Session T4: Focus Session: Kagome Antiferromagnets II

A Spin-1 Kagom\'{e} Heisenberg Antiferromagnet

We study a spin-1 Heisenberg antiferromagnet on a $2D$ kagom\'{e} lattice by projecting the Heisenberg Hamiltonian onto a restricted subspace of the full Hilbert space. This subspace consist of AKLT-like valence bound states described by closed loops. While not orthogonal, these singlet states are linearly independent; we derive the overlap between them and show that it is non-local and depends on the topology of nested loops. All of these states are characterized by the exponential decay of spin-spin correlations. Within this subspace, we identify lowest energy states which can be thought of as variational candidates for the ground states of the spin-1 kagom\'{e} Heisenberg and compare them with previous numerical studies.   

Identifying the nature of various quantum spin liquids on kagome lattice

We develop the density matrix renormalization group approach to systematically identify the topological order of the quantum spin liquid (QSL) through adiabatically obtaining different topological sectors of the QSL on an infinite cylinder. As an application, we study the easy axis anisotropic kagome Heisenberg model known for hosting a Z2 QSL, however no numerical simulations have been able to access all four sectors before. We obtain the complete set of four topological degenerate ground states distinguished by the presence or absence of the spinon and vison quasiparticle line, which fully characterizes the topological nature of the quantum phase. Using the four topological degenerate sectors, we calculate the modular matrix, which gives the braiding statistics that fits the Z2 QSL. We also find other type of QSL on kagome lattice, its nature has been identified through the modular matrix, etc. Finally, we study the kagome Heisenberg model, where our results have the potential to solve many mysteries and non-consistencies of former study on this model. \\[4pt] [1] Yin-Chen He, D. N. Sheng, and Yan Chen, arXiv: 1309.5669\\[0pt] [2] Yin-Chen He, D. N. Sheng, and Yan Chen (in preparation).   

Dipolar order by disorder in the classical Heisenberg antiferromagnet on the kagome lattice

The first experiments on the ``kagome bilayer'' SCGO triggered a wave of interest in kagome antiferromagnets in particular, and frustrated systems in general. A cluster of early seminal theoretical papers established kagome magnets as model systems for novel ordering phenomena, discussing in particular spin liquidity, partial order, disorder-free glassiness and order by disorder. Despite significant recent progress in understanding the ground state for the quantum $S=1/2$ model, the nature of the low-temperature phase for the classical kagome Heisenberg antiferromagnet has remained a mystery: the non-linear nature of the fluctuations around the exponentially numerous harmonically degenerate ground states has not permitted a controlled theory, while its complex energy landscape has precluded numerical simulations at low temperature. Here we present an efficient Monte Carlo algorithm which removes the latter obstacle. Our simulations detect a low-temperature regime in which correlations saturate at a remarkably small value. Feeding these results into an effective model and analyzing the results in the framework of an appropriate field theory implies the presence of long-range dipolar spin order with a tripled unit cell.\\[4pt] [1] G.-W. Chern and R. Moessner, Phys. Rev. Lett. {\bf 110}, 077201 (2013).   

Spin dynamics in the classical kagome antiferromagnet: Theory versus experiments

We investigate numerically the dynamical properties of the classical antiferromagnetic Heisenberg model on the kagome lattice using a combination of Monte Carlo method and molecular dynamics. We find that order from disorder induces a distribution of timescales in the cooperative paramagnetic regime (ie far above the transition toward coplanarity), as recently reported experimentally in the deuterium jarosite. At lower temperature, when the octupolar order is well established, we show that the weathervane loop fluctuations control the system relaxation : the time distribution observed at higher temperatures splits into two distinct time scales associated with fluctuations in the plane and out of the plane of coplanarity. The temperature and wave vector dependences of these two components are qualitatively consistent with loops diffusing in the entropically induced energy landscape. Numerical results are discussed and compared within analytical models and recent experiments obtained in both classical and quantum realisations of the kagome lattice.   

Composition dependence of magnetic order and spin chirality of Kagom\'{e} lattices in BaMn$_{\mathrm{1+x}}$Ru$_{\mathrm{5-x}}$O$_{11}$ R-type ferrites

The effects of atomic disorder on magnetic frustration have not been extensively studied. Single-crystal BaMn$_{2.49}$Ru$_{3.51}$O$_{11}$ exhibits three closely-spaced anomalies in the magnetization at temperatures T$_{1} =$ 183 K, T$_{2} =$ 171 K and T$_{3}$ $=$ 128 K, signaling complex magnetic/chiral ordering, due to an interplay between antiferromagnetic correlations, magnetic frustration and non-zero scalar chirality (induced by spin canting) within the hexagonal (Kagom\'{e}) ab-plane [1]. We observe that small increases in Ru content change the temperature and nature of the anomalies: A single crystal of composition BaMn$_{1.915}$Ru$_{4.085}$O$_{11}$ exhibits anomalies shifted to lower temperatures T$_{1} =$ 149 K, T$_{2} =$ 90 K and T$_{3} =$ 48 K. The anomaly at T$_{3}$ is rapidly weakened by fields H \textgreater 25 Oe applied parallel to the Kagom\'{e} plane for both compositions studied; whereas further field increases shift the onset of magnetic order substantially upward to T$_{1} =$ 175 K for the higher Ru concentration. \\[4pt] [1] L. Shlyk et al., Phys. Rev. B 81, 014413 (2010).   

Entanglement spectroscopy of SU(2)-broken phases in two dimensions

In magnetically ordered systems the breaking of SU(2) symmetry in the thermodynamic limit is associated with the appearance of a special type of low-lying excitations in finite size energy spectra, the so called tower of states (TOS). In the present work we numerically demonstrate that there is a correspondence between the SU(2) tower of states and the lower part of the ground state entanglement spectrum (ES). Using state-of-the-art DMRG calculations, we examine the ES of the 2D antiferromagnetic $J_1$-$J_2$ Heisenberg model on both the triangular and kagom\'e lattice. At large ferromagnetic $J_2$ the model exhibits a magnetically ordered ground state. Correspondingly, its ES contains a family of low-lying levels that are reminiscent of the energy tower of states. Their behavior (level counting, finite size scaling in the thermodynamic limit) sharply reflects tower of states features, and is characterized in terms of an effective entanglement Hamiltonian that we provide. At large system sizes TOS levels are divided from the rest by an entanglement gap. Our analysis suggests that (TOS) entanglement spectroscopy provides an alternative tool for detecting and characterizing SU(2)-broken phases using DMRG.   

Optical Conductivity of Valence Bond Solid Phases on the Kagome Lattice

We propose that optical responses below the Mott gap can be used to obtain useful information about excitation spectra in valence bond solid phases in Mott insulators. The optical conductivity in this regime arises due to the electronic polarization mechanism via virtual electron hopping processes. We apply this mechanism to the Hubbard model with spin-orbit coupling and/or the corresponding spin model with significant Dzyaloshinskii-Moriya interaction, and compute the optical conductivity. Our results are discussed in light of the existing and future experiments on the deformed Kagome lattice material, Rb2Cu3SnF12 with the pinwheel valence bond solid state, and other valence bond solid phases proposed for the ideal Kagome lattice.   

Vanishing spin gap in a competing spin-liquid phase in the kagome Heisenberg antiferromagnet

We provide strong numerical evidence, using improved variational wave functions, for a ground state with vanishing spin gap in the spin-1/2 quantum Heisenberg model on the kagome lattice. Starting from the algebraic U(1) Dirac spin liquid state proposed by Y. Ran {\it et al.} {Phys. Rev. Lett. {\bf 98}, 117205 (2007)}] and iteratively applying a few Lanczos steps, we compute the lowest S=2 excitation constructed by exciting spinons close to Dirac nodes. Our results are compatible with a vanishing spin gap in the thermodynamic limit and in consonance with a power-law decay of long distance spin-spin correlations in real space. The competition with a gapped (topological) spin liquid is discussed.   

A unification of Z2 spin liquids on Kagome lattice

While there is mounting numerical evidence for the existence of a gapped Z2 spin liquid in the Kagome Heisenberg model, a complete characterization of this topological phase remains to be accomplished. A defining property, the projective symmetry group (PSG) which fixes how the emergent excitations of the spin liquid phase transform under symmetry, remains to be determined. Two popular mean field approaches, based on a fermionic or bosonic representation of spinons, provide seemingly disparate classifications. Here we discuss a duality relation that pairs a fermionic spinon ansatz to a bosonic one, which unifies these classifications, and provides concrete predictions for identifying the spin liquid state on the Kagome lattice.   

Ground state uniqueness of the twelve site RVB spin-liquid parent Hamiltonian on the kagome lattice

Anderson's idea of a (short-ranged) resonating valence bond (RVB) spin liquid has been the first ever proposal of what we now call a topologically ordered phase. Since then, a wealth of exactly solvable lattice models have been constructed with topologically ordered ground states. For a long time, however, it has been difficult to realize Anderson's original vision in such solvable models, according to which the ground state has an unbroken SU(2) spin rotational symmetry and is dominated by fluctuation of singlet bonds. The kagome lattice is the simplest lattice geometry for which parent Hamiltonians stabilizing a prototypical spin-1/2 short-ranged RVB wave function have been constructed and strong evidence has been given that this state belongs to a topological phase. The uniqueness of the desired RVB-type ground states has, however, not been rigorously proven for the simplest possible such Hamiltonian, which acts on 12 spins at a time. Rather, this uniqueness has been demonstrated for a longer ranged (19-site) variant of this Hamiltonian by Schuch et al., using powerful approach of projected entangled-pair states. In this talk, we report on a ``ground state intersection property'' implying the ground state uniqueness of the 12-spin Hamiltonian for lattices of arbitrary size.   

Spinon Excitations and Entanglement Spectra of Z2 quantum spin liquids

Z2 quantum spin liquids are topologically ordered states endowed with spin rotational symmetry and lattice symmetry. The entanglement spectra of Z2 quantum spin liquids exhibit rich structure. Using short-range resonate-valence-bond (RVB) states as toy models [1], we show that the entanglement spectra contain signatures of spinons and their physical properties such as the quantum statistics and symmetry fractionalization pattern [2] could be extracted from the Schmidt states. References: [1] Didier Poilblanc, Norbert Schuch, David Perez-Garcia, and J. Ignacio Cirac, Phys. Rev. B 86, 014404 (2012). [2] Andrew M. Essin and Michael Hermele, Phys. Rev. B 87, 104406 (2013).