Quantum Magnetism - Neutrons in the Quasi-particle Zoo Henrik Moodysson Rønnow Laboratory for Quantum Magnetism (LQM), EPFL, Switzerland Niels Bohr Institute, University of Copenhagen, Denmark Henrik M. Ronnow – HERCULES 2015 Slide 1 Outline • Quantum Magnetism – Arena for many-body physics and highly correlated materials – Models-Materials-Measurements • Neutron scattering – Basics, uniqueness, and a bright future – The quasi-particle zoo • Selected examples – Multi-spinons in one-dimensional chains – Spin-wave anomaly and quest for pairing in 2D Henrik M. Ronnow – HERCULES 2015 Slide 2 Complexity of many-body systems CuO S=1/2 • Structure of a protein Ground state of a magnet 2 H = J Si Sj 1 spin: trivial 2 spins: singlet state |↑↓ - |↓↑ 4 spins: back-of-the-envelope calc. 16 spins:10 seconds on computer (4GB) 40 spins: World record:1’099’511’627’776 coefficients needed to describe a state Classical: 3N Quantum: 2N 1023 spins: 1D: analytic solution (Bethe 1931) • Pop2p-subunit Jonstrup et al (2007) 2D: antiferromagnet (Néel 1932) or • Mega-Dalton: fluctuating singlets? (Anderson 1973,1987) ~1’000’000 atoms (5 colors?) 23 ~3’000’000 numbers needed 10 ±some electrons:High-Tc superconductivity to describe the structure – THE enigma of modern solid state physics Henrik M. Ronnow – HERCULES 2015 Slide 3 Spin – the drosophila of quantum physics Spin: an atomic scale magnetic moment • Quantization: S=0, 1/2, 1, 3/2,…..∞ +1/2 S = 1/2 - 1/2 • Superposition: |ψ = |↑ + |↓ likelihood of up: ρ(↑) = |↑|ψ|2 = 2 • Quantum fluctuations average moment Sz = 0 imagine that spins fluctuate in ‘imaginary time’ • Quantum correlations e.g. two spins ‘entangled’ |ψ = ( |↑↓ - |↓↑ ) /√2 this is why 2N, not N Henrik M. Ronnow – HERCULES 2015 Slide 4 Quantum Magnetism – an arena for quantum phenomena 1) Model and Materials 3) Experimental tools: Spin, interactions Bulk probes The dimension Neutron scattering frustration Phenomena: Order, phase transitions, quantum fluctuations, collective excitations, 2) Theoretical methods of many body physics entanglement … analytic approximations numerical simulations Henrik M. Ronnow – HERCULES 2015 Slide 5 Magnetic measurements CuGeO Hsat (d6-5CAP)2CuCl4 3 2DHAF CuGeO3 (Hpip)2CuBr4 SR etc. Specific Specific heat SR etc. μ Magnetization Susceptibility Magnetization NMR, Henrik M. Ronnow – HERCULES 2015 Slide 6 A unique tool: Neutron scattering scattering and conservation rules Conservation rules: Sample Neutron Momentum ħQ = ħki – ħkf 2 2 Energy ħω = Ei – Ef = ħ(ki – kf )/2mn Spin S = σi – σf We can control and measure these quantities ! Henrik M. Ronnow – HERCULES 2015 Slide 7 Large scale instruments and facilities Ch. Ruegg, PSI Ch. Ruegg, Sample Source CuSO4·5D2O ILL - France MAPS 16m2 detector bank Henrik M. Ronnow – HERCULES 2015 Slide 8 Neutron scattering – an intense future • 1st generation facilities: European Spallation Source (ESS) – General purpose research reactors - All eyes are on Lund – congratulations ! nd • 2 generation facilities: Switzerland will contribute 3-4% – Dedicated to neutron scattering: CH-DK collaboration on 5 instrument – ILL, France, FRM2 Munich, SINQ CH, ISIS, UK etc. design workpackages 2 4 • 3rd generation facilities: CAMEA: 10 -10 over current instr. – SNS, US 1.4b$, commission 2006 – J-Parc, Japan 150b¥, commission 2008 – ESS, Sweden 1.4b€, start 2013, commission 2019 – China Spallation, start 2011*, commission 2018 • 2nd to 3rd generation gains of 10-1000 times ! – Faster experiments, smaller samples, better data – Time resolved physics, new fields of science – New instrument concepts SNS Henrik M. Ronnow – HERCULESJ-Parc 2015 ESS Slide 9 Neutron scattering cross-section – the power of simplicity From initial state i to final state f of neutron k and sample λ Neutrons treated as plane waves: Fourier transform in - space/momentum Energy conservation integral rep.: - time/energy Henrik M. Ronnow – HERCULES 2015 Slide 10 Magnetic neutron scattering Dipole interaction – electron spin and orbit moment spin-spin magnetic pre factor dipole factor Fourier transform cross-section form factor correlation function Henrik M. Ronnow – HERCULES 2015 Slide 11 Dynamic structure factor Spin-spin correlation function Dynamic structure factor Theory ! Fluctuation dissipation theorem gen. susceptibility intrinsic dynamics response to perturbation Henrik M. Ronnow – HERCULES 2015 Slide 12 Structure factors – time and energy • Dynamic structure factor: inelastic Elastic ) w Inelastic Quasi-elastic Q, – periodic: sin(ω0t) peak: δ(ω0-ω) S( decay: exp(-t/) Lorentzian: 1/(1+ω22) • Static structure factor: elastic Energy Transfer – Bragg peaks at ω = 0 • Instantaneous structure factor - integrate over energy – Finite time/length scale of correlations Henrik M. Ronnow – HERCULES 2015 Slide 13 Remark: instantaneous correlations Width Correlation length ξ Softening Damping v Z s c Γ = vs / ξ vs → Correlations and fluctuations on the 2D ξ square lattice Heisenberg antiferromagnet: Life-time J. Mag. Mag. Mat. 236, 4 (2001) PRL 82, 3152 (1999); 87, 037202 (2001) τ = ξ / vs =1/ G Henrik M. Ronnow – HERCULES 2015 Slide 14 Inelastic magnetic scattering: Lets take the scenic route… Between long range ordered states Selected examples – the zoo : • Spin-flip, singlet-triplet, dispersive triplets • 1D spin chain – spinons vs spin waves • 2D HAF zone boundary anomaly – as instability of spin waves ? – the smoking gun of RVB ? Aim: • Show the many types of quasiparticles • Show quantitativeness of neutron scattering … and spin liquids Henrik M. Ronnow – HERCULES 2015 Slide 15 paramagnetic spins S=1/2 Spin-flip excitation 5 T • Two states |↑, |↓, can be magnetized 1 • Zeemann-split energy of the levels gμBH=0.7 meV • A gap for transitions 0 E 1 |↓> 3 T gμBH gμBH=0.4 meV H Energy Energy [meV] 0 |↑> 0.8 1 T • Local excitation gμ H=0.13 meV no Q-dependence 0 B 0 Q π Henrik M. Ronnow – HERCULES 2015 Slide 16 Take two – the spin pair H = J Si Sj J No magnetization or Antiferromagnetic: J > 0 susceptibility up to critical field |↑↑, |↓↓, |↑↓ + |↓↑ E |1,1> E = 3/4J Stot=1 J triplets |1,0 > |0,0 >H -1/4J S =0 singlet |1,-1 tot Hc1 > |↑↓ - |↓↑ z z Singlet ground state: S 1 = S 2 = 0 Henrik M. Ronnow – HERCULES 2015 Slide 17 Singlet-Triplet excitations E Structure factor along pairs: S(q) ~ cos(d||·q) + cos(d ·q) |1,1> |1,0 > et al. |0,0 >H |1,-1 Demmel H > 2 c1 ) 3 , HMR, HMR, , Cu(BO 2 Ba Rüegg E = 3/4J Stot=1 triplets et al. singlet HMR HMR , 2 ) 3 Rüegg (BO 2 SrCu Zayed, Henrik M. Ronnow – HERCULES 2015 Slide 18 The spin-ladder – array of spin pairs k Perturbation from isolated rungs: Jr J l Ground state ≈ product of singlets E Excited states are triplets t±(r), t0(r) 2Jl ≈Jr-Jl Leg coupling Jl makes triplets move k 1/2 Create Bloch-waves of triplets t(k) 0 T < H < Hc1= ∆ spin liquid phase Dispersion Ek = Jr+Jl cos kd Jl Jr Real ground state has singlet fluctuations Jl – renormalisation of above result Henrik M. Ronnow – HERCULES 2015 Slide 19 1- and 2-triplet dispersion in Sr14Cu24O41 Different structure- triplet E=150 meV E=240 meV factor perp to ladder Braden, Regnault, Eccleston, Bewley Eccleston, Regnault, Braden, Henrik M. Ronnow – HERCULES 2015 Slide 20 Spin ladders • Sr14Cu24O41: Cuprate ladders, 1- and 2-triplons Neutrons Resonant Inelastic X-ray Scattering RIXS Henrik M. Ronnow – HERCULES 2015 Slide 21 Dynamics Luttinger-Liquid E m Ladder → Chain mapping 1 |1,1> ~ J /J >> 1 ~ band- 1 ( ) r l width ½ 2 | H Heff 1,0> H | H 0,0> H c1 c2 F. Mila, Eur. Phys. J. B 6 (1998) |1,-1> Jl Jr Jl Hc1 < H < Hc2 : Luttinger liquid no 3D long-range order Search Ch Ruegg et al. unique 1D dynamics for refs on beautiful ladder expts Henrik M. Ronnow – HERCULES 2015 Slide 22 Quasi-particle zoo in one-dimension Electronic states of matter: Metal / Semiconductor / Insulator Single particle picture Superconductors: Cooper-pairs, Majorana fermions Correlated electron states fractional Quantum Hall effect: fractional charges Magnetic states and excitations: Magnetic order semiclassical spin-wave magnon excitations single particle picture Quantum ‘disordered’ states (quantum spin liquids) Multi-magnon excitations collective quantum states Fractionalized excitations Possibly simplest example: 1D Heisenberg chain Analytic solution by Bethe in 1931: ‘domain wall quantum soup’ Henrik M. Ronnow – HERCULES 2015 Slide 23 Ferromagnets are simple (classical) z z + - - + H=-∑rr’Jrr’Sr·Sr’ = -J ∑<r,r’=r+d> S rS r’+ ½(S rS r’+S rS r’) nearest neighbour 2 Ordered ground state, all spin up: H|g> = Eg|g>, Eg=-zNS J -½ - - + Single spin flip not eigenstate: |r> = (2S) S r|g>, S rS r’|r> = 2S|r’> 2 H|r>=(-zNS J+2zSJ)|r> - 2SJ∑d |r+d> flipped spin moves to neighbors -½ ikr Periodic linear combination: |k> = N Σre |r> plane wave ikd Is eigenstate: H|k> = Eg+Ek|k>, Ek=SJΣd1-e dispersion = 2SJ (1-cos(kd)) in 1D iHt iE t Time evolution: |k(t)> = e |k> = e k |k> sliding wave Dispersion: relation between time- and space- modulation period Same result in classical calculation precession: Henrik M. Ronnow – HERCULES 2015 Slide 24 Ferromagnetic model is simple: Solution: Spin waves sharp dispersion Picture: easy cartoon https://www.ill.eu/?id=11644 Henrik M. Ronnow – HERCULES 2015 Slide 25 Spin waves in a “ferromagnet” CuSO45D2O dispersion = 2SJ (1-cos(kd)) Actually it is an antiferromagnet polarized by 5T field Henrik M. Ronnow – HERCULES 2015 Slide 26 Antiferromagnets are tricky Fluctuations stronger for fewer neighbours 1D: Ground state ‘quantum disordered’ spin liquid of S=1/2 spinons. Bethe ansatz ‘solves’ the model 2D: Ground state ordered at T=0 <S> = 60% of 1/2 (although not rigorously proven). 3D: Ground state long range ordered, weak quantum-effects Henrik M. Ronnow – HERCULES 2015 Slide 27 antiferromagnetic spin chain = 0 Ferro <<S2 Classical AF ~ S2 Quantum AF Ground state (Bethe 1931) – a soup of domain walls Henrik M.