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arXiv:1508.00113 · Nature Communications

A kagome map of spin liquids from XXZ to Dzyaloshinskii–Moriya ferromagnet

Karim Essafi, Owen Benton, Ludovic D. C. Jaubert

Published Jan 22, 2016• 41 citations
arXivHTMLPDF

Abstract

Despite its deceptive simplicity, few concepts have more fundamental implications than chirality, from the therapeutic activity of drugs to the fundamental forces of nature. In magnetic materials, chirality gives rise to unconventional phenomena such as the anomalous Hall effect and multiferroicity, taking an enhanced flavour in the so-called spin-liquid phases where magnetic disorder prevails. Kagome systems sit at the crossroad of these ideas. Motivated by the recent synthesis of rare-earth kagome materials and the progresses in optical-lattice experiments, we bring together an entire network of spin liquids with anisotropic and Dzyaloshinskii-Moriya interactions. This network revolves around the Ising antiferromagnet and ends on (ferromagnetic) chiral spin liquids with spontaneously broken time-reversal symmetry. As for the celebrated Heisenberg antiferromagnet, it now belongs to a triad of equivalently disordered phases. The present work provides a unifying theory of kagome spin liquids with time-reversal invariant nearest-neighbour Hamiltonians.

DOI
10.1038/ncomms10297

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Despite its deceptive simplicity, few concepts have more fundamental implications than chirality, from the therapeutic activity of drugs to the fundamental forces of nature. In magnetic materials, chirality gives rise to unconventional phenomena such as the anomalous Hall effect and multiferroicity, taking an enhanced flavour in the so-called spin-liquid phases where magnetic disorder prevails. Kagome systems sit at the crossroad of these ideas. Motivated by the recent synthesis of rare-earth kagome materials and the progresses in optical-lattice experiments, we bring together an entire network of spin liquids with anisotropic and Dzyaloshinskii-Moriya interactions. This network revolves around the Ising antiferromagnet and ends on (ferromagnetic) chiral spin liquids with spontaneously broken time-reversal symmetry. As for the celebrated Heisenberg antiferromagnet, it now belongs to a triad of equivalently disordered phases. The present work provides a unifying theory of kagome spin liquids with time-reversal invariant nearest-neighbour Hamiltonians.
By using the density matrix renormalization group approach, we study spin-liquid phases of spin-1/2 XXZ kagome antiferromagnets. We find that the emergence of the spin-liquid phase is independent of the anisotropy of the XXZ interaction. In particular, the two extreme limits-the Ising (a strong S^{z} interaction) and the XY (zero S^{z} interaction)-host the same spin-liquid phases as the isotropic Heisenberg model. Both a time-reversal-invariant spin liquid and a chiral spin liquid with spontaneous time-reversal symmetry breaking are obtained. We show that they evolve continuously into each other by tuning the second- and the third-neighbor interactions. And last, we discuss possible implications of our results for the nature of spin liquid in nearest-neighbor XXZ kagome antiferromagnets, including the nearest-neighbor spin-1/2 kagome antiferromagnetic Heisenberg model.
We study the nearest-neighbor XXZXXZXXZ Heisenberg quantum antiferromagnet on the kagome lattice. Here we consider the effects of several perturbations: (a) a chirality term, (b) a Dzyaloshinski-Moriya term, and (c) a ring-exchange type term on the bowties of the kagome lattice, and inquire if they can support chiral spin liquids as ground states. The method used to study these Hamiltonians is a flux attachment transformation that maps the spins on the lattice to fermions coupled to a Chern-Simons gauge field on the kagome lattice. This transformation requires us to consistently define a Chern-Simons term on the kagome lattice. We find that the chirality term leads to a chiral spin liquid even in the absence of an uniform magnetic field, with an effective spin Hall conductance of ${\ensuremath{\sigma}}_{xy}^{s}=\frac{1}{2}$ in the regime of XYXYXY anisotropy. The Dzyaloshinski-Moriya term also leads a similar chiral spin liquid but only when this term is not too strong. An external magnetic field also has the possibility of giving rise to additional plateaus which also behave like chiral spin liquids in the XYXYXY regime. Finally, we consider the effects of a ring-exchange term and find that, provided its coupling constant is large enough, it may trigger a phase transition into a chiral spin liquid by the spontaneous breaking of time-reversal invariance.
By using the density matrix renormalization group, we study the spin-liquid phases of spin-1/21/21/2 XXZ kagome antiferromagnets. We find that the emergence of spin liquid phase does not depend on the anisotropy of the XXZ interaction. In particular, the two extreme limits---Ising (strong SzS^zSz interaction) and XY (zero SzS^zSz interaction)---host the same spin-liquid phases as the isotropic Heisenberg model. Both the time-reversal-invariant spin liquid and the chiral spin liquid with spontaneous time-reversal symmetry breaking are obtained. We show they evolve continuously into each other by tuning the second- and third-neighbor interactions. To understand the continuous phase transition between these two spin liquids, we construct a phenomenological theory, and propose the double-semion spin liquid as the candidate ground state of nearest-neighbor XXZ kagome antiferromagnets, including the most studied nearest-neighbor spin-1/21/21/2 kagome antiferromagnetic Heisenberg model.
We investigate the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg model with a Dzyaloshinskii-Moriya interaction on kagome lattice, making use of the variational Monte Carlo technique. An exotic quantum spin state is found to arise from a melting of the $\mathbit{Q}=0$ long-range magnetic order by a topological transition, when a small anisotropic third-nearest-neighbor antiferromagnetic Heisenberg interaction is turned on. This state is a gapped quantum spin liquid, characterized by a topological order with ground-state degeneracy ng=4{n}_{g}=4ng​=4 and topological entanglement entropy $\ensuremath{\gamma}=ln2$, suggesting it is an Abelian topological phase. Furthermore, the Chern numbers of the spin-up (-down) spinon occupied bands of this state are ${C}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}=\ifmmode\pm\else\textpm\fi{}1$, respectively. From this perspective, this state is also a time-reversal symmetric (total Chern number Ctotal=0{C}_{\text{total}}=0Ctotal​=0) topological insulator with spinons as the chiral edge states, which carry opposite spin and move in the opposite direction. It is analogous to the quantum spin Hall state but the spin current is carried by deconfined spinons in a quantum spin liquid, and so is dubbed as the spinon quantum spin Hall state.
Motivated by recent experiments on the Heisenberg S=1/2S=1/2S=1/2 quantum spin liquid candidate material kapellasite, we classify all possible chiral (time-reversal symmetry breaking) spin liquids with fermionic spinons on the kagome lattice. We obtain the phase diagram for the physically relevant extended Heisenberg model, comparing the energies of a wide range of microscopic variational wave functions. We propose that, at low temperature, kapellasite exhibits a gapless chiral spin liquid phase with spinon Fermi surfaces. This two-dimensional state inherits many properties of the nearby one-dimensional phase of decoupled antiferromagnetic spin chains, but also shows some remarkable differences. We discuss the spin structure factors and other physical properties.
Motivated by the search for chiral spin liquids (CSL), we consider a simple model defined on the kagome lattice of interacting SU(3) spins (in the fundamental representation) including two-site and three-site permutations between nearest neighbor sites and on triangles, respectively. By combining analytical methods and various numerical techniques, namely, exact Lanczos diagonalizations and tensor network variational approaches, we find a rich phase diagram with short-range entangled (i.e., nontopological) and topological (possibly chiral) gapped spin liquids (SLs). Short-range entangled spin liquids include an Affleck-Kennedy-Lieb-Tasaki (AKLT)-like phase---argued to be a Symmetry-Protected Topological (SPT) phase---and a trimerized phase breaking the inversion center between the up and down triangles of the kagome lattice. A topological SL is stabilized in a restricted part of the phase diagram by the time-reversal symmetry breaking (complex) three-site permutation term. Analyzing the chiral edge modes of this topological SL on long cylinders or on finite disks, we have come up with two competing scenarios, either a CSL or a double Chern-Simons SL characterized by a single or by two counter-propagating Wess-Zumino-Witten SU(3)1\mathrm{SU}{(3)}_{1}SU(3)1​ chiral mode(s), respectively. In the vicinity of the extended ferromagnetic region, we have found a magnetic phase corresponding either to a modulated canted ferromagnet or to a uniform partially magnetized ferromagnet.