We present an effective ${\mathbb{Z}}_{2}$ gauge theory that captures various competing phases in spin-1/2 kagome lattice antiferromagnets: the topological ${\mathbb{Z}}_{2}$ spin liquid (SL) phase, and the 12-site and 36-site valence bond solid (VBS) phases. Our effective theory is a generalization of the recent ${\mathbb{Z}}_{2}$ gauge theory proposed for SL phases by Wan and Tchernyshyov. In particular, we investigate possible VBS phases that arise from vison condensations in the SL. In addition to the 12-site and 36-site VBS phases, there exists 6-site VBS that is closely related to the symmetry-breaking valence bond modulation patterns observed in the recent density matrix renormalization group simulations. We find that our results have remarkable consistency with a previous study using a different ${\mathbb{Z}}_{2}$ gauge theory. Motivated by the lattice geometry in the recently reported vanadium oxyfluoride kagome antiferromagnet, our gauge theory is extended to incorporate lowered symmetry by inequivalent up and down triangles. We investigate effects of this anisotropy on the 12-site, 36-site, and 6-site VBS phases. The 12-site VBS is stable to anisotropy while the 36-site VBS undergoes severe dimer melting. Interestingly, any analog of the 6-site VBS is not found in this approach. We discuss the implications of these findings and also compare the results with a different type of ${\mathbb{Z}}_{2}$ gauge theory used in previous studies.We present an effective ${\mathbb{Z}}_{2}$ gauge theory that captures various competing phases in spin-1/2 kagome lattice antiferromagnets: the topological ${\mathbb{Z}}_{2}$ spin liquid (SL) phase, and the 12-site and 36-site valence bond solid (VBS) phases. Our effective theory is a generalization of the recent ${\mathbb{Z}}_{2}$ gauge theory proposed for SL phases by Wan and Tchernyshyov. In particular, we investigate possible VBS phases that arise from vison condensations in the SL. In addition to the 12-site and 36-site VBS phases, there exists 6-site VBS that is closely related to the symmetry-breaking valence bond modulation patterns observed in the recent density matrix renormalization group simulations. We find that our results have remarkable consistency with a previous study using a different ${\mathbb{Z}}_{2}$ gauge theory. Motivated by the lattice geometry in the recently reported vanadium oxyfluoride kagome antiferromagnet, our gauge theory is extended to incorporate lowered symmetry by inequivalent up and down triangles. We investigate effects of this anisotropy on the 12-site, 36-site, and 6-site VBS phases. The 12-site VBS is stable to anisotropy while the 36-site VBS undergoes severe dimer melting. Interestingly, any analog of the 6-site VBS is not found in this approach. We discuss the implications of these findings and also compare the results with a different type of ${\mathbb{Z}}_{2}$ gauge theory used in previous studies.

arXiv:1507.04747Physical Review BNov 30, 201519 citations

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.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.

arXiv:0902.1752Physical Review BJun 12, 200919 citations

Quantum Simulation of a $\mathbb {Z}_2$ -Lattice Gauge TheoryQuantum Simulation of a $$\mathbb {Z}_2$$-Lattice Gauge Theory

Sebastian Saner

Springer thesesJan 1, 20261 citation

_{\mathrm{2}}

gauge theory for valence bond solids on the kagome lattice

We present an effective

{\mathbb{Z}}_{2}

gauge theory that captures various competing phases in spin-1/2 kagome lattice antiferromagnets: the topological

{\mathbb{Z}}_{2}

spin liquid (SL) phase, and the 12-site and 36-site valence bond solid (VBS) phases. Our effective theory is a generalization of the recent

{\mathbb{Z}}_{2}

gauge theory proposed for SL phases by Wan and Tchernyshyov. In particular, we investigate possible VBS phases that arise from vison condensations in the SL. In addition to the 12-site and 36-site VBS phases, there exists 6-site VBS that is closely related to the symmetry-breaking valence bond modulation patterns observed in the recent density matrix renormalization group simulations. We find that our results have remarkable consistency with a previous study using a different

{\mathbb{Z}}_{2}

gauge theory. Motivated by the lattice geometry in the recently reported vanadium oxyfluoride kagome antiferromagnet, our gauge theory is extended to incorporate lowered symmetry by inequivalent up and down triangles. We investigate effects of this anisotropy on the 12-site, 36-site, and 6-site VBS phases. The 12-site VBS is stable to anisotropy while the 36-site VBS undergoes severe dimer melting. Interestingly, any analog of the 6-site VBS is not found in this approach. We discuss the implications of these findings and also compare the results with a different type of

{\mathbb{Z}}_{2}

gauge theory used in previous studies.

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.

Quantum Simulation of a

\mathbb {Z}_2

-Lattice Gauge Theory