quantum kagome spin liquids · quantum kagome spin liquids philippe mendels - univ paris-sud orsa...
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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.
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Quantum Spin liquids
F. Bert, E. Kermarrec, A. Olariu, J. Quilliam, M. Jeong, A. Zorko
Laboratoire de Physique des Solides,
Université Paris-Sud, Orsay, France
Quantum Kagome Antiferromagnets
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0 ,. <−= ijjiij JSSJrr
H
ΤΝ∼J
Antiferromagnetism, Classical state ‘à la Néel’
-excitations = magnons (S=1)
- 1/S quantum corrections (fluctuations) -> moments reduction
+∞→>=<−−
ξξ;.
/ji rr
ji eSSrrrr
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Néel state, any alternative?
?Geometrical Frustration of
magnetic interactions
0rrrr
=++ CBA SSS
A
C
B
Class.
1972
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……=RVB
Néel AF state, any alternative?
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A B
C
A
A
A
A
A
A
C
C
C
C
C
C
B
B
B
B
B
B
C
C
A
A
C
A
B C
C
A
BC
A
C
B A
A
C
A B
...
...
...
...
0rrrr
=++ CBA SSS
Corner sharing: classical kagomé lattice
Soft modesMacroscopic degeneracy
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Corner sharing: classical vs quantum kagomé lattice
Classical soft modes Triangular ≠≠≠≠ Kagome
• Δ<J/20
• No gap in the singlet sector
•Lecheminant, PRB 56, 2521 (1997)
•Waldtmann et al., EPJB 2, 501 (1998).
<J/20
Back to 90’s
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Spinon continuum≠
Spin waves branches« Fractionalexcitations
fractionnaires »(∆S ≠ 1)
Other stream: can one stabilize a spin liquidin dimension >1?
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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
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Kagome Heisenberg antiferromagnet, a model of geometrically "frustrated " magnetism
2011
S. Yan, D. Huse and S.R. White, Science 332 (2011)
1972
Néel Anderson
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diamond-pattern valence bond crystal
honeycomb valence bond crystal
Quantum Spin Liquid
S. Yan, D. Huse and S.R. White, Science 332 (2011)
Kagome Heisenberg antiferromagnet, a model of geometrically "frustrated " magnetism
Chiral spin liquidMessio et al., PRL 2012
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- Quantum spin liquid?
A state without any spontaneously broken symmetry
Kagome ground state?
Short range RVB Long range RVB
Z2 QSL
Gapped magnetic excitations (S=1/2)
Gapped non-magnetic excitations
Cv~e-∆/T ; χ~e-∆’/T
Algebraic/Critical/Dirac/U(1) QSL
Gapless excitations
Cv~T2 ; χ~T
Hastings, PRB 63, 2000Ran et al, PRL 98, 2007Ryu et al, PRB 75, 2007Yan et al, science (2011)
=
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The ground-state of QKHA would be
a gapped spin-liquid (short-range RVB)
S. Yan et al, Science 332 (2011)
Quantum Spin Liquid
singlet
triplet
valence bond crystal
Large size numerical study (DMRG)
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The Cu2+ world (S=1/2, ~Heisenberg)
b
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Kapellasite
VolborthiteHerbertsmithite
Materials are all existing minerals !
Cu2+ S=1/2
Haydeeite Vesignieite
Ross H Colman, David Boldrin, Andrew S Wills, UCL, UK
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Sciences, perspectives sept 2008
2005
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From C. Broholm (2010)
Kagome stats
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2nd n.n. interactions| J2 | / J1=0.5, J1 (AF)J2
J1
?
NETWORK GEOMETRY :
•Triangular based lattice•AF coupling
?
?
INTERACTIONS GEOMETRY:
• Disorder : Spin glasses (e.g. CuMn)
• No disorder : Geometric Frustration
FRUSTRATION: unsatisfied pairwise interactions
(Carretta; Geibel)
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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
- local susceptibility (NMR)- heterogeneous frozen ground state (µSR+NMR)
-Kapellasite Cu3Zn(OH)6Cl2- competing exchange interactions:
Collaborations (Herbertsmithite, Vesignieite, Kapellasite)
Ross H Colman, David Boldrin, Andrew S Wills, UCL, UKP. Strobel, Institut Neel, Grenoble, FranceA. Harrisson, M. de Vries, Edinburgh, UKF. Duc, Toulouse
Quantum criticalityDzyaloshinky-Moriya interactions
J1-J2 model on kagome latticeNovel spin liquid?
samples
µSR @ ISIS, PSI ⊕ NMR
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Cu
Zn
Cl
OH
Herbertsmithite:ZnCu3(OH)6Cl2
Cu2+, S=1/2
M.I.T., 2005
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µSR : ZnCu3(OH)6Cl2: zero field
P. Mendels et al, PRL 98, 077204 (2007)
0 5 100.0
0.2
0.4
0.6
0.8
1.0
P
ola
risatio
n
time (µs)
50 mK
H
Deuterated
HerbertsmithiteZero Field µSR
upper limit of a frozen moment for Cu2+, if any : 6x10-4 µB
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
OH-µ : 85%Cl- : 15%
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µSR : phase diagram of paratacamites ZnxCu4-x(OH)6Cl2
- Paramagnetic (x=1 type) component
- Oscillations smeared out
- x=0 : fully ordered below ~18KX.G. Zheng et al, PRL 95, 057201 (2005)
Large dynamical domain 0.66<x<1
-> small influence of interlayer coupling
P. Mendels et al, PRL 98, 077204 (2007)
time (µs)
Polariza
tion
0.0 0.3 0.6
0.3
0.6
0.9
x = 0
x = 0.33
x =0.5
x = 0.66
x = 1.0
x = 0.15
T=1.5K, Zero Field
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Susceptibility: χmuons, D NMR vs 17O NMR (intrinsic, defects)
OCu
Zn
Cl
µSR D NMR
0 100 200 3000
1
2
0
10
20
1
7O
lin
eshift
(%)
T (K)
χSQUID (10
-4 cm
3/m
ol Cu)
17O NMR
χac
O. Ofer et al., ArXiv (2010) T. Imai, Phys. Rev. B (2011)
A. Olariu et al., Phys. Rev. Lett (2008)
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1 10 1000
1
2
0
10
20
17O
lin
eshift
(%)
T (K)
χSQUID (10
-4 cm
3/m
ol Cu)
J/20
Gapless spin liquid
Olariu et al, PRL 100, 087202 (2008)
See also Imai et al, PRL 100, 077208 (2008)
0 1 2
0.1
1
T1
-1 (
ms
-1)
T-1 (K
-1)
T1
-1 ~ T
0.7+/-0.1
χ’’~ω-0.7Inelastic Neutron scattering: Helton et al, PRL 98 107204 (2007)
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Very Low T Spin Dynamics
M. Jeong et al,Phys. Rev. Lett 107 (2011)
0.01 0.1 1 10
1E-5
1E-4
1E-3
0.01
0.1
1/T
1 (
ms
-1)
T (K)
6.8 T
?
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Very Low T Spin Dynamics: freezing under a field
M. Jeong et al,Phys. Rev. Lett 107 (2011)
0.01 0.1 1 10
1E-5
1E-4
1E-3
0.01
0.1
1/T
1 (
ms
-1)
T (K)
6.8 T
6.7 T
Tc
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Small frozen moment ~ 0.1µBA Quantum Critical Point ?
Algebraic critical spin liquid ?
• No gap (H = 0)• Instability ( H≠ 0) M ~ H
ααααbut Tc ~ H
• DM interaction -> L.R.O.Ran et al, PRL 98 117205 (2007)
µSR
Tc ~ (H-Hc)0.65
Hc = 1.5 T
∆ ~ 2.3 kBTc
µB Hc ~ J/180
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Ideally….
… Nobody is perfect
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Magnetic defects : Zn/Cu intersite mixing
Zn in the kagome plane-> magnetic vacancy
Cu on the Zn siteNearly free ½ spins
Cu
Zndefects
(~ 1 - 7%)
J (AF) ~180 K
J’ (AF), F ~ few K
Cu
Cl
OH
119°
97°Zn/Cu
(~ 15-25 %)
?
For a review and discussion, seeP. Mendels and F. Bert, J. Phys. Soc. Jpn 79 011001 (2010)
J. Phys. Conf. Series (2011)
Freedman et al, JACS, 132 (2010)
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0 5 100.0
0.2
0.4
0.6
0.8
1.0 2500G
80G
10G
Pola
risation
time (µs)
50 mK
ZF
Electronic spin signal
Dynamical relaxation (T1 process)
)exp( tP λ−=
- Weak decrease of the electronicspin fluctuation rate when T->0
- ”Dynamical plateau” vanishes when x ZnCu3(OH)6Cl2 : x=1
Energy scale ~1K in the defect channel0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.2
0.4
0.6
0.8
1.0
P21/n
Zn content
Static
Dynamic
Fro
ze
n f
ractio
n
R3m
- Hµ ~ 20 G very small->relaxation involves few Cu2+ (defects?)
Out-of-plane (?) defects (2)
νγλ µµ /2 22H=
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Mg- Herbertsmithite: control of inter-site defects
E. Kermarrec, Phys. Rev. B (R) (2011)
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Dzyaloshinskii-Moriya interactions: quantum criticality
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)
M. Elhajal et al, PRB 66, 014422 (2002)
For classical spins, DM stabilizes ordered phases (cf jarosites)
In the quantum case,a moment free phase survives up to D/J~0.1
Herbertsmithite
Disordered
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Dzyaloshinskii-Moriya interactions: quantum criticality
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)
M. Elhajal et al, PRB 66, 014422 (2002)
For classical spins, DM stabilizes ordered phases (cf jarosites)
In the quantum case,a moment free phase survives up to D/J~0.1
Herbertsmithite
Hc ~ Dc - Dherb
Disordered
ESR: A. Zorko et al., PRL 101 (2008)
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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
- 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, UKL. Messio, C. Lhuillier, B. Bernu, Paris, FranceB. Fak, CENG, Grenoble, France
Quantum criticalityDzyaloshinky-Moriya interactions
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a weakly distorted kagome lattice 0.1 % (Volborthite : 3%)
Cu3Ba(VO5H)2
Coll. UCL LondonR.C. Colman et al., Phys Rev B, Rapid Com 2011
VesignieiteY. Okamoto et al, JPSJ 78, 33701 (2009)
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V NMR (T>9K)
Shift ≡ Herbertsmithite
V @ Cu hexagon
0 1 2 30
1
2
3
4
5
χin
tJ =
(K
-σ)J
/A (
10
-2 K
µB /
kO
e)
T/J
Vesignieite (J = 53 K)
Herbertsmithite (Olariu2008, J=170 K)
Volborthite (Hiroi2001, J=84 K)
Theory (Misguich2007, adjusted)
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)
susceptibility
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µSR: two componentsOne static – disordered- one dynamical
R.C. Colman et al., Phys Rev B, Rapid Com (2011) Coll. UCL London
T < 9K decoupling µstat ~ 0.1 µBP(t) = ffPf(t) + (1-ff)Ppara(t)
At least 4 muon sites
Not zeroStatic component!
Vesignieite
Not zero slopeDynamical component!
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µSR: two components
• No macroscopic phase separation.• Spin dynamics of all Cu2+ suppressed down to
spin freezing for 40%.
Coupled
slow dynamics non frozen fraction
Vesignieite
Frozen fraction
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V NMR T<9K
Static component below 9 K ~ 0.2 µB
Vesignieite
J. Quilliam et al, PRB (R), 2011
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NMR -> two componentsStatic + dynamic (lost)
NMR µSR
J. Quilliam et al. R.C. Colman et al
Coll. Orsay / UCL London
Lost in NMR
Good agreement
Vesignieite
Seen in NMR
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Dzyaloshinskii-Moriya induced quantum criticality
VesignieiteHerbertsmithite
A. Zorko et al, PRL 101, 026405 (2008)
|Dz|=0.08J, |Dp|~0.01J Jvesi ~ Jherb/3; ∆Hvesi ~ ∆Hvolb
D/Jvesi ~ 0.1 - 0.17
W. Zhang, H. Ohta et al., JSPJ (2010)
ESR: ∆H~D2/J
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• Perfect kagome
• No freezing at H=0
• Field induced freezing
• 0.44 < D/J < 0.08
• Close to perfect kagome
• Partial freezing @ J/6
• D/J > 0.1
Conclusions
Herbertsmithite Vesignieite
QCP
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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
- 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, UKL. Messio, C. Lhuillier, B. Bernu, Paris, FranceB. Fak, CENG, Grenoble, France
Quantum criticalityDzyaloshinky-Moriya interactions
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Quantum Kagome Antiferromagnets ZnCu3(OH)6Cl2
B. Fak, E. Kermarrec et al, PRL, 2012
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20 µµµµm
119° 105°
Herbertsmithite KapellasiteCu3Zn(OH)6Cl2
Kapellasite : a polymorph of Herbertsmithite
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Cu
Cl-146 %
Zn
Cl-237 % Cl-315 % 2 % Cl-4
-0.0100 -0.0075 -0.0050 -0.0025 0.0000 0.00250.00.20.40.60.81.0 NMR lines T=78 K Cl-1 Cl-2 Cl-3 Cl-4
NMR intensity (a.u.)
NMR shift
35Cl NMR
Cu2.3Zn1.7(OH)6Cl2Chemical analysis (ICP)
Neutron diffraction (Cu0.73Zn0.27)3 (Zn0.88Cu0.12) (OH)6Cl2
Kagome site pc=0.65
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High-Temperature series expansion analysis
J1 ferro -15 K ~ further neighbor J2,d 13 KB. Fak, E. Kermarrec et al, PRL, 2012
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L. Messio, et. al, PRB 83, 184401 (2011)
Interactions scheme
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Classical ordered states on the Kagome latticeL. 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)
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µSR
-no freezing (no decoupling)-Persistent slow fluctuations
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Cuboc-2 correlations survive up to 100 K (also specific heat)
Neutron scattering (B. Fak, ILL)
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µSR vs NMR
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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 understandingof the relaxation plateaus?
� Other materials: S=1/2 Vanadates
Thank you to ISIS team!