Yong Baek Kim- Emergent Quantum Phases in Frustrated Magnets

8/3/2019 Yong Baek Kim- Emergent Quantum Phases in Frustrated Magnets

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Yong Baek KimUniversity of Toronto

CIFAR Spring School, May 5, 2010


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Emergent Quantum Phases in

Frustrated Magnets

Yong Baek KimUniversity of Toronto

CIFAR Spring School, May 5, 2010


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Emergent Quantum Phases

What are good places to look for Emergent

Behavior of correlated many-body system ?

Competing interactions

No separation of energy scales

Competing phases almostdegenerate in energy

Many-Body Frustration


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Emergent Quantum Phases

What are good places to look for Emergent

Behavior of correlated many-body system ?

Competing interactions

No separation of energy scales

Competing phases almostdegenerate in energy

Many-Body Frustration

Quantum fluctuations delicately

lift the many-body degeneracy -Possible Emergence of New Phases

Fluctuation leadsto Novel Phases


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Outline

1. Classical and Quantum Theory ofFrustrated Magnets

2. Novel Magnetic Order in Diamond Lattice Spinel

3. Spin Liquid on Hyper-Kagome Lattice

Na4Ir3O8

Quantum Order by Disorder

Disorder by Disorder


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Introduction to Frustrated Magnets

Geometric Frustration:The arrangement of spins on alattice precludes (fully) satisfyingall interactions at the same time

eNLarge degeneracy of the

(classical) ground state manifold

Consequence:No energy scale of its own; any perturbation is strong

Mother of the many conventional and exotic phases

?

J1J2


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Kagome Pyrochlore

Hyper-Kagome

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 simplex

Si

2

is a vector with a fixed lengthSi


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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 simplex

Si

2

is a vector with a fixed lengthSi

Classical ground state should satisfy

i simplex

Si = 0

These constraints are not independent; counting is subtle

Nonetheless there exists macroscopic degeneracy


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Susceptibility fingerprint:CW: Curie-Weiss temperature;

Mean-field Ordering Temperature;interaction energy scale

strong frustration

useful diagnosticof frustration

f 1


TCW

T |CW|

f= |CW|TF

| || |


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Susceptibility fingerprint:CW: Curie-Weiss temperature;

Mean-field Ordering Temperature;interaction energy scale

strong frustration

useful diagnosticof frustration

f 1


TCW

T |CW|

f= |CW|TF

Cooperative paramagnet:correlations remain weak

more universal

T < TF

Magnetically ordered ?

Spin liquid ?

not universal

Glassy ?

TF < T < |CW|

| || |


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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.


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Order by Disorder




33

q = 0

Classical Heisenberg Model on the Kagome lattice

Consider co-planar states

i

Si = 0


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Order by Disorder

3

3

q = 0





A

A

AA

A

AA

BB

BB

B

B

B

C

CC

C

C

A

A A

A

A

A

A

B B B

B B B

C C C

C C C

=

A

B C

Three-StatePotts spins

Degeneracy = e0.379N


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Order by Disorder





Weathervane loop

Non-planar states can begenerated by continuous

distortions of a planar state

Planar ground states have more soft modes(introduction of defect removes certain soft modes)


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Order by Disorder





3

3 : favored asSofter fluctuations for T 0

3

3 q = 0 Arbitrary


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Order by Disorder

Quantum zero point energy and further quantumfluctuations may select an ordered ground state.

Order by Disorder via Quantum Fluctuations:


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Order by Disorder

Quantum zero point energy and further quantumfluctuations may select an ordered ground state.

Disorder by Disorder

Sufficiently strong quantum fluctuations (S=1/2 for example),

however, may destabilize any ordered phase;possible quantum spin liquid - Disorder by Disorder

Order by Disorder via Quantum Fluctuations:


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Order by Disorder

Order by Disorder via Thermal/Quantum Fluctuations

Novel Magnetic Order and/or Valence Bond Solid States+ + + +

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Valence Bond Solid


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Square lattice

Large-S - Neel Order

Small-S - Valence Bond Solid (VBS)

Translational symmetry is broken

' '' ' ' ' ' '

' '

' ' ' ' ' '

'

' ' '

' ' '

Quantum Fluctuations


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. . .

.

.

. . .

.

.

. . . . . . .

. .. . . . .

Breakup and Separation

Spinons are confined by a linearly confining potential

' '' ' ' ' ' '

' ' ' ' ' ' ' ' ' '

' ' ' ' '

' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' ' '' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' '

' ' ' ' ' ' '

Valence Bond Solid (VBS)

Largest number of singlet pairs can resonate


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Frustrated Lattices

Increasing frustration in the large-S or semiclassical limit

Neel order Spiral order

increasing J/J


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Frustrated Lattices


Geometric frustration + quantum fluct. in the small-S limit Suppression of magnetic long range order


increasing J/J


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Frustrated Lattices


Geometric frustration + quantum fluct. in the small-S limit Suppression of magnetic long range order


increasing J/J

RVB (resonating valence bond) state on frustrated lattices

|RV B =

vbAvb|vb |vb =


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- - - - -

Breakup and Separation

Spinons are deconfined

Spinons: Q=0, S=1/2 excitations


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Quantum Phases of Frustrated Mott Insulator


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Spin Liquids come in Many Varieties

Excitations with a Gap

Gapless Excitations

Short-Range Valence Bond

Long-Range Valence Bond


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Global property unaffected by local dynamics

Red line intersects an even number of bonds

Recall the cartoon of the RVB state


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Two quantum states:


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and describe TWO topologically distinct

valence bond coverings

|vbeven |vbodd

intersectingeven number

of dimers

intersectingodd number

of dimers

Topological Order ?

|RV Beven =

even

Aeven

vb|vbeven |RV Bodd =

odd

Aodd

vb|vbodd

On the torus, there are FOUR topological sectors

(even,even), (even,odd), (odd,even), (odd,odd)


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Resonating Valence Bond:

Simplest quantum state

with topological particles!

A quantum system having particles with topological

character would be automatically protected against

errors caused by local disturbances

Alexei Kitaev (1995)


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More exotic topological states

Quantum states supporting particles with

Non Abelian particles

Braid 1 and 2

1

2 3

Order of braids matters!

Multiple braids:


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Non-Abelianstatistics

Glitteringequations, plusgreathandwavings-

the bestofthephysical reviewletterarticles.AlbertEinstein

Now updated with

new guidelines

for topological quantum

computing

Masaki OshikawaTokyo Institute of Technology

physicist and worstselling joker


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Quantum Order by Disorder:

Frustrated Diamond LatticeSpinel Antiferromagnet


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Frustrated Diamond Spinel Antiferromagnets

Spinel Structure

B2X4

A-site (Yellow) - diamond lattice

B-site (Blue) - pyrochlore lattice

X-site - vertices

Our interest: Spinel compounds with magnetic A-sites only

1 10 900

CoRh2O4

Co3O4

MnAl2O4

FeAl2O4

MnSc2S4

CoAl2O4

FeSc2S4

f = |!CW|/Tc


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Frustrated Diamond Spinel Antiferromagnets

Spinel Structure

B2X4

A-site (Yellow) - diamond lattice

B-site (Blue) - pyrochlore lattice

X-site - vertices

Our interest: Spinel compounds with magnetic A-sites only

1 10 900

CoRh2O4

Co3O4

MnAl2O4

FeAl2O4

MnSc2S4

CoAl2O4

FeSc2S4

f = |!CW|/Tc

magnetically ordered


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H = J1ij

Si

Sj + J2ij

Si

Sj

Heisenberg Model on the Diamond Lattice

J1

J2

J1

J2

Not frustrated

Frustrated !

favors Neel ordering


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H = J1ij

Si S

j + J2ij

Si S

j


J1

J2

Not frustrated

Frustrated !



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H = J1ij

Si S

j + J2ij

Si S

j


J1

J2

Not frustrated

Frustrated !



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Classical Heisenberg Model

J2/J10 1/8Nel Highly degenerate coplanar spirals

J2 J1 > 1 8Highly degenerate coplanar spiral states for at T=0

J2/J1=0.2

Deformed sphere:

1/8 < J2/J1< 1/4

2!/|k|

Surface of equal energy in momentum space

J2/J1=0.6

J2/J1=0.85

Punctured deformed sphere:

J2/J1> 1/4


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Quantum Order by Disorder

Expand the ground state energy in 1/

E= Ec +1

E1 + ...

E1 quantum zero-point fluctuation energy

Ec Classical energy

Energy landscape in

momentum space:

deformed sphere

Unfolded sphere

One point

One point0

!

2!

"k

#k

Equivalent

points

Represent the resulting energy change/difference

lifts the ground state degeneracy

= 2S controls quantumfluctuations


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Phase Diagram

Quantum2J2/J1

Nel

Degenerate

(k,k,k)

Circlearound(k,k,k)

6 points around (k,k,k)

including (k,k,0)

Cross at (k,0,0)

(k,0,0)

0

1/4

1/2

3/4

1


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Disorder by Disorder:

Spin Liquid on Hyper-Kagome Lattice


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Three-dimensional S=1/2 Frustrated Magnet

has a Hyper-Kagome sublattice of Ir ionsNa4Ir3O8

Pyrochlore Hyper-Kagome

3/4 Ir, 1/4 Na

Ir Na

Ir 4+ (5d )5All Ir-Ir bonds are equivalent

carries S=1/2 moment ?

Y. Okamoto, M. Nohara, H. Agura-Katrori, and H. Takagi, PRL 99, 137207 (2007)


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2000

1500

1000

500

(a) Na4Ir3O8

#

1(molIr/emu)

1.6

1.4

1.2

1.0

0.8

0.6(10

-3emu/molIr)

0.1 1 10 100T(K)

0.01 T0.1 T1 T

5 T

300250200150100500

T (K)

Inverse Spin Susceptibility; Strong Spin Frustration

CW = 650K

No magnetic ordering

down to

Curie-Weiss fit

Large Window ofCooperative Paramagnet

!"'&*

!+*,-

5

5

5 5 5 5

5 5 5

5 5 5

5 5

CW|/300


Strong Frustration - Macroscopicdegeneracy of classical ground states


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Specific Heat; Low Energy Excitations ?

6

4

2

0Sm(

J/KmolIr)

60

40

20

0

Cm/T

(mJ/K

2molIr)

300250200150100500

T (K)

(c)

(b)

1

10

100

Cm

/T(mJ/K

2molIr)

1 10 100T (K)

Cm$T2Cm$T3

12 T8 T0 T

4 T

Gapless Excitationsin an Insulator ?

No Magnetic Ordering

Field-independentup to 12T

Is the T=0 Ground State a Spin Liquid ?



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Theory of Spin Liquid with Fermionic Spinons

, = {, }

with the constraint

fifi = 1Si =

1

2fifi

Fermionic representation of the spin operator


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, = {, }

with the constraint

fifi = 1Si =

1

2fifi

Si Sj fermion-fermion interaction



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, = {, }

with the constraint

fifi = 1Si =

1

2fifi



Mean-Field Theory

ij = fifj

ij = fifj fermion kinetic energy dynamically generated

possible pairing correlation

The constraint is ONLY imposed on average !


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, = {, }

with the constraint

f

ifi = 1Si =

1

2fifi


Projected Wave Function Approach

= PGMF

Project out unphysical Hilbert space in

the mean-field ground states

Impose the constraint exactly


Mean-Field Theory

ij = fifj

ij = fifj fermion kinetic energy dynamically generated

possible pairing correlation

The constraint is ONLY imposed on average !


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, = {, }Fermionic representation of the spin operator

with the constraint

f

ifi = 1Si =

1

2fifi

In the spin liquid phases, the fermionic spinonsare liberated (deconfined) and become

emergent excitations


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Theory of Spin Liquid with Gapless Fermionic Spinons

The lowest energy state has uniform andij ij = 0

The resulting spin liquid has a spinon Fermi surface

MEAN FEILD THEORY + PROJECTION


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The lowest energy state has uniform andij ij = 0

The resulting spin liquid has a spinon Fermi surface

MEAN FEILD THEORY + PROJECTION

3 X


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Theory of Spin Liquid with Spinon Fermi Surface

Re-normalized Mean Field Theory with J = 304K

C/T looks linear for 5K < T < 20K

0

10

20

30

40

50

60

70

80

0 5 10 15 20 250

0.5

1

1.5

2

CV

/T

[mJ/K2

molIr]

Entropy[J/KmolIr]

Temperature [K]

Uniform State Cv/T

Exp. Cv/T

Uniform state entropy

Exp. entropy


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The spin liquid with a spinon Fermi surface is theground state ofthe nearest-neighbor Heisenberg model

At low temperatures, other perturbations may selectone of the competing phases as the ground state

If it happens, the resulting phase may be understood asa Fermi surface instability of the spinon Fermi surface

of the spin liquid

(just like in metals, where magnetic ordering or superconductivity canbe understood as an instability of the electron Fermi surface)

Spinon Fermi-liquid Theory of Correlated Insulators ?


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Lev Landau

Louis Neel

Neel Order v.s. Spin Liquid

Landau: Quantum Fluctuations will destroy Neel order

Strasbourg meetingin 1939

e.g. 1D antiferromagnetic Heisenberg model - Spin Liquid with S=1/2 spinons

... leads to his skepticism about the usefulness ofquantum mechanics; this was one of the few limitationsof this superior mind.(Jacques Friedel, Physics Today)

Yong Baek Kim- Emergent Quantum Phases in Frustrated Magnets

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