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arXiv:2610.30290

$π$-Flux Tuning of Quantum Paramagnetism in the Triangular-Lattice Hubbard Model

Zhaochen Liu, Andreas Feuerpfeil, Ronny Thomale

Published Oct 21, 2026• 0 citations
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Abstract

We compare two time-reversal-symmetric triangular-lattice Hubbard models at half filling: the zero-flux model and a staggered $π$-flux model with zero and $π$ 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-$π$-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.

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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 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 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 $\ensuremath{\pi}$ 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-UUU spin liquid ground states amenable to numerical investigation.
We study the effects of a periodically varying electric field on the Hubbard model at half filling on a triangular lattice. The electric field is incorporated through the phase of the nearest-neighbor hopping amplitude via the Peierls prescription. When the on-site interaction UUU is much larger than the hopping, the effective Hamiltonian Heff{H}_{\text{eff}}Heff​ describing the spin sector can be found using a Floquet perturbation theory. To third order in the hopping, Heff{H}_{\text{eff}}Heff​ is found to have the form of a Heisenberg antiferromagnet with three different nearest-neighbor couplings $({J}_{\ensuremath{\alpha}},{J}_{\ensuremath{\beta}},{J}_{\ensuremath{\gamma}})$ on bonds lying along the different directions. Remarkably, when the periodic driving does not have time-reversal symmetry, Heff{H}_{\text{eff}}Heff​ can also have a chiral three-spin interaction in each triangle, with the coefficient CCC of the interaction having opposite signs on up- and down-pointing triangles. Thus periodic driving which breaks time-reversal symmetry can simulate the effect of a perpendicular magnetic flux which is known to generate such a chiral term in the spin sector, even though our model does not have a magnetic flux. The four parameters $({J}_{\ensuremath{\alpha}},{J}_{\ensuremath{\beta}},{J}_{\ensuremath{\gamma}},C)$ depend on the amplitude, frequency, and direction of the oscillating electric field. We then study the spin model as a function of these parameters using exact diagonalization and find a rich phase diagram of the ground state with seven different phases consisting of two kinds of ordered phases (collinear and coplanar) and disordered phases. Thus periodic driving of the Hubbard model on the triangular lattice can lead to an effective spin model whose couplings can be tuned over a range of values thereby producing a variety of interesting phases.
We investigate the ground state properties of the two-dimensional half-filled one band Hubbard model in the strong (large-UUU) to intermediate coupling limit (i.e., away from the strict Heisenberg limit) using an effective spin-only low-energy theory that includes nearest-neighbor exchange, ring exchange, and all other spin interactions to order t(t∕U)3t{(t∕U)}^{3}t(t∕U)3. We show that the operator for the staggered magnetization, transformed for use in the effective theory, differs from that for the order parameter of the spin model by a renormalization factor accounting for the increased charge fluctuations as t∕Ut∕Ut∕U is increased from the $t∕U\ensuremath{\rightarrow}0$ Heisenberg limit. These charge fluctuations lead to an increase of the quantum fluctuations over and above those for an S=1∕2S=1∕2S=1∕2 antiferromagnet. The renormalization factor ensures that the zero temperature staggered moment for the Hubbard model is a monotonously decreasing function of t∕Ut∕Ut∕U, despite the fact that the moment of the spin Hamiltonian, which depends on transverse spin fluctuations only, in an increasing function of t∕Ut∕Ut∕U. We also comment on quantitative aspects of the t∕Ut∕Ut∕U and 1∕S1∕S1∕S expansions.
Motivated by experimental studies that have found signatures of a quantum spin liquid phase in organic crystals whose structure is well described by the two-dimensional triangular lattice, we study the Hubbard model on this lattice at half filling using the infinite-system density matrix renormalization group (iDMRG) method. On infinite cylinders with finite circumference, we identify an intermediate phase between observed metallic behavior at low interaction strength and Mott insulating spin-ordered behavior at strong interactions. Chiral ordering from spontaneous breaking of time-reversal symmetry, a fractionally quantized spin Hall response, and characteristic level statistics in the entanglement spectrum in the intermediate phase provide strong evidence for the existence of a chiral spin liquid in the full two-dimensional limit of the model.
Chirality and flux correlations are investigated by the exact diagonalization and the quantum Monte Carlo method. In all the cases of the triangular and the square lattice with and without the next nearest-neighbor exchange, the extrapolation to the thermodynamic limit indicates the absence of uniform and staggered chirality in the spin-1/2 Heisenberg models. The spin-1/2 models with the 4-spin and the 6-spin ring exchange show similar behavior to the ordinary Heisenberg models. The quantum Monte Carlo results on the Hubbard model also show no indication of the uniform or the staggered flux order.
We study the interplay of competing interactions in spin-12\frac{1}{2}21​ triangular Heisenberg model through tuning the first- (J1{J}_{1}J1​), second- (J2{J}_{2}J2​), and third-neighbor (J3{J}_{3}J3​) couplings. Based on a large-scale density-matrix renormalization group calculation, we identify a quantum phase diagram of the system and discover a gapless chiral spin-liquid (CSL) phase in the intermediate J2{J}_{2}J2​ and J3{J}_{3}J3​ regime. This CSL state spontaneously breaks time-reversal symmetry with finite scalar chiral order, and it has gapless excitations implied by a vanishing spin triplet gap and a finite central charge on the cylinder. Moreover, the central charge grows rapidly with the cylinder circumference, indicating emergent spinon Fermi surfaces. To understand the numerical results we propose a parton mean-field spin-liquid state, the U(1)U(1)U(1) staggered flux state, which breaks time-reversal symmetry with chiral edge modes by adding a Chern insulator mass to Dirac spinons in the U(1)U(1)U(1) Dirac spin liquid. This state also breaks lattice rotational symmetries and possesses two spinon Fermi surfaces driven by nonzero J2{J}_{2}J2​ and J3{J}_{3}J3​, which naturally explains the numerical results. This realizes an example of a gapless CSL state with coexisting spinon Fermi surfaces and chiral edge states, demonstrating the rich family of interesting quantum phases emergent from competing interactions in triangular-lattice magnets.