Holes in a Quantum Spin Liquid

1
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

Ca2
Ca2
Y2-xCaxBaNiO5
Viewgraphs posted at http//www.pha.jhu.edu/broh
olm/canada/index.htm
2
Collaborators
Guangyong Xu JHU -gt University of Chicago G.
Aeppli NEC J. F. DiTusa Louisiana State
University I. A. Zaliznyak JHU -gt 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 Publication
on Y2-xCaxBaNiO5 to appear in Science (2000)
3
Quantum effects in atoms and in solids
Atom (He) Line spectrum
G. Dieke et al. (1968)
Solid (Mo) continuous spectrum if Q not specified
Christensen (1980)
4
Quantum effects from a gap
Empty
Band gap only possible for even number of
electrons per cell
Filled
Gap Insulator
No gap Metal
5
Impurities create states in the gap
Cleaner sample
Insulator
Resistivity (ohm-cm)

Metal
Dirty sample
0 0.2 0.4 0.6
0.8 1/T (K-1)
From Aschroft Mermin
6
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 TltltD
• Quantum Hall effects
• Uniform integer spin chains

7
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)
8
Spin systems of recent interest
9
Varying the dimensionality of spin systems

• The crystals are actually three dimensional
• Exchange links spins on low-D network only

Chain direction
Non-magnetic spacer molecules in NDMAP
Anisotropic bonding in cubic KCuF3
10
Susceptibility of gapped spin system
Measure magnetization versus T in infinitesimal
field
Tatsuo et al. (1995)
Renard et al (1988)
11
Magnetization of gapped spin system
Ajiro et al. (1989)
12
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)
13
Magnetic Neutron Scattering
The scattering cross section is proportional to
the Fourier transformed dynamic spin correlation
function
14
NIST Center for Neutron Research
15
(No Transcript)
16
Detection system on SPINS neutron spectrometer
17
Neutron scattering from gapped spin chain
Copper nitrate T0.3 K
gap
Xu et al. IRIS, ISIS
Nuclear incoherent scattering
Triplet creation peak
Proton extraction pulse
18
Dispersion relation for triplet waves
Dimerized spin-1/2 system copper nitrate
Xu et al PRL May 2000
19
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

20
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 2.1/21
  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
21
Low T excitations in spin-1 AFM chain
pure

• Haldane gap D8 meV
• Coherent mode
• S(q,w)-gt0 for Q-gt2np

22
AKLT state for spin-1 chain

• Magnets with 2Snz 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
23
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)
24
Single mode approximation for spin-1 chain
Dispersion relation
Equal time correlation function
25
Haldane mode in Y2BaNiO5 at finite T

• Relaxation rate increases
• with T due to
• triplet interactions
• Resonance energy increases
• with T due to
• decreasing correlation length

Low T fine structure from spin anisotropy
26
T-dependence of relaxation rate and resonance
energy

• Parameter free comparison
• Semi-classical theory of triplet
• scattering by Damle and Sachdev

ö
æ
D
T
k
3
(
)

ç
-

G
B
0
T
exp

ç
T
k
p
ø
è
B

• Quantum non linear s model

ö
æ
D
(
)

ç
-
D

D

D
p
T
0
T
k
exp
2

ç
B
0
0
T
k
ø
è
B
27
Q-scans versus T energy resolved and energy
integrated
D
³
w
h
Probing equal time correlation length
D

w
h
Probing spatial coherence of Haldane mode
28
Coherence and correlation lengths versus T
Equal-time correlation length saturates at x8.

(Solid line from Quantum non linear s model)
29
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
  Tlt D/kB

30
Impurities in Y2BaNiO5

• Mg2on Ni2 sites finite length
  chains
• Ca2 on Y3 sites mobile bond defects

Mg
Ca2
Ni
Y3
Kojima et al. (1995)
31
Zeeman resonance of chain-end spins
g2.16
20
hw (meV)
15
0 2 4 6 8
H (Tesla)
10
I(H9 T)-I(H0 T) (cts. per min.)
0
-5
0 0.5 1 1.5
2
32
Form factor of chain-end spins
Y2BaNi1-xMgxO5 x4
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
33
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

34
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

x
q
µ
d
x
35
Does dq vary with calcium concentration?
dq not strongly dependent on x
Double peak is single impurity effect
36
Bond Impurities in a spin-1 chain Y2-xCaxBaNiO5
Y
Ba
(a)
O
Ni
37
Form-factor for FM-coupled chain-end spins
A symmetric AFM droplet
Ensemble of independent randomly truncated AFM
droplets
38
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 0.04ltxlt0.14
  (and Tlt100 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

39
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
5
?

• How to distinguish
• Neutron scattering in an applied field
• Transport measurements
• Theory

0
40
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/broh
olm/canada/index.htm