Chern-Simons theory of the magnetization plateaus of the spin-1/2 quantum XXZ Heisenberg model on Kagome Lattice

Abstract

Frustrated spin systems on Kagome lattices have long been considered to be a promising candidate for realizing exotic spin liquid phases. Recently, there has been a lot of renewed interest in these systems with the discovery of materials such as Volborthite and Herbertsmithite that have Kagome like structures. In the presence of an external magnetic field, these frustrated systems can give rise to magnetization plateaus of which the plateau at $m=\frac{1}{3}$ is considered to be the most prominent. Here we study the problem of the antiferromagnetic spin-1/2 quantum XXZ Heisenberg model on a Kagome lattice by using a Jordan-Wigner transformation that maps the spins onto a problem of fermions coupled to a Chern-Simons gauge field. This mapping relies on being able to define a consistent Chern-Simons term on the lattice. Using a recently developed method to rigorously extend the Chern-Simons term to the frustrated Kagome lattice we can now formalize the Jordan-Wigner transformation on the Kagome lattice. We then discuss the possible phases that can arise at the mean-field level from this mapping and focus specifically on the case of $\frac{1}{3}$-filling ($m=\frac{1}{3}$ plateau) and analyze the effects of fluctuations in our theory. We show that in the regime of $XY$ anisotropy the ground state at the $1/3$ plateau is equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction $1/2$ and that at the $5/9$ plateau it is equivalent to the first bosonic Jain daughter state at filling fraction $2/3$.

arXiv comment21 pages, 4 figures (fig. 3 is new) and 35 references (ref. 28 and 35 are new). Improved presentation, new section on the Jordan-Wigner transformation (II E), old sections II and IV are merged into section III, expanded conclusions section

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• Topological orders and Edge excitations in FQH states
• Projected wavefunction study of Spin-1/2 Heisenberg model on the Kagome lattice
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Cited by in corpus (34)

• Spontaneous fractional Chern insulators in transition metal dichalcogenides Moire superlattices
• Spin-S Kagome quantum antiferromagnets in a field with tensor networks
• Band structure engineering of ideal fractional Chern insulators
• Coupled wire construction of chiral spin liquids
• Nature of chiral spin liquids on the kagome lattice
• Chiral Spin Liquids in Arrays of Spin Chains
• A discretized Chern-Simons gauge theory on arbitrary graphs
• Haldane-Hubbard Mott Insulator: From Tetrahedral Spin Crystal to Chiral Spin Liquid
• The mother of all states of the kagome quantum antiferromagnet
• Frustration-induced emergent Hilbert space fragmentation
• Chiral spin liquids on the kagome Lattice
• Emergence of Chiral Spin Liquids via Quantum Melting of Non-Coplanar Magnetic Orders
• Chern-Simons Composite Fermion Theory of Fractional Chern Insulators
• Lattice effects on Laughlin wave functions and parent Hamiltonians
• Interaction-driven fractional quantum Hall state of hard-core bosons on kagome lattice at one-third filling
• Fermionization of Bosons in a Flat Band
• The spin-1/2 Kagome XXZ model in a field: competition between lattice nematic and solid orders
• Resonating quantum three-coloring wavefunctions for the kagome quantum antiferromagnet
• Staircase of crystal phases of hard-core bosons on the Kagome lattice
• Coexistence of non-Abelian chiral spin liquid and magnetic order in a spin-1 antiferromagnet
• Numerical evidence for a chiral spin liquid in the XXZ antiferromagnetic Heisenberg model on the kagome lattice at $m=\frac{2}{3}$ magnetization
• Correlated spin liquids in the quantum kagome antiferromagnet at finite field: a renormalisation group analysis
• Topological phase transiton of anisotropic XY model with Dzyaloshinskii-Moriya interaction
• Lattice realization of compact $U(1)$ Chern-Simons theory with exact 1-symmetries
• Is the composite fermion state of Graphene a doped Chern insulator?
• Nematic quantum liquid crystals of bosons in frustrated lattices
• Anatomy of Z2 fluxes in anyon Fermi liquids and Bose condensates
• Magnetisation plateaux of the quantum pyrochlore Heisenberg antiferromagnet
• From Fractional Quantum Anomalous Hall Smectics to Polar Smectic Metals: Nontrivial Interplay Between Electronic Liquid Crystal Order and Topological Order in Correlated Topological Flat Bands
• Properties and application of the SO(3) Majorana representation of spin: equivalence with the Jordan-Wigner transformation and exact $Z_2$ gauge theories for spin models
• Protection against Spin Gap in 2-d Insulating Antiferromagnets with a Chern-Simons Term
• Jordan-Wigner fermionization of quantum spin systems on arbitrary 2D lattices: A mutual Chern-Simons approach
• Jordan-Wigner composite-fermion liquids in 2D quantum spin-ice
• Microscopic theory of field-tuned topological transitions in the Kitaev honeycomb model