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arXiv:1106.6134

The spin - 1/2 Heisenberg model on Kagome lattice as a quantum critical system

Tao Li

Published Jun 30, 2011• 1 citation
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Abstract

Through exact diagonalization study of the spin - 1/2 Heisenberg model on Kagome lattice with ring-exchange coupling $J_{r}$, we find the pure Heisenberg model with $J_{r}=0$ stands as a quantum critical point, as evidenced by avoided level crossing and divergence of the second derivative of the ground state energy with respect to $J_{r}$ at $J_{r}=0$. The pure Heisenberg model should thus be gapless in the thermodynamic limit, contrary to common beliefs. At the same time, the ring exchange coupling is found to drive the system into a state with more short ranged spin correlation and with a local spin correlation pattern equivalent to that of the antiferromagnetic Heisenberg model on triangular lattice(the peak of spin structure factor moves to the momentum $\vec{q}=(\frac{4π}{3},0)$). The resemblance of state with the Marston-Zeng spin Pereils solid state(in terms of dimer-dimer correlation) is also much enhanced by the ring exchange coupling, although it is unclear if such correlation would solidify into static order in the thermodynamic limit.

DOI
10.48550/arxiv.1106.6134

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Through exact diagonalization study of the spin - 1/2 Heisenberg model on Kagome lattice with ring-exchange coupling JrJ_{r}Jr​, we find the pure Heisenberg model with Jr=0J_{r}=0Jr​=0 stands as a quantum critical point, as evidenced by avoided level crossing and divergence of the second derivative of the ground state energy with respect to JrJ_{r}Jr​ at Jr=0J_{r}=0Jr​=0. The pure Heisenberg model should thus be gapless in the thermodynamic limit, contrary to common beliefs. At the same time, the ring exchange coupling is found to drive the system into a state with more short ranged spin correlation and with a local spin correlation pattern equivalent to that of the antiferromagnetic Heisenberg model on triangular lattice(the peak of spin structure factor moves to the momentum q⃗=(4π3,0)\vec{q}=(\frac{4π}{3},0)q​=(34π​,0)). The resemblance of state with the Marston-Zeng spin Pereils solid state(in terms of dimer-dimer correlation) is also much enhanced by the ring exchange coupling, although it is unclear if such correlation would solidify into static order in the thermodynamic limit.
We study the spin-1∕21∕21∕2 Heisenberg model in a lattice that interpolates between the triangular and the kagome lattices. The exchange interaction along the bonds of the kagome lattice is JJJ, and the one along the bonds connecting kagome and nonkagome sites is ${J}^{\ensuremath{'}}$, so that ${J}^{\ensuremath{'}}=J$ corresponds to the triangular limit and ${J}^{\ensuremath{'}}=0$ to the kagome one. We use variational and exact diagonalization techniques. We analyze the behavior of the order parameter for the antiferromagnetic phase of the triangular lattice, the spin gap, and the structure of the spin excitations as functions of ${J}^{\ensuremath{'}}∕J$. Our results indicate that the antiferromagnetic order is not affected by the reduction of ${J}^{\ensuremath{'}}$ down to ${J}^{\ensuremath{'}}∕J\ensuremath{\simeq}0.2$. Below this value, antiferromagnetic correlations grow weaker, a description of the ground state in terms of a N\'eel phase renormalized by quantum fluctuations becomes inadequate, and the finite-size spectra develop features that are not compatible with antiferromagnetic ordering. However, this phase does not appear to be connected to the kagome phase as well, as the low-energy spectra do not evolve with continuity for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}0$ to the kagome limit. In particular, for any nonzero value of ${J}^{\ensuremath{'}}$, the latter interaction sets the energy scale for the low-lying spin excitations, and a gapless triplet spectrum, destabilizing the kagome phase, is expected.
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A spin-1 Heisenberg model on trimerized kagome lattice is studied by doing a low-energy bosonic theory in terms of plaquette triplons defined on its triangular unit cells. The model considered has an intratriangle antiferromagnetic exchange interaction JJJ (set to 1) and two intertriangle couplings ${J}^{\ensuremath{'}}>0$ (nearest neighbor) and ${J}^{\ensuremath{''}}$ (next nearest neighbor; of both signs). The triplon analysis performed on this model investigates the stability of the trimerized singlet ground state (which is exact in the absence of intertriangle couplings) in the ${J}^{\ensuremath{'}}\ensuremath{-}{J}^{\ensuremath{''}}$ plane. It gives a quantum phase diagram that has two gapless antiferromagnetically ordered phases separated by the spin-gapped trimerized singlet phase. The trimerized singlet ground state is found to be stable on ${J}^{\ensuremath{''}}=0$ line (the nearest-neighbor case), and on both sides of it for ${J}^{\ensuremath{''}}\ensuremath{\ne}0$, in an extended region bounded by the critical lines of transition to the gapless antiferromagnetic phases. The gapless phase in the negative ${J}^{\ensuremath{''}}$ region has a coplanar ${120}^{\ensuremath{\circ}}$ antiferromagnetic order with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ structure. In this phase, all the magnetic moments are of equal length, and the angle between any two of them on a triangle is exactly ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this case has a unit cell consisting of three triangles. The other gapless phase, in the positive ${J}^{\ensuremath{''}}$ region, is found to exhibit a different coplanar antiferromagnetic order with ordering wave vector q=(0,0)\mathbf{q}=(0,0)q=(0,0). Here, two magnetic moments in a triangle are of the same magnitude, but shorter than the third. While the angle between two short moments is ${120}^{\ensuremath{\circ}}\ensuremath{-}2\ensuremath{\delta}$, it is ${120}^{\ensuremath{\circ}}+\ensuremath{\delta}$ between a short and the long one. Only when ${J}^{\ensuremath{''}}={J}^{\ensuremath{'}}$, their magnitudes become equal and the relative angles ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this q=(0,0)\mathbf{q}=(0,0)q=(0,0) phase has the translational symmetry of the kagome lattice with triangular unit cells of reduced (isosceles) symmetry. This reduction in the point-group symmetry is found to show up as a difference in the intensities of certain Bragg peaks, whose ratio ${I}_{(1,0)}/{I}_{(0,1)}=4{sin}^{2}(\frac{\ensuremath{\pi}}{6}+\ensuremath{\delta})$ presents an experimental measure of the deviation $\ensuremath{\delta}$ from the ${120}^{\ensuremath{\circ}}$ order.
We derive exact results for close-packed dimers on the triangular kagome lattice (TKL), formed by inserting triangles into the triangles of the kagome lattice. Because the TKL is a nonbipartite lattice, dimer-dimer correlations are short ranged so that the ground state at the Rokhsar-Kivelson (RK) point of the corresponding quantum dimer model on the same lattice is a short-ranged spin liquid. Using the Pfaffian method, we derive an exact form for the free energy, and we find that the entropy is 13ln 2\frac{1}{3}\text{ln}\text{ }231​ln 2 per site, regardless of the weights of the bonds. The occupation probability of every bond is 14\frac{1}{4}41​ in the case of equal weights on every bond. Similar to the case of lattices formed by corner-sharing triangles (such as the kagome and squagome lattices), we find that the dimer-dimer correlation function is identically zero beyond a certain (short) distance. We find in addition that monomers are deconfined on the TKL, indicating that there is a short-ranged spin liquid phase at the RK point. We also find exact results for the ground-state energy of the classical Heisenberg model. The ground state can be ferromagnetic, ferrimagnetic, locally coplanar, or locally canted, depending on the couplings. From the dimer model and the classical spin model, we derive upper bounds on the ground-state energy of the quantum Heisenberg model on the TKL.
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Using Lanczos exact diagonalization (ED), we study the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model with the four-spin ring-exchange interaction Jr{J}_{r}Jr​ on a triangular lattice. We mainly use the level spectroscopic technique of two 36-site tori to investigate the ground-state phase diagram, and further characterize phases by spin, dimer, and chiral correlation functions. The ground state has rich phases including several magnetic-ordered phases like the zigzag phase and tetrahedral phase, as well as several novel nonmagnetic phases, some of which exhibit valence bond solid behavior in their dimer correlation functions. However, we do not find direct evidence of a quantum spin liquid phase with spinon Fermi surface in this model. Our results can give a better understanding of the ground-state properties of the triangular Heisenberg model with ring-exchange interaction and help to understand the relevant triangular materials.