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arXiv:1411.4649 · Physical Review Letters

Quantum Paramagnet in a π Flux Triangular Lattice Hubbard Model

Stephan Rachel, Manuel Laubach, Johannes Reuther, Ronny Thomale

Published Apr 23, 2015• 20 citations
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Abstract

We propose the $\ensuremath{\pi}$ flux triangular lattice Hubbard model ($\ensuremath{\pi}$ THM) as a prototypical setup to stabilize magnetically disordered quantum states of matter in the presence of charge fluctuations. The quantum paramagnetic domain of the $\ensuremath{\pi}$ THM that we identify for intermediate Hubbard $U$ is framed by a Dirac semimetal for weak coupling and by 120\ifmmode^\circ\else\textdegree\fi{} N\'eel order for strong coupling. Generalizing the Klein duality from spin Hamiltonians to tight-binding models, the $\ensuremath{\pi}$ THM maps to a Hubbard model which corresponds to the $({J}_{\mathrm{H}},{J}_{\mathrm{K}})=(\ensuremath{-}1,2)$ Heisenberg-Kitaev model in its strong coupling limit. The $\ensuremath{\pi}$ THM provides a promising microscopic testing ground for exotic finite-$U$ spin liquid ground states amenable to numerical investigation.

DOI
10.1103/physrevlett.114.167201

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We propose the \ensuremathπ\ensuremath{\pi}\ensuremathπ flux triangular lattice Hubbard model (\ensuremathπ\ensuremath{\pi}\ensuremathπ THM) as a prototypical setup to stabilize magnetically disordered quantum states of matter in the presence of charge fluctuations. The quantum paramagnetic domain of the \ensuremathπ\ensuremath{\pi}\ensuremathπ THM that we identify for intermediate Hubbard UUU is framed by a Dirac semimetal for weak coupling and by 120\ifmmode^\circ\else\textdegree\fi{} N'eel order for strong coupling. Generalizing the Klein duality from spin Hamiltonians to tight-binding models, the \ensuremathπ\ensuremath{\pi}\ensuremathπ THM maps to a Hubbard model which corresponds to the (JH,JK)=(\ensuremath−1,2)({J}_{\mathrm{H}},{J}_{\mathrm{K}})=(\ensuremath{-}1,2)(JH​,JK​)=(\ensuremath−1,2) Heisenberg-Kitaev model in its strong coupling limit. The \ensuremathπ\ensuremath{\pi}\ensuremathπ THM provides a promising microscopic testing ground for exotic finite-UUU spin liquid ground states amenable to numerical investigation.
πππ-Flux Tuning of Quantum Paramagnetism in the Triangular-Lattice Hubbard Model
We compare two time-reversal-symmetric triangular-lattice Hubbard models at half filling: the zero-flux model and a staggered πππ-flux model with zero and πππ flux through oppositely oriented triangles. Although both reduce to the same Heisenberg model at leading order in the strong-coupling limit, their distinct charge fluctuation profiles yield different quantum paramagnetic regimes. Combining a strong-coupling expansion with infinite density matrix renormalization group (iDMRG) calculations, we find that the positive ring-exchange of the zero-flux model promotes spontaneous chirality, consistent with its notion of a chiral spin liquid at intermediate coupling. In the staggered-πππ-flux model, whose noninteracting limit is a Dirac semimetal, the ring exchange changes sign and disfavors chiral order and we find no resolvable scalar chirality. This renders flux-tuned ring-exchange an efficient discriminator between chiral and non-chiral quantum paramagnets at finite coupling.
We study quantum disordered ground states of the two-dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg and Kitaev couplings and sufficiently small spin SSS, we find three different symmetric Z2{Z}_{2}Z2​ spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well-known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops ${120}^{\ensuremath{\circ}}$ order in the semiclassical limit at large SSS. In the opposite Kitaev limit, we find a different spin liquid ground state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings, we find a spin liquid state with unusual spin correlations. Upon spinon condensation, this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works.
We investigate the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the triangular lattice with an additional scalar chirality term and show that a chiral spin liquid is stabilized in a sizable region of the phase diagram. This topological phase is situated in between a coplanar ${120}^{\ensuremath{\circ}}$ N\'eel ordered and a noncoplanar tetrahedrally ordered phase. Furthermore we discuss the nature of the spin-disordered intermediate phase in the ${J}_{1}\ensuremath{-}{J}_{2}$ model. We compare the ground states from exact diagonalization with a Dirac spin liquid wave function and propose a scenario where this wave function describes the quantum critical point between the ${120}^{\ensuremath{\circ}}$ magnetically ordered phase and a putative Z2{\mathbb{Z}}_{2}Z2​ spin liquid.
Motivated by recent progress in moiré superlattices and spin-1/2 triangular-lattice antiferromagnets, we study how orbital magnetic flux and Zeeman coupling compete or cooperate in generating internal U(1) gauge flux in a triangular spin liquid. We show that orbital flux favors a chiral spin liquid with staggered internal flux, whereas Zeeman coupling destabilizes the spinon Fermi-pocket state toward a Landau-level state with spontaneous uniform internal flux and conical spin order. We demonstrate this mechanism in a spin-1/2 J_{1}-J_{2}-J_{χ} model and further identify thermal Hall and magnetic signatures that distinguish these regimes, with potential applications to moiré Hofstadter-Hubbard systems and triangular-lattice antiferromagnets.
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.
We investigate ground state properties of the half-filled staggered-flux Hubbard model on a square lattice. Energy gaps to charge and spin excitations and magnetic as well as dimer orders are calculated as a function of interaction strength U/t by means of a constrained-path quantum Monte Carlo method. It is found that the system is a semimetal at U/t≲5.6 and a Mott insulator, with long-range antiferromagnetic order, at U/t≳6.6. In the range 5.6≲U/t≲6.6, the ground state is a correlated insulator where both magnetic and dimer orders are absent. Furthermore, spin excitation in the intermediate phase appears to be gapless, and the measured spin-spin correlation function exhibits power-law decaying behavior. The data suggest that the nonmagnetic ground state is a possible candidate for the putative algebraic spin liquid.