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Physical review. B./Physical review. B

1-form symmetry and the selection rule of the plaquette valence bond solid phase on the kagome lattice

Zijian Xiong

Published Nov 6, 2023• 3 citations
Publisher

Abstract

We study the plaquette valence bond solid phase in an XXZ-type spin-1/2 model in the kagome lattice. The low-energy theory for this phase is a U(1) lattice gauge theory on the honeycomb lattice. We find that there is an emergent 1-form U(1) symmetry at low energy and there is a mixed anomaly. We also show that this 1-form symmetry restricts the longitudinal dynamical structure factor and leads to the selection rule relating to the vanishing intensity along some high-symmetry momentum paths (e.g., the $\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}M$ path). Since this emergent 1-form symmetry is robust in the presence of local UV perturbation, the selection rule will also apply to the model which is obtained by perturbing the classical limit of our model.

DOI
10.1103/physrevb.108.195107

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We study the plaquette valence bond solid phase in an XXZ-type spin-1/2 model in the kagome lattice. The low-energy theory for this phase is a U(1) lattice gauge theory on the honeycomb lattice. We find that there is an emergent 1-form U(1) symmetry at low energy and there is a mixed anomaly. We also show that this 1-form symmetry restricts the longitudinal dynamical structure factor and leads to the selection rule relating to the vanishing intensity along some high-symmetry momentum paths (e.g., the \ensuremathΓ\ensuremath-M\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}M\ensuremathΓ\ensuremath-M path). Since this emergent 1-form symmetry is robust in the presence of local UV perturbation, the selection rule will also apply to the model which is obtained by perturbing the classical limit of our model.
We study the plaquette valence bond solid phase in a XXZ type spin-1/2 model in the kagome lattice. The low energy theory for this phase is a U(1) lattice gauge theory on the honeycomb lattice. We find that there is an emergent 1-form U(1) symmetry in low energy, and there is a mixed anomaly. We also show that this 1-form symmetry constraints the longitudinal dynamical structure factor and leads to the selection rule relating to the vanishing intensity along some high symmetry momentum paths (e.g. Γ−MΓ-MΓ−M path). We point out that this emergent 1-form symmetry is robust against the translation symmetry preserving UV perturbation, thus the selection rule will also apply to the model which is obtained by perturbing the classical limit of our model.
While the existence of a chiral spin liquid (CSL) on a class of spin-1/2 kagome antiferromagnets is by now well established numerically, a controlled theoretical path from the lattice model leading to a low-energy topological field theory is still lacking. This we provide via an explicit construction starting from reformulating a microscopic model for a CSL as a lattice gauge theory and deriving the low-energy form of its continuum limit. A crucial ingredient is the realization that the bosonic spinons of the gauge theory exhibit a U(1) symmetry protected topological (SPT) phase, which upon promoting its U(1) global symmetry to a local gauge structure ("gauging"), yields the CSL. We suggest that such an explicit lattice-based construction involving gauging of a SPT phase can be applied more generally to understand topological spin liquids.
Using large-scale quantum Monte Carlo simulations we show that a spin-12XXZ\frac{1}{2}\phantom{\rule{1em}{0ex}}XXZ21​XXZ model on a two-dimensional anisotropic kagome lattice exhibits a tripartite entangled plaquette state that preserves all of the Hamiltonian symmetries. It is connected via phase boundaries to a ferromagnet and a valence-bond solid that break U(1)U(1)U(1) and lattice translation symmetries, respectively. We study the phase diagram of the model in detail, in particular the transitions to the tripartite entangled plaquette state, which are consistent with conventional order-disorder transitions. Our results can be interpreted as a description of the charge sector dynamics of a Hubbard model applied to the spin liquid candidate LiZn2Mo3O8{\mathrm{LiZn}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{8}LiZn2​Mo3​O8​, as well as a model of strongly correlated bosonic atoms loaded onto highly tunable trimerized optical kagome lattices.
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.
We map out the ground state phase diagram of the spin-12\frac{1}{2}21​ isotropic Kekul\'e-Kitaev model on the honeycomb lattice in the presence of the Heisenberg exchange couplings. Our study relies on large-scale tensor network simulations based on a graph-based projected entangled pair state approach and simple tensor update in the thermodynamic limit. We find that, on top of the quantum spin liquid and conventional magnetically ordered phases which are typical of the Kitaev-Heisenberg model, the Kekul\'e-Heisenberg phase diagram hosts two plaquette valence bond solid (VBS) phases with vanishing magnetic order. While the VBS phases preserve the symmetries of the original Hamiltonian, they differ markedly from the Kitaev spin liquid by having decorated plaquette ordering which is distinguished by a plaquette order parameter.
We study the plaquette valence-bond solid phase of the spin-1/21/21/2J1{J}_{1}J1​-J2{J}_{2}J2​ antiferromagnet Heisenberg model on the square lattice within the bond-operator theory. We start by considering four S=1/2S=1/2S=1/2 spins on a single plaquette and determine the bond operator representation for the spin operators in terms of singlet, triplet, and quintet boson operators. The formalism is then applied to the J1{J}_{1}J1​-J2{J}_{2}J2​ model and an effective interacting boson model in terms of singlets and triplets is derived. The effective model is analyzed within the harmonic approximation and the previous results of Zhitomirsky and Ueda [Phys. Rev. B 54, 9007 (1996)] are recovered. By perturbatively including cubic (triplet-triplet-triplet and singlet-triplet-triplet) and quartic interactions, we find that the plaquette valence-bond solid phase is stable within the parameter region 0.34<J2/J1<0.590.34<{J}_{2}/{J}_{1}<0.590.34<J2​/J1​<0.59, which is narrower than the harmonic one. Differently from the harmonic approximation, the excitation gap vanishes at both critical couplings J2=0.34J1{J}_{2}=0.34{J}_{1}J2​=0.34J1​ and J2=0.59J1{J}_{2}=0.59{J}_{1}J2​=0.59J1​. Interestingly, for J2<0.48J1{J}_{2}<0.48{J}_{1}J2​<0.48J1​, the excitation gap corresponds to a singlet-triplet excitation at the $\ensuremath{\Gamma}$ point while, for J2>0.48J1{J}_{2}>0.48{J}_{1}J2​>0.48J1​, it is related to a singlet-singlet excitation at the $\mathbf{X}=(\ensuremath{\pi}/2,0)$ point of the tetramerized Brillouin zone.