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m wavelength data. Blue, green, bright green, red, and g Zn, Cu, Cl, O, and H atoms, respectively. (a) Local c Downloaded www.sciencemag.org on Junein 7, a 2009 ment of thefrom intersite is highlighted projection perpe Thermal ellipsoids are drawn at the 70% probability le of the kagomé layer is highlighted, in a projection c-axis. All atoms outside the kagomé plane have been (c) Photograph of a crystal from the batch used for interatomic distances (Å) and angles (deg): Zn-O 1.9821(3); Cu-Cl, 2.7676(2); Zn · · · Cu, 3.0560; Japanese name for the weave pattern of a basket (see the second figure, top panel). The structure consists of corner-sharing triangles and is even more frustrated than the triangular lattice considered by Anderson. Last year saw the synthesis of such a solid-state Kagome system: ZnCu3(OH)6Cl2, where a single electron spin resides on the Cu (10) (see the second figure, bottom panel). Although the exchange energy is ~200 K, this material does not show any magnetic ordering down to millikelvin temperatures. The magnetic excitations are apparently gapless, but unlike the organic compound, the large specific heat at low temperatures is sensitive to magnetic field, which suggests that the low-temperature properties may be dominated by a few percent of local moment defects. Spin liquids are not limited to twoA R T Isystems. C L E S A newly synthesized dimensional material, Na4Ir3O8, has Ir ions that form a three-dimensional network of corner-sharing triangles, termed a hyper-Kagome structure (11). Despite an exchange energy of ~300 K, no magnetic order was found down to 1 K and below. It is an exciting time in the history of antiferromagnetism. After decades of searching, three examples of the defeat of Néel order by quantum fluctuations have been discovered in quick succession. There are good reasons to believe that fermionic spinons will emerge as the low-energy excitations, but more work will be needed to confirm this. Figure 1. Crystal structure of herbertsmithite obtain gredients are dimers of an organic molecule, T [bis(ethylenedithio) - tetrathiafulvalene]. A ngle electron is localized on each dimer, hich forms layers of approximately triangular ttices. Despite an exchange energy of ~250 K, o magnetic order was detected down to 30 K. This material is an insulator but becomes a uperconductor (critical temperature Tc = 3.5 K) nd then a metal under pressure. It is believed at the proximity to an insulator-to-metal trantion implies that the spins interact with a more omplicated Hamiltonian than the Heisenberg odel and allows the spin liquid state to form , 9). Remarkably, the spin susceptibility goes a constant at low temperatures and the spefic heat is linear in temperature (7). These operties are normally associated with metals, eing consequences of the electron Fermi surce. The linear specific heat is particularly nusual for an insulator that is relatively defectee. Furthermore, the ratio of magnetic suseptibility to the linear temperature coefficient the specific heat is close to that of free fermi- ARTICLES Spinons in the S=1/2 Heisenberg antiferromagnet on kagome Tuesday, April 23, 13 Oleg Tchernyshyov Johns Hopkins University Papers • Zhihao Hao and Oleg Tchernyshyov: Tuesday, April 23, 13 • Spin-1/2 Heisenberg antiferromanget on kagome: a Z2 spin liquid with fermionic spinons. arxiv:1301.3261 • "Structure factor of low-energy spin excitations in a S=1/2 kagome antiferromagnet," Phys. Rev. B 81, 214445 (2010). • "Fermionic spin excitations in two and three-dimensional antiferromagnets," Phys. Rev. Lett. 103, 187203 (2009). Long story 1-body physics Spinons on kagome resemble strongly spinons of the Heisenberg model on the Delta/sawtooth chain. Both come in two flavors and have similar kinematics. Tuesday, April 23, 13 Long story 2-body physics Spinons exhibit Fermi statistics: upon exchange of two spinons, the 2-spinon wavefunction changes sign. They feel strong attraction in the S=0 channel mediated by the exchange interaction. This attraction binds two spinons into a small pair with a binding energy of 0.06J. Tuesday, April 23, 13 Long story Many-body physics We find a finite concentration of spinons in the ground state, one spinon per unit cell (3 sites). Their ground state is not a Fermi sea (as in large-N inspired approaches), for two reasons. First, strong attraction in the S=0 channel (see 2-body physics). Second, strong interaction with an emergent U(1) gauge field. Spinons carry the U(1) charge +1. They also have a small electric charge and couple to the real electric field. Spin pairs are bosons with charge +2. Their condensation may create a Z2 spin liquid with vison excitations. Tuesday, April 23, 13 Spinons on kagome: The gory details Tuesday, April 23, 13 Kagome lattice lattice of corner-sharing triangles Test H=J ⇥ij⇤ Tuesday, April 23, 13 J Si · Sj = 2 S + const. 2 Single triangle of s=1/2 hw02, problem 4 E= -3/4 J is the energy of a spin singlet: the lowest energy state of the single triangle is made of a singlet bond and a free unpaired spin OS addition Tuesday, April 23, 13 Construct a ground state? Place a valence bond on every triangle. Every triangle minimizes its exchange energy and the entire lattice is in a state of lowest possible energy. Tuesday, April 23, 13 Construct a ground state? Place a valence bond on every triangle. Every triangle minimizes its exchange energy and the entire lattice is in a state of lowest possible energy. Tuesday, April 23, 13 Construct a ground state? Place a valence bond on every triangle. Every triangle minimizes its exchange energy and the entire lattice is in a state of lowest possible energy. Tuesday, April 23, 13 Construct a ground state? Place a valence bond on every triangle. Every triangle minimizes its exchange energy and the entire lattice is in a state of lowest possible energy. No matter how hard we try, we always end up with a few defect triangles containing no valence bond. Tuesday, April 23, 13 Defects are inevitable! • N unit cells. • 2N triangles (up and down). • 3N sites. • 3N/2 valence bonds. • 3N/2 have valence bonds. • N/2 triangles are empty. • That is one in four triangles. V. Elser, PRL 62, 2405 (1989). Tuesday, April 23, 13 Focus on defect triangles Defect triangles are sources of quantum fluctuations. Without them, the system would have valence bonds frozen in place. Tuesday, April 23, 13 Focus on defect triangles Defect triangles are sources of quantum fluctuations. Without them, the system would have valence bonds frozen in place. Tuesday, April 23, 13 Focus on defect triangles Defect triangles are sources of quantum fluctuations. Without them, the system would have valence bonds frozen in place. Tuesday, April 23, 13 Focus on defect triangles Defect triangles are sources of quantum fluctuations. Without them, the system would have valence bonds frozen in place. Tuesday, April 23, 13 Focus on defect triangles Defect triangles are sources of quantum fluctuations. Without them, the system would have valence bonds frozen in place. Quantum fluctuations of valence bonds near a defect triangle. No fluctuations away from the defects. Tuesday, April 23, 13 Look at defect triangles Can’t study isolated defects on kagome. Instead, use the Husimi cactus: same local connectivity, but no loops apart from triangles. Tuesday, April 23, 13 Look at defect triangles Can’t study isolated defects on kagome. Instead, use the Husimi cactus: same local connectivity, but no loops apart from triangles. The dual lattice is a tree Tuesday, April 23, 13 Husimi cactus (kagome in a hyperbolic plane) V. Elser and C. Zeng (1993). P. Chandra and B. Douçot (1994). Tuesday, April 23, 13 Husimi cactus (kagome in a hyperbolic plane) A ground state. V. Elser and C. Zeng (1993). P. Chandra and B. Douçot (1994). Tuesday, April 23, 13 A single defect triangle The triangle without a bond is not in a stationary state. Nearby bonds fluctuate, those farther away remain static. Tuesday, April 23, 13 Ground state without defects V. Elser and C. Zeng (1993). P. Chandra and B. Douçot (1994). Tuesday, April 23, 13 Ground state without defects A ground state. V. Elser and C. Zeng (1993). P. Chandra and B. Douçot (1994). Tuesday, April 23, 13 S=1 excitations Tuesday, April 23, 13 S=1 excitations H Tuesday, April 23, 13 = 5/4 1/2 1/2 Displaced bonds are shown in blue. Tuesday, April 23, 13 Displaced bonds are shown in blue. Tuesday, April 23, 13 Displaced bonds are shown in blue. kink antikink The free spin can freely travel along a particular line. It shifts bonds along its path (cf. the sawtooth chain). Tuesday, April 23, 13 It’s a Δ/sawtooth chain embedded in the Husimi cactus. kink antikink The rest of the lattice remains frozen in time. Tuesday, April 23, 13 Spin excitations on the cactus • are spinons: quasiparticles with S=1/2; • come in two flavors: • kinks are completely localized, • antikinks are able to move along a line, rearranging bonds along the way; • resemble closely spinons on the Δ/sawtooth chain. Tuesday, April 23, 13 Defect triangle revisited The triangle without a bond is not in a stationary state. Tuesday, April 23, 13 State with a 3rd-neighbor singlet bond. Tuesday, April 23, 13 Long-range bond = spins ● and ● with total S=0. Tuesday, April 23, 13 Long-range bond = spins ● and ● with total S=0. Tuesday, April 23, 13 He II at low temperatu α ⌅S1 · S2 ⇥ S3⇧ = ⇤ 0. a a = 2S. † β S=a † ⇥ 2 Long-range bond = spins ● and ● with total S=0. Consider a general state of the two spins, | Tuesday, April 23, 13 • ⇥• ⇧ a⇥ . Spin ● is able to move around. Tuesday, April 23, 13 Spin ● is able to move around. Tuesday, April 23, 13 So is spin ●. Tuesday, April 23, 13 The spinons are antikinks traveling along three branches of the cactus. Tuesday, April 23, 13 We can now probe their exchange statistics. Tuesday, April 23, 13 We can now probe their exchange statistics. Tuesday, April 23, 13 An important technical point Arrows indicate a sign convention for singlet bonds. He II at low temperatures | ⇥• ⇤ ⌅ | ⇤• ⇥ ⌅ ⌃ = 2 Tuesday, April 23, 13 An important technical point Arrows indicate a sign convention for singlet bonds. He II at low temperatures | ⇥• ⇤ ⌅ | ⇤• ⇥ ⌅ ⌃ He II at low temperatures= 2 | ⇥• ⇤ ⌅ | ⇤• ⇥ ⌅ ⌃ = 2 Tuesday, April 23, 13 | ⇥ ⇤• ⌅ | ⇤ ⇥• ⌅ ⌃ = 2 An important technical point Arrows indicate a sign convention for singlet bonds. He II at low temperatures | ⇥• ⇤ ⌅ | ⇤• ⇥ ⌅ ⌃ He II at low temperatures= 2 | ⇥• ⇤ ⌅ | ⇤• ⇥ ⌅ ⌃ = 2 Tuesday, April 23, 13 | ⇥ ⇤• ⌅ | ⇤ ⇥• ⌅ ⌃ = −= 2 Tuesday, April 23, 13 Tuesday, April 23, 13 Tuesday, April 23, 13 Tuesday, April 23, 13 Tuesday, April 23, 13 Tuesday, April 23, 13 Initial state Tuesday, April 23, 13 State with spinons exchanged Initial state Tuesday, April 23, 13 State with spinons exchanged Initial state State with spinons exchanged Spinon exchange flips one valence bond. That produces a minus sign. Tuesday, April 23, 13 Forget about the singlet signs and treat valence bonds as passive dimers. Then the extra minus sign must be assigned to the spinons themselves. Tuesday, April 23, 13 Forget about the singlet signs and treat valence bonds as passive dimers. Then the extra minus sign must be assigned to the spinons themselves. In this sense, spinons are fermions. Tuesday, April 23, 13 On a more technical level • Obtain an orthogonal basis (similar to the sawtooth chain). • Find the ground state of two spinons with a given total spin S and check the spatial wavefunction. Fermi statistics requires that • Ψ(r , r ) = +Ψ(r , r ) for total S=0. ✔ • Ψ(r , r ) = −Ψ(r , r ) for total S=1. ✔ ● ● ● ● ● ● ● ● • Spinons are fermions. Tuesday, April 23, 13 Finite-size scaling of the ground-state energies S=1: no bound state. E1 = bottom of the 2-spinon continuum E1 + π2/2(R+2)2 E 0.65 0.6 S=0: bound state with binding energy Δ = 0.06J and radius ξ = 1.4 spacings of the dual lattice. Tuesday, April 23, 13 S=1 0.55 0.5 0.45 0.4 Δ S=0 0 5 10 15 20 R What about kagome? • Antikinks are still fermions. • 1 in 4 triangles contains a spinon pair, • That translates to 1 spinon per unit cell. Tuesday, April 23, 13 Low-energy spin excitations S=0 Tuesday, April 23, 13 S=1 Low-energy spin excitations S=0 Tuesday, April 23, 13 S=1 Low-energy spin excitations S=0 S=1 Away from the pair: S=0 bond → S=1 bond → kink + antikink. E ≈ 0.25J. Tuesday, April 23, 13 Low-energy spin excitations S=0 S=1 Away from the pair: S=0 bond → S=1 bond → kink + antikink. E ≈ 0.25J. Near the pair: S=0 bond → S=1 bond → 2 antikinks. E ≈ 0.06J. Tuesday, April 23, 13 Inelastic neutron scattering just above the spin-gap threshold N=5 Single dimer 2 Ke 1 K 0.4 Ke 0.3 K 0 qy 0.2 -1 -2 -2 -1 0 qx 1 2 0.1 0 -2 -1 0 qx 1 2 Left: I(Q) of inelastic neutron scattering just above the spin gap. Right: same for isolated dimers of three different orientations. Z. Hao and O.T., PRB 81, 214454 (2010); arXiv:1004.2293. R. R. P. Singh, Phys. Rev. Lett. 104, 177203 (2010). A. Läuchli and C. Lhuillier, arXiv:0901.1065. L. Messio, O. Cépas, and C. Lhuillier, Phys. Rev. B 81, 064428 (2010). Tuesday, April 23, 13 1.4 2 1.2 1 1 0.8 0 qy 0.6 -1 0.4 0.2 -2 0 LETTER Experiment Nature v.492, p.406 (2012) doi:10.1038/nature11659 Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet Tian-Heng Han1, Joel S. Helton2, Shaoyan Chu3, Daniel G. Nocera4, Jose A. Rodriguez-Rivera2,5, Collin Broholm2,6 & Young S. Lee1 LETTER RESEARCH The experimental realization of quantum spin liquids is a long- Cu21 ions (,5% of the total) substituting for Zn21 ions in the intersought goal in physics, as they represent new states of matter. Quan- layer sites, the kagome planes contain only Cu21 ions10. Measurements 0 5 10 20 011–13 0.15 0.3 0.45 tum spin liquids cannot be described by the broken symmetries on15 powder samples indicate strong antiferromagnetic superassociated with conventional ground states. In fact, the interacting exchange (J < 17 meV, where J is the exchange coupling that appears a 3 magnetic moments in these systems do not order, but are highly in the nearest-neighbour Heisenberg Hamiltonian) and the absence d3 entangled with one another over long ranges1. Spin liquids have a of long-range magnetic order or spin freezing down to temperaprominent role in theories describing high-transition-temperature tures of T 5 0.05 K. The bulk magnetic properties reveal a small 2 superconductors2,3, and the topological properties of these states Dzyaloshinskii–Moriya interaction and an easy-axis exchange 2 may have applications in quantum information4. A key feature of anisotropy14,15, both of order J/10. Despite these small imperfections, 1 exotic spin excitations carrying the nearest-neighbour Heisenberg model on a kagome lattice is still an spin liquids is that they support 1 fractional quantum numbers. However, detailed measurements of excellent approximation of the spin Hamiltonian for herbertsmithite. these ‘fractionalized excitations’ 0have been lacking. Here we report This is especially important, because recent calculations on record neutron scattering measurements on single-crystal samples of the lattice sizes indicate that the ground state of this model is in fact a b spin-1/2 kagome-lattice antiferromagnet ZnCu3(OD)6Cl2 (also called quantum spin liquid016. Thus, experiments to probe the spin correla2 herbertsmithite), which provide striking evidence for this characte- tions in herbertsmithite are all the more urgent. ristic feature of spin liquids. At low temperatures, we find that the To this end, we recently succeeded in developing a technique for the e3 spin excitations form a continuum, single crystals of herbertsmithite17, and 1 in contrast to the conventional growth of large, high-quality spin waves expected in ordered antiferromagnets. The observation of small pieces have been used in studies involving local probes18,19, such a continuum is noteworthy because, so far, this signature of anomalous X-ray diffraction10, susceptibility15 and Raman scattering20. 0 2 inelastic neutron scattering measurements on fractional spin excitations has been c 2 observed only in one-dimensional In this Letter, we report systems. The results also serve as a hallmark of the quantum spin- a large, deuterated, single-crystal sample of herbertsmithite. The neu20 liquid state in herbertsmithite. tron scattering cross-section is directly proportional to the dynamic 1 1 In a spin liquid, the atomic magnetic moments are strongly corre- structure10 factor Stot(Q, v) (where Q and v stand for the momentum lated but do not order or freeze even in the limit as the temperature, T, and energy transferred to the sample, respectively), which includes goes to zero. Although many types0 of quantum spin-liquid states exist both the nuclear 0 and0 magnetic signals. The magnetic part, Smag(Q, v), in theory, a feature that is expected –2 to be common is the correlation –1to all is the presence 0 1 Fourier transform 2 –2 (in time and–1space) of the spin–spin 0 1 2 of deconfined spinons as an elementary excitation from the ground function and can be obtained by subtracting the nuclear scattering as (H, H, 0) (H, H, 0) state1. Spinons are spin-half (S 5 1/2) quantum excitations into which described in the Supplementary Information. After calibration with conventional spin-wave excitations with S 5 1 fractionalize. In one respectplotted to a vanadium the9 meV. measured structure factors are 1 # Bv # e, Calculation of the equal-time structure factor, Figure 1 | Inelastic neutron scattering from the spin excitations, in overstandard, dimension, this phenomenon well established the atS T 55 1/21.6 Kexpressed in absoluteSunits. reciprocal space. a–c,isMeasurements werefor made on a singlemag(Q), for a model of uncorrelated nearest-neighbour dimers. The intensity Heisenberg antiferromagnetic chain, where may bestructure thought factor,Contour plotsis of Scorresponds T 5 1.6sum K and The dynamic Stot(Q, v), to 1/8 in of Fig. the 1a–c total for moment rule S(S 1 1) for the spins on the crystal sample of ZnCu tot(Q, v) are shown 3(OD) 6Cl2.spinons of as magnetic domain thatand disrupt Néel order(b) and are Efree three energy transfers BvThe (B data denotes Planck’s constant meVdifferent and plotted for Bvboundaries 5 6 meV (a) Bv 5 2 meV with kagome lattice. presented in a–c are expressed in barn sr21 eV21 per f 5 5.1 to propagate from each the3.0 one-dimensional compound divided by 2p). 1a shows for Bv 6 meV. Surprisingly, Bv away 5 0.75 meV (c)other. with EInf 5 formula unit,data as shown by5the left colour bars. The data presented in parts d and meV. The background was measured with anFigure KCuF3, a empty continuum of spinon has been characterized the scattered intensity is exceedingly diffuse, spanning e are dimensionless, with the scalea large givenfraction by the right colour bar. The Brillouin sample holder excitations and subtracted. Thewell diffuse scattering is mostly magnetic 5 using neutron scattering . Inthe twophonon dimensions, the nature thesignal spinon of the hexagonal zone. A similar pattern zone boundaries are drawn in of thediffuse figurescatterfor clarity; they correspond to the in origin, because contribution toofthe is small (except near Brillouin excitations is less First,positions, the existence of two-dimensional ingare is strong). observed for Bv 5 2 meV (Fig. The diffuse nature ofbthe conventional unit 1b). cell with parameters a5 5 6.83 Å, c 5 14.05 Å, the (2, 2,clear. 0)-type where the fundamentalmagnets Bragg peaks with a quantum spin-liquid ground state is still a matter of great debate. scattering at a temperature that is two orders of magnitude below the a 5 b 5 90u and c 5 120u. d, The magnetic part of the dynamic structure factor, Smag(Q, v), integrated Second, the various spin-liquid states which are proposed in theory exchange energy scale, J, is in strong contrast to observations in nongive rise to a variety of spinon excitation spectra, which may be either frustrated quantum magnets. The S 5 1/221 antiferromagsites, which are believed to affect the low-energy The observed Q dependence of the scattered intensity provides the interlayer Zn square-lattice gapped or gapless. net La2CuO4 develops substantial antiferromagnetic correlations for 25 . scattering important information on the ground-state spin correlations. The The S 5 1/2 kagome-lattice Heisenberg antiferromagnet has long T , J/2 (ref. 21), temperatures at which the low-energy scattering is scattering in reciprocal hastothe shape of broadened hexagonal The scattering pattern’s space. overall insensitivity to energy transfer is been recognized as a promising systemspace in which search for quantum strongly peaked near the (p, p) point in reciprocal In herbertrings centred (0,kagome 0, 0)- and (2, 0, positions. of thethe scans another remarkable feature of theatdata. spin-wave excitaspin-liquid states, becauseatthe network of0)-type corner-sharing tri- All smithite, scattered intensity is not strongly peaked any Conventional speangles frustrates magnetic from order6–8 . We syn-show cificsimilar point, patand thistions remains for allofenergies measuredoffrom that welong-range have performed Bv 5 have 1.5 todevised 11 meV taketrue the form sharp surfaces dispersion in Q–v space. Such thetic methods produce (ZnCu3(OH) whichenergy-integrated Bv 5 0.25 to 11 meV. This behaviour is also markedly 6Cl2) inThe terns to for the herbertsmithite scattered magnetic intensity. spin-wave excitations were indeeddifferent observed in the S 5 5/2 kagome 21 the S 5 1/2 Cu moments are arranged on a structurally perfect from that observed in the larger, S 5 5/2 kagome antiferromagnet dynamic structure factor over the integration range 1 # Bv # 9 meV (OH) (SO ) (ref. 26). In herbertsmithite, no antiferromagnet KFe 3 6 4 2 21 kagome lattice9 and nonmagnetic Zn ions separate the lattice planes. KFe3(OH)6(SO4)2 which becomes magnetically ordered at low temis plotted in Fig. 1d. This quantity serves as an approximation of the surfaces of dispersion are observable in the low-temperature data. A depiction of the crystal structure is shown in Supplementary Fig. 1. peratures and has magnetic peaks at q 5 0 wavevectors above the equal-time structure factor. For comparison, a calculation of the equal(Q, v) on Bv and Q is plotted in Fig. 2 for The dependence of S 22 tot Whereas herbertsmithite typically contains a small percentage of excess ordering temperature . (–K, K, 0) ARTICLES Figure 1. Crystal structure of herbertsmithite obtained OS withaddition 0.41 × 10-10 time structure factor for a collection of uncorrelated nearest-neighbour singlets on a kagome lattice is shown in Fig. 1e. To a first approximation, the observed magnetic signal resembles this calculation. Therefore, the ground-state wavefunction of herbertsmithite has a Tuesday, April 23, 13 1 m wavelength data. Blue, green, bright green, red, and gray spheres represent Zn, Cu, Cl, O, and H atoms, respectively. (a) Local coordination environment of the intersite is highlighted in a projection perpendicular to the c-axis. Thermal ellipsoids are drawn at the 70% probability level. (b) The geometry of the kagomé layer is highlighted, in a projection perpendicular to the two high-symmetry directions in reciprocal space: the (H, 0, 0) direction (Fig. 2a) All and theatoms (H, H, 0) direction (Fig. 2b). Thesekagomé directions are plane have been removed for clarity. c-axis. outside the indicated by thick black lines in Fig. 2d. These plots show that the spin (c) Photograph of abandcrystal from the batch used for these data. Selected excitations form a broad, continuous (or a continuum), extending Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 2NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA. 3Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 4Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 5Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742, USA. 6Institute for Quantum Matter and Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA. excitation continuum is observed over the entire range measured. The colour bar shows the magnitude of Stot(Q, v) in barn sr21 eV21 per formula unit. c, Energy dependence of Stot(Q, v) measured at high-symmetry reciprocal normalized to have units of eV21 per formula unit, consistent with the magnetic structure factor defined in Supplementary Information. d, The integrated areas in reciprocal space referred to a–c. the free-Cu21 magnetic form factor. Here the measured data indicate longer-range correlations than the nearest-neighbour singlet model. Figure 3c depicts a line scan of the dynamic structure factor (integrated over 1 # Bv # 7 meV) along the (0, K, 0) direction. The nearestneighbour singlet model does not account for the observed scattering intensity at the (0, 2, 0)-type positions. 30 2 meV 6 meV 10 meV 0.2 20 10 0 0.3 0.1 Integrated over 1 to 7 meV Dimer calculation 0.0 0 Integrated over 1 to 11 meV Dimer calculation 1 2 (0, K, 0) d 3 0.2 (–K, K, 0) Integrated Smag(Q, ω) b c Integrated Smag(Q, ω) Smag(Q, ω) (eV–1 form. unit–1) a 0.1 0.0 –1 0 (–2, 1 + K, 0) Figure 3 | The measured dynamic structure factor along specific directions in reciprocal space with comparison to the nearest-neighbour singlet model. a, Smag(Q, v) along the (22, 1 1 K, 0) direction, indicated by the thick red line on the reciprocal space map in d. Three energy transfers, Bv 5 2, 6 and 10 meV, are shown. b, Smag(Q, v) along the (22, 1 1 K, 0) direction integrated over 1 2 ut )c ,0 , (–2 1 0 –2 K 1+ –1 0 (H, H, 0) ut )c ,0 K (0, 1 2 1 # Bv # 11 meV. c, Smag(Q, v) along the (0, K, 0) direction, indicated by the thick orange line on the reciprocal space map in d, integrated over 1 # Bv # 7 meV. The solid lines in b and c are the calculated equal-time structure factors for uncorrelated nearest-neighbour singlets multiplied by | F(Q) | 2. d, The trajectories in reciprocal space referred to in a–c. Error bars, 1 s.d. 4 0 8 | N AT U R E | VO L 4 9 2 | 2 0 / 2 7 D E C E M B E R 2 0 1 2 ©2012 Macmillan Publishers Limited. All rights reserved OS addition Tuesday, April 23, 13 Spin gap • Our calculation: 0.06 to 0.10. • less than 0.1, exact diagonalization*. • 0.06 to 0.10, perturbation series . • 0.05 to 0.06, DMRG (Sheng). • Steve White’s DMRG: 0.14 or so? † *C. Waldtmann et al., Eur. Phys. J. B 2, 501 (1998). †R.R.P. Singh and D. A. Huse, arXiv:0801.2735. H. C. Jiang, Z.Y. Weng, and D. N. Sheng, Phys. Rev. Lett. 101, 117203 (2008). Tuesday, April 23, 13 Spinons in d=1: Test Δ/sawtooth chain H=J ⇥ij⇤ J Si · Sj = 2 S2 + const. Ground state: every triangle has total spin 1/2. T. Nakamura and K. Kubo, Phys. Rev. B 53, 6393 (1996). D. Sen, B. S. Shastry, R. E. Walstedt, and R. Cava, ibid., 6401 (1996). Tuesday, April 23, 13 Spinons in d=1: Test Δ/sawtooth chain H=J ⇥ij⇤ J Si · Sj = 2 S2 + const. Ground state: every triangle has total spin 1/2. T. Nakamura and K. Kubo, Phys. Rev. B 53, 6393 (1996). D. Sen, B. S. Shastry, R. E. Walstedt, and R. Cava, ibid., 6401 (1996). Tuesday, April 23, 13 Spinons in d=1: Test Δ/sawtooth chain H=J ⇥ij⇤ J Si · Sj = 2 S2 + const. Ground state: every triangle has total spin 1/2. T. Nakamura and K. Kubo, Phys. Rev. B 53, 6393 (1996). D. Sen, B. S. Shastry, R. E. Walstedt, and R. Cava, ibid., 6401 (1996). Tuesday, April 23, 13 Spin excitations Tuesday, April 23, 13 Spin excitations Tuesday, April 23, 13 Spin excitations Tuesday, April 23, 13 Spin excitations Tuesday, April 23, 13 Spin excitations Tuesday, April 23, 13 Spinons kink antikink • Serve as domain walls between the 2 vacua. • Carry spin S=1/2. • Kinks are localized in this model. • Antikinks are mobile. • Their quantum statistics is undefined: can’t exchange particles in 1 spatial dimension. Tuesday, April 23, 13 Test Energetics J H=J S ·S = i j ⇤ij⌅ H=J ⇤ij⌅ J Si · Sj = 2 ⇥ kink 2 S⇥ 2 ⇥ 2 S⇥ + const. E (k) = 0. antikink E (k) = 0. 3 1 E+(k) = + 4 2 1 k2 cos k ⇤ + . 4 2 1 = min(E + E+) = . 4 2 1 k cos k ⇤ + . 4 2 – Typeset by FoilTEX – 3 1 E+(k) = + 4 2 – Tuesday, April 23, 13 + const. 1 Virtual excitations +1 (b) Tuesday, April 23, 13 T T Virtual excitations +1 T (b) T 1 band 2 bands ∆ = 0.250 in 1-band approximation. ∆ = 0.219 in 2-band approximation. ∆ = 0.215 exact diag (Kubo 1996). Fast convergence. Not much room for improvement. 2J J 0 /2 Tuesday, April 23, 13 0 /2 k