Classical and Quantum Frustrated Magnets Yong Baek Kim University - - PowerPoint PPT Presentation

Classical and Quantum Frustrated Magnets

Yong Baek Kim University of Toronto University of Virginia, November 1, 2007

• J. Hopkinson (T
• ronto), S. Isakov (Toronto), M. Lawler (Toronto), H.
• Y. Kee (Toronto)
• F. Wang (Berkeley), A.

Vishwanath (Berkeley),

• S. Sachdev (Harvard), L. Friz (Harvard)

Collaborators:

Outline

Introduction to Frustrated Magnets Review of the Heisenberg model

• n the Kagome lattice

Spin-1/2 Quantum Magnets; Real Materials Distorted Kagome (Volborthites) Lattices Hyper-Kagome (Na4Ir3O8) Lattice Zn-Paratacamite Lattice

Introduction to Frustrated Magnets

Geometric Frustration: the arrangement of spins on a lattice precludes (fully) satisfying all interactions at the same time

∼ eαN

Modern: large degeneracy of the (classical) ground state manifold Consequence: No energy scale of its own; any perturbation is strong (reminiscent of the lowest Landau level physics) Mother of the conventional and exotic phases

Susceptibility ‘fingerprint’:

Introduction to Frustrated Magnets

ΘCW Curie-Weiss temperature

mean-field ordering temp.

TF /ΘCW ≪ 1 strong frustration

Cooperative paramagnet: correlations remain weak interaction energy scale more universal

T < TF

Magnetically ordered ? Spin liquid ? not universal Glassy ?

TF < T < ΘCW

Origin of Classical Ground State Degeneracy

Classical nearest-neighbor antiferromagnetic Heisenberg model on lattices with corner-sharing simplexes (simplex = triangle, tetrahedron)

H = J

• i,j

Si · Sj = J 2

• simplex

 

• i ǫ simplex

Si  

2

Classical ground state should satisfy

• i ǫ simplex

Si = 0

These constraints are not independent; counting is subtle Nonetheless there exists macroscopic degeneracy is a vector with a fixed length

Si

Kagome Pyrochlore Hyper-Kagome

Order by Disorder

Order by Disorder via Thermal Fluctuations: Different entropic weighting to each ground state Softer the fluctuations around a particular ground state, more likely this ground state will be entropically favored. Order by Disorder via Quantum Fluctuations: Quantum zero point energy and anharmonic contributions may select an ordered ground state. Sufficiently strong quantum fluctuations (S=1/2 for example), however, may destabilize any ordered phase; possible quantum spin liquid - Disorder by Disorder

S=1/2 Frustrated Magnets; Real Materials

t ZnCu3(OH)6Cl2.

Herbertsmithite

ΘCW = −300 ± 20K

J ≈ 197K (fit to series expansion)

Ideal Kagome lattice

• J. S. Helton et al, PRL 98, 107204 (2007)

S=1/2 Frustrated Magnets; Real Materials

Fig.2a, Hiroi et al. 1000 800 600 400 200

1/ ( mol Cu/cm3)

400 300 200 100

T (K)

H = 7 T

Curie-Weiss fitting peff = 1.95 µB (g = 2.26) = -115 K

(b)

Volborthite

• ets. In this context, the r

te Cu3V2O7(OH)2 2H2O

J1 J1 J2

Cu2 Cu1

ΘCW = −115K

Distorted Kagome lattice

• Z. Hiroi et al, JPSJ 70, 3377 (2001); F. Bert et al, PRL 95, 087203 (2005)

S=1/2 Frustrated Magnets; Real Materials

6 4 2

Sm (J/Kmol Ir)

60 40 20

Cm/T (mJ/K

2mol Ir)

300 250 200 150 100 50

T (K)

2000 1500 1000 500 (a) Na4Ir3O8 (c) (b)

1(mol Ir/emu)

1.6 1.4 1.2 1.0 0.8 0.6

(10

• 3emu/mol Ir)

0.1 1 10 100

T (K) 0.01 T 0.1 T 1 T 5 T

1 10 100

Cm/T (mJ/K

2mol Ir)

1 10 100

T (K) Cm T

2

Cm T

3

12 T 8 T

0 T 4 T

e, Na4Ir3O8, Hyper-Kagome

!" % '&*

ΘCW = −650K

• Y. Okamoto, M. Nohara, H. Agura-Katrori,

and H. Takagi, arXiv:0705.2821 (2007)

Classical Heisenberg Model on Kagome Lattice

Consider first co-planar states = A B C 3-state Potts spins

• i ǫ △

Si = 0

A A A A A A A B B B B B B B C C C C C A A A A A A A B B B B B B C C C C C C

√ 3 × √ 3

q = 0

Non-planar states can be generated by continuous distortions of a planar state All ground states can be generated by repeated introduction

• f ‘defects’ into the different parent planar states

Weathervane loop Degeneracy of the co-planar states is extensive; e0.379N (from 3-state Potts antiferromagnet)

• D. A. Huse, A. Rutenberg, PRB 45, 7536 (1992)

Planar ground states have more soft modes (introduction of ‘defect’ removes certain soft modes) Planar ground states are favored at small temperatures Expect growing correlation of nematic order (broken spin-rotation symmetry);

g(ra − rb) = 3 2(na · nb)2 − 1 2, na = 2 3 √ 3(S1 × S2 + S2 × S3 + S3 × S1),

N --- nearest-neighbor triangles NN --- next-nearest-neighbor triangles

g(r) → 1 for coplanar ground states

• J. T. Chalker, P

. C. W.Holdsworth, E. F. Shender, PRL 68, 855 (1992)

• D. A. Huse, A. Rutenberg, PRB 45, 7536 (1992)

T → 0

Nematic long range order as

Small distortions from an arbitrary coplanar ground state

H = H0 +

• n

Hn(ǫn)

quadratic potential; identical for all coplanar states

H2 :

H3 + H4 : Quartic potential not the same for different

coplanar states

√ 3 × √ 3 favored as

Boltzman weight not the same;

T → 0

• D. A. Huse, A. Rutenberg, PRB 45, 7536 (1992)
• A. Chubukov, PRL 69, 832 (1992)

zero mode (for all q)

What about spin-1/2 quantum model ? Exact Diagonalization Singlet Ground State ? Nature of the ground state not understood ... Series Expansion

Quantum Heisenberg Model on Kagome Lattice

Spin Liquid ? Effective Field Theory Valence Bond Solid ?

Distorted Kagome (Volborthite) Lattice

B C J J J’ A L

H =

• triangles

(SA · SB + αSB · SC + SC · SA) = α 2

• triangles

[(1/α)SA + SB + SC]2 − constant consider mainly th JAB = JCA = JBC.

t JAB = JCA = 1

B CA

d JBC = α.

Volborthite;

t α > 1

Constraint on classical ground states [(1/α)SA + SB + SC = 0

• M

K

angle between A-site s s θ0 = arccos(−1/2α) (

Single triangle

(3-state Potts does not apply)

< 1/2

• α ≤ 1/2. ‘cluster spin’ cannot be zero

collinear ground state; no degeneracy; ferrimagnet

r α > 1/2 ccoplanar ground states

chirality variables are more useful

η = ±1

Classical Heisenberg Model

Constraints on the chirality variables

1 6

3

• 4
• 5

2 B D A F C E

to 6

i=1 ηi = ±6 or 0.

ingle hexagon out of 2 mensurate to 2π and e: 6

i=1 ηi = ±6 or,

e last equation is the nd the constraint is more restric- r, 6

i=1 ηi = 0 and η1 + η4 = 0.

Isotropic Kagome Volborthite Kagome Degeneracy of coplanar ground states Isotropic Kagome

• magnet (or 3-

e, exp(0.379N),

Volborthite Kagomet exp(2.2L). Direct enumeration Transfer Matrix Method Sub-extensive ! Classical Ground State Degeneracy

No local weather-vane mode not equivalent cannot be in the same direction Non-local weather-vane modes exist - the number do not scale as the area of the system Classical ground state manifold of the Volborthite Kagome is much less connected than the isotropic case requires moving an infinite number of spins; large kinetic barriers; may expect freezing at low temperatures Consequence of Sub-extensive Degeneracy

Low temperature NMR experiments on Volborthite; spin freezing below 1.5 K (J/60) (isotropic Herbertsmithite; no freezing observed)

51V NMR;

V atoms at the hexagon centers

1/T1 rises rapidly through the glass transition temperature

two distinct local environments for the

51V

higher static field - 20% lower static field - 80% Assume that the glassy state locally resembles certain classical ground state Volume average of a local quantity in the glassy state = ensemble average over classical ground states Application to Volborthtite

• F. Bert et al, Phys. Rev. Lett. 95, 087203 (2005)

α ≈ 1 case: three different field values are possible

≈ , H ≈ 3Hcu,

, three different , H ≈ √ 3HCu

eld values d H ≈ 0, re HCu : the field from a single spin

( √ 3 × √ 3)

In the experiment, assume

≈ , H ≈ 3Hcu, for high static field

copper moment per site = Constraint on the classical ground states (via transfer matrix method) leads to

, three different , H ≈ √ 3HCu

eld values d H ≈ 0,

• 25%
• 75%

copper moment per site =

√ 3 × 0.4 µB = 0.7 µB

0.4 µB

Summary

Distorted Kagome Lattice Classical Heisenberg model: sub-extensive degeneracy of the classical ground states much less connected than the isotropic case; glassy behavior ? thermal fluctuations favor Chirality Stripe state Quantum spin-1/2 Heisenberg model singlet ground state with a spin gap; spin liquid ?

B C J J J’ A L

• F. Wang, A.

Vishwanath,

• Y. B. Kim, Phys. Rev. B 76, 094421(2007)

Three-dimensional S=1/2 Frustrated Magnet

has a Hyper-Kagome sublattice of Ir ions e, Na4Ir3O8, Pyrochlore Hyper-Kagome 3/4 Ir, 1/4 Na Ir Na Ir 4+ (5d ) 5 All Ir-Ir bonds are equivalent carries S=1/2 moment (low spin state)

• Y. Okamoto, M. Nohara, H. Agura-Katrori, and H. Takagi, arXiv:0705.2821 (2007)

Hyper-Kagome Lattice

2000 1500 1000 500 (a) Na4Ir3O8

1(mol Ir/emu)

1.6 1.4 1.2 1.0 0.8 0.6

(10

• 3emu/mol Ir)

0.1 1 10 100

T (K) 0.01 T 0.1 T 1 T 5 T mol Ir)

300 250 200 150 100 50

T (K)

Inverse Spin Susceptibility; Strong Spin Frustration

ΘCW = −650K

No magnetic ordering down to Curie-Weiss fit Large Window of Cooperative Paramagnet

!" % '&* !+*,-

• Y. Okamoto, M. Nohara, H. Agura-Katrori, and H. Takagi, arXiv:0705.2821 (2007)

|ΘCW|/300

Specific Heat; Low Energy Excitations ?

6 4 2

Sm (J/Kmol Ir)

60 40 20

Cm/T (mJ/K

2mol Ir)

300 250 200 150 100 50

T (K)

(c) (b)

1 10 100

Cm/T (mJ/K

2mol Ir)

1 10 100

T (K) Cm T

2

Cm T

3

12 T 8 T

0 T 4 T

Gapless Excitations

• r Small Gap ?

No Magnetic Ordering Field-independent up to 12T Is the T=0 Ground State a Spin Liquid ?

• Y. Okamoto, M. Nohara, H. Agura-Katrori, and H. Takagi, arXiv:0705.2821 (2007)

Classical Model

• d Si = (S1

i , ..., SN i )

• Si · Si = N

Classical Antiferromagnetic O(N) Model

H = J

• ij

Si · Sj

N-component spins with fixed length N

e N → ∞

Large-N limit: The lowest eigenvalue (4-fold) is independent of wavevector macroscopic degeneracy Describes physics in the Cooperative Paramagnet regime

Thermal Order by Disorder; Monte Carlo

First order transition to a nematic order

g(ra − rb) = 3 2(na · nb)2 − 1 2,

na = 2 3 √ 3(S1 × S2 + S2 × S3 + S3 × S1),

First order transition to a nematic order First order transition to a nematic or First order transition to a nematic or coplanar favored

L x L x L x 12 lattice (L = 3, 4, 6, 8, 9)

T < 1 ∼ 5 × 10−3J

0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 1-g(r) (T/J)1/2

(b)

L=3 L=4 L=6 L=8 0.2 0.4 0.6 0.8 1 1-g(r)

(a)

L=9 down L=9 up

. g(r) = 1 ideal coplanar state d g(r) = 0 non-coplanar state

Comparison with Experiment ?

5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1/ T/J L=6 L=8 5.557+2.413T/J

• θCW = −2.303(5)J Monte Carlo

θexp

CW = −650K

s J ≈ 280K.

Experimental data exist only down to 2-3 K ... Magnetic Order ? not found, but cannot reliably be determined below Tn

Tn ∼ 0.3 − 1.4 K

Thermal Order by Disorder; Monte Carlo

Cooperative Paramagnetic Regime

T > J/100 ∼ 2 − 3K

Physics is dominated by Cooperative Paramagnet behavior Behavior of the Large-N O(N) model ≈ O(3) Monte Carlo for T > J/100 ∼ 2 − 3K Spin Structure Factor in the [hhl] plane

(a) (b)

O(N) large-N O(3) Monte Carlo

r, S(q) =

µνSq,µ·S−q,ν

rocal space is shown in Fig.

Dipolar Spin Correlations in the Cooperative Paramagnet Regime

Sα

i Sβ j ∝ δαβ

3(ei · rij)(ej · rij) |rij|5 − ei · ej |rij|3

• e1

1

e’ e2 e3 e’

2

e’

2 c.f. Pyrochlore lattice: S.

• V. Isakov, K. Gregor, R. Moessner, S. L. Sondhi, PRL 93, 167204 (2004)

Quantum Heisenberg Model

s, Si = 1

2b†α i σβ αbiβ,

spin states of each

e α, β =↑, ↓

• ssible spin states of

t nb = b†α

i biα = 2S

H = −1 2

• ij

Jij(αβb†α

i b†β j )(γδbiγbjδ)

SU(2) Heisenberg Model

ǫαβ antisymmetic tensor of SU(2)

Quantum Heisenberg Model

s, Si = 1

2b†α i σβ αbiβ,

spin states of each

e α, β =↑, ↓

• ssible spin states of

t nb = b†α

i biα = 2S

H = −1 2

• ij

Jij(αβb†α

i b†β j )(γδbiγbjδ)

SU(2) Heisenberg Model

ǫαβ antisymmetic tensor of SU(2)

Sp(N) generalized model; N flavors of bosons on each site

α i biαe α = 1, ..., 2N

bosons on each site. T to nb = b†α

i biα = 2NS,

• index. For the physical

H = − 1 2N

• ij

Jij(Jαβb†α

i b†β j )(J γδbiγbjδ),

−Jβα is th a 2N × 2N

e J αβ = Jαβ = −Jβα r of SU(2); it is a 2N ×

matrix = blockdiag[ǫ, ǫ, ...] N=1 is the physical limit (S = half-integer); Sp(1)=SU(2)

ere. e N → ∞ r, nb/N = 2S = κ,

r S = κ/2

Large-N limit: with fixed Mean-field theory for well controlled Non-perturbative in the coupling constant and S

Qij = 1 N

• J αβb†

iαb† jβ

• bα

i = xα i .

Valence Bond Singlet Magnetic Order Large Small

κ κ

Magnetic Order Disordered (Spin Liquid, Valence Bond Solid) Finite-N fluctuations; Compact U(1) gauge theory

Sp(N) model - quantum spin

κ < κc = 0.4

κ > κc = 0.4

“S” = κ/2

Large-N limit Z2 spin liquid 8-fold degenerate ground states with p.b.c. Topological order Coplanar magnetically ordered state (finite spin gap)

• 120o Co-planer ordering

• 120o Co-planer ordering
• no local weather-vane mode

fluctuations can only occur along an infinitely long thread with pattern BCBC... thermal fluctuations may not select this state

Spin Structure Factor for Z2 Spin Liquid is very different from that of Cooperative Paramagnet

• 2
• 1

1 2

hh0

• 2
• 1

1 2

00l

0.00 0.35

(a)hhl-plane

(b)

Quantum Spin Liquid at T=0 Cooperative Paramagnet for T > J/100

Two-Spinon Continuum in the Z2 Spin Liquid

Exact Diagonalization on a unit cell (12, 24 sites) High temperature results for susceptibility and specific heat with J = 300K compare well with experimental data Small spin gap ... Alternative Approaches/Possibilities Gapless U(1) spin liquid with fermionic spinons Projected Variational Wavefunction (in progress) Role of DM interaction ?

Spin-1/2 Hyper-Kagome Lattice - Na4Ir3O8

Summary

Z2 Spin Liquid or Coplanar order ? Nematic Order Cooperative Paramagnet { }

|ΘCW| = 650K J/100 2 − 3K

J/1000 − J/200 0.3 − 1.5K Thermal order by disorder

J ≈ 280 K

• 2
• 1

1 2

• 2
• 1

1 2

(b)

• J. M. Hopkinson, S.
• V. Isakov, H.-Y. Kee,
• Y. B. Kim, Phys. Rev. Lett. 99, 037201 (2007)
• M. J. Lawler, H.-Y. Kee,
• Y. B. Kim, A.

Vishwanath, arXiv:0705.0990

Zn-paratacamitete ZnxCu4−x(OH)6Cl2

monoclinic rhombohedral distorted kagome lattices coupled by triangular sites ideal kagome lattice

Zn mostly goes to triangular sites spin-1/2 moment

Cu2+

x < 0.33 x > 0.33

no magnetic ordering

• r spin freezing

down to

ΘCW ∼ −300 K 50 mK

(µSR, NMR)

Cm ∼ T α

gapless excitations ?

• J. S. Helton et al, arXiv:0610539

P . Mendels et al, arXiv:0610565

• O. Ofer et al, arXiv:0610540
• M. A. de

Vries et al, arXiv:0705.0654

• J. S. Helton et al, arXiv:0610539

x = 1

herbertsmithite:

S.-H. Lee et al, Nature Materials, Aug 26, 2007 (arXiv:0705.2279)

T (K) Zn concentration (x) Neel + VBS SG VBL VBS Data of this work Data from ref. 14 ZnxCu4–x(OD)6Cl2 RVB (?) 1.0 0.8 0.6 0.4 0.2 20 10

d

'

c

Suggested Phase Diagram Neutron Scattering: Collinear Magnetic Order Valence Bond Solid

u v w w u

Order pattern (|<S>|=m):

mu = − ¯ mei(π,0)·R

mv = − ¯ m, mw = ¯ m

Distorted Kagome Lattice

J J′

bond length

3.41˚ A

3.42˚ A

Classical O(N) model; Large-N limit

J′/J < 0.5

collinear magnetic ordering for with moderate can be stabilized up to

J′/J ∼ 1

J3 < 0

J′/J ≈ 1/3

Goodenough-Kanamori rule

Quantum Sp(N) model; Large-N limit larger quantum fluctuations

↑

Quantum Sp(N) model; Instanton (via Berry phase) analysis

(a)Pin-wheel state (b)Columnar state

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6

h00qx a 2Π

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6

0k0qy a2Π

(c)Pin-wheel triplons

• 1
• 0.5

0.5 1

h00qx a 2Π

• 1
• 0.5

0.5 1

0k0qy a2Π

(d)Columnar triplons

favored

Lattice Distortion and X-Ray Scattering

• FIG. 4: X-ray structure factor: circles represent Bragg peaks
• f the ideal kagome lattice; triangles arise from the structural

distortion shown in Fig. 1. These are the only Bragg peaks in the columnar state. In the pin-wheel state, additional Bragg peaks (hexagons) appear due to further lattice distortion.

Summary

Zn-paratacamitete ZnxCu4−x(OH)6Cl2

x < 0.33 J

J′

u v w w u

distorted kagome lattice collinear order VBS phase triplon dispersion

• M. J. Lawler, L. Fritz,
• Y. B. Kim, S. Sachdev, arXiv:0709.4489 (2007)
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6

h00qx a 2Π

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6

0k0qy a2Π