Holes in a Quantum Spin Liquid Collin Broholm Johns Hopkins University and NIST Center for Neutron Research Gapped phases in condensed matter Gapped spin chains Pure systems Doped systems Conclusions Y3+ I am grateful for opportunity to speak today An unusual type of magnetism: magnetic ions but large non-magnetic molecule The new results : impurity induced magnetism Diverse audience so develop an analogy to doped semiconductors I encourage you to stop me with questions viewgraphs are posted on the web Ca2+ Y2-xCaxBaNiO5 Viewgraphs posted at http://www.pha.jhu.edu/~broholm/jhuseminar/index.htm

Collaborators Y2BaNiO5 Copper Nitrate [Cu(NO3)2.2.5D2O] Guangyong Xu JHU -> University of Chicago G. Aeppli NEC J. F. DiTusa Louisiana State University I. A. Zaliznyak JHU -> BNL C. D. Frost ISIS T. Ito Electro-Technical Lab Japan K. Oka Electro-Technical Lab Japan H. Takagi ISSP and CREST-JST M. E. Bisher NEC M. M. J. Treacy NEC R. Paul NIST Center for Neutron Research “Holes in a Quantum Spin Liquid” to appear in Science (2000) Copper Nitrate [Cu(NO3)2.2.5D2O] Daniel Reich JHU M. A. Adams ISIS facility “Triplet Waves in a Quantum Spin Liquid” to appear in PRL May 1 (2000) Acknowledge competent collaborators former student Guangyong Xu who did most of the analysis that you shall see Ito, Oka, and Takagi that grew the crystals that made the experiments possible A paper describing the main results presented will appear in Science magazine shortly

Quantum effects in atoms and in solids Atom (He) Line spectrum G. Dieke et al. (1968) Solid (Mo) continuous spectrum if Q not specified Long before my time a proud tradition for atomic spectroscopy in the department of physics and astronomy at Johns Hopkins University Henry Rowland who founded our department invented the diffraction grating used to resolve these spectra. Later experimentalist such as Gerhard Dieke with the help I am told of 30 graduate students at a time collected spectra from the far infrared to the far ultraviolet for atoms in the gas phase or disolved in various solid matrixes. In these spectra we see direct evidence for the quantum nature of matter: discrete transitions between atomic levels that produce light of specific or wavelength. Shown in the figure is the spectrum produced by a helium lamp recorded on a photographic plate using a quartz prizm as monochromator. Forming a solid the energy levels for outer electrons acquires dispersion as a function of wave vector transfer. Essentially this comes about because of Bragg diffraction of electrons in the periodic lattice structure. Q must be specified to appreciate these quantum levels and this means that bulk measurements probe crude averages of this intricate structure. While measurements exist that are sensitive to these details they are not versatile so all I can show is a numerical calculation. But there are circumstances where the nature of the bandstructure has profound effects on the properties of solids. Christensen (1980) JHU 4/26-13/00

Quantum effects from a gap Empty Band gap only possible for even number of electrons per cell Filled No gap Metal Gap Insulator If you ever wondered what the gap means here is the explanation. It happens in many cases that an energy gap separates filled from empty levels. Such a gap has direct and profound effect on all materials properties. As an example I show electrical transport properties for a material with and without a gap. When there is no gap in the spectrum an external electric field can shift the population of these states ever slightly in an amount that depends on the life time of these states. The shifted occupation carries electrical current and we end up with a metal. We show the resistivity of copper versus temperature and see that the resistance decreases to values less than 1uW cm for low T and low level of defect. In contrast if there is a gap in the spectrum the state is robust and unaffected by a small external electric field. So the zero field state persists. That state is invarient under time reversal symmetry and so carries no electrical current, that is we have an insulator. The figure shows the conductivity versus inverse temperature in so-called Arhenius plot. The ability of the material to conduct depends entirely upon thermal excitation of electrons across the gap and so is exponentially activated. The gap has profound consequences for all materials property that one might choose to consider. Because each band can accommodate two electrons per unit cell can only have a band gap when there are an even number of electrons per cell JHU 4/26-13/00

Impurities create states in the gap Cleaner sample 10 5 10 7 10 9 1011 Insulator Resistivity (ohm-cm) 10-3 10-1 10 10 3 Impurities may not be something you normally would like to think about but in gapped systems they lead to interesting and very useful properties. Impure semiconductors are the basis for the information technological revolution that we are in the midst of. I am showing electrical transport data for doped silicon as resistivity versus inverse temperature. In clean samples we see the exponentially activated behavior. Perversely the low T resistivity is less for the dirtier samples. Suitable impurities eventually drive the material from insulator to metal. In the pure system for every electron in the filled band there is another electron moving in the exact opposite direction. Impurities create states in the gap that can donate or accept electrons from the band such that this delicate balance is destroyed and the material can conduct. All gapped systems have dramatic impurity effects because their unusual properties are a result of perfect cancellation of certain electronic properties in the pure and symmetric state. Metal “Dirty” sample 0 0.2 0.4 0.6 0.8 1/T (K-1) From Aschroft & Mermin JHU 4/26-13/00

Other gapped phases in condensed matter High energy physics yields low energy gap Band insulator Band semiconductor Dimerized spin systems Phase transition to gapped phase Superconductor Rotons in superfluid 4He Mott Metal-Insulator transition Spin Peierls transition Cross over to gapped phase for T<<D Quantum Hall effects Uniform integer spin chains In this talk I am trying to draw out the analogies between a vast range of different systems where an energy gap dominates bulk properties. Looking closer these systems are of course really very different. Specifically there are very different ways that a gap can come about that I have tried to classify in this transparency. In blue I have pointed out the magnetic analogues of each of these types of gapped phases. In the remainder of the talk we shall focus on the magnetic systems. In doing so we will get tied up in their specific details but it is important to bear in mind the connections to these other physical systems. JHU 4/26-13/00

The beauty of magnetic dielectrics Well defined low energy Hamiltonian Chemistry provides qualitatively different H Vary H with pressure, magnetic field Efficient experimental techniques Exchange interaction Single ion anisotropy Dipole in magnetic field (Zeeman) Insulating magnets are unique model systems for a very wide range of collective phenomena in physics. Here are reasons for this…. JHU 4/26-13/00

Spin systems of recent interest To give you a feel for the range of physics that can be explored through experiments in magnetic materials I provide this list that summarizes key properties. Lattice dimensionality, lattice symmetry, spin quantum number, and spin space anisotropy are the factors that lead to a variety of different ground states and collective behavior in these systems. You will notice that the dominant exchange constant varies over two or three orders of magnitude in these systems. While the magnitude of J is unimportant for the physics of these materials it has a profound effect on how we must carry out the experiments. By selecting materials with specific values we can facilitate experiments that probe specific aspects of the system. I would like to dwell for a moment on what we mean when we say that a spin system has a dimensionality less than the three spatial dimensions that we live in. JHU 4/26-13/00

Varying the dimensionality of spin systems The crystals are actually three dimensional Exchange links spins on low-D network only The materials are real three dimensional solids but the magnetic interactions link lower dimensional networks of spins. As a result in what we would call a one dimensional magnet we really have an ensemble of chains of magnetic spins that to a good approximation act independently. However we need them all to build enough signal for our experiments to be successful. Spin chains can be separated by non-magnetic ions in low symmetry structures such as orthorhombic NDMAP shown to the left. But the isolation of spin chains can occur in materials that are almost cubic such as KCuF3 where the anisotropic electronic orbital of the hole on the copper atom twists and turns leaving only one contiguous path for magnetic interactions. Now we specialize within this class of materials on those magnets that have a gap in their excitation spectra and we would like see what we can learn about such systems by studying magnetic dielectrics. So that you can understand the new results that we have obtained concerning impurities in these systems, I will start off by summarizing key experimental properties of spin systems with a gap in their excitation spectra. Chain direction Non-magnetic “spacer” molecules in NDMAP JHU 4/26-13/00 Anisotropic bonding in cubic KCuF3

Susceptibility of gapped spin system Measure magnetization versus T in infinitesimal field: JHU 4/26-13/00 Renard et al (1988) Tatsuo et al. (1995)

Magnetization of gapped spin system Ajiro et al. (1989) JHU 4/26-13/00

Specific heat of gapped spin chain A way to probe the low energy density of states NENC Orendac et al. (1995) NENP Tatsuo et al. (1995) JHU 4/26-13/00

Magnetic Neutron Scattering The scattering cross section is proportional to the Fourier transformed dynamic spin correlation function JHU 4/26-13/00

NIST Center for Neutron Research JHU 4/26-13/00

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Detection system on SPINS neutron spectrometer JHU 4/26-13/00

Neutron scattering from gapped spin chain Copper nitrate T=0.3 K gap Xu et al. IRIS, ISIS Nuclear incoherent scattering Proton extraction pulse Triplet creation peak JHU 4/26-13/00

Dispersion relation for triplet waves Dimerized spin-1/2 system: copper nitrate JHU 4/26-13/00 Xu et al PRL May 2000

Why a gap in spectrum of dimerized spin system A spin-1/2 pair has a singlet - triplet gap: Weak inter-dimer coupling cannot close gap Bond alternation is relevant operator for quantum critical uniform spin chain infinitesimal bond alternation yields gap J JHU 4/26-13/00

Gapped phases in isotropic spin systems? n = number of spins per primitive unit cell S = the spin quantum number m = the magnetization per spin n(S-m)= Oshikawa, Yamanaka, and Affleck (1997) and Oshikawa (2000) gaps in non-magnetized spin chains? Alternating spin-1/2 chain 2.1/2=1 perhaps Uniform spin-1/2 chain 1.1/2 =1/2 no Uniform spin-1 chain 1.1 =1 perhaps Integer: gap possible Non-Integer: gap impossible JHU 4/26-13/00

Low T excitations in spin-1 AFM chain pure Y2BaNiO5 T=10 K MARI chain ki Haldane gap D=8 meV Coherent mode S(q,w)->0 for Q->2np JHU 4/26-13/00

AKLT state for spin-1 chain Magnets with 2S=nz have a nearest neighbor singlet covering with full lattice symmetry. This is exact ground state for spin projection Hamiltonian Excited states are propagating bond triplets separated from the ground state by an energy gap Haldane PRL 1983 Affleck, Kennedy, Lieb, and Tasaki PRL 1987 JHU 4/26-13/00

Sum rules and the single mode approximation The dynamic spin correlation function obeys sum-rules: When a coherent mode dominates the spectrum: Sum-rules link S(q) and e(q) JHU 4/26-13/00

Single mode approximation for spin-1 chain Dispersion relation Equal time correlation function JHU 4/26-13/00

Haldane mode in Y2BaNiO5 at finite T Relaxation rate increases with T due to triplet interactions Resonance energy increases decreasing correlation length Low T fine structure from spin anisotropy JHU 4/26-13/00

T-dependence of relaxation rate and “resonance” energy Parameter free comparison: Semi-classical theory of triplet scattering by Damle and Sachdev 3 k T æ D ö ( ) G T = B exp ç - ÷ ç ÷ p è k T ø B Quantum non linear s model æ D ö ( ) D T = D + 2 p D k T exp ç - ÷ ç ÷ B è k T ø B JHU 4/26-13/00

Q-scans versus T: energy resolved and energy integrated h w ³ D Probing equal time correlation length h w = D Probing spatial coherence of Haldane mode JHU 4/26-13/00

Coherence and correlation lengths versus T Coherence length exceeds correlation length for kBT<D becoming very long as T 0 Equal-time correlation length saturates at x=8. JHU 4/26-13/00 (Solid line from Quantum non linear s model)

Pure quantum spin chains - at zero and finite T Gap is possible when n(S-m) is integer gapped systems: alternating spin-1/2 chain, integer chain,… gapless systems: uniform spin-1/2 chain gapped spin systems have coherent collective mode For appreciable gap SMA applies: S(q) ~ 1/e(q) Thermally activated relaxation due to triplet interactions Thermally activated increase in resonance energy Coherence length exceeds correlation length for T< D/kB JHU 4/26-13/00

Impurities in Y2BaNiO5 Mg2+on Ni2+ sites finite length chains Ca2+ on Y3+ sites mobile bond defects Mg Ca2+ Ni Y3+ JHU 4/26-13/00 Kojima et al. (1995)

Zeeman resonance of chain-end spins 20 g=2.16 hw (meV) 15 0 2 4 6 8 H (Tesla) 10 I(H=9 T)-I(H=0 T) (cts. per min.) -5 0 0.5 1 1.5 2 JHU 4/26-13/00

Form factor of chain-end spins Y2BaNi1-xMgxO5 x=4% Q-dependence reveals that resonating object is AFM. The peak resembles S(Q) for pure system. Chain end spin carry AFM spin polarization of length x back into chain JHU 4/26-13/00

New excitations in Ca-doped Y2BaNiO5 Pure 9.5% Ca Y2-xCaxBaNiO5: Ca-doping creates states below the gap sub-gap states have doubly peaked structure factor JHU 4/26-13/00

Why a double ridge below the gap in Y2-xCaxBaNiO5 ? Charge ordering yields incommensurate spin order Quasi-particle Quasi-hole pair excitations in Luttinger liquid Anomalous form factor for independent spin degrees of freedom associated with each donated hole d q µ x q d is single impurity prop. Indep. of x JHU 4/26-13/00

Does dq vary with calcium concentration? dq not strongly dependent on x Double peak is single impurity effect JHU 4/26-13/00

Bond Impurities in a spin-1 chain: Y2-xCaxBaNiO5 (f) JHU 4/26-13/00

Form-factor for FM-coupled chain-end spins A symmetric AFM droplet Ensemble of independent randomly truncated AFM droplets

Calcium doping Y2BaNiO5 Experimental facts: Ca doping creates sub-gap excitations with doubly peaked structure factor and bandwidth The structure factor is insensitive to concentration and temperature for 0.04<x<0.14 (and T<100 K) Analysis: Ca2+ creates FM impurity bonds which nucleate AFM droplets with doubly peaked structure factor AFM droplets interact through intervening chain forming disordered random bond 1D magnet JHU 4/26-13/00

Incommensurate modulations in high TC superconductors YBa2Cu3O6.6 T=13 K E=25 meV h (rlu) k (rlu) Hayden et al. 1998 JHU 4/26-13/00

What sets energy scale for sub gap scattering ? Possibilities: Residual spin interactions through Haldane state. A Random bond AFM. Hole motion induces additional interaction between static AFM droplets AFM droplets move with holes: scattering from a Luttinger liquid of holes. 10 hw (meV) 5 ? How to distinguish: Neutron scattering in an applied field Transport measurements Theory JHU 4/26-13/00

Conclusions: Dilute impurities in the Haldane spin chain create sub-gap composite spin degrees of freedom. Edge states have an AFM wave function that extends into the bulk over distances of order the Haldane length. Holes in Y2-xCaxBaNiO5 are surrounded by AFM spin polaron with central phase shift of p Neutron scattering can detect the structure of composite impurity spins in gapped quantum magnets. The technique may be applicable to probe impurities in other gapped systems eg. high TC superconductors. Microscopic details of gapped spin systems may help understand related systems where there is no direct info. Viewgraphs posted at http://www.pha.jhu.edu/~broholm/jhuseminar/index.htm JHU 4/26-13/00