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arXiv:2103.10714 · Physical review. B./Physical review. B

Bosonic resonating valence bond theory of the possible chiral spin-liquid state in the triangular-lattice Hubbard model

Qiu Zhang, Tao Li

Published Aug 3, 2021• 11 citations
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Abstract

Recent numerical simulations find a possible chiral spin-liquid state in the intermediate coupling regime of the triangular lattice Hubbard model. Here we provide a simple picture for its origin in terms of a bosonic resonating valence bond (RVB) description. More specifically, we show that such a chiral spin-liquid state can be understood as a quantum disordered tetrahedral spin state stabilized by the four spin ring exchange coupling, which suppresses the order-by-disorder effect toward a stripy spin state. Such a chiral spin-liquid state features a spin Berry phase of $\frac{\ensuremath{\pi}}{2}$ per triangle. However, we show that the topological property of such a state is totally missed in the Schwinger boson mean-field description as a result of the lack of boson rigidity caused by the no double occupancy constraint.

DOI
10.1103/physrevb.104.075103

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A gapped Z2Z_{2}Z2​ spin liquid phase with a U(1)U(1)U(1) mean field ansatz: a Bosonic resonating valence-bond description
Gapped Z2Z_{2}Z2​ spin liquid as the simplest spin liquid has been proved to be the most difficult to realize in realistic models. Here we show that the frustration from a third-neighbor exchange J3J_{3}J3​ on the spin-12\frac{1}{2}21​ J1−J2J_{1}-J_{2}J1​−J2​ model on the square lattice may serve to stabilize such a long-sought state. We argue that a Bosonic RVB description is more appropriate than a Fermonic RVB description for such a gapped spin liquid phase. We show that while the mean field ansatz of the proposed state has a U(1)U(1)U(1) gauge symmetry, the gauge fluctuation spectrum on it is actually gapped and the state should be understood rather as a Z2Z_{2}Z2​ spin liquid state with topological order. The state is thus locally stable with respect to gauge fluctuation and can emerge continuously from a collinear Neel ordered phase.
The weak Mott insulating regime of the triangular lattice Hubbard model exhibits a rich magnetic phase diagram as a result of the ring exchange interaction in the spin Hamiltonian. These phases include the Kalmeyer-Laughlin type chiral spin liquid (CSL) and a valence bond solid (VBS). A natural question arises regarding the robustness of these phases in the presence of a weak spin-orbit coupling (SOC). In this study, we derive the effective spin model for the spin-orbit coupled triangular lattice Hubbard model in the weak Mott insulting regime, including all SOC-mediated spin-bilinear and ring-exchange interactions. We then construct a simplified spin model keeping only the most relevant SOC-mediated spin interactions. Using infinite density matrix renormalization group (iDMRG), we show that the CSL and VBS phases of the triangular lattice Hubbard model can be stabilized in the presence of a weak SOC. The stabilization results from a compensation between the Dzyaloshinskii-Moriya interaction and a SOC-mediated ring-exchange interaction. We also provide additional qualitative arguments to intuitively understand the compensation mechanism in the iDMRG quantum phase diagrams. This mechanism for stabilization can potentially be useful for the experimental realization of quantum spin liquids.
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We study the ground-state phase diagram of the spin-1/21/21/2${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice with an accurate bosonic resonating-valence-bond (RVB) wave function. In contrast to the RVB ansatz based on Schwinger fermions, the representation based on Schwinger bosons, supplemented by a variational Monte Carlo technique enforcing the exact projection onto the physical subspace, is able to describe a fully gapped spin liquid in the strongly frustrated regime. In particular, a fully symmetric Z2{Z}_{2}Z2​ spin liquid is stable between two antiferromagnetic phases; a continuous transition at J2=0.4J1{J}_{2}=0.4{J}_{1}J2​=0.4J1​, when the Marshall sign rule begins to be essentially violated, and a first-order transition around J2=0.6J1{J}_{2}=0.6{J}_{1}J2​=0.6J1​ are present. Most importantly, the triplet gap is found to have a nonmonotonic behavior, reaching a maximum around J2=0.51J1{J}_{2}=0.51{J}_{1}J2​=0.51J1​, when the lowest spinon excitation moves from the $\ensuremath{\Gamma}$ to the MMM point, i.e., $k=(\ensuremath{\pi},0)$.
We investigate the ground state of the d1{d}^{1}d1 spin-orbital model for triply degenerate t2g{t}_{2g}t2g​ orbitals on a triangular lattice that unifies intrinsic frustration of spin and orbital interactions with geometrical frustration. Using full or Lanczos exact diagonalization of finite clusters, we establish that the ground state of the spin-orbital model that interpolates between superexchange and direct exchange interactions on the bonds is characterized by valence-bond correlations. In the absence of Hund's exchange the model describes a competition between various possible valence-bond states. By considering the clusters with open boundary conditions we demonstrate that orbital interactions are always frustrated, but this frustration is removed by pronounced spin singlet correlations that coexist with dimer orbital correlations supporting them. Such local configurations contribute to the disordered ground states found for the clusters with periodic boundary conditions that interpolate between a highly resonating, dimer-based, entangled spin-orbital liquid phase and a valence-bond state with completely static spin-singlet states. We argue that these states are also realized for the infinite lattice and anticipate that pronounced transitions between different regimes found for particular geometries will turn out to smooth crossovers in the properties of the spin-orbital liquid in the thermodynamic limit. Finally, we provide evidence that the resonating spin-orbital liquid phase involves entangled states on the bonds. In such a phase classical considerations based on the mean-field theory cannot be used, spin exchange interactions do not determine spin bond correlations, and quantum fluctuations play a crucial role in the ground states and magnetic transitions.
A chiral spin liquid (CSL) phase has been recently reported in the Hubbard model on a triangular lattice at half-filling. It emerges in an intermediate coupling regime, which is sandwiched between a 120∘120^\circ120∘ antiferromagnetic (AFM) phase and a metallic phase as a function of on-site repulsion UUU. In this work, we examine how the charge fluctuation may cause the CSL and complex phase diagram. We first identify an exact sign structure of the model at arbitrary temperature, sample size, and doping, which is reduced from the original Fermi exchange signs of the electrons by finite UUU. In particular, the spin and charge degrees of freedom are generally entangled via the phase-string in the sign structure. By precisely switching off such a phase-string, the CSL phase is shown to disappear, with only the 120∘120^\circ120∘ AFM order remaining down to weak UUU by a density matrix renormalization group calculation. Here the charge fluctuation is effectively decoupled from the AFM background, which is a 120∘120^\circ120∘ AFM order due to a geometric Berry phase in the sign structure of the triangular lattice. General implications for the Mott physics will also be discussed.