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 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

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

Néel state, any alternative?

?Geometrical Frustration of

magnetic interactions

0rrrr

=++ CBA SSS

A

C

B

Class.

1972

……=RVB

Néel AF state, any alternative?

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

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

Spinon continuum≠

Spin waves branches« Fractionalexcitations

fractionnaires »(∆S ≠ 1)

Other stream: can one stabilize a spin liquidin dimension >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

S. Yan, D. Huse and S.R. White, Science 332 (2011)

1972

Néel Anderson

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

- 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)

=

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)

The Cu2+ world (S=1/2, ~Heisenberg)

b

Kapellasite

VolborthiteHerbertsmithite

Materials are all existing minerals !

Cu2+ S=1/2

Haydeeite Vesignieite

Ross H Colman, David Boldrin, Andrew S Wills, UCL, UK

Sciences, perspectives sept 2008

2005

From C. Broholm (2010)

Kagome stats

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)

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

Cu

Zn

Cl

OH

Herbertsmithite:ZnCu3(OH)6Cl2

Cu2+, S=1/2

M.I.T., 2005

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%

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

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)

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)

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

?

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

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

Ideally….

… Nobody is perfect

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)

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=

Mg- Herbertsmithite: control of inter-site defects

E. Kermarrec, Phys. Rev. B (R) (2011)

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

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)

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

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)

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

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!

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

V NMR T<9K

Static component below 9 K ~ 0.2 µB

Vesignieite

J. Quilliam et al, PRB (R), 2011

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

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

• 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

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

Quantum Kagome Antiferromagnets ZnCu3(OH)6Cl2

B. Fak, E. Kermarrec et al, PRL, 2012

20 µµµµm

119° 105°

Herbertsmithite KapellasiteCu3Zn(OH)6Cl2

Kapellasite : a polymorph of Herbertsmithite

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

High-Temperature series expansion analysis

J1 ferro -15 K ~ further neighbor J2,d 13 KB. Fak, E. Kermarrec et al, PRL, 2012

L. Messio, et. al, PRB 83, 184401 (2011)

Interactions scheme

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)

µSR

-no freezing (no decoupling)-Persistent slow fluctuations

Cuboc-2 correlations survive up to 100 K (also specific heat)

Neutron scattering (B. Fak, ILL)

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 understandingof the relaxation plateaus?

� Other materials: S=1/2 Vanadates

Thank you to ISIS team!