We provide strong numerical evidence, using improved variational wave functions, for a ground state with vanishing spin gap in the spin-$1/2$ quantum Heisenberg model on the kagome lattice. Starting from the algebraic $U(1)$ Dirac spin liquid state proposed by Ran et al. [Phys. Rev. Lett. 98, 117205 (2007)] and iteratively applying a few Lanczos steps, we compute the lowest $S=2$ excitation constructed by exciting spinons close to the Dirac nodes. Our results are compatible with a vanishing spin gap in the thermodynamic limit and in consonance with a power-law decay of long distance spin-spin correlations in real space. The competition with a gapped (topological) spin liquid is discussed.
We provide strong numerical evidence, using improved variational wave functions, for a ground state with vanishing spin gap in the spin-1/2 quantum Heisenberg model on the kagome lattice. Starting from the algebraic U(1) Dirac spin liquid state proposed by Ran et al. [Phys. Rev. Lett. 98, 117205 (2007)] and iteratively applying a few Lanczos steps, we compute the lowest S=2 excitation constructed by exciting spinons close to the Dirac nodes. Our results are compatible with a vanishing spin gap in the thermodynamic limit and in consonance with a power-law decay of long distance spin-spin correlations in real space. The competition with a gapped (topological) spin liquid is discussed.
We study the energy and the static spin structure factor of the ground state of the spin-1/2 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) Dirac spin liquid is competitive with previously proposed gapped (topological) 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.
The nature of the ground state of the spin S=1/2 Heisenberg antiferromagnet on the kagome lattice with breathing anisotropy (i.e., with different superexchange couplings J▵ and J▿ within elementary up- and down-pointing triangles) is investigated within the framework of Gutzwiller projected fermionic wave functions and Monte Carlo methods. We analyze the stability of the U(1) Dirac spin liquid with respect to the presence of fermionic pairing that leads to a gapped Z2 spin liquid. For several values of the ratio J▿/J▵, the size scaling of the energy gain due to the pairing fields and the variational parameters are reported. Our results show that the energy gain of the gapped spin liquid with respect to the gapless state either vanishes for large enough system size or scales to zero in the thermodynamic limit. Similarly, the optimized pairing amplitudes (responsible for opening the spin gap) are shown to vanish in the thermodynamic limit. Our outcome is corroborated by the application of one and two Lanczos steps to the gapless and gapped wave functions, for which no energy gain of the gapped state is detected when improving the quality of the variational states. Finally, we discuss the competition with the ``simplex'' Z2 resonating-valence-bond spin liquid, valence-bond crystal, and nematic states in the strongly anisotropic regime, i.e., ${J}_{▿}\ensuremath{\ll}{J}_{▵}$.
Competing Spin Liquid Phases in the S=21 Heisenberg Model on the Kagome Lattice
The properties of ground state of spin- \frac{1}{2} 1 2 kagome antiferromagnetic Heisenberg (KAFH) model have attracted considerable interest in the past few decades, and recent numerical simulations reported a spin liquid phase. The nature of the spin liquid phase remains unclear. For instance, the interplay between symmetries and Z_2 Z 2 topological order leads to different types of Z_2 Z 2 spin liquid phases. In this paper, we develop a numerical simulation method based on symmetric projected entangled-pair states (PEPS), which is generally applicable to strongly correlated model systems in two spatial dimensions. We then apply this method to study the nature of the ground state of the KAFH model. Our results are consistent with that the ground state is a U(1) U ( 1 ) Dirac spin liquid rather than a Z_2 Z 2 spin liquid.
Z2 spin liquid ground state of the spin-21 Kagome antiferromagnetic Heisenberg model
We show that the best variational ground state of the spin-21 Kagome antiferromagnetic Heisenberg model with nearest-neighboring exchange coupling(NN-KAFH) is a Z2 spin liquid state, rather than the widely believed U(1) Dirac spin liquid state. The spinon excitation in the Z2 spin liquid state has a small gap of about 1/40 of the spinon band width. We find that while the Z2 and the U(1) spin liquid state have a large overlap on finite clusters and are thus very close in energy, they host totally different spinon excitation spectrum. The strength of the RVB theory becomes particularly clear in such a situation, as it provides not only a variational understanding of the ground state structure, but also a comprehensive picture for the excitation spectrum of the system. Our result indicates that the spin-21 NN-KAFH should be better understood as a nearly critical system, rather than a prototypical gapped Z2 spin liquid system.
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.
Due to strong geometric frustration and quantum fluctuation, the S=1/2 quantum Heisenberg antiferromagnet on the kagome lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg S=1/2,kagome lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of Z2 topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of Z2 SLs on the kagome lattice. Among them we found only one gapped Z2 SL (which we call the ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state to be the numerically discovered SL ground state of the HKLM.