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arXiv:1507.01278 · Physical Review B

Chiral spin liquids on the kagome lattice

Krishna Kumar, Kai Sun, Eduardo Fradkin

Published Sep 18, 2015• 52 citations
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Abstract

We study the nearest-neighbor $XXZ$ 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 $XY$ 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 $XY$ 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.

DOI
10.1103/physrevb.92.094433

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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 \ensuremathσxys=12{\ensuremath{\sigma}}_{xy}^{s}=\frac{1}{2}\ensuremathσxys​=21​ 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.
We perform an exact-diagonalization study of the spin-12\frac{1}{2}21​ XXZ Heisenberg antiferromagnet on the kagome lattice at finite magnetization m=23m=\frac{2}{3}m=32​ with an emphasis on the XYXYXY point (Jz=0)({J}_{z}=0)(Jz​=0) and in the presence of a small chiral term. Recent analytic work by Kumar et al. [K. Kumar, K. Sun, and E. Fradkin, Phys. Rev. B 90, 174409 (2014)] on the same model, using a newly developed flux attachment transformation, predicts a plateau at this value of the magnetization described by a chiral spin liquid (CSL) with a spin Hall conductance of ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$. Such a state is topological in nature, has a ground-state degeneracy, and exhibits fractional excitations. We analyze the degeneracy structure in the low-energy manifold, identify the candidate topological states, and use them to compute the modular matrices and Chern numbers, all of which strongly agree with expected theoretical behavior for the ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$ CSL. In the limit of zero chirality, we find on most (not all) clusters that the topological invariants are still those of a CSL.
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, 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.
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 experimental materials such as Volborthite and Herbertsmithite that have Kagome like structures. In this thesis I will focus on studying frustrated spin systems on the Kagome lattice using a spin-1/2 antiferromagnetic XXZ Heisenberg model in the presence of an external magnetic field as well as other perturbations. Such a system is expected to give rise to magnetization platueaus which can exhibit topological characteristics in certain regimes. We will first develop a flux-attachment transformation that maps the Heisenberg spins (hard-core bosons) 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 this newly developed mapping we analyse the phases/magnetization plateaus that arise at the mean-field level and also consider the effects of adding fluctuations to various mean-fi eld states. Along the way, we show how to discretize an abelian Chern-Simons gauge theory on generic 2D planar lattices that satisfy certain conditions. We find that as long as there exists a one-to-one correspondence between the vertices and plaquettes defined on the graph, one can write down a discretized lattice version of the abelian Chern-Simons gauge theory. Using the newly developed flux attachment transformation, we show the existence of chiral spin liquid states for various magnetization plateaus for certain range of parameters in the XXZ Heisenberg model in the presence of an external magnetic field. Speci cally, in the regime of XY anisotropy the ground states at the 1/3 and 2/3 plateau are equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction 1/2 and that the 5/9 plateau is equivalent to the first bosonic Jain daughter state at filling fraction 2/3. Next, we also consider the effects of several perturbations: a) a chirality term, b) a Dzyaloshinskii-Moriya term, and c) a ring-exchange type term on the bowties of the kagome lattice, and inquire if they can also support chiral spin liquids as ground states. 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 1/2 in the regime of XY anisotropy. The Dzyaloshinkii-Moriya term also leads a similar chiral spin liquid but only when this term is not too strong. An external magnetic field when combined with some of the above perturbations also has the possibility of giving rise to additional plateaus which also behave like chiral spin liquids in the XY regime. Under the in influence of a ring-exchange term we 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. Finally, we also present some numerical results based on some exact diagonalization studies. Here, we specifically focus on the 2/3-magnetization plateau which we previously argued should be a chiral spin liquid with a spin hall conductance of 1/2 . Such a topological state has a non-trivial ground state degeneracy and it excitations are described by semionic quasiparticles. In the numerical analysis, we analyse the ground state degeneracy structure on various Kagome clusters of different sizes. We compute modular matrices from the resultant minimally entangled states as well as the Chern numbers of various eigenstates all of which provide strong evidence that the 2/3-magnetization plateau very closely resembles a chiral spin liquid state with the expected characteristics.
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 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.