Quantum kagome spin liquids

Quantum kagome spin liquids
Philippe Mendels - Univ Paris-Sud Orsa
The discovery of Herbertsmithite, ZnCu3(OH)6Cl2, which features a perfect kagome geometry, has
been coined as the "end to the drought of spin liquids" [1,2]. It has triggered an intense activity on
new kagome materials and related theories for the ground state of the quantum kagome Heisenberg
antiferromagnet which has eluded any definitive conclusion for the last twenty years.
This is indeed the first experimental example of a kagome lattice where no order has been found at
any temperature well below J through all experimental techniques, among which µSR [3] has been by
far the most accurate. This illustrates the power of the association of an enhancement of quantum
fluctuations for S=1/2 spins with the frustration of antiferromagnetic interactions on the loosely
connected kagome lattice to stabilize novel ground states of magnetic matter, namely quantum spin
liquids.
I will review and discuss our contribution to this field through µSR, in connection with our NMR work.
Through the various Cu2+ S=1/2 materials which we have studied, I’ll illustrate some of the research
thrusts in tracking down quantum spin liquid physics. This will encompass Zn and Mg-paratacamites,
(Zn,Mg)xCu4-x(OH)6Cl2 [), their polymorphs kapellasite [4] and haydeite, vesignieite [5] as well as the
recently discovered triangular-based lattice Ba3CuSb2O9 [6] compound.
[1] P.A.Lee, Science, Perspectives 321, 1306 (2008).
[2] For a review, see P. Mendels and F.Bert, Special Topics Section on "Novel States of Matter
Induced by Frustration", J. Phys. Soc. Jpn 1, 011001 (2010).
[3] Phys. Rev. Lett. 98, 077204 (2010).
[4] Phys. Rev. Lett. 109, 037208 (2012).
[5] Phys. Rev. B 84, 180401 (2011).
[6] Phys. Rev. Lett., in press.
Quantum
liquids
Quantum
KagomeSpin
Antiferromagnets
F. Bert, E. Kermarrec, A. Olariu, J. Quilliam, M. Jeong, A. Zorko
Laboratoire de Physique des Solides,
Université Paris-Sud, Orsay, France
Antiferromagnetism, Classical state ‘à la Néel’
r r
H = − J ij S i .S j , J ij < 0
ΤΝ∼J
-excitations = magnons (S=1)
- 1/S quantum corrections (fluctuations)
-> moments reduction
r r
r r
− ri − r j / ξ
< Si .S j >= e
; ξ → +∞
Néel state, any alternative?
1972
Geometrical Frustration of
magnetic interactions
?
C
Class.
A
B
r
r
r
r
S A + S B + SC = 0
Néel AF state, any alternative?
RVB =
……
Corner sharing: classical kagomé lattice
C
r r r r
S A + SB + SC = 0
A B C
A
B C
C
A B C
A
...
C
A B C
A
C B
C
A B
...
C
A B A
C
B A
C
A B A
C
...
C
A B A
C
A B
...
Macroscopic degeneracy
Soft modes
Corner sharing: classical vs quantum kagomé lattice
Classical soft modes
Triangular ≠ Kagome
<J/20
Back to 90’s
• Δ<J/20
• No gap in the singlet sector
•Lecheminant, PRB 56, 2521 (1997)
•Waldtmann et al., EPJB 2, 501 (1998).
Other stream: can one stabilize a spin liquid
in dimension >1?
Spinon continuum
≠
Spin waves branches
« Fractional
excitations
fractionnaires »
(∆S ≠ 1)
The IDEAL Highly Frustrated Magnet
- Heisenberg or Ising spins
- S=1/2 (quantum spins)
- Corner sharing geometry:
Kagomé (2D) , hyperkagome or pyrochlore (3D) lattice
- No « perturbation » (anisotropy, dipolar interaction,
n.n. interactions, dilution)
- For kagome: stacking should keep the 2D kagome planes
uncoupled
Kagome Heisenberg antiferromagnet, a model of
geometrically "frustrated " magnetism
2011
Néel
Anderson
1972
S. Yan, D. Huse and S.R. White, Science 332 (2011)
Kagome Heisenberg antiferromagnet, a model of
geometrically "frustrated " magnetism
Chiral spin liquid
Messio et al., PRL 2012
Quantum
Spin Liquid
diamond-pattern
valence bond crystal
honeycomb valence
bond crystal
S. Yan, D. Huse and S.R. White, Science 332 (2011)
Kagome ground state?
- Quantum spin liquid?
A state without any spontaneously broken symmetry
Z2 QSL
Gapped magnetic excitations (S=1/2)
Gapped non-magnetic excitations
Cv~e-∆/T ; χ~e-∆’/T
Short range RVB
Yan et al, science (2011)
Algebraic/Critical/Dirac/U(1) QSL
Gapless excitations
Cv~T2 ; χ~T
Long range RVB
=
Hastings, PRB 63, 2000
Ran et al, PRL 98, 2007
Ryu et al, PRB 75, 2007
Large size numerical study (DMRG)
The ground-state of QKHA would be
a gapped spin-liquid (short-range RVB)
Quantum
Spin Liquid
triplet
valence bond crystal
singlet
S. Yan et al, Science 332 (2011)
The Cu2+ world (S=1/2, ~Heisenberg)
b
Materials are all existing minerals !
Cu2+ S=1/2
Herbertsmithite
Haydeeite
Volborthite
Kapellasite
Vesignieite
Ross H Colman, David Boldrin, Andrew S Wills, UCL, UK
2005
Sciences, perspectives sept 2008
Kagome stats
From C. Broholm (2010)
FRUSTRATION: unsatisfied pairwise interactions
• Disorder : Spin glasses (e.g. CuMn)
• No disorder : Geometric Frustration
NETWORK GEOMETRY :
•Triangular based lattice
•AF coupling
?
J1
INTERACTIONS GEOMETRY:
2nd n.n. interactions
| J2 | / J1=0.5, J1 (AF)
J2
?
?
(Carretta; Geibel)
µSR @ ISIS, PSI ⊕ NMR
Outline
- Zn and Mg Herbertsmithite Cu3(Zn,Mg)(OH)6Cl2
- a gapless quantum spin liquid (summary)
- the fate of defects
- field induced solidification of the QSL
- Vesignieite Cu3Ba(VO5H)2
Quantum criticality
Dzyaloshinky-Moriya interactions
- local susceptibility (NMR)
- heterogeneous frozen ground state (µSR+NMR)
-Kapellasite Cu3Zn(OH)6Cl2
- competing exchange interactions:
Collaborations
J1-J2 model on kagome lattice
Novel spin liquid?
(Herbertsmithite, Vesignieite, Kapellasite)
samples
Ross H Colman, David Boldrin, Andrew S Wills, UCL, UK
P. Strobel, Institut Neel, Grenoble, France
A. Harrisson, M. de Vries, Edinburgh, UK
F. Duc, Toulouse
M.I.T., 2005
Cu
Herbertsmithite:
ZnCu3(OH)6Cl2
Zn
OH
Cu2+, S=1/2
Cl
µSR : ZnCu3(OH)6Cl2: zero field
1.0
Herbertsmithite
Zero Field µSR
Polarisation
0.8
0.6
Deuterated
0.4
upper limit of a frozen moment for
Cu2+, if any : 6x10-4 µB
0.2
H
0.0
OH-µ : 85%
Cl: 15%
50 mK
0
5
10
time ( µ s)
P. Mendels et al, PRL 98, 077204 (2007)
No order or frozen disorder down to 50 mK despite J=190 K !
Liquid down to J/4000 under ZF
Also: ac-χ Helton et al, PRL 98 107204 (2007)
µSR O. Ofer et al, cond-mat/0610540
µSR : phase diagram of paratacamites ZnxCu4-x(OH)6Cl2
x = 1.0
0.9
Polarization
x =0.5
x = 0.66
0.6
- Paramagnetic (x=1 type) component
x = 0.33
- Oscillations smeared out
x = 0.15
0.3
x=0
T=1.5K, Zero Field
0.0
0.3
- x=0 : fully ordered below ~18K
X.G. Zheng et al, PRL 95, 057201 (2005)
0.6
time (µs)
Large dynamical domain 0.66<x<1
-> small influence of interlayer coupling
P. Mendels et al, PRL 98, 077204 (2007)
Susceptibility: χmuons, D NMR vs 17O NMR (intrinsic, defects)
µSR
D NMR
χ
2
20
ac
10
1
17O
0
0
A. Olariu et al., Phys. Rev. Lett (2008)
100
NMR
200
T (K)
300
0
χSQUID (10-4 cm3/mol Cu)
T. Imai, Phys. Rev. B (2011)
17
O lineshift (%)
O. Ofer et al., ArXiv (2010)
Cu
Zn O
Cl
10
1
-1
-1
T1 ~ T
0.7+/-0.1
-1
J/20
1
T1 (ms )
2
17
O lineshift (%)
20
χSQUID (10-4 cm3/mol Cu)
Gapless spin liquid
0.1
0
1
10
100
0
0
1
-1
2
-1
T (K )
T (K)
Olariu et al, PRL 100, 087202 (2008)
See also Imai et al, PRL 100, 077208 (2008)
Inelastic Neutron scattering: Helton et al, PRL 98 107204 (2007)
χ’’~ω-0.7
Very Low T Spin Dynamics
6.8 T
0.1
M. Jeong et al,
Phys. Rev. Lett 107 (2011)
-1
1/T1 (ms )
0.01
1E-3
?
1E-4
1E-5
0.01
0.1
1
T (K)
10
Very Low T Spin Dynamics: freezing under a field
Tc
6.8 T
6.7 T
0.1
M. Jeong et al,
Phys. Rev. Lett 107 (2011)
-1
1/T1 (ms )
0.01
1E-3
1E-4
1E-5
0.01
0.1
1
T (K)
10
Small frozen moment ~ 0.1µB
A Quantum Critical Point ?
Tc ~ (H-Hc)0.65
Hc = 1.5 T
∆ ~ 2.3 kBTc
µSR
Algebraic critical spin liquid ?
• No gap (H = 0)
α
• Instability ( H≠ 0) M ~ H but Tc ~ H
• DM interaction -> L.R.O.
Ran et al, PRL 98 117205 (2007)
µB Hc ~ J/180
Ideally….
… Nobody is perfect
Magnetic defects : Zn/Cu intersite mixing
Cu
Zn
Cu on the Zn site
Nearly free ½ spins
(~ 15-25 %)
Freedman et al, JACS, 132 (2010)
defects
Zn in the kagome plane
-> magnetic vacancy
(~ 1 - 7%)
?
Cu
J’ (AF), F ~ few K
J (AF) ~180 K
For a review and discussion, see
97°
Zn/Cu
OH
119°
P. Mendels and F. Bert, J. Phys. Soc. Jpn 79 011001 (2010)
J. Phys. Conf. Series (2011)
Cl
Out-of-plane (?) defects (2)
Electronic spin signal
Dynamical relaxation (T1 process)
2500G
1.0
P = exp(−λt )
Polarisation
0.8
80G
0.6
0.4
0.0
- Hµ ~ 20 G very small
->relaxation involves few Cu2+ (defects?)
10G
0.2
λ = 2γ µ2 H µ2 /ν
ZF
50 mK
0
5
10
time (µs)
ZnCu3(OH)6Cl2 : x=1
- Weak decrease of the electronic
spin fluctuation rate
when T->0
- ”Dynamical plateau” vanishes when x
1.0
Frozen fraction
0.8
Dynamic
0.6
0.4
Static
0.2
0.0
P21/n R3m
0.2
0.4
0.6
Zn content
0.8
1.0
1.2
Energy scale ~1K in the defect channel
Mg- Herbertsmithite: control of inter-site defects
E. Kermarrec, Phys. Rev. B (R) (2011)
Dzyaloshinskii-Moriya interactions: quantum criticality
For classical spins, DM stabilizes
ordered phases (cf jarosites)
Disordered
M. Elhajal et al, PRB 66, 014422 (2002)
In the quantum case,
a moment free phase
survives up to D/J~0.1
O. Cepas et al, PRB 78, 140405 (R) (2008)
Y. Huh et al, PRB 81, 144432 (2010)
L. Messio et al, PRB 81 , 064428 (2010)
Herbertsmithite
Dzyaloshinskii-Moriya interactions: quantum criticality
For classical spins, DM stabilizes
ordered phases (cf jarosites)
Disordered
M. Elhajal et al, PRB 66, 014422 (2002)
In the quantum case,
a moment free phase
survives up to D/J~0.1
ESR: A. Zorko et al., PRL 101 (2008)
O. Cepas et al, PRB 78, 140405 (R) (2008)
Y. Huh et al, PRB 81, 144432 (2010)
L. Messio et al, PRB 81 , 064428 (2010)
S. El Shawish et al, PRB 81, 224421 (2010)
Herbertsmithite
Hc ~ Dc - Dherb
Outline
- Zn and Mg Herbertsmithite Cu3(Zn,Mg)(OH)6Cl2
- a gapless quantum spin liquid (summary)
- field induced solidification of the QSL
- Vesignieite Cu3Ba(VO5H)2
Quantum criticality
Dzyaloshinky-Moriya interactions
- local susceptibility (NMR)
- heterogeneous frozen ground state (µSR+NMR)
-Kapellasite Cu3Zn(OH)6Cl2
- competing exchange interactions
Collaborations
Ross H Colman, David Boldrin, Andrew S Wills, UCL, UK
L. Messio, C. Lhuillier, B. Bernu, Paris, France
B. Fak, CENG, Grenoble, France
Vesignieite
Cu3Ba(VO5H)2
Y. Okamoto et al, JPSJ 78, 33701 (2009)
a weakly distorted kagome lattice 0.1 %
(Volborthite : 3%)
R.C. Colman et al., Phys Rev B, Rapid Com 2011
Coll. UCL London
susceptibility
V NMR (T>9K)
-2
χintJ = (K-σ)J/A (10 K µB / kOe)
V @ Cu hexagon
5
4
3
2
Vesignieite (J = 53 K)
Herbertsmithite (Olariu2008, J=170 K)
Volborthite (Hiroi2001, J=84 K)
Theory (Misguich2007, adjusted)
1
0
0
1
2
3
T/J
Shift ≡ Herbertsmithite
J. Quilliam et al, arXiv:1105.4338
- J ~ 50 K
- Curie tail ~ 7% S=1/2
- Kink at T~9K
R.C. Colman et al (2011)
Vesignieite
µSR: two components
One static – disordered- one dynamical
Not zero slope
Dynamical component!
Not zero
Static component!
At least 4 muon sites
P(t) = ffPf(t) + (1-ff)Ppara(t)
T < 9K decoupling µstat ~ 0.1 µB
R.C. Colman et al., Phys Rev B, Rapid Com (2011)
Coll. UCL London
Vesignieite
µSR: two components
Frozen fraction
Coupled
slow dynamics
non frozen fraction
• No macroscopic phase separation.
• Spin dynamics of all Cu2+ suppressed down to
spin freezing for 40%.
Vesignieite
V NMR T<9K
Static component below 9 K ~ 0.2 µB
J. Quilliam et al, PRB (R), 2011
Vesignieite
NMR -> two components
Static + dynamic (lost)
Seen in NMR
Lost in NMR
NMR
µSR
Good agreement
Coll. Orsay / UCL London
J. Quilliam et al.
R.C. Colman et al
Dzyaloshinskii-Moriya induced quantum criticality
|Dz|=0.08J, |Dp|~0.01J
Jvesi ~ Jherb/3; ∆Hvesi ~ ∆Hvolb
D/Jvesi ~ 0.1 - 0.17
ESR: ∆H~D2/J
A. Zorko et al, PRL 101, 026405 (2008)
W. Zhang, H. Ohta et al., JSPJ (2010)
Herbertsmithite
Vesignieite
Conclusions
Herbertsmithite
Vesignieite
• Perfect kagome
• Close to perfect kagome
• No freezing at H=0
• Partial freezing @ J/6
• Field induced freezing
• 0.44 < D/J < 0.08
• D/J > 0.1
QCP
Outline
- Zn and Mg Herbertsmithite Cu3(Zn,Mg)(OH)6Cl2
- a gapless quantum spin liquid (summary)
- field induced solidification of the QSL
- Vesignieite Cu3Ba(VO5H)2
Quantum criticality
Dzyaloshinky-Moriya interactions
- local susceptibility (NMR)
- heterogeneous frozen ground state (µSR+NMR)
-Kapellasite Cu3Zn(OH)6Cl2
- competing exchange interactions
Collaborations
Ross H Colman, David Boldrin, Andrew S Wills, UCL, UK
L. Messio, C. Lhuillier, B. Bernu, Paris, France
B. Fak, CENG, Grenoble, France
Quantum Kagome Antiferromagnets ZnCu3(OH)6Cl2
B. Fak, E. Kermarrec et al, PRL, 2012
Kapellasite : a polymorph of Herbertsmithite
Cu3Zn(OH)6Cl2
Herbertsmithite
Kapellasite
20 µm
119°
105°
35Cl
NMR
1.0
NMR lines T=78 K
).
.ua
(
yit
sn
et
ni
R
M
N
0.8
Cl-1
Cl-2
Cl-3
0.6
Cl-4
0.4
0.2
0.0
-0.0100
-0.0075
-0.0050
-0.0025
0.0000
0.0025
NMR shift
Zn
Cu
46 %
Cl-1
37 %
Chemical analysis (ICP)
Neutron diffraction
Cl-2
15 %
Cl-3
2%
Cl-4
Cu2.3Zn1.7(OH)6Cl2
(Cu0.73Zn0.27)3 (Zn0.88Cu0.12) (OH)6Cl2
Kagome site pc=0.65
High-Temperature series expansion analysis
J1 ferro -15 K ~ further neighbor J2,d 13 K
B. Fak, E. Kermarrec et al, PRL, 2012
Interactions scheme
L. Messio, et. al, PRB 83, 184401 (2011)
Classical ordered states on the Kagome lattice
L. Messio, C. Lhuillier, G. Misguich, LPTMC Paris, CEA
8 Classical long-range ordered states allowed by symmetries
Non coplanar state
• 12 sublattice spins order : « cuboc2 »
• Neutrons experiment shows correlations
reminiscent of this peculiar state
L. Messio, et. al, PRB 83, 184401 (2011)
µSR
-no freezing (no
decoupling)
-Persistent slow
fluctuations
Neutron scattering (B. Fak, ILL)
Cuboc-2 correlations survive up to 100 K (also specific heat)
µSR vs NMR
Conclusions
Experimentally:
two quantum spin liquids on the kagome lattice:
Heisenberg + anisotropy and J1-J2
Fragility of the QSL:
• Defects: no
• Anisotropy, field: yes
Is the gapless behavior intrinsic or not?
Can we achieve a better understanding
of the relaxation plateaus?
Other materials: S=1/2 Vanadates
Thank you to ISIS team!