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Physics

Closing in on the Kagome Magnet

Bryan K. Clark

Published Mar 29, 2017• 4 citations
Publisher

Abstract

The defining problem in frustrated quantum magnetism, the ground state of the nearest-neighbor S ¼ 1=2 antiferromagnetic Heisenberg model on the kagome lattice, has defied all theoretical and numerical methods employed to date.We apply the formalism of tensor-network states, specifically the method of projected entangled simplex states, which combines infinite system size with a correct accounting for multipartite entanglement.By studying the ground-state energy, the finite magnetic order appearing at finite tensor bond dimensions, and the effects of a next-nearest-neighbor coupling, we demonstrate that the ground state is a gapless spin liquid.We discuss the comparison with other numerical studies and the physical interpretation of this result.

DOI
10.1103/physics.10.33

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The defining problem in frustrated quantum magnetism, the ground state of the nearest-neighbor S ¼ 1=2 antiferromagnetic Heisenberg model on the kagome lattice, has defied all theoretical and numerical methods employed to date.We apply the formalism of tensor-network states, specifically the method of projected entangled simplex states, which combines infinite system size with a correct accounting for multipartite entanglement.By studying the ground-state energy, the finite magnetic order appearing at finite tensor bond dimensions, and the effects of a next-nearest-neighbor coupling, we demonstrate that the ground state is a gapless spin liquid.We discuss the comparison with other numerical studies and the physical interpretation of this result.
The defining problem in frustrated quantum magnetism, the ground state of the nearest-neighbor S=1/2 antiferromagnetic Heisenberg model on the kagome lattice, has defied all theoretical and numerical methods employed to date. We apply the formalism of tensor-network states, specifically the method of projected entangled simplex states, which combines infinite system size with a correct accounting for multipartite entanglement. By studying the ground-state energy, the finite magnetic order appearing at finite tensor bond dimensions, and the effects of a next-nearest-neighbor coupling, we demonstrate that the ground state is a gapless spin liquid. We discuss the comparison with other numerical studies and the physical interpretation of this result.
We use the density matrix renormalization group to perform accurate calculations of the ground state of the nearest-neighbor quantum spin S = 1/2 Heisenberg antiferromagnet on the kagome lattice. We study this model on numerous long cylinders with circumferences up to 12 lattice spacings. Through a combination of very-low-energy and small finite-size effects, our results provide strong evidence that, for the infinite two-dimensional system, the ground state of this model is a fully gapped spin liquid.
We apply the symmetric tensor network state (TNS) to study the nearest-neighbor spin-1/2 antiferromagnetic Heisenberg model on the kagome lattice. Our method keeps track of the global and gauge symmetries in the TNS update procedure and in tensor renormalization group (TRG) calculations. We also introduce a very sensitive probe for the gap of the ground state---the modular matrices, which can also determine the topological order if the ground state is gapped. We find that the ground state of the Heisenberg model on the kagome lattice is a gapped spin liquid with the Z2{\mathbb{Z}}_{2}Z2​ topological order (or toric code type), which has a long correlation length $\ensuremath{\xi}\ensuremath{\sim}10$ unit cells. We justify that the TRG method can handle very large systems with thousands of spins. Such a long $\ensuremath{\xi}$ explains the gapless behaviors observed in simulations on smaller systems with less than 300 spins or shorter than the length of 10 unit cells. We also discuss experimental implications of the topological excitations encoded in our symmetric tensors.
We apply the symmetric tensor network state (TNS) to study the nearest-neighbor spin-1/2 antiferromagnetic Heisenberg model on the kagome lattice. Our method keeps track of the global and gauge symmetries in the TNS update procedure and in tensor renormalization group (TRG) calculations. We also introduce a very sensitive probe for the gap of the ground state—the modular matrices, which can also determine the topological order if the ground state is gapped. We find that the ground state of the Heisenberg model on the kagome lattice is a gapped spin liquid with the Z[subscript 2] topological order (or toric code type), which has a long correlation length ξ∼10 unit cells. We justify that the TRG method can handle very large systems with thousands of spins. Such a long ξ explains the gapless behaviors observed in simulations on smaller systems with less than 300 spins or shorter than the length of 10 unit cells. We also discuss experimental implications of the topological excitations encoded in our symmetric tensors.
We perform a systematic study of the antiferromagnetic Heisenberg model on the Husimi lattice using numerical tensor-network methods based on projected entangled simplex states. The nature of the ground state varies strongly with the spin quantum number SSS. For S=12S=\frac{1}{2}S=21​, it is an algebraic (gapless) quantum spin liquid. For S=1S=1S=1, it is a gapped, nonmagnetic state with spontaneous breaking of triangle symmetry (a trimerized simplex-solid state). For S=2S=2S=2, it is a simplex-solid state with a spin gap and no symmetry breaking; both integer-spin simplex-solid states are characterized by specific degeneracies in the entanglement spectrum. For S=32S=\frac{3}{2}S=23​, and indeed for all spin values $S\ensuremath{\ge}\frac{5}{2}$, the ground states have ${120}^{\ensuremath{\circ}}$ antiferromagnetic order. In a finite magnetic field, we find that, irrespective of the value of SSS, there is always a plateau in the magnetization at m=13m=\frac{1}{3}m=31​.
Gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids have been proposed as candidates for the ground state of the S=1/2S=1/2S=1/2 quantum antiferromagnet on the kagome lattice. We extend the use of projected entangled pair states to construct (on the cylinder) resonating valence bond (RVB) states including both nearest-neighbor and next-nearest-neighbor singlet bonds. Our ansatz---dubbed ``simplex spin liquid''---allows for an asymmetry between the two types of triangles (of order 22%--3%2 in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological Z2{\mathbb{Z}}_{2}Z2​ structure is still preserved and, by considering the presence or the absence of spinon and vison lines along an infinite cylinder, we explicitly construct four orthogonal RVB minimally entangled states. The spinon and vison coherence lengths are extracted from a finite size scaling with regard to the cylinder perimeter of the energy splittings of the four sectors and are found to be of the order of the lattice spacing. The entanglement spectrum of a partitioned (infinite) cylinder is found to be gapless, suggesting the occurrence, on a cylinder with real open boundaries, of gapless edge modes formally similar to Luttinger liquid (nonchiral) spin and charge modes. When inversion symmetry is spontaneously broken, the RVB spin liquid exhibits an extra Ising degeneracy, which might have been observed in recent exact diagonalization studies.