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The quasiparticle spectrum termination in a quantum spin liquid

The quasiparticle spectrum termination in a quantum spin liquid. Igor Zaliznyak Neutron Scattering Group, Brookhaven National Laboratory. Particles. GeV. MeV. Quasiparticle: phonon, magnon q = k i - k f. meV, μ eV. neutron out k f. neutron in k i. Quasiparticles. 1 meV = 11.6 K.

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The quasiparticle spectrum termination in a quantum spin liquid

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  1. The quasiparticle spectrum termination in a quantum spin liquid Igor Zaliznyak Neutron Scattering Group, Brookhaven National Laboratory

  2. Particles GeV MeV

  3. Quasiparticle: phonon, magnon q = ki - kf meV, μeV neutron out kf neutron in ki Quasiparticles 1 meV = 11.6 K

  4. Undamped quasiparticle  delta-function singularity in cross-section nuclear scattering length, b ~ 10-13cm magnetic scattering length, rm = -5.39*10-13 cm Neutron scattering: how neutrons measure excitations.

  5. Gain up to factor 10 Gain up to factor 5 How neutrons measure excitations now. B. Brokhouse (1961) I. A. Zaliznyak and S.-H. Lee, in Modern Techniques for Characterizing Magnetic Materials, Ed. Y. Zhu, Springer (2005)

  6. How do neutrons measure excitations.

  7. What is liquid? • no shear modulus • no elastic scattering = no static correlation of density fluctuations ‹ρ(r1,0)ρ(r2,t)›→ 0 t → ∞ What is quantum liquid? • What is quantum liquid? • all of the above at T→ 0 (i.e. at temperatures much lower than inter-particle interactions in the system) • Elemental quantum liquids: • H, He and their isotopes • made of light atoms  strong quantum fluctuations

  8. ε(q) (Kelvin) whatsgoingon? q (Å-1) Excitations in quantum Bose liquid: superfluid 4He maxon roton phonon Woods & Cowley, Rep. Prog. Phys. 36 (1973)

  9. The “cutoff point” of the quasiparticle spectrum in the quantum Bose-liquid

  10. Montfrooij & Svensson, J. Low Temp. Phys. (2000) Fak & Bossy, J. Low Temp. Phys. (1998) Graf, Minkiewicz, Bjerum Moller & Passell, Phys. Rev. A (1974) Spectrum termination in 4He: experiments H = Sqε (q) aq+aq + Sq,q′Vq,q′(aqa+q′a+q-q′ + H.c.) + …

  11. Spectrum termination in 4He: experiment

  12. Coupled planes J||/J<<1 • no static spin correlations ‹Siα(0)Sjβ(t)›→ 0, i.e. ‹Siα(0)Sjβ (t)›= 0 • hence, no elastic scattering (e.g. no magnetic Bragg peaks) Coupled chains J||/J>> 1 t → ∞ What is quantum spin liquid? • Quantum liquid state for a system of Heisenberg spins H = J||SSiSi+||+ JS SiSi+D • Exchange couplings J||, Jthrough orbital overlaps may be different • J||/J >> 1 (<<1) parameterize quasi-1D (quasi-2D) case

  13. J0 > 0 triplet 0 = J0 singlet Simple example: coupled S=1/2 dimers Single dimer: antiferromagnetically coupled S=1/2 pair H = J0 (S1S2) = J0/2 (S1 + S2)2 + const.

  14. J0 J1 e(q) 0 = J0 q/(2p) Simple example: coupled S=1/2 dimers Chain of weakly coupled dimers H = J0 S(S2iS2i+1) + J1S (S2iS2i+2) Dispersione(q) ~ J0 + J1cos(q) triplet 

  15. Dimers in 1D (aka alternating chain) Chains of weakly interacting dimers in Cu(NO3)2x2.5D2O Cu2+ 3d9 S=1/2 E (meV)

  16. Weakly interacting dimers in Cu(NO3)2x2.5D2O D. A. Tennant, C. Broholm, et. al. PRB 67, 054414 (2003) Spin excitations never cross into 2-particle continuum and live happily ever after

  17. ground state has static Neel order (spin density wave with propagation vector q = p) • quasiparticles are gapless Goldstone magnons e(q) ~ sin(q) Sn = S0 cos(p n) n n+1 e(q) • elastic magnetic Bragg scattering at q = p q/(2p) What would be a “spin solid”? Heisenberg antiferromagnet with classical spins, S >> 1

  18. short-range-correlated “spin liquid” Haldane ground state • quasiparticles with a gap  ≈ 0.4J at q = p e2 (q) = D2 + (cq)2 Quantum Monte-Carlo for 128 spins. Regnault, Zaliznyak & Meshkov, J. Phys. C (1993) e(q) 2  q/(2p) 1D quantum spin liquid: Haldane spin chain Heisenberg antiferromagnetic chain with S = 1

  19. Ni2+ 3d8 S=1 chains J = 2.3 meV = 26 K J = 0.03 meV = 0.37 K = 0.014 J D = 0.002 meV = 0.023 K = 0.0009 J 3D magnetic order below TN = 4.84 K unimportant for high energies Spin-quasiparticles in Haldane chains in CsNiCl3

  20. Spin-quasiparticles in Haldane chains in CsNiCl3

  21. Spectrum termination point in CsNiCl3 I. A. Zaliznyak, S.-H. Lee, S. V. Petrov, PRL 017202 (2001)

  22. strong interaction weak interaction   2D quantum spin liquid: a lattice of frustrated dimers M. B. Stone, I. Zaliznyak, et. al. PRB (2001) (C4H12N2)Cu2Cl6 (PHCC) Cu2+ 3d9 S=1/2 • singlet disordered ground state • gapped triplet spin excitation

  23. Single dispersive mode along h • Single dispersive mode along l • Non-dispersive mode along k PHCC: a two-dimensional quantum spin liquid • gap D = 1 meV • bandwidth = 1.8 meV

  24. Quasiparticle spectrum termination line in PHCC max{E2-particle (q)} min{E2-particle (q)} Spectrum termination line E1-particle(q)

  25. Q = (0.15,0,-1.15) 200 resolution-corrected fit 800 150 600 100 400 50 200 150 Q = (0.1,0,-1.1) 0 resolution-corrected fit 0 100 400 50 300 0 200 200 Q = (0,0,1) 120 resolution-corrected fit 100 150 80 100 0 40 50 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 PHCC: dispersion along the diagonal Q = (0.5,0,-1.5) resolution-corrected fit Intensity (counts in 1 min) Q = (0.25,0,-1.25) resolution-corrected fit Q = (0.15,0,-1.15) resolution-corrected fit E (meV) E (meV)

  26. 1.0 1.5 2.0 2.5 3.0 log(intensity) 7 6 5 E (meV) 4 3 2 1 0 0.20 6 Total a 5 Triplon 4 Continuum 0.15 3 Integrated int (arb.) (meV) 0.10 2 G 0.05 100 9 0 8 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.4 0.3 0.2 0.1 0 (0.5,0,-1-l) (h,0,-1-h) (h 0 -1-h) 2D map of the spectrum along both directions

  27. Compare: spectrum end point in helium-4

  28. Summary and conclusions • Quasiparticle spectrum termination at E > 2 is a generic property of quantum Bose (spin) fluids • observed in the superfluid 4He • observed in the Haldane spin chains in CsNiCl3 • observed in the 2D frustrated quantum spin liquid in PHCC • A real physical alternative to the ad-hoc “excitation fractionalization” explanation of scattering continua • Implications for the high-Tc cuprates: spin gap implies disappearance of coherent spin modes at high E

  29. OAK RIDGE NATIONAL LABORATORY / U. Virginia Thanks to collaborators • M. B. Stone • C. Broholm, D. Reich, T. Hong • S.-H. Lee • S. V. Petrov

  30. Temperature dependence in PHCC

  31. Temperature dependence in copper nitrate

  32. 800 Q = (0.5,0,-1.5) resolution-corrected fit 600 400 200 0 Q = (0.5,0,-1.15) 400 Q = (0.5 0 -1) 300 resolution-corrected fit resolution-corrected fit 300 200 200 100 100 0 0 1 2 3 4 5 6 7 400 Q = (0.5,0,-1.1) resolution-corrected fit 300 200 100 0 Dispersion along the side (l) in PHCC Intensity (counts in 1 min) E (meV)

  33. (h, 0, 1.5) (0.5, 0, l) T=1.4K h = 0.6 l = 1.5 (0, k, 0.5) Intensity (counts/min) h = 0.7 l = 1.6 k = 0.5 Intensity (counts/min) h = 0.8 l = 1.8 k = 0.75 k = 1.0 PHCC: a two-dimensional quantum spin liquid • D = 1 meV, bandwidth = 1.8 meV • Single dispersive mode along H • Single dispersive mode along L • Non-dispersive mode along K

  34. nuclear scattering length, b ~ 10-13cm magnetic scattering length, rm = -5.39*10-13cm Neutron scattering cross-section

  35. Quasiparticle (undamped)  singularity in cross-section (delta-function) Quasiparticle cross-section

  36. I. A. Zaliznyak and S.-H. Lee, in Modern Techniques for Characterizing Magnetic Materials, Ed. Y. Zhu, Springer (2005) How do neutrons measure quasiparticles. B. Brokhouse (1961)

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