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arXiv:1103.1159 · Physical Review B

Ground-state energy and spin gap of spin- 1 2 Kagomé-Heisenberg antiferromagnetic clusters: Large-scale exact diagonalization results

Andreas Martin Läuchli, Julien Sudan, Erik Schwartz Sørensen

Published Jun 3, 2011• 130 citations
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Abstract

We present a comprehensive list of the ground-state energies and the spin gaps of finite Kagom\'e clusters with up to 42 spins obtained using large-scale exact diagonalization techniques. This represents the current limit of this exact approach. For a fixed number of spins $N$, we study several cluster shapes under periodic boundary conditions in both directions resulting in a toroidal geometry. The clusters are characterized by their side length and diagonal as well as the shortest ``Manhattan'' diameter of the torii. A finite-size scaling analysis of the ground-state energy as well as the spin gap is then performed in terms of the shortest toroidal diameter as well as the shortest ``Manhattan'' diameter. The structure of the spin-spin correlations further supports the importance of short loops wrapping around the torii.

DOI
10.1103/physrevb.83.212401

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•Nov 8, 2017•0 citations
We present a comprehensive list of the ground-state energies and the spin gaps of finite Kagom'e clusters with up to 42 spins obtained using large-scale exact diagonalization techniques. This represents the current limit of this exact approach. For a fixed number of spins NNN, we study several cluster shapes under periodic boundary conditions in both directions resulting in a toroidal geometry. The clusters are characterized by their side length and diagonal as well as the shortest Manhattan'' diameter of the torii. A finite-size scaling analysis of the ground-state energy as well as the spin gap is then performed in terms of the shortest toroidal diameter as well as the shortest Manhattan'' diameter. The structure of the spin-spin correlations further supports the importance of short loops wrapping around the torii.
We study the low-energy properties of Heisenberg antiferromagnetic spin-1/2 and spin-1 systems on various clusters made up of triangles. Some of the clusters have a geometry similar to representative pieces of the Kagome lattice, while others have the geometry of a sawtooth chain. While the ground state always has the lowest possible spin (0 or 1/2), the nature of the low-energy excitations depends on the geometry and the site spin. For the Kagome clusters with spin-1 sites, the lowest excitations are gapped, with singlet and triplet excitations having similar gaps. This is in contrast to Kagome clusters with spin-1/2 sites where there are several low-energy singlet excitations (possibly gapless in the thermodynamic limit), while triplet excitations have a gap. For the sawtooth chain with spin-1 sites, the lowest excitations are triplets with a gap; the gap to singlet excitations is about twice the triplet gap.
Using series expansions around the dimer limit, we find that the ground state of the spin-1/2 Heisenberg antiferromagnet on the kagome lattice appears to be a valence bond crystal (VBC) with a 36 site unit cell, and ground-state energy per site $E=\ensuremath{-}0.433\ifmmode\pm\else\textpm\fi{}0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}$. It consists of a honeycomb lattice of ``perfect hexagons.'' The energy difference between the ground state and other ordered states with the maximum number of perfect hexagons, such as a stripe-ordered state, is of order 0.001J0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}0.001J. The expansion is also done for the 36 site system with periodic boundary conditions; its energy per site is $0.005\ifmmode\pm\else\textpm\fi{}0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}$ lower than the infinite system, consistent with exact diagonalization results. Every unit cell of the VBC has two singlet states whose degeneracy is not lifted to sixth order in the expansion. We estimate this energy difference to be less than 0.001J0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}0.001J. The dimerization order parameter is found to be robust. Two leading orders of perturbation theory give lowest triplet excitations to be dispersionless and confined to the perfect hexagons.
We examine the perennial quantum spin liquid candidate S=12S=\frac{1}{2}S=21​ Heisenberg antiferromagnet on the kagome lattice. Our paper is based on achieving Lanczos diagonalization of the Hamiltonian on a 48 site cluster in sectors with dimensions as large as $5\ifmmode\times\else\texttimes\fi{}{10}^{11}$. The results reveal intricate structures in the low-lying energy spectrum. These structures by no means unambiguously support a Z2{\mathbb{Z}}_{2}Z2​ spin liquid ground state, but instead appear compatible with several scenarios, including fourfold topological degeneracy, inversion symmetry breaking, and a combination thereof. We discuss finite-size effects, such as the apparent absence of the eigenstate thermalization hypothesis, and note that, while considerably reduced, some are still present for the largest cluster. Finally, we observe that an XXZ model in the Ising limit reproduces remarkably well the most striking features of finite-size spectra.
We study the energy and the static spin structure factor of the ground state of the spin-1/21/21/2 quantum Heisenberg antiferromagnetic model on the kagome lattice. By the iterative application of a few Lanczos steps on accurate projected fermionic wave functions and the Green's function Monte Carlo technique, we find that a gapless (algebraic) U(1)U(1)U(1) Dirac spin liquid is competitive with previously proposed gapped (topological) Z2{\mathbb{Z}}_{2}Z2​ spin liquids. By performing a finite-size extrapolation of the ground-state energy, we obtain an energy per site $E/J=\ensuremath{-}0.4365(2)$, which is equal, within three error bars, to the estimates given by the density-matrix renormalization group (DMRG). Our estimate is obtained for a translationally invariant system, and, therefore, does not suffer from boundary effects, like in DMRG. Moreover, on finite toric clusters at the pure variational level, our energies are lower compared to those from DMRG calculations.
Frustrated quantum spin systems are investigated by the large-scale numerical exact diagonalization based on the Lanczos algorithm. Our recent results based on the calculation for finite-size clusters up to 45 spins are introduced about the following topics; (1) spin gap issue of the kagome-lattice antiferromagnet[1.2], (2) magnetization plateaux of the triangular- and kagome-lattice antiferromagnets[3,4]. (3) quatnum phase transitions of the Shastry-Sutherland model[5]. References [1]H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 80 (2011)053704;arXiv:1103.5829 [2]T. Sakai and H. Nakano, Physica B 536 (2018)85; arXiv:1801.04458 [3]H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 87 (2018)063706. [4]H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 86 (2017)114705;arXiv:1708.07248. [5]H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 87 (2018) 123702; arXiv: 1810.11533.
The S=1/2 kagome-lattice antiferromagnet is one of interesting frustrated quantum spin systems. The system is supposed to exhibit the quantum spin liquid in the ground state, which was proposed as a possible origin of the high-temprature superconductivity. The spin gap is an important physical quantity to characterize the spin liquid behavior. Whether the S=1/2 kagome-lattice antiferromagnet is gapless or has a finite spin gap, is still an unsolved issue. Because any recently developped numerical calculation methods are not enough to determine it in the thermodynamic limit. Our large-scale numerical diazonalization up to 42-spin clusters and a finite-size scaling analysis indicated that the S=1/2 kagome-lattice antiferromagnet is gapless in the thremodynamic limit[1]. It is consistent with the U(1) Dirac spin liquid theory of the kagome-lattice antiferromagnet[2,3]. On the other hand, some density matrix renormalization group (DMRG) calculations supported the gapped Z2 topological spin liquid theory[4,5]. Our recent numerical diagonalization analysis on the magnetization process of a distorted kagome-lattice antiferromagnet indicated that the perfect kagome-lattice system is just on a quantum critical point[6]. It would be a possible reason why it is difficult to determine whether the perfect kagome-lattice antiferromagnet is gapless or gapped. In the present study, a field derivative analysis is applied for the spin gap issue of the kagome-lattice antiferromanget, using the numerical diagonalization data up to 42-spin clusters. It is consistent with our previous conclusion that it is gapless[7]. [1]H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 80 (2011) 053704. [2]Y. Ran, M. Hermele, P. A. Lee and X. -G. Wen: Phys. Rev. Lett. 98 (2007) 117205. [3]Y. Iqbal, F. Becca, S. Sorella and D. Poilblanc: Phys. Rev. B 87 (2013) 060405(R). [4]S. Yan, D. A. Huse and S. R. White: Science 332 (2011) 1173. [5]S. Nishimoto, N. Shibata and C. Hotta: Nat. Commun. 4 (2013) 2287. [6]H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 83 (2014) 104710. [7]H. Nakano and T. Sakai: in preparation.