II. Dynamically generated quantum degrees of freedom and their resonance & coherence properties

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1 II. Dynamically generated quantum degrees of freedom and their resonance & coherence properties

2 collaborators G-Y Xu (BNL) C.Broholm (Hopkins) J.F.diTusa(LSU) H. Takagi (Tokyo) Y. Itoh(Tsukuba) Y-A Soh (Dartmouth) M. Treacy (Arizona) D. Reich (Hopkins) D. Dender (NIST) Xu et al, Science 2000 and 2007; Kenzelmann et al, PRL 2003

3 outline Why one dimension is special Clean S=1 chain Isolated S=1 chain segments Coupled S=1 chain segments Conclusions

4 For the envelope function the Hamiltonian is Hydrogenic: 2 1 H = + 2m * ε r r For Si, εr=11.4, m*=0.3. Si Si P Si Si

5 Heisenberg antiferromagnet H=ΣJS i S j with J>0 classical ground state

6 But there is a problem with this picture

7 Consider just a spin pair

8 Exchange interaction E = - J S 1. S 2

9 Exchange interaction J>0 Ferromagnetic E = - J S 1. S 2

10 Exchange interaction J<0 Anti-Ferromagnetic E = - J S 1. S 2

11 Exchange interaction + quantum mechanics J>0 Ferromagnet Triplet Ground State Classical intuition OK

12 Exchange interaction + quantum mechanics J<0 Antiferromagnet Singlet entangled Ground State Classical intuition not OK

13 Exchange interaction + quantum mechanics J<0 Antiferromagnet Singlet entangled Ground State Classical intuition not OK

14 >, < > + > J > - >

15 Does this matter for chains, planes etc?

16 1. Consider commutator M fm =ΣS z l (ferromagnet) M af =Σ(-1) l S z l (antiferromagnet) [M,H]=... (-1) l ([S z l,s l ](S l-1 +S l+1 ) -([S z l-1,s l-1 ]+[Sz l-1,s l-1 ])S l ) for FM, [M,H]=0 while not so for AFM

17 2. Consider spin waves

18 2. Consider spin waves

19 spin wave amplitudes <Q S + 0> 2 Diverge as 1/Q when Qà magnetic zone center for AFM ~ constant for FM

20 Break-down of S-W theory total moment sum rule argument <M 2 >=S(S+1)=static piece + fluctuating piece <M 2 >= M o2 + δ(e-eo(q)) <Q S + 0> 2 ded d Q=M o2 + (1/Q)d d Q(AFM) (M o =ordered moment) clearly a problem for AFM in d=1 where (1/Q)d d Q~log(L) where L=system size

21 Experimental program Observe dynamics Is there anything other than Neel state and spin waves? Over what length scale do quantum degrees of freedom matter?

22 Pictures are essential can t understand nor use what we can t visualizedifficulty is that antiferromagnet has no external fieldneed atomic-scale object which interacts with spins Subatomic bar magnet neutron

23 Scattering experiments" k i,e i,σ i k f,e f,σ f Q=k i -k f hω=e i -E f Measure differential cross-section=ratio of outgoing flux" per unit solid angle and energy to ingoing flux = δ 2 σ/δωδω

24 Neutron scattering Fermi s Golden Rule at T=0, δ 2 σ/δωδω=σ f <f S(Q) + 0> 2 δ(ω-e 0 +E f ) where S(Q) + =Σ m S m+ expiq.r m for finite T δ 2 σ/δωδω= k f /k i S(Q,ω) where S(Q,ω)=(n(ω)+1)Imχ(Q,ω) S(Q,ω)=Fourier transform in space and time of 2-spin correlation function <S i (0)S j (t)>= dt Σ ij expiq(r i -r j )expiωt <S i (0)S j (t)>

25 Real space a Reciprocal Space S(Q) 0 π/a 2π/a

Neutron generation by proton accelerator Original Nucleus Excited Nucleus E p Proton Recoiling particles remaining in nucleus Emerging Cascade Particles (high energy, ~ E < E p ) (n, p.

26 Neutron generation by proton accelerator Original Nucleus Excited Nucleus E p Proton Recoiling particles remaining in nucleus Emerging Cascade Particles (high energy, ~ E < E p ) (n, p. π, ) (These may collide with other nuclei with effects similar to that of the original proton collision.) ~10 20 sec γ Evaporating Particles (Low energy, E ~ 1 10 MeV); (n, p, d, t, (mostly n) and γ rays and electrons.) Residual Radioactive Nucleus > 1 sec ~ e γ e γ Electrons (usually e + ) and gamma rays due to radioactive decay.

27 ISIS Spallation Neutron Source

28 ISIS - UK Pulsed Neutron Source

29 Moderator MAPS Anatomy t=0 Nimonic Chopper Fermi Chopper Sample Low Angle 3º-20º High Angle 20º-60º

30 Information 576 detectors 147,456 total pixels 36,864 spectra 0.5Gb Typically collect 100 million data points

31 S=1 AFM chain compound YBaNiO5

33 width =1/x o where x o ~7a S(Q)=S<S l S m >expi l-m Q equal-time correlation function = liquid structure factor no AFM order, only fluctuations

34 An unstable antiferromagnet

35 60 hω (mev) q ( π )

36 a gapped spin liquid (Haldane) Why? rationalization #1 S z = Zeros ignored

37 a gapped spin liquid (Haldane) Why? rationalization #2( valence bond solid )- consider J Hund in addition to J Ni-Ni J Ni-Ni J Hund Ni +2

38 J hund <<J Ni-Ni dispersionless VB state

39 VB state J Ni-Ni J Hund

40 VB state J Ni-Ni J Hund J hund <<J Ni-Ni J hund >>J Ni-Ni

41 Just a simple liquid? secret order(quantum coherence) in explanations, but apparently not visible in the equal-time two-spin correlation function <0 S - -q S+ q 0>= S(q,ω)δω can we measure coherence length for this new state?

42 60 hω (mev) q ( π )

43 S(q,ω)δω=FT <S i (t)s j (t)> S(q,ω=7.5meV)

44 Theory by Sachdev et al

46 Oscillator Q s > 100 dephasing time > 100 psecs Oscillator Q s > 100 dephasing time > 100 psecs

47 Mesoscopic phase(>15nm) phase coherence in quantum spin fluid as T 0, <triplet S + q collective singlet ground state> 2 δ(q -π) even while the 2-spin correlations in ground state are short-ranged: <0 S i S j 0>=exp- i-j /ξ where ξ~7 T=0 quantum coherence limited only by inter-impurity spacing dephasing at finite T observed

48 S=0 S=1 T=0 S=1 T>0

49 What happens when we cut the chains? via Mg substitution for Ni Ni 2+ - O 2- - Ni 2+ - O 2- - Mg 2+ - O 2- - Ni 2+...

50 S=1/2 Mg 2+

51 Can we detect this spin??! Magnetic resonance Magnetic field polarizes unpaired spins two energy levels for electron spin S = ½ Energy anti-aligned Zeeman term in Hamiltonian: H Zeeman = g m B B S aligned When microwave cavity energy hν = g µ B B, resonant absorportion B-Field

hν/gµb 2600 2800 3000 3200 30000 3400 3600 3800 CuPc

52 hν/gµb CuPc 100% parallel Empty EPR tube Intensity a) b) 3200 Field (G)

53 Can we detect this spin??! Magnetic resonance Magnetic field polarizes unpaired spins two energy levels for electron spin S = ½ Energy anti-aligned Zeeman term in Hamiltonian: H Zeeman = g m B B S aligned When microwave cavity energy hν = g µ B B, resonant absorportion B-Field

54 Can we detect this spin??! Magnetic resonance Magnetic field polarizes unpaired spins two energy levels for electron spin S = ½ Energy anti-aligned Zeeman term in Hamiltonian: H Zeeman = g m B B S aligned When neutron energy transfer hν = g µ B B, resonant absorportion B-Field

55 gµ B H

56 Structure Factor κ -1 =ξ=ξ Haldane ~8a o M. Kenzelmann et al. Physical Review Letters, 90, /1-4, (2003)

57 But J Hund >>J Ni-Ni antiferromagnetism survives on a length scale >lattice spacing edge states are more extended than single lattice spacing 1/κ=ξ Mg 2+ ξ=ξ Haldane ~8a o A dynamically generated S=1/2 nanomagnet!!

58 Subgap bound states in Ca-doped YBaNiO 5

59 G. Xu et al., Science, 289(5478), pp , (2000)

60 Ca-doping induces subgap resonance incommensurability which does not seem to depend on x sharper at low x net spectral weight well in excess(~4 times larger) of spectral Weight for S=1/2 one might associate with added hole

61 O - S=1/2 X S=1/2 X S=1/2 Strong coupling J O-Ni between oxygen & nickel spins à net ferromagnetic(no matter what is sign of J O-Ni ) bond of strength J O-Ni 2

62 S(Q)=cos 2 (Q) peaks at 2nπ, nodes at (2n+1)π

63 But J Hund >>J Ni-Ni 1/κ=ξ S( Q) = F ( Q) (1 + e κ ) cos Q / 2 cosh κ + cos Q 2

64 interference between left and right hand side of bound state wavefunction produces two incommensurate peaks centered around π κ=2/ξ

65 for finite(rather than infinitesimal) impurity density, interference effect no longer perfect, and node at π partially relieved

66 Immobile holes in 1-d quantum spin liquid nucleate subgap edge states Incommensurate structure factor - not from charge ordering Fermi surface etc. - but from delocalized quantum spin degree of freedom which extends over several Ni-Ni spacings into QSF and accounts for large spectral weight

67 Higher energies triplet sector

69 Finite size (box) confinement effects -limit spatial coherence induced both by static impurities as well as thermal excitations - a quantum effect seen on warming 2ξ Δ= Δ ο + (v/ξ) 2 ξ / a = (x + ρ T ( )) 1 / 2 ρ( T) = 3 k TΔ B 2π v exp & ( Δ 2 ' kb T ) + *

70 summary Antiferromagnets in 1d avoid classical order & display mesoscopic quantum effects with consequences for correlation functions directly measurable 1d magnets a good experimental laboratory for edge states in quantum systems Dynamically generated effective molecular nanomagnets from edge states in quantum spin fluids Phase coherence concepts valid even for these many-body wavefunctions

72 hω q A B C D E F

Quantum spin systems - models and computational methods

Summer School on Computational Statistical Physics August 4-11, 2010, NCCU, Taipei, Taiwan Quantum spin systems - models and computational methods Anders W. Sandvik, Boston University Lecture outline Introduction