Three-sublattice order in triangular- and Kagomé-lattice spin-half antiferromagnets

Abundance of Hard-Hexagon Crystals in the Quantum Pyrochlore Antiferromagnet

We propose a simple family of valence-bond crystals as potential ground states of the S=1/2 and S=1 Heisenberg antiferromagnet on the pyrochlore lattice. Exponentially numerous in the linear size of the system, these can be visualized as hard-hexagon coverings, with each hexagon representing a resonating valence-bond ring. This ensemble spontaneously breaks rotation, inversion, and translation symmetries. A simple, yet accurate, variational wave function allows a precise determination of the energy, confirmed by the density matrix renormalization group and numerical linked cluster expansion, and extended by an analysis of excited states. The identification of the origin of the stability indicates applicability to a broad class of frustrated lattices, which we demonstrate for the checkerboard and ruby lattices. Our work suggests a perspective on such quantum magnets, in which unfrustrated motifs are effectively uncoupled by the frustration of their interactions.

Frustration- and doping-induced magnetism in a Fermi-Hubbard simulator

Geometrical frustration in strongly correlated systems can give rise to a plethora of novel ordered states and intriguing magnetic phases, such as quantum spin liquids1-3. Promising candidate materials for such phases4-6 can be described by the Hubbard model on an anisotropic triangular lattice, a paradigmatic model capturing the interplay between strong correlations and magnetic frustration7-11. However, the fate of frustrated magnetism in the presence of itinerant dopants remains unclear, as well as its connection to the doped phases of the square Hubbard model12. Here we investigate the local spin order of a Hubbard model with controllable frustration and doping, using ultracold fermions in anisotropic optical lattices continuously tunable from a square to a triangular geometry. At half-filling and strong interactions U/t ≈ 9, we observe at the single-site level how frustration reduces the range of magnetic correlations and drives a transition from a collinear Néel antiferromagnet to a short-range correlated 120° spiral phase. Away from half-filling, the triangular limit shows enhanced antiferromagnetic correlations on the hole-doped side and a reversal to ferromagnetic correlations at particle dopings above 20%, hinting at the role of kinetic magnetism in frustrated systems. This work paves the way towards exploring possible chiral ordered or superconducting phases in triangular lattices8,13 and realizing t-t' square lattice Hubbard models that may be essential to describe superconductivity in cuprate materials14.

Gapless Quantum Spin Liquid in the Triangular System Sr_{3}CuSb_{2}O_{9}

We report gapless quantum spin liquid behavior in the layered triangular Sr_{3}CuSb_{2}O_{9} system. X-ray diffraction shows superlattice reflections associated with atomic site ordering into triangular Cu planes well separated by Sb planes. Muon spin relaxation measurements show that the S=1/2 moments at the magnetically active Cu sites remain dynamic down to 65 mK in spite of a large antiferromagnetic exchange scale evidenced by a large Curie-Weiss temperature θ_{CW}≃-143  K as extracted from the bulk susceptibility. Specific heat measurements also show no sign of long-range order down to 0.35 K. The magnetic specific heat (C_{m}) below 5 K reveals a C_{m}=γT+αT^{2} behavior. The significant T^{2} contribution to the magnetic specific heat invites a phenomenology in terms of the so-called Dirac spinon excitations with a linear dispersion. From the low-T specific heat data, we estimate the dominant exchange scale to be ∼36  K using a Dirac spin liquid ansatz which is not far from the values inferred from microscopic density functional theory calculations (∼45  K) as well as high-temperature susceptibility analysis (∼70  K). The linear specific heat coefficient is about 18  mJ/mol K^{2} which is somewhat larger than for typical Fermi liquids.

Anisotropic Triangular Lattice Realized in Rhenium Oxychlorides A3ReO5Cl2 (A = Ba, Sr)

We report the synthesis, crystal structure, and magnetic properties of the two new quantum antiferromagnets A₃ReO₅Cl₂ (A = Sr, Ba). The crystal structure is isostructural with the mineral pinalite Pb₃WO₅Cl₂, in which the Re⁶⁺ ion is square pyramidally coordinated by five oxide atoms and forms an anisotropic triangular lattice (ATL) made of S = 1/2 spins. The magnetic interactions J and J' in the ATL are estimated from magnetic susceptibilities to be 19.5 (44.9) and 9.2 (19.3) K, respectively, with J'/J = 0.47 (0.43) for A = Ba (Sr). For each compound, the heat capacity at low temperatures shows a large T-linear component with no signature of long-range magnetic order above 2 K, which suggests a gapless spin liquid state of one-dimensional character of the J chains in spite of the significantly large J' couplings. This is a consequence of one-dimensionalization by geometrical frustration in the ATL magnet; a similar phenomenon has been observed in two compounds with slightly smaller J'/J values: Cs₂CuCl₄ (J'/J = 0.3) and the related compound Ca₃ReO₅Cl₂ (0.32). Our findings demonstrate that 5d mixed-anion compounds provide a unique opportunity to explore novel quantum magnetism.

Thermal evolution of quasi-one-dimensional spin correlations within the anisotropic triangular lattice of α - NaMnO 2

Magnetic order on the spatially anisotropic triangular lattice of α - NaMnO 2 is studied via neutron diffraction measurements. The transition into a commensurate, collinear antiferromagnetic ground state with k = ( 0.5 , 0.5 , 0 ) was found to occur belowT N = 22 K . Above this temperature, the transition is preceded by the formation of a coexisting, short-range ordered, incommensurate state belowT IC = 45 K whose two-dimensional propagation vector evolves toward k = ( 0.5 , 0.5 ) as the temperature approachesT N . At high temperatures( T > T IC ) , quasielastic scattering reveals one-dimensional spin correlations along the nearest-neighbor Mn-Mn "chain direction" of the MnO6 planes. Our data are consistent with the predictions of a mean-field model of Ising-like spins on an anisotropic triangular lattice, as well as the predominantly one-dimensional Heisenberg spin Hamiltonian reported for this material.

Structure of the magnetic excitations in the spin-1/2 triangular-lattice Heisenberg antiferromagnet Ba3CoSb2O9

A spin-1/2 triangular-lattice Heisenberg antiferromagnet (TLHAF) is a prototypical frustrated quantum magnet, which exhibits remarkable quantum many-body effects that arise from the synergy between spin frustration and quantum fluctuation. The ground-state properties of a spin-1/2 TLHAF are theoretically well understood. However, the theoretical consensus regarding the magnetic excitations is limited. The experimental study of the magnetic excitations in spin-1/2 TLHAFs has also been limited. Here we show the structure of magnetic excitations in the spin-1/2 TLHAF Ba₃CoSb₂O₉ investigated by inelastic neutron scattering. Significantly different from theoretical expectations, the excitation spectrum has a three-stage energy structure. The lowest-energy first stage is composed of dispersion branches of single-magnon excitations. The second and third stages are dispersive continua accompanied by a columnar continuum extending above 10 meV, which is six times larger than the exchange interaction J = 1.67 meV. Our results indicate the shortcomings of the current theoretical framework.

Two-dimensional frustrated magnets are heavily studied because theories predict that quantum effects may lead to the emergence of fractionalized excitations. Ito et al. use inelastic neutron scattering to show that the excitation spectrum of Ba₃CoSb₂O₉ disagrees with current theoretical expectations.

Muon Spin Relaxation Evidence for the U(1) Quantum Spin-Liquid Ground State in the Triangular Antiferromagnet YbMgGaO_{4}

Muon spin relaxation (μSR) experiments on single crystals of the structurally perfect triangular antiferromagnet YbMgGaO_{4} indicate the absence of both static long-range magnetic order and spin freezing down to 0.048 K in a zero field. Below 0.4 K, the μ^{+} spin relaxation rates, which are proportional to the dynamic correlation function of the Yb^{3+} spins, exhibit temperature-independent plateaus. All these μSR results unequivocally support the formation of a gapless U(1) quantum spin liquid ground state in the triangular antiferromagnet YbMgGaO_{4}.

Exactly solvable antiferromagnetic Blume-Capel model on a sawtooth chain

The geometrically frustrated spin-1 Blume-Capel model on an infinite sawtooth chain is exactly solved by the transfer matrix method. The magnetization, ground-state phase diagram, magnetocaloric properties, and specific heat of the system are investigated. The results indicate that: (i) Magnetization plateaus appear at zero temperature. Their number depends on the sign of the crystal field D. For D≥0 there are two magnetization plateaus; however, for D<0 five plateaus exist. At a finite temperature, thermal excitation will destroy these plateaus completely. (ii) Phase transition between any two long-range-ordered ground states, whose spin configurations are given in phase diagram, is the first-order one. The macroscopic degeneracy of the ground states described by the entropy only exists at phase coexistence points. (iii) As temperature approaches zero, magnetocaloric properties and entropy change sharply near phase coexistence points. (iv) The crossovers of the specific heat from a single-peak structure to double-peak ones can signal the phase coexistence points in ground-state phase diagram.

Two-Step Antiferromagnetic Transitions and Ferroelectricity in Spin-1 Triangular-Lattice Antiferromagnetic Sr3NiTa2O9

We report the low-temperature characterizations on structural, specific heat, magnetic, and ferroelectric behaviors of transition metal oxide compound Sr3NiTa2O9. It is suggested that Sr3NiTa2O9 is a spin-1 triangular lattice Heisenberg quantum antiferromagnet which may have weak easy-axis anisotropy. At zero magnetic field, a two-step transition sequence at T(N1) = 3.35 K and T(N2) = 2.74 K, respectively, is observed, corresponding to the up-up-down (uud) spin ordering and 120° spin ordering, respectively. The two transition points shift gradually with increasing magnetic field toward the low temperature, accompanying an evolution from the 120° spin structure (phase) to the normal oblique phases. Ferroelectricity in the 120° phase is clearly identified. The first-principles calculations confirm the 120° phase as the ground state whose ferroelectricity originates mainly from the electronic polarization.

Gapless quantum spin liquid ground state in the two-dimensional spin-1/2 triangular antiferromagnet YbMgGaO4

Quantum spin liquid (QSL) is a novel state of matter which refuses the conventional spin freezing even at 0 K. Experimentally searching for the structurally perfect candidates is a big challenge in condensed matter physics. Here we report the successful synthesis of a new spin-1/2 triangular antiferromagnet YbMgGaO₄ with symmetry. The compound with an ideal two-dimensional and spatial isotropic magnetic triangular-lattice has no site-mixing magnetic defects and no antisymmetric Dzyaloshinsky-Moriya (DM) interactions. No spin freezing down to 60 mK (despite θ_w ~ −4 K), the power-law temperature dependence of heat capacity and nonzero susceptibility at low temperatures suggest that YbMgGaO₄ is a promising gapless (≤|θ_w|/100) QSL candidate. The residual spin entropy, which is accurately determined with a non-magnetic reference LuMgGaO₄, approaches zero (<0.6%). This indicates that the possible QSL ground state (GS) of the frustrated spin system has been experimentally achieved at the lowest measurement temperatures.

Quantum selection of order in an XXZ antiferromagnet on a Kagome lattice

Selection of the ground state of the kagome-lattice XXZ antiferromagnet by quantum fluctuations is investigated by combining nonlinear spin-wave and real-space perturbation theories. The two methods unanimously favor q=0 over sqrt[3]×sqrt[3] magnetic order in a wide range of the anisotropy parameter 0≤Δ≲0.72. Both approaches are also in accord on the magnitude of the quantum order-by-disorder effect generated by topologically nontrivial, looplike spin-flip processes. A tentative S-Δ phase diagram of the model is proposed.

Experimental realization of a spin-1/2 triangular-lattice Heisenberg antiferromagnet

We report the results of magnetization and specific heat measurements on Ba{3}CoSb{2}O{9}, in which the magnetic Co{2+} ion has a fictitious spin 1/2, and provide evidence that a spin-1/2 Heisenberg antiferromagnet on a regular triangular lattice is actually realized in Ba{3}CoSb{2}O{9}. We found that the entire magnetization curve including the one-third quantum magnetization plateau is in excellent quantitative agreement with the results of theoretical calculations. We also found that Ba{3}CoSb{2}O{9} undergoes a three-step transition within a narrow temperature range.

Evolution of the commensurate and incommensurate magnetic phases of the S = 3/2 kagome staircase Co₃V₂O₈ in an applied field

Single crystal neutron diffraction studies have been performed on the S = 3/2 kagome staircase compound Co(3)V(2)O(8) with a magnetic field applied along the magnetization easy-axis ([Formula: see text]). Previous zero-field measurements (Chen Y et al 2006 Phys. Rev. B 74 014430) reported a rich variety of magnetic phases, with a ferromagnetic ground state as well as incommensurate, transversely polarized spin density wave (SDW) phases (with a propagation vector of [Formula: see text]) interspersed with multiple commensurate lock-in transitions. The magnetic phase diagram with [Formula: see text] adds further complexity. For small applied fields, μ(0)H ≈ 0.05 T, the commensurate lock-in phases are destabilized in favor of the incommensurate SDW ones, while slightly larger applied fields restore the commensurate lock-in phase with δ = 1/2 and yield a new commensurate phase with δ = 2/5. For measurements in an applied field, higher-order scattering is observed that corresponds to the second harmonic.

Magnetic properties of S = 1/2 quasi-triangular lattice materials: Cu(2(1-x))Zn(2x)(OH)3NO3/(C7H15COO)·mH2O

We have investigated the structural and magnetic properties of two classes of spin S = 1/2 antiferromagnetic quasi-triangular lattice materials: Cu(2(1-x))Zn(2x)(OH)(3)NO(3) (0 ≤ x ≤ 0.65) and its long chain organic derivatives Cu(2(1-x))Zn(2x)(OH)(3)(C(7)H(15)COO)·mH(2)O (0 ≤ x ≤ 0.29). The series of layered structure compounds constitute a substitutional magnetic system, in which spin S = 1/2Cu(2+) ions and nonmagnetic Zn(2+) ions are arranged on a two-dimensional quasi-triangular lattice. For the nitrate compounds we found that the substitution of Zn(2+) ions can continuously decrease the Néel temperature, T(N), but never completely remove the magnetic order. In addition, the frustration effect in these materials is suppressed by a three-dimensional interlayer interaction. On the other hand, the corresponding long chain alkyl carboxylic acid group of intercalated materials, Cu(2(1-x))Zn(2x)(OH)(3)(C(7)H(15)COO)·mH(2)O, show spin-glass-like behavior, which is caused by the interplay of geometric frustration and mixed sign interactions. A tentative explanation for these findings is proposed in terms of a cluster-glass picture.

Dynamic scaling in the susceptibility of the spin-1/2 kagome lattice antiferromagnet herbertsmithite

The spin-1/2 kagome lattice antiferromagnet herbertsmithite, ZnCu(3)(OH)(6)Cl(2), is a candidate material for a quantum spin liquid ground state. We show that the magnetic response of this material displays an unusual scaling relation in both the bulk ac susceptibility and the low energy dynamic susceptibility as measured by inelastic neutron scattering. The quantity chiT(alpha) with alpha approximately 0.66 can be expressed as a universal function of H/T or omega/T. This scaling is discussed in relation to similar behavior seen in systems influenced by disorder or by the proximity to a quantum critical point.

A constrained stochastic state selection method applied to quantum spin systems

We describe a further development of the stochastic state selection method, which is a kind of Monte Carlo method we have proposed in order to numerically study large quantum spin systems. In the stochastic state selection method we make a sampling which is simultaneous for many states. This feature enables us to modify the method so that a number of given constraints are satisfied in each sampling. In this paper we discuss this modified stochastic state selection method that will be called the constrained stochastic state selection method in distinction from the previously proposed one (the conventional stochastic state selection method) in this paper. We argue that, by virtue of the constrained sampling, some quantities obtained in each sampling become more reliable, i.e. their statistical fluctuations are less than those from the conventional stochastic state selection method. In numerical calculations of the spin- 1/2 quantum Heisenberg antiferromagnet on a 36-site triangular lattice we explicitly show that data errors in our estimation of the ground state energy are reduced. Then we successfully evaluate several low-lying energy eigenvalues of the model on a 48-site lattice. Our results support that this system can be described by the theory based on the spontaneous symmetry breaking in the semiclassical Néel ordered antiferromagnet.

Dynamics and transport of the Z2 spin liquid: application to kappa-(ET)2Cu2(CN)3

We describe neutron scattering, NMR relaxation, and thermal transport properties of Z2 spin liquids in two dimensions. Comparison to recent experiments on the spin S=1/2 triangular lattice antiferromagnet in kappa-(ET)2Cu2(CN)3 shows that this compound may realize a Z2 spin liquid. We argue that the topological "vison" excitations dominate thermal transport, and that recent thermal conductivity experiments by M. Yamashita et al. have observed the vison gap.

Numerical linked-cluster algorithms. I. Spin systems on square, triangular, and kagomé lattices

We discuss recently introduced numerical linked-cluster (NLC) algorithms that allow one to obtain temperature-dependent properties of quantum lattice models, in the thermodynamic limit, from exact diagonalization of finite clusters. We present studies of thermodynamic observables for spin models on square, triangular, and kagomé lattices. Results for several choices of clusters and extrapolations methods, that accelerate the convergence of NLCs, are presented. We also include a comparison of NLC results with those obtained from exact analytical expressions (where available), high-temperature expansions (HTE), exact diagonalization (ED) of finite periodic systems, and quantum Monte Carlo simulations. For many models and properties NLC results are substantially more accurate than HTE and ED.

Spin dynamics of the spin-1/2 kagome lattice antiferromagnet ZnCu3(OH)6Cl2

We have performed thermodynamic and neutron scattering measurements on the S=1/2 kagomé lattice antiferromagnet ZnCu3(OH)6Cl2. The susceptibility indicates a Curie-Weiss temperature of theta CW approximately = -300 K; however, no magnetic order is observed down to 50 mK. Inelastic neutron scattering reveals a spectrum of low energy spin excitations with no observable gap down to 0.1 meV. The specific heat at low-T follows a power law temperature dependence. These results suggest that an unusual spin liquid state with essentially gapless excitations is realized in this kagomé lattice system.

Numerical linked-cluster approach to quantum lattice models

We present a novel algorithm that allows one to obtain temperature dependent properties of quantum lattice models in the thermodynamic limit from exact diagonalization of small clusters. Our numerical linked-cluster approach provides a systematic framework to assess finite-size effects and is valid for any quantum lattice model. Unlike high temperature expansions, which have a finite radius of convergence in inverse temperature, these calculations are accurate at all temperatures provided the range of correlations is finite. We illustrate the power of our approach studying spin models on kagomé, triangular, and square lattices.

Anomalous excitation spectra of frustrated quantum antiferromagnets

We use series expansions to study the excitation spectra of spin-1/2 antiferromagnets on anisotropic triangular lattices. For the isotropic triangular lattice model (TLM), the high-energy spectra show several anomalous features that differ strongly from linear spin-wave theory (LSWT). Even in the Néel phase, the deviations from LSWT increase sharply with frustration, leading to rotonlike minima at special wave vectors. We argue that these results can be interpreted naturally in a spinon language and provide an explanation for the previously observed anomalous finite-temperature properties of the TLM. In the coupled-chains limit, quantum renormalizations strongly enhance the one-dimensionality of the spectra, in agreement with experiments on Cs2CuCl4.

Role of the spin magnitude of the magnetic ion in determining the frustration and low-temperature properties of kagome lattices

In view of the variety of low-temperature magnetic properties reported recently for kagome lattices with transition-metal ions in different oxidation states, we have investigated the low-energy spectrum and low-temperature thermodynamic properties of antiferromagnetic kagome lattices with varying magnitudes of site spins, employing quantum many-body Heisenberg models. The ground state and the low-lying excitation spectrum are found to depend strongly on the nature of the spin magnitude of the magnetic ions. The system remains highly frustrated if spins are half-odd-integer in magnitude, while the frustration is very weak or almost absent for integer spins or mixed-spin systems. In fact, for a mixed-spin kagome system with a certain magnitude, the whole system behaves as a classical magnet with a ferrimagnetic ground state without any frustration. These theoretical findings are consistent with a few experimental observations recently reported in the literature and would be of value in designing new kagome systems with unusual and interesting low-temperature magnetic properties.