Conservation laws and magnon decay in quantum spin liquids

1. Conservation laws and magnon decay in quantum spin liquids Igor Zaliznyak Neutron Scattering Group, Brookhaven National Laboratory

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

3. Particles in the Universe GeV MeV

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

5. Long-lived quasiparticle (magnon)  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 quasiparticles.

6. 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

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

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

9. Breakdown of the excitations in 4He: experiment H = Sqε (q) aq+aq + Sq,q′Vq,q′(aqa+q′a+q-q′ + H.c.) + …

10. q’ q q” Roton decays and conservation laws • Breakdown of roton quasiparticle spectrum at E > 2 due to pair decays satisfies: • Particle non-conservation: cubic terms in the boson Hamiltonian => Vq,q′(aqa+q′a+q-q′ + H.c.) • Energy-momentum conservation q = q’ + q” e(q) = e(q’) + e(q”)

11. 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 → ∞ Quantum spin liquid: what is it? • 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

12. 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.

13. 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 

14. 1D array of dimers (aka alternating chain) Chains of weakly interacting dimers in Cu(NO3)2x2.5D2O Cu2+ 3d9 S=1/2 E (meV)

15. 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

16. 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

17. 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

18. Spin-quasiparticles in Haldane chains in CsNiCl3

19. Magnon quasiparticle breakdown in CsNiCl3 I. A. Zaliznyak, S.-H. Lee, S. V. Petrov, PRL 017202 (2001)

20. Spectrum termination in the dimer-chain material IPA-CuCl3 T. Masuda, A. Zheludev, et. al., PRL 96 047210 (2006)

21. 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

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

23. 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)

24. 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 M. B. Stone, I. Zaliznyak, T. Hong, C. L. Broholm, D. H. Reich, Nature 440 (2006)

25. Magnon breakdown: theory Zhitomirsky, PRB 73 100404R (2006) Kolezhuk and Sachdev, PRL 96 087203 (2006) Width appears at the crossing point Coherent magnon disappears

26. Spectrum end point in helium-4 and quantum spin liquid in PHCC M. B. Stone, I. Zaliznyak, T. Hong, C. L. Broholm, D. H. Reich, Nature 440 (2006)

27. Spectrum breakdown in quantum spin liquid in PHCC in magnetic field I. Zaliznyak, T. Hong, M. B. Stone, C. L. Broholm, D. H. Reich, unpublished gmBH Sz=+1 gmBH Sz=0 Sz=-1

28. Spectrum end point in PHCC in magnetic field: spin conservation

29. Summary • Quasiparticle spectrum breakdown at E > 2 is a generic property of quantum Bose (spin) fluids • Governed by conservation laws • Roton breakdown in He-4 • particle non-conservation • energy-momentum conservation • Magnon breakdown in quantum magnets • particle non-conservation • energy-momentum conservation • spin angular momentum conservation => apparent in magnetic field

30. How do neutrons measure excitations.

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

32. 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

33. Temperature dependence in copper nitrate

34. Temperature dependence in PHCC

35. 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

36. 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)

37. (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

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

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

40. 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)

41. 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)