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arXiv:2308.06711 · Physical review. B./Physical review. B

Instability of the U (1) spin liquid with a large spinon Fermi surface in the Heisenberg-ring exchange model on the triangular lattice

Jianhua Yang, Tao Li

Published Dec 1, 2023• 6 citations
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Abstract

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 ${J}_{4}$ 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-${\mathrm{TTF})}_{2}{\mathrm{Cu}}_{2}{(\mathrm{CN})}_{3}$ and ${\mathrm{EtMe}}_{3}\mathrm{Sb}{[\mathrm{Pd}{(\mathrm{dmit})}_{2}]}_{2}$, which are systems close to their Mott transition and thus have large ${J}_{4}$. 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.

DOI
10.1103/physrevb.108.235105

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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.
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We study the spin-12\frac{1}{2}21​ Heisenberg model on the triangular lattice with antiferromagnetic first- (J1)({J}_{1})(J1​) and second- (J2)({J}_{2})(J2​) nearest-neighbor interactions using density matrix renormalization group. By studying the spin correlation function, we find a ${120}^{\ensuremath{\circ}}$ magnetic order phase for ${J}_{2}\ensuremath{\lesssim}0.07{J}_{1}$ and a stripe antiferromagnetic phase for ${J}_{2}\ensuremath{\gtrsim}0.15{J}_{1}$. Between these two phases, we identify a spin-liquid region characterized by exponential decaying spin and dimer correlations, as well as large spin singlet and triplet excitation gaps on finite-size systems. We find two near degenerating ground states with distinct properties in two sectors, which indicates more than one spin-liquid candidate in this region. While the sector with spinons is found to respect time reversal symmetry, the even sector without spinons breaks such a symmetry for finite-size systems. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. Moreover, by tuning the anisotropic bond coupling, we explore the nature of the spin-liquid phase and find the optimal parameter region for gapped Z2{Z}_{2}Z2​ spin liquids.