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arXiv:2110.08198 · Physical review. B./Physical review. B

Pinwheel valence bond crystal ground state of the spin- 1 2 Heisenberg antiferromagnet on the shuriken lattice

Nikita Astrakhantsev, Francesco Ferrari, Nils Niggemann, Tobias Müller, Aishwarya Chauhan, Augustine Kshetrimayum, Pratyay Ghosh, Nicolas Regnault, Ronny Thomale, Johannes Reuther, Titus Neupert, Yasir Iqbal

Published Dec 22, 2021• 37 citations
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Abstract

Quantum spin models on two-dimensional corner-sharing triangular arrangements, e.g., the kagome lattice, are fertile playgrounds for realizing exotic phases. Following the recent report of a gapless spin liquid in KCu${}_{6}$AlBiO${}_{4}$(SO${}_{4}$)${}_{5}$Cl, wherein the Cu${}^{2+}$ spin-\textonehalf{} ions form a perfect two-dimensional shuriken lattice, the authors employ state-of-the-art quantum many-body approaches to reveal a pinwheel valence-bond-crystal ground state of the isotropic Heisenberg antiferromagnet. This work thus points to the important role of spatial anisotropy in triggering the gapless spin liquid, in contrast to the kagome material herbertsmithite.

DOI
10.1103/physrevb.104.l220408

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Valence bond solid phases on deformed kagome lattices: Application to Rb2Cu3SnF12

Bohm‐Jung Yang, Yong Baek Kim

Motivated by a recent experiment on ${\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}$, where spin-1/2 ${\text{Cu}}^{2+}$ moments reside on the layers of kagome-like lattices, we investigate quantum ground states of the antiferromagnetic Heisenberg model on a series of deformed kagome lattices. The deformation is characterized by a weaker exchange coupling $\ensuremath{\alpha}J$ on certain lattice links appropriate for ${\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}$ with $\ensuremath{\alpha}=1$ corresponding to the ideal kagome lattice. In particular, we study possible valence bond solid phases using the perturbation theory around isolated dimer limits, dimer series expansion, and self-consistent bond operator mean-field theory. It is shown that the valence bond solid phase with a 36-site unit cell of the ideal kagome lattice is quite sensitive to a small lattice distortion as the kind discovered in ${\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}$. As a result, we find that a more likely quantum ground state in ${\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}$ is the valence bond solid phase with a 12-site unit cell, where six dimers form a pinwheel structure, leading to strong modification of the elementary triplet and singlet excitation spectra in the deformed kagome lattices.

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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.

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The spin-$\frac{1}{2}$ Heisenberg model on the kagome lattice, which is closely realized in layered Mott insulators such as ${\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}{\mathrm{Cl}}_{2}$, is one of the oldest and most enigmatic spin-$\frac{1}{2}$ lattice models. While the numerical evidence has accumulated in favor of a quantum spin liquid, the debate is still open as to whether it is a ${Z}_{2}$ spin liquid with very short-range correlations (some kind of resonating valence bond spin liquid), or an algebraic spin liquid with power-law correlations. To address this issue, we have pushed the program started by Rokhsar and Kivelson in their derivation of the effective quantum dimer model description of Heisenberg models to unprecedented accuracy for the spin-$\frac{1}{2}$ kagome, by including all the most important virtual singlet contributions on top of the orthogonalization of the nearest-neighbor valence bond singlet basis. Quite remarkably, the resulting picture is a competition between a ${Z}_{2}$ spin liquid and a diamond valence bond crystal with a 12-site unit cell, as in the density-matrix renormalization group simulations of Yan et al. Furthermore, we found that, on cylinders of finite diameter $d$, there is a transition between the ${Z}_{2}$ spin liquid at small $d$ and the diamond valence bond crystal at large $d$, the prediction of the present microscopic description for the two-dimensional lattice. These results show that, if the ground state of the spin-$\frac{1}{2}$ kagome antiferromagnet can be described by nearest-neighbor singlet dimers, it is a diamond valence bond crystal, and, a contrario, that, if the system is a quantum spin liquid, it has to involve long-range singlets, consistent with the algebraic spin liquid scenario.

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Emergent quasi-one-dimensionality in a kagome magnet: A simple route to complexity

Shou-Shu Gong, Wei Kang Zhu, Kun Yang +3 more

We study the ground-state phase diagram of the quantum $\text{spin-}1/2$ Heisenberg model on the kagome lattice with first- $({J}_{1}<0)$, second- $({J}_{2}<0)$, and third-neighbor interactions $({J}_{d}>0)$ by means of analytical low-energy field theory and numerical density-matrix renormalization group (DMRG) studies. The results offer a consistent picture of the ${J}_{d}\text{-dominant}$ regime in terms of three sets of spin chains weakly coupled by the ferromagnetic interchain interactions ${J}_{1,2}$. When either ${J}_{1}$ or ${J}_{2}$ is much stronger than the other one, the model is found to support one of two cuboctohedral phases, cuboc1, and cuboc2. These cuboc states host noncoplanar long-ranged magnetic order and possess finite scalar spin chirality. However, in the compensated regime ${J}_{1}\ensuremath{\simeq}{J}_{2}$, a valence bond crystal phase emerges between the two cuboc phases. We find excellent agreement between an analytical theory based on coupled spin chains and unbiased DMRG calculations, including at a very detailed level of comparison of the structure of the valence bond crystal state. To our knowledge, this is the first such comprehensive understanding of a highly frustrated two-dimensional quantum antiferromagnet. We find no evidence of either the one-dimensional gapless spin liquid or the chiral spin liquids, which were previously suggested by parton mean-field theories.

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Spin- 12 Heisenberg antiferromagnet on an anisotropic kagome lattice

P. H. Y. Li, R. F. Bishop, C. E. Campbell +3 more

We use the coupled-cluster method to study the zero-temperature properties of an extended two-dimensional Heisenberg antiferromagnet formed from spin-$\frac{1}{2}$ moments on an infinite spatially anisotropic kagome lattice of corner-sharing isosceles triangles, with nearest-neighbor bonds only. The bonds have exchange constants ${J}_{1}>0$ along two of the three lattice directions and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$ along the third. In the classical limit, the ground-state (GS) phase for $\ensuremath{\kappa}<\frac{1}{2}$ has collinear ferrimagnetic (N\'eel${}^{\ensuremath{'}}$) order where the ${J}_{2}$-coupled chain spins are ferromagnetically ordered in one direction with the remaining spins aligned in the opposite direction, while for $\ensuremath{\kappa}>\frac{1}{2}$ there exists an infinite GS family of canted ferrimagnetic spin states, which are energetically degenerate. For the spin-$\frac{1}{2}$ case, we find that quantum analogs of both these classical states continue to exist as stable GS phases in some regions of the anisotropy parameter $\ensuremath{\kappa}$, namely, for $0<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}$ for the N\'eel${}^{\ensuremath{'}}$ state and for (at least part of) the region $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}$ for the canted phase. However, they are now separated by a paramagnetic phase without either sort of magnetic order in the region ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$, which includes the isotropic kagome point $\ensuremath{\kappa}=1$ where the stable GS phase is now believed to be a topological (${\mathbb{Z}}_{2}$) spin liquid. Our best numerical estimates are ${\ensuremath{\kappa}}_{{c}_{1}}=0.515\ifmmode\pm\else\textpm\fi{}0.015$ and ${\ensuremath{\kappa}}_{{c}_{2}}=1.82\ifmmode\pm\else\textpm\fi{}0.03$.

arXiv:1208.3989•Physical Review B•Dec 4, 2012•29 citations
Quantum spin models on two-dimensional corner-sharing triangular arrangements, e.g., the kagome lattice, are fertile playgrounds for realizing exotic phases. Following the recent report of a gapless spin liquid in KCu6{}_{6}6​AlBiO4{}_{4}4​(SO4{}_{4}4​)5{}_{5}5​Cl, wherein the Cu2+{}^{2+}2+ spin-\textonehalf{} ions form a perfect two-dimensional shuriken lattice, the authors employ state-of-the-art quantum many-body approaches to reveal a pinwheel valence-bond-crystal ground state of the isotropic Heisenberg antiferromagnet. This work thus points to the important role of spatial anisotropy in triggering the gapless spin liquid, in contrast to the kagome material herbertsmithite.
We study the ground-state phase diagram of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the square-kagome lattice using infinite projected entangled-pair states (iPEPS). By systematically varying the ratio of exchange couplings on triangular and square plaquettes, we establish a complete quantum phase diagram in the thermodynamic limit. In the intermediate-coupling regime, we identify four distinct nonmagnetic phases that are unambiguously characterized as valence-bond crystals (VBCs) by their symmetry-inequivalent patterns of strong and weak spin-spin correlations. These include a plaquette crossed-dimer VBC, a loop-six VBC stabilized around the isotropic point, a generalized pinwheel VBC with reduced rotational symmetry, and a decorated loop-six VBC proximate to ferrimagnetic order. We determine the phase boundaries using a combination of bond-resolved correlation functions, entanglement entropy, and magnetization. For transitions not accompanied by sharp entanglement signatures, we extract the spin gap from finite-field simulations, allowing us to distinguish gapped and gapless VBC phases. At larger coupling ratios, the system undergoes transitions into imperfect and perfect ferrimagnetic states. Our results resolve long-standing ambiguities in the square-kagome Heisenberg model and provide a quantitatively reliable reference for future theoretical and experimental studies of frustrated quantum magnets.
Motivated by a recent experiment on Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​, where spin-1/2 Cu2+{\text{Cu}}^{2+}Cu2+ moments reside on the layers of kagome-like lattices, we investigate quantum ground states of the antiferromagnetic Heisenberg model on a series of deformed kagome lattices. The deformation is characterized by a weaker exchange coupling $\ensuremath{\alpha}J$ on certain lattice links appropriate for Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​ with $\ensuremath{\alpha}=1$ corresponding to the ideal kagome lattice. In particular, we study possible valence bond solid phases using the perturbation theory around isolated dimer limits, dimer series expansion, and self-consistent bond operator mean-field theory. It is shown that the valence bond solid phase with a 36-site unit cell of the ideal kagome lattice is quite sensitive to a small lattice distortion as the kind discovered in Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​. As a result, we find that a more likely quantum ground state in Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​ is the valence bond solid phase with a 12-site unit cell, where six dimers form a pinwheel structure, leading to strong modification of the elementary triplet and singlet excitation spectra in the deformed kagome lattices.
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.
The spin-12\frac{1}{2}21​ Heisenberg model on the kagome lattice, which is closely realized in layered Mott insulators such as ZnCu3(OH)6Cl2{\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}{\mathrm{Cl}}_{2}ZnCu3​(OH)6​Cl2​, is one of the oldest and most enigmatic spin-12\frac{1}{2}21​ lattice models. While the numerical evidence has accumulated in favor of a quantum spin liquid, the debate is still open as to whether it is a Z2{Z}_{2}Z2​ spin liquid with very short-range correlations (some kind of resonating valence bond spin liquid), or an algebraic spin liquid with power-law correlations. To address this issue, we have pushed the program started by Rokhsar and Kivelson in their derivation of the effective quantum dimer model description of Heisenberg models to unprecedented accuracy for the spin-12\frac{1}{2}21​ kagome, by including all the most important virtual singlet contributions on top of the orthogonalization of the nearest-neighbor valence bond singlet basis. Quite remarkably, the resulting picture is a competition between a Z2{Z}_{2}Z2​ spin liquid and a diamond valence bond crystal with a 12-site unit cell, as in the density-matrix renormalization group simulations of Yan et al. Furthermore, we found that, on cylinders of finite diameter ddd, there is a transition between the Z2{Z}_{2}Z2​ spin liquid at small ddd and the diamond valence bond crystal at large ddd, the prediction of the present microscopic description for the two-dimensional lattice. These results show that, if the ground state of the spin-12\frac{1}{2}21​ kagome antiferromagnet can be described by nearest-neighbor singlet dimers, it is a diamond valence bond crystal, and, a contrario, that, if the system is a quantum spin liquid, it has to involve long-range singlets, consistent with the algebraic spin liquid scenario.
We study the ground-state phase diagram of the quantum spin-1/2\text{spin-}1/2spin-1/2 Heisenberg model on the kagome lattice with first- (J1<0)({J}_{1}<0)(J1​<0), second- (J2<0)({J}_{2}<0)(J2​<0), and third-neighbor interactions (Jd>0)({J}_{d}>0)(Jd​>0) by means of analytical low-energy field theory and numerical density-matrix renormalization group (DMRG) studies. The results offer a consistent picture of the Jd-dominant{J}_{d}\text{-dominant}Jd​-dominant regime in terms of three sets of spin chains weakly coupled by the ferromagnetic interchain interactions J1,2{J}_{1,2}J1,2​. When either J1{J}_{1}J1​ or J2{J}_{2}J2​ is much stronger than the other one, the model is found to support one of two cuboctohedral phases, cuboc1, and cuboc2. These cuboc states host noncoplanar long-ranged magnetic order and possess finite scalar spin chirality. However, in the compensated regime ${J}_{1}\ensuremath{\simeq}{J}_{2}$, a valence bond crystal phase emerges between the two cuboc phases. We find excellent agreement between an analytical theory based on coupled spin chains and unbiased DMRG calculations, including at a very detailed level of comparison of the structure of the valence bond crystal state. To our knowledge, this is the first such comprehensive understanding of a highly frustrated two-dimensional quantum antiferromagnet. We find no evidence of either the one-dimensional gapless spin liquid or the chiral spin liquids, which were previously suggested by parton mean-field theories.
We use the coupled-cluster method to study the zero-temperature properties of an extended two-dimensional Heisenberg antiferromagnet formed from spin-12\frac{1}{2}21​ moments on an infinite spatially anisotropic kagome lattice of corner-sharing isosceles triangles, with nearest-neighbor bonds only. The bonds have exchange constants J1>0{J}_{1}>0J1​>0 along two of the three lattice directions and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$ along the third. In the classical limit, the ground-state (GS) phase for $\ensuremath{\kappa}<\frac{1}{2}$ has collinear ferrimagnetic (N\'eel${}^{\ensuremath{'}}$) order where the J2{J}_{2}J2​-coupled chain spins are ferromagnetically ordered in one direction with the remaining spins aligned in the opposite direction, while for $\ensuremath{\kappa}>\frac{1}{2}$ there exists an infinite GS family of canted ferrimagnetic spin states, which are energetically degenerate. For the spin-12\frac{1}{2}21​ case, we find that quantum analogs of both these classical states continue to exist as stable GS phases in some regions of the anisotropy parameter $\ensuremath{\kappa}$, namely, for $0<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}$ for the N\'eel${}^{\ensuremath{'}}$ state and for (at least part of) the region $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}$ for the canted phase. However, they are now separated by a paramagnetic phase without either sort of magnetic order in the region ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$, which includes the isotropic kagome point $\ensuremath{\kappa}=1$ where the stable GS phase is now believed to be a topological (Z2{\mathbb{Z}}_{2}Z2​) spin liquid. Our best numerical estimates are ${\ensuremath{\kappa}}_{{c}_{1}}=0.515\ifmmode\pm\else\textpm\fi{}0.015$ and ${\ensuremath{\kappa}}_{{c}_{2}}=1.82\ifmmode\pm\else\textpm\fi{}0.03$.