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arXiv:cond-mat/0604389 · Physical Review B

Gapless quantum spin liquid, stripe, and antiferromagnetic phases in frustrated Hubbard models in two dimensions

T. Mizusaki, Masatoshi Imada

Published Jul 18, 2006• 107 citations
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Abstract

Unique features of the nonmagnetic insulator phase are revealed, and the phase diagram of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model containing the diagonal transfers ${t}^{\ensuremath{'}}$ on a square lattice is presented. Using the path-integral renormalization group method, we find an antiferromagnetic phase for small next-nearest-neighbor transfer ${t}^{\ensuremath{'}}$ and a stripe (or collinear) phase for large ${t}^{\ensuremath{'}}$ in the Mott insulating region of the strong on-site interaction $U$. For intermediate ${t}^{\ensuremath{'}}∕t\ensuremath{\sim}0.7$ at large $U∕t>7$, we find a longer-period antiferromagnetic-insulator phase with $2\ifmmode\times\else\texttimes\fi{}4$ structure. In the Mott insulating region, we also find a quantum spin liquid (in other words, a nonmagnetic insulator) phase near the Mott transition to paramagnetic metals for the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model on the square lattice as well as on the anisotropic triangular lattice. Correlated electrons often crystallize to the Mott insulator usually with some magnetic orders, whereas the ``quantum spin liquid" has been a long-sought issue. We report numerical evidence that a nonmagnetic insulating phase gets stabilized near the Mott transition with remarkable properties: The two-dimensional Mott insulators on geometrically frustrated lattices contain a phase with gapless spin excitations and degeneracy of the ground state in the whole Brillouin zone of the total momentum. The obtained vanishing spin renormalization factor suggests that spin excitations do not propagate coherently in contrast to conventional phases, where there exist either magnons in symmetry-broken phases or particle-hole excitations in paramagnetic metals. It imposes a constraint on the possible pictures of quantum spin liquids and supports an interpretation for the existence of an unconventional quantum liquid. The present concept is useful in analyzing a variety of experimental results in frustrated magnets including organic BEDT-TTF compounds and $^{3}\mathrm{He}$ atoms adsorbed on graphite.

DOI
10.1103/physrevb.74.014421

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Unique features of the nonmagnetic insulator phase are revealed, and the phase diagram of the t\ensuremath-t\ensuremath′t\text{\ensuremath{-}}{t}^{\ensuremath{'}}t\ensuremath-t\ensuremath′ Hubbard model containing the diagonal transfers t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ on a square lattice is presented. Using the path-integral renormalization group method, we find an antiferromagnetic phase for small next-nearest-neighbor transfer t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ and a stripe (or collinear) phase for large t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ in the Mott insulating region of the strong on-site interaction UUU. For intermediate t\ensuremath′∕t\ensuremath∼0.7{t}^{\ensuremath{'}}∕t\ensuremath{\sim}0.7t\ensuremath′∕t\ensuremath∼0.7 at large U∕t>7U∕t>7U∕t>7, we find a longer-period antiferromagnetic-insulator phase with 2\ifmmode×\else\texttimes\fi42\ifmmode\times\else\texttimes\fi{}42\ifmmode×\else\texttimes\fi4 structure. In the Mott insulating region, we also find a quantum spin liquid (in other words, a nonmagnetic insulator) phase near the Mott transition to paramagnetic metals for the t\ensuremath-t\ensuremath′t\text{\ensuremath{-}}{t}^{\ensuremath{'}}t\ensuremath-t\ensuremath′ Hubbard model on the square lattice as well as on the anisotropic triangular lattice. Correlated electrons often crystallize to the Mott insulator usually with some magnetic orders, whereas the ``quantum spin liquid" has been a long-sought issue. We report numerical evidence that a nonmagnetic insulating phase gets stabilized near the Mott transition with remarkable properties: The two-dimensional Mott insulators on geometrically frustrated lattices contain a phase with gapless spin excitations and degeneracy of the ground state in the whole Brillouin zone of the total momentum. The obtained vanishing spin renormalization factor suggests that spin excitations do not propagate coherently in contrast to conventional phases, where there exist either magnons in symmetry-broken phases or particle-hole excitations in paramagnetic metals. It imposes a constraint on the possible pictures of quantum spin liquids and supports an interpretation for the existence of an unconventional quantum liquid. The present concept is useful in analyzing a variety of experimental results in frustrated magnets including organic BEDT-TTF compounds and 3He^{3}\mathrm{He}3He atoms adsorbed on graphite.
We study the ground-state properties of the geometrically frustrated Hubbard model on the anisotropic checkerboard lattice with nearest-neighbor hopping ttt and next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. By using the path-integral renormalization-group method, we study the phase diagram in the parameter space of the Hubbard interaction UUU and the frustration-control parameter ${t}^{\ensuremath{'}}/t$. Close examinations of the effective hopping, the double occupancy, the momentum distribution, and the spin- and charge-correlation functions allow us to determine the phase diagram at zero temperature, at which the plaquette-singlet insulator emerges besides the antiferromagnetic insulator and the paramagnetic metal. Spin-liquid insulating states without any kind of symmetry breaking cannot be found in our frustrated model.
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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.
The nature of quantum spin liquids is studied for the spin-1/21/21/2 antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites, J1J_1J1​, and those between next-nearest-neighbor sites, J2J_2J2​. We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region 0.4<J2/J1≤0.60.4< J_2/J_1\le 0.60.4<J2​/J1​≤0.6 sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of 0.4<J2/J1≤0.50.4 < J_2/J_1 \le 0.50.4<J2​/J1​≤0.5, while the gapped valence-bond-crystal (VBC) phase is stabilized for 0.5<J2/J1≤0.60.5 < J_2/J_1 \le 0.60.5<J2​/J1​≤0.6. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the C4vC_{4v}C4v​ symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation ⟨S(0)S(r)⟩∝1/rz+η\langle S(0)S(r)\rangle \propto 1/r^{z+η}⟨S(0)S(r)⟩∝1/rz+η at a long distance rrr appears to increase from z+η∼1z+η\sim 1z+η∼1 at J2/J1∼0.4J_2/J_1\sim0.4J2​/J1​∼0.4 toward the continuous transition to the VBC phase at J1/J1∼0.5J_1/J_1\sim0.5J1​/J1​∼0.5. Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.
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