Frustration and Field-driven Quantum Criticality
Collin Broholm
Johns Hopkins University and NIST Center for Neutron Research, USA
 Frustrated magnets close to a QCP
—Lattice of triangles
—Kagome sandwich
—Spinel AFM
 Field driven QCP in Gapped Spin Systems
— Spin-1 chain
— Frustrated AFM bi-layer
Thanks to many Collaborators (yesterday’s talk)
G. Aeppli
Y. Chen
P. Hammar
M. Kenzelmann
C. P. Landee
Seunghun Lee
K. Lefmann
Y. Qiu
D. H. Reich
C. Rische
M. B. Stone
H. Takagi
M. M. Turnbull
G. Xu
ICTP 10/20-26/03
UCL
LANL
formerly at JHU
JHU & NIST
Clarke University
NIST
Risø National Lab
NIST & Univ. Maryland
JHU
Univ. of Copenhagen
Penn State University
ISSP
Clarke University
BNL
Thanks to many Collaborators (today’s talk)
M. Azuma
R. Bewley
W. J. L. Buyers
Y. Chen
S. W. Cheong
D. V. Ferraris
G. Gasparovic
Q. Huang
S. Ishiwata
M. Kibune
T. Lectka
ICTP 10/20-26/03
Kyoto
ISIS Facility
Chalk River
JHU
Rutgers
JHU
JHU
NIST
Kyoto
Tokyo
JHU
S. H. Lee
M. Nohara
Y. Qiu
W. Ratcliff
D. H. Reich
J. Rittner
M. B. Stone
H. Takagi
M. Takano
H. Yardimci
I. A. Zaliznyak BNL
NIST
Tokyo
JHU
NIST
JHU
JHU
JHU
Tokyo
Kyoto
JHU
T/J
Si
0
Quantum Critical
Conceptual Phase Diagram for Quantum Magnets
Si
0
1/S, frustration, 1/z, H, P, x, …
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Magnetic Frustration
Interacting spin pairs cannot simultaneously
be in their lowest energy configuration
Frustrated
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Progression of near quantum critical models
La4Cu3MoO12
Spinel AFM
Kagome Slab
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La4Cu3MoO12: A lattice of spin-1/2 trimers
Magnetic susceptibility
Crystal Structure
z=3/4 CuMoO plane
(Azuma et. al., PRB 62 R3588)
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Frustrated quantum spin triangles
J
J2
J1
J
3
J3
J
J
2
  J S 1  S 2  S 2  S 3  S 3  S 1 
3
  J 1S 1  S 2  J 2 S 2  S 3  J 3S 3  S 1
3
J
J 
J
2
2
2
J
Yiming
Qiu et al. cond-mat/0205018
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J 
1
2
 J
 J 2   J 2  J 3   J 3  J1 
2
1
2
2

Spectroscopy of spin trimers
0.2
10 K
Transition to quartet
0.1
0.0
70 K
0.1
0.0
Yiming
Qiu et al. cond-mat/0205018
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J
Magnetic Ordering of Composite spin-1/2
Q 
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 12 00 
Strongly fluctuating spin trimer AFM
300K
Yiming
Qiu et al. cond-mat/0205018
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2.6K
AFM on kagome’ sandwich
I. S. Hagemann et al. PRL (2001)
QCW=-500 K but no phase transition for T>4 K
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Decoupled Q- and w-dependence
S Q w  
1
 g B 

2
 Q   A T  1 

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 Q n ( w )  1 
sin Qd 

Qd 
w
 w
2
2
AFM interactions satisfied w/o LRO
 

JS i  S
j
ij

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1
2

JS
2
 cst
Local Spin relaxation rate
T
→0
a
SCGO
QS Ferrite
SCGO:
a=0.71(2)
QS-Ferrite: a=0.67(2)
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I. S. Hagemann et al. PRL (2001)
What side of the QCP?
C~T2
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Linearly Dispersive mode in 2D
Clandestine Long Range Order?
Low T Spin freezing
DC susceptibility
“AC susceptibility”
~250 GHz
~250 MHz
C~T2 may come from “spin-waves” formed by
twisting frozen spins of satisfied tetrahedra
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AFM on lattice of corner-sharing tetrahedra
Q CW
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Compound Spin QCW (K) Order
1
MgTi2O4
Singlet?
2
MgV2O4
ZnV2O4
CdCr2O4
1
1
MgCr2O4
ZnCr2O4
TC (K)
260
-750
-600
-83
Orbital/AFM -/45
Orbital/AFM -/40
AFM
9
3
2
-350
AFM
15
3
2
-392
AFM
12.5
3
2
Interactions satisfied w/o LRO
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Average form factor for AFM hexagons
+
▬
nˆ
▬
+
▬
+
I Q  

Fnˆ Q 
2
nˆ  111

 

 
  sin
h  cos
k  cos l  
2 
2
2 

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S.-H. Lee et al. Nature (2002)
2

 

 
  sin
k  cos l  cos
h 
2
2
2



2

 

 
  sin
l  cos
h  cos
k 
2 
2
2 

2
Tchernyshyov et al. PRL (2001)
A possible interpretation
 Physics at the scale of |QCW| order spins
antiferromagnetically on hexagons
 Staggered magnetization of hexagons is
effective low energy degree of freedom
 System is transformed from strongly
correlated spins to weakly correlated
“hexagon directors”
 Neutrons scatter from hexagon directors not
individual spins
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Why AFM hexagons?
 Low energy manifold has
zero spin tetrahedra
 Spins on tetrahdra form
hinged parallelograms
 Spins on hexagons form
cart-wheel
 Hexagons decouple when
Antiferromagnetic
 AFM hexagons account
for 1/6 of spin entropy
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Instabilities close to QCP
T/J
Si
0
Si
?
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0
1/S, frustration, H, P, x, …
TN<T<|QCW| : Dynamic Short Range Order
Points of interest:
• 2/Qr0=1.4
⇒ nn. AFM correlations
• No scattering at low Q
⇒ satisfied tetrahedra
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S.-H. Lee et al. PRL (2000)
T<TN : Resonant mode and spin waves
Points of interest:
• 2/Qr0=1.4
⇒ nn. AFM correlations
• No scattering at low Q
⇒ satisfied tetrahedra
• Resonance for ħw ≈ J
• Low energy spin waves
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S.-H. Lee et al. PRL (2000)
Magneto-elastic first order transition
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Straining to order
Edge sharing n-n exchange in ZnCr2O4 depends on Cr-Cr distance, r .
J
  40 meV

/A
r
The implication is that there are forces between Cr3+ atoms
F12    J 12 S 1  S 2
Cr3+
O2-
   rˆ
12
 J 12
r
S1  S 2
Cr3+
O2-
These magneto-elastic interactions destabilize QC spin system on compliant lattice
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Tchernyshyov et al. PRL (2001) and PRB (2002)
Sensitivity to impurities near quantum criticality
TN
Tf
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10/20-26/03
Ratcliff
et al. PRB (2002)
Conclusions part#1
 Frustration and weak connectivity can greatly
suppress TN in real materials
 AFM interactions satisfied to the extent
possible without LRO
 The local spin relaxation   T a , a  0 .7  0 .8
 A description in terms of fluctuating composite
degrees of freedom appears to be relevant
 High sensitivity to various perturbations:
- Impurities yield spin glass type state
- Lattice distortions can induce Neel order
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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) =
Integer:
gap possible
Non-Integer: gap impossible
Oshikawa, Yamanaka, and Affleck (1997) and Oshikawa (2000)
 gaps in non-magnetized spin chains?
-
Uniform spin ½ chain
1.½= ½
Alternating spin ½ chain 2.½= 1
(2n+1) leg spin ½ ladder (2n+1).½ = n+½
2n leg spin ½ ladder
2n.½ = n
Uniform spin 1 chain
1.1= 1
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no gap
perhaps
no gap
perhaps
perhaps
Haldane gap in Y2BaNiO5
MAPS (ISIS)
hw (meV)
60
40
1-cosq
S(qw)~
(w-e(q))
e(q)
Impure
20
Pure
0
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0
0.5
1
q ()
1.5
2
Two length scales in a quantum magnet
Equal time correlation length
S  q~     S  q~ , w  dw
hw (meV)
60
1
~
S q  
N

S0Sl 
1
40
exp   l 


l
Triplet Coherence length :
length of coherent triplet
wave packet
20
0
ll 
S l S l  exp i q~ l  l  
0
0.5
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1
q ()
1.5
2
Coherence in a fluctuating system
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w  D
Short range G.S.
spin correlations
w  D
Coherent triplet
propagation
Coherence and correlation lengths versus T
Damle and Sachdev
theory of triplon scattering
Including impurity scattering
Jolicoeur and Golinelly
Quantum non-linear s model
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Macroscopic singlet ground state of S=1 chain
• Magnets with 2S=nz have a nearest neighbor singlet covering
with full lattice symmetry.
• This is exact ground state for spin projection Hamiltonian

H   Pi  S tot  2    S i  S i  1 
i
i
1
3
S i  S i  1 
2
  S
i
 S i 1
i
• Excited states are propagating bond triplets separated from
the ground state by an energy gap D  J .
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Haldane PRL 1983
Affleck, Kennedy, Lieb, and Tasaki PRL 1987
Form factor for chain-end spin
Kenzelmann et al. PRL (2003)
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Spin-1 chains a’la carte
H  J
 D


nm
S n , m  S n  1, m
nm
S 
2
 E
nm
 S
x
n ,m
  S 
2
y
n ,m
J 'm m ' S n , m  S n , m '
nm m '
Chemical Formula
AgVP2S6
Y2BaNiO5
Ni(C3H10N2)2N3(ClO4)
Ni(C3H10N2)2NO2(ClO4)
Ni(C2H8N2)2NO2(ClO4)
Ni(C5D14N2)2N3(PF6)
CsNiCl3
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z
n ,m
Intra-chain exchange
Common name
NINAZ
NINO
NENP
NDMAP
2
Anisotropy

Inter-chain exchange
J
meV
58(4)
21
10.7
4.5
4.1(3)
2.85
2.275
D/J
10-3
5.8
-39
170
250
180
250
-1.9
E/J
10-3
12
gBH/J
0.020
0.055
0.11
0.21
0.28
0.41
0.45
J’/J
10-3
0.01
< 0.5
<0.7
0.8
0.6
17
TN
K
<2
< 0.05
<0.06
<1.2
<0.0003
<0.25
4.9
Spin-1 chains that can be magnetized
NENP=Ni(C2H8N2)2NO2ClO4
h eff 
NDMAP=Ni(C5H14N2)2N3(PF6)
g B H
J
[ClO4]-
b
c
Ni(en)2
NO2
Staggered g-tensor
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No staggered g-tensor
q= excitations versus H in NENP
NENP
0T
Enderle et al. Physica B (2000)
T=35 mK
12 T
13 T
14.5 T
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Data from BENSC, Hahn-Meittner Institute
NENP with staggered g-tensor: Statics
Intensity (103 cts/min.)
3
(110) B=2 T
2
1
0
0
2
4
6
8
10
T (K)
Applied field breaks translational symmetry
Cross-over instead of Quantum Phase Transition
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True Critical Point in NDMAP
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NDMAP without staggered field: Statics
Chen et al., PRL (2001)
Quasi 2D
H||a
H||b
3D LRO
Haldane Singlet
Data10/20-26/03
from NIST
ICTP
Center for neutron research
/max
Singlet Ground state in PHCC
J1=12.5 K
a=0.6
T J1
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Daoud et al., PRB (1986).
Structure of C4H12N2Cu2Cl6=PHCC
N
c
c
C
a
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Cl
b
Cu
w (meV)
2D dispersion relation
1
0
h
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1
0
l
Zeeman splitting of cooperative triplet
PHCC T=60 mK
GS-level crossing for H8 T
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Quantum phase transition
Non-linear Magnetization Curve
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H-T Phase Diagram from Magnetization
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Intensity
Field-induced AFM Order
H=14.5 T
T=1.77 K
>300 c
Q  ( 12 , 0 , l )
cˆ
aˆ
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Bragg Intensity  M2
Order Parameter Critical Exponent
=0.4 (1)
Compare to =0.355 for 3D X-Y model
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Gapless paramagnetic phase
Gap closes
Gapless paramagnet?
Onset of 3D LRO
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H-T phase diagram
2D Gapped FM
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PHCC
Scaling analysis near QCP
Low T data consistent with
500
500
T =2K
T =2K
T =3K
T =3K
T =4K
T =4K
400
400
  w   vq 
 G 

S q w  
F 
k BT  k BT   k BT 
1
PHCC
H=Hc
300
300
200
200
100
Intensity * T
Intensity * T
T =5K
100
PHCC
H = 7 .5 T
0
1 .0
E n e rg y /T
1 .5
0 .5
1 .0
150
T = 4 K E = 0 .5 m e V
1 .5
E n e rg y /T
In te n s ity * T
0 .5
T = 2 K E = 0 .2 5 m e V
0
100
50
0
-2
-1
0
(h -0 .5 )/h 0
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1
2
H-T phase diagram
PHCC
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Reentrant low T transition?
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- Or some form of cross over close to QCP?
gapless
3 D long range order
Spin gap
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Possible effects that cut off critical behavior
 Effective staggered field from alternating
coordination and DM interaction
- Caused transition from LL to QsG for spin-1/2 chain
 Magneto-elastic effects
- like spin-Peierls transition for spin-1/2 chain
 Coupling to copper nuclear spin system
- As for transverse field Ising model in LiHoF4
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Conclusions part #2
 Haldane spin-1 chain:
- Coherent triplet carries most spectral weight for T → 0
- Evidence for spatially distributed chain end spins
- Field induced Quantum phase transition for uniform spin
chain only
 Spin-1/2 bi-layer system PHCC
- Frustration helps to stabilize gapped phase
- 3D LRO phase surrounded by gapless paramagnetic phase
- Anomalies in phase boundary close to QCP common in real
material due to high susceptibility to small perturbations
ICTP 10/20-26/03