order and no special symmetry is required.
This paper reports the effect of randomness on a quantum spin liquid state of 1T-TaS
2 with a two-dimensional perfect triangular lattice. Systematic measurements of heat capacity and thermal conductivity in pure, Se-substituted, and electron-irradiated crystals reveal the microscopic coexistence of localized orphan spins that form random valence bonds and itinerant spinons that appear to form a Fermi surface.
Motivated by the recent progress in the spin-orbit-coupled triangular lattice spin liquid candidate
YbMgGaO4, we carry out a systematic projective symmetry group analysis and mean-field study of candidate
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) 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. Further connection of this spinon Fermi surface
U(1) spin liquid with
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.
It is widely believed that the
U(1) spin liquid with a large spinon Fermi surface (SFS state) can be realized in the spin-$\frac{1}{2}\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{4}$ model on the triangular lattice, when the ring exchange coupling
J4 is sufficiently strong to suppress the ${120}^{\ensuremath{\circ}}$ magnetic ordered state. This belief is supported by many variational studies on this model and seems to be consistent with the observations on the organic spin liquid materials such as $\ensuremath{\kappa}$-(BEDT-
TTF)2Cu2(CN)3 and
EtMe3Sb[Pd(dmit)2]2, which are systems close to their Mott transition and thus have large
J4. Here we show through systematic variational search that such a state is never favored in the ${J}_{1}\text{\ensuremath{-}}{J}_{4}$ model on the triangular lattice. Instead, a state with broken spatial symmetry is favored in the most probable parameter regime for the SFS state and has an energy much lower than that of the SFS state and other proposed variational states. More specifically, we find that for ${J}_{4}\ensuremath{\ge}0.09{J}_{1}$, the model favors a valence bond solid state with a $4\ifmmode\times\else\texttimes\fi{}6$ period in its local spin correlation pattern and has a variational energy that is about
5 lower than that of the SFS state. This state is separated from the $\ensuremath{\pi}$-flux state for ${J}_{4}\ensuremath{\le}0.045{J}_{1}$ by an intermediate symmetry-breaking phase with a zigzag pattern in its local spin correlation. We find that the variational phase diagram we got is in qualitative agreement with that obtained from exact diagonalization on a $6\ifmmode\times\else\texttimes\fi{}6$ cluster.
Mott insulators with localized magnetic moments will exhibit a quantum spin liquid state when the quantum fluctuations are strong enough to suppress the ordering of the spins. Such an entangled state will give rise to collective excitations, in which spin and charge information are carried separately. Our angle-resolved photoemission spectroscopy measurements on single-layer 1T-TaS_{2} show a flat band around the zone center and a gap opening of about 200 meV in the low temperature, indicating 2D Mott insulating nature in the system. This flat band is dispersionless in momentum space but shows anomalously broad width around the zone center and the spectral weight decays rapidly as momentum increases. The observation is described as a spectral continuum from electron fractionalization, corroborated by a low energy effective model. The intensity of the flat band is reduced by surface doping with magnetic adatoms and the gap is closing, a result from the interaction between spin impurities coupled with spinons and the chargons, which gives rise to a charge redistribution. Doping with nonmagnetic impurities behaves differently as the chemical potential shift dominates. These findings provide insight into the quantum spin liquid states of strongly correlated electrons on 2D triangular lattices.