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arXiv:1105.0581 · Physical Review B

Doping on the kagome lattice: A variational Monte Carlo study of the t - J model

Siegfried Guertler, H. Monien

Published Nov 10, 2011• 14 citations
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Abstract

We study doping on the kagome lattice by exploring the $t$-$J$ model with variational Monte-Carlo. We use a number of Gutzwiller projected spin-liquid and valence-bond-crystal states and compare their energies at several system sizes. We find that introducing mobile holes drives the system away from the spin-liquid state proposed by Ran et al. [Y. Ran, M. Hermele, P. A. Lee, and X. G. Wen, Phys. Rev. Lett. 98, 117205 (2007)] for the undoped system, toward a uniform state with zero flux. On top of the uniform state a valence-bond crystal of the Hastings type is formed for low doping. The results are compared and agree well with exact diagonalization on small clusters.

DOI
10.1103/physrevb.84.174409

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We study doping on the kagome lattice by exploring the ttt-JJJ model with variational Monte-Carlo. We use a number of Gutzwiller projected spin-liquid and valence-bond-crystal states and compare their energies at several system sizes. We find that introducing mobile holes drives the system away from the spin-liquid state proposed by Ran et al. [Y. Ran, M. Hermele, P. A. Lee, and X. G. Wen, Phys. Rev. Lett. 98, 117205 (2007)] for the undoped system, toward a uniform state with zero flux. On top of the uniform state a valence-bond crystal of the Hastings type is formed for low doping. The results are compared and agree well with exact diagonalization on small clusters.
We investigate the kagome lattice Hubbard model at half filling using the variational Monte Carlo method by testing the U(1)U(1)U(1)-Dirac spin-liquid, uniform, and valence bond crystal states. Even for the large-UUU case, the U(1)U(1)U(1) Dirac state, which is the optimal state in the Heisenberg model, cannot be recovered. While the finite-UUU Hubbard model allows for the introduction of vacancies in a different manner compared to the $t\text{\ensuremath{-}}J$ model, the physics appears to have many similarities. In particular, a valence bond crystal is formed in the intermediate-UUU regime. We observe an impact of the formation of this valence bond crystal on a possible Mott transition in this model and discuss the properties of the wave function.
We investigate the impact of spin-phonon coupling on the S=1/2S=1/2S=1/2 Heisenberg model on the kagome lattice. For the pure spin model, there is increasing evidence that the low-energy properties can be correctly described by a Dirac spin liquid, in which spinons with a conical dispersion are coupled to emergent gauge fields. Within this scenario, the ground-state wave function is well approximated by a Gutzwiller-projected fermionic state [Ran, Hermele, Lee, and Wen, Phys. Rev. Lett. 98, 117205 (2007)]. However, the existence of U(1)U(1)U(1) gauge fields may naturally lead to instabilities when small perturbations are included. Phonons are ubiquitous in real materials and may play a relevant role in the determination of the actual physical properties of the kagome antiferromagnet. Therefore, we perform a step forward in this direction, including phonon degrees of freedom (at the quantum level) and applying a variational approach based upon Gutzwiller-projected fermionic Ans\"atze. Our results suggest that the Dirac spin liquid is stable for small spin-phonon couplings, while valence-bond solids are obtained at large couplings. Even though different distortions can be induced by the spin-phonon interaction, the general aspect is that the energy is lowered by maximizing the density of perfect hexagons in the dimerization pattern.
We investigate the ground state properties of the kagome lattice t - J model at low doping by variational Monte Carlo calculations. The resulting state possesses an interesting balance of spin exchange and kinetic exchange through the building blocks of stars which are linked by triangles and their internal hexagons. While the spin exchange is taking place mainly on the stars, hopping is favored on the hexagons. There is a density modulation, resulting in the holes having an effective static contribution. From this observation, how holes lead to dimerization in this model and why a particular valence bond crystal pattern is formed can be understood. Furthermore, we argue the optimal doping for this state. We discuss our result in connection with static impurities, and show the likely relevance to the diluted kagome lattice Heisenberg model, describing actual compounds.
The doping of quantum Heisenberg antiferromagnets on the kagome lattice by nonmagnetic impurities is investigated within the framework of a generalized quantum dimer model (QDM) describing (a) the valence-bond crystal (VBC), (b) the dimer liquid, and (c) the critical region on equal footing. Following the approach by Ralko et al. [Phys. Rev. Lett. 101, 117204 (2008)] for the square and triangular lattices, we introduce the (minimal) extension of the QDM on the kagome lattice to account for spontaneous creation of mobile S=1/2S=1/2S=1/2 spinons at finite magnetic field. Modulations of the dimer density (at zero or finite magnetic field) and of the local field-induced magnetization in the vicinity of impurities are computed using Lanczos exact diagonalization techniques on small clusters (48 and 75 sites). The VBC is clearly revealed from its pinning by impurities, while, in the dimer liquid, crystallization around impurities involves only two neighboring dimers. We also find that a next-nearest-neighbor ferromagnetic coupling favors VBC order. Unexpectedly, a small-size spinon-impurity bound state appears in some region of the dimer-liquid phase. In contrast, in the VBC phase the spinon delocalizes within a large region around the impurity, revealing the weakness of the VBC confining potential. Lastly, we observe that an impurity concentration as small as 4% enhances dimerization substantially. These results are confronted to the valence-bond glass scenario [R. R. P. Singh, Phys. Rev. Lett. 104, 177203 (2010)] and implications to the interpretation of the nuclear magnetic resonance spectra of the Herbertsmithite compound are outlined.
In this paper, we do a complete classification of valence-bond crystals (VBCs) on the kagomé lattice based on general arguments of symmetry only and thus identify many new VBCs for different unit cell sizes. For the spin-1/2 Heisenberg antiferromagnet, we study the relative energetics of competing gapless spin liquids (SLs) and VBC phases within the class of Gutzwiller-projected fermionic wave functions using variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. By using a state-of-the-art optimization method, we conclusively show that the U(1) Dirac SL is remarkably stable towards dimerizing into all 6-, 12- and 36-site unit cell VBCs. This stability is also preserved on addition of a next-nearest-neighbor super-exchange coupling of both antiferromagnetic and ferromagnetic (FM) type. However, we find that a 36-site unit cell VBC is stabilized on addition of a very small next-nearest-neighbor FM super-exchange coupling, i.e. | J 2 | ≈ 0.045, and this VBC is the same in terms of space-group symmetry as that obtained in an effective quantum dimer model study. It breaks reflection symmetry, has a nontrivial flux pattern and is a strong dimerization of the uniform RVB SL.
It has long been believed that doped quantum spin liquids (QSLs) can give rise to fascinating quantum phases, including the possibility of high-temperature superconductivity (SC) as proposed by P. W. Anderson's resonating valence bond (RVB) scenario. The Kagome lattice ttt-JJJ model is known to exhibit spin liquid behavior at half-filling, making it an ideal system for studying the properties of doped QSL. In this study, we employ the fermionic projected entangled simplex state (PESS) method to investigate the ground state properties of the Kagome lattice ttt-JJJ model with t/J=3.0t/J = 3.0t/J=3.0. Our results reveal a phase transition from charge density wave (CDW) states to uniform states around a critical doping level δc≈0.27δ_c \approx 0.27δc​≈0.27. Within the CDW phase, we observe different types of Wigner crystal (WC) formulated by doped holes that are energetically favored. As we enter the uniform phase, a non-Fermi liquid (NFL) state emerges within the doping range 0.27<δ<0.320.27 < δ< 0.320.27<δ<0.32, characterized by an exponential decay of all correlation functions. With further hole doping, we discover the appearance of a pair density wave (PDW) state within a narrow doping region 0.32<δ<1/30.32 < δ< 1/30.32<δ<1/3. We also discuss the potential experimental implications of our findings.