The linear dispersion of the low-energy electronic structure of monolayer graphene supports chiral quasiparticles that obey the relativistic Dirac equation and have a Berry phase of π (refs 1,2). In bilayer graphene 3 , the shape of the energy bands is quadratic, and its quasiparticles have a chiral degree, l = 2, and a Berry phase of 2π. These characteristics are usually determined from quantum Hall effect (QHE) measurements in which the Berry phase causes shifts in Shubnikov-de Haas (SdH) resistance oscillations. The QHE in graphene also exhibits an unconventional sequence of plateaux of Hall conductivity, σ xy , with quantized steps of 4e 2 /h, except for the first plateau, where it is governed by the Berry phase. Here, we report magnetotransport measurements in ABC-stacked trilayer graphene, and their variation with carrier density, magnetic field and temperature. Our results provide the first evidence of the presence of l = 3 chiral quasiparticles with cubic dispersion, predicted to occur in ABC-stacked trilayer graphene [4][5][6][7][8][9][10][11][12] . The SdH oscillations we observe suggest Landau levels with four-fold degeneracy, a Berry phase of 3π, and the marked increase of cyclotron mass near charge neutrality. We also observe the predicted unconventional sequence of QHE plateaux, σ xy = ±6e 2 /h, ±10e 2 /h, and so on. Despite significant interest in studying layered graphene systems with more than two layers, experimental progress has been limited [13][14][15][16][17][18] . Low-energy electronic properties depend crucially on the stacking order of graphene layers [4][5][6][7][8][9][10][11][12]18 , and therefore such studies require samples with a well-defined stacking sequence. In a bilayer, two honeycomb nets of carbon atoms are positioned with half of the atoms of the top layer (B) right above the atoms of the bottom layer (A) and the other half at the centres of the hexagonal voids in the bottom layer. The third carbon net in a trilayer can either be placed with its atoms above the atoms of the bottom layer A, as in the Bernal structure of crystalline graphite 19 , or with its voids above the lined-up atom pairs in layers A and B, thus breaking the reflection symmetry (Fig. 1a). The latter, ABC stacking, is found in the metastable rhombohedral modification of graphite 19 . The electronic structure of graphene multilayers is derived from the hybridization of monolayer states through interlayer hopping. Its main features are captured already by only considering hopping between the nearest-neighbour carbons, which are stacked above each other in two adjacent layers, γ 1 ∼ 0.1γ 0 , as shown in Fig. 1a (γ 0 ≈ 3.16 eV is the intralayer hopping, in bulk graphite γ 1 ≈ 0.4 eV, and there are also further-neighbour hoppings, γ 2 -γ 5 , which are not shown) [9][10][11][12] . In a bilayer, low-energy electronic states retain is the velocity of linear dispersion in the monolayer, p = (p x ,p y ) = p(cosϕ p ,sinϕ p ) is the 2D momentum, π = p x + iξ p y ,σ x,y are the pseudo-spin Pauli matrices, and ξ = ±1 is ...

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Theories involving highly energetic spin fluctuations are among the leading contenders for explaining high-temperature superconductivity in the cuprates 1 . These theories could be tested by inelastic neutron scattering (INS), as a change in the magnetic scattering intensity that marks the entry into the superconducting state provides a precise quantitative measure of the spin-interaction energy involved in the superconductivity 2-11 . However, the absolute intensities of spin fluctuations measured in neutron scattering experiments vary widely, and are usually much smaller than expected from fundamental sum rules, resulting in 'missing' INS intensity 2-5,12,13 . Here, we solve this problem by studying magnetic excitations in the one-dimensional related compound, Sr 2 CuO 3 , for which an exact theory of the dynamical spin response has recently been developed. In this case, the missing INS intensity can be unambiguously identified and associated with the strongly covalent nature of magnetic orbitals. We find that whereas the energies of spin excitations in Sr 2 CuO 3 are well described by the nearestneighbour spin-1/2 Heisenberg Hamiltonian, the corresponding magnetic INS intensities are modified markedly by the strong 2p-3d hybridization of Cu and O states. Hence, the ionic picture of magnetism, where spins reside on the atomic-like 3d orbitals of Cu 2+ ions, fails markedly in the cuprates.Over the past 20 years, the magnetic properties of cuprates have been studied extensively by theorists and experimentalists alike. These systems are usually described within the antiferromagnetic Mott insulator model, in which the unpaired electrons are localized on the Cu 2+ ions because of the overwhelming cost in the on-site Coulomb interaction energy, U , associated with the charge transfer between the Cu sites, a strong correlation phenomenon. Virtual electron hopping, which in the one-band Hubbard model of a Mott insulator often adopted for cuprates 14 is quantified by the transition matrix element, t , results in antiferromagnetic exchange. For t U , electronic spins form the only low-energy electronic degrees of freedom. Their properties are well approximated by the spin-1/2 Heisenberg Hamiltonian on the lattice 15 , H = J i(nn)j S i S j , with the nearest-neighbour exchange coupling J ≈ 4t 2 /U . This description conveniently splits the problem of electronic magnetism in the Mott insulator into two parts 15 . The first deals with the electron transfer between the neighbouring sites of the crystal lattice, which is determined by the overlap integral (∼t ) of the wavefunctions occupied by the unpaired electrons, and leads to the Hubbard model, or the Heisenberg spin Hamiltonian. The second concerns the form of the electronic Wannier wavefunctions, that is, the shape of the spin magnetization cloud associated with 1 ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UK, 2 Department of Physics, University College London, Gower Street, London WC1E 6BT, UK, 3 London Centre for Nanotechnology, 17-19 Go...

Much of modern condensed matter physics is understood in terms of elementary excitations, or quasiparticles-fundamental quanta of energy and momentum 1,2 . Various strongly interacting atomic systems are successfully treated as a collection of quasiparticles with weak or no interactions. However, there are interesting limitations to this description: in some systems the very existence of quasiparticles cannot be taken for granted. Like unstable elementary particles, quasiparticles cannot survive beyond a threshold where certain decay channels become allowed by conservation laws; their spectrum terminates at this threshold. Such quasiparticle breakdown was first predicted for an exotic state of matter-super-fluid 4 He at temperatures close to absolute zero, a quantum Bose liquid where zero-point atomic motion precludes crystallization 1-4 . Here we show, using neutron scattering, that quasiparticle breakdown can also occur in a quantum magnet and, by implication, in other systems with Bose quasiparticles. We have measured spin excitations in a two-dimensional quantum magnet, piperazinium hexachlorodicuprate (PHCC) 5 , in which spin-1/2 copper ions form a non-magnetic quantum spin liquid, and find remarkable similarities with excitations in superfluid 4 He. We observe a threshold momentum beyond which the quasiparticle peak merges with the two-quasiparticle continuum. It then acquires a finite energy width and becomes indistinguishable from a leading-edge singularity, so that excited states are no longer quasiparticles but occupy a wide band of energy. Our findings have important ramifications for understanding excitations with gapped spectra in many condensed matter systems, ranging from band insulators to high-transition-temperature superconductors 6 . Although of all the elements only liquid helium fails to crystallize at temperature T ¼ 0, quantum liquids are quite common in condensed matter. Metals host electron Fermi liquids, and superconductors contain Bose liquids of Cooper pairs. Trapped ultracold atoms can also form quantum liquids, and some remarkable new examples were recently identified in magnetic crystals 5,7-10 . The organometallic material PHCC is an excellent physical realization of a quantum spin liquid (QSL) in a two-dimensional (2D) Heisenberg antiferromagnet (HAFM). Its Cu 2þ spins are coupled through a complex network of orbital overlaps, and form an array of slightly skewed anisotropic spin-1/2 ladders 10 in the crystalline a-c plane with highly frustrated super-exchange interactions 5 . The spin excitations in PHCC have a spectral gap D s < 1 meVand nearly isotropic 2D dispersion in the (h0l) plane with a bandwidth slightly larger than D s . In the absence of a magnetic field, only the short-range dynamic spin correlations typical of a liquid exist: the spin gap precludes longrange magnetic order down to T ¼ 0. Here we explore magnetic excitations in PHCC via inelastic neutron scattering and compare the results with similar measurements in the quantum fluid 4 He, emphasizing the effects ...

We have investigated the spin dynamics in the strongly correlated chain copper oxide SrCuO2 for energies up to greater, similar 0.6 eV using inelastic neutron scattering. We observe a gapless continuum of magnetic excitations, which is well described by the "Müller ansatz" for the two-spinon continuum in the S=1/2 antiferromagnetic Heisenberg spin chain. The lower boundary of the continuum extends up to approximately 360 meV, which corresponds to an exchange constant J=226(12) meV.

We present elastic and quasielastic neutron scattering measurements characterizing peculiar short-range charge-orbital and spin order in the layered perovskite material La1.5Sr0.5CoO4. We find that below T(c) approximately 750 K holes introduced by Sr doping lose mobility and enter a statically ordered charge glass phase with loosely correlated checkerboard arrangement of empty and occupied d(3z(2)-r(2)) orbitals ( Co3+ and Co2+). The dynamics of the resultant mixed spin system is governed by the anisotropic nature of the crystal-field Hamiltonian and the peculiar exchange pattern produced by the orbital order. It undergoes a spin freezing transition at a much lower temperature, T(s) less, similar30 K.

High-temperature superconductivity in both the copper-oxide and the iron-pnictide/chalcogenide systems occurs in close proximity to antiferromagnetically ordered states. Neutron scattering has been an essential technique for characterizing the spin correlations in the antiferromagnetic phases and for demonstrating how the spin fluctuations persist in the superconductors. While the nature of the spin correlations in the superconductors remains controversial, the neutron scattering measurements of magnetic excitations over broad ranges of energy and momentum transfer provide important constraints on the theoretical options. We present an overview of the neutron scattering work on high-temperature superconductors and discuss some of the outstanding issues.

Piperazinium hexachlorodicuprate is shown to be a frustrated quasi-two-dimensional quantum Heisenberg antiferromagnet with a gapped spectrum. Zero-field inelastic neutron scattering and susceptibility and specificheat measurements as a function of applied magnetic field are presented. At Tϭ1.5 K, the magnetic excitation spectrum is dominated by a single propagating mode with a gap, ⌬ϭ1 meV, and bandwidth of Ϸ1.8 meV in the (h0l) plane. The mode has no dispersion along the b* direction indicating that neighboring a-c planes of the triclinic structure are magnetically decoupled. The heat capacity shows a reduction of the gap as a function of applied magnetic field in agreement with a singlet-triplet excitation spectrum. A field-induced ordered phase is observed in heat capacity and magnetic susceptibility measurements for magnetic fields greater than H c1 Ϸ7.5 T. Analysis of the neutron-scattering data reveals the important exchange interactions and indicates that some of these are highly frustrated.