wing-ho ko- novel magnetic and superconducting states in spin-1/2 kagome lattice and its doped...

8/3/2019 Wing-Ho Ko- Novel Magnetic and Superconducting States in Spin-1/2 Kagome Lattice and its Doped Variant

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Novel Magnetic and Superconducting States inSpin-1/2 Kagome Lattice and its Doped Variant

Wing-Ho Ko

MIT

July 7, 2010
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Acknowledgments

Acknowledgments

Patrick A. Lee

Xiao-Gang WenSenthil TodadriTai-Kai NgYoung S. Lee

John D. JoannopoulosRobert JaffeMehran KardarWolfgang KetterleIssac ChuangLeonid Levitov

Washington TaylorIain StewartJohn McGreevyEdward Farhi

Margaret OMeara

Michael Hermele

Cody NaveYing RanZheng-Cheng GuK. T. LawSaeed SaremiMaissam Barkeshli

Nan GuJamal RahiBrian SwingleTarun GroverFa WangTim Chen

Abolhassan VaeziZheng-Xin LiuRahul NandkishoreJuven WangAlejandro Rodriguez

Anjan Soumyanarayanan

Wing-Ho Ko (MIT) Novel states in spin-1/2 kagome July 7, 2010 2 / 41
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Outline

Outline

1 Quantum Spin Liquid

2 The Spin-1/2 Kagome Lattice and the U(1) DSL State

3 Raman Scattering

4 External Magnetic Field

5 Hole Doping

Reference: Ran, Ko, Lee, & Wen, PRL 102, 047205 (2009)Ko, Lee, & Wen, PRB 79, 214502 (2009)Ko, Liu, Ng, & Lee, PRB 81, 024414 (2010)

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Quantum Spin Liquid

Outline



3 Raman Scattering


5 Hole Doping


Q S i Li id
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Quantum Spin Liquid

Strong Coulomb repulsion leads to Mott insulators withantiferromagnetic interactions.

Hubbard modelthe archetypal strongly correlated system:HHb = t

ij, c

icj + U

i nini

U large and

ni

1 =

t-J model:

HtJ =

ij JSi Sj 14 ninj

t

cicj + h.c.

;

ci ci 1

Half-filling (ni = 1) = antiferromagnetic spin system(Heisenberg model):

HHsb =

ij JSi Sj

Kinetic energy Coulomb interaction

Constraint

virtual hopping


Q t S i Li id
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Quantum Spin Liquid

Under normal circumstances, the classical Neel order wins.

HHsb =

ij JSi Sj

The classical Neel state is not the true quantum ground state. But quantum fluctuation is usually not strong enough to destroy the

classical order.

Neel1970

Shull1994


Quantum Spin Liquid
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Quantum Spin Liquid

However, the Neel order is not the only possibility.

In 1973, Anderson proposed the resonating valence

bond (RVB) state, which is formed by superposingsinglets.

Unlike the Neel state, the elementary excitation in theRVB state are spin-1/2 spinons.

And in the doped case, an electron is fractionalizedinto a spinon and a holon. Anderson

= + +

= ||2


Quantum Spin Liquid
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Quantum Spin Liquid






vs.


Quantum Spin Liquid
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Quantum Spin Liquid






vs.


Quantum Spin Liquid
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Quantum Spin Liquid

Two major classes of singlet states: valence bond solids(VBS) and spin liquid (SL)

In a SL, different bond configurationssuperpose, which results in a state that breaks

no lattice symmetry.

In a VBS, certain singlet bonds are preferred,which results in a symmetry-broken state.

A VBS state generally has a spin gap, while aSL state can be gapped or gapless.

vs.


Quantum Spin Liquid
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Q p q


Quantum Spin Liquid
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Q p q

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Quantum Spin Liquid


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Theory of spin liquid: emergent gauge field

HMF = i

f

i(ii

0

F)f

i J

ij,(eiijf

ifj

+h.c.)

+

ihi (i

i0 B)hi t

ij

(eiijhi hj + h.c.)

The

field is an emergent gauge field, corresponding to gaugesymmetry f eif, h eih. At the lattice level is a compact gauge field (i.e., monopoles are

allowed).

But with Dirac fermions, the system can be in a deconfined phase(i.e., monopoles can be neglected) [Hermele et al., PRB 70, 214437 (2004)].

Neglecting fluctuation of, spinons and holons are decoupled. Mean-field ansatz for SL state can be specified by pattern of flux.


Quantum Spin Liquid
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Factors favoring singlet states: small spin, lowdimensionality, and geometrical frustration

1D chain:

ENeel = S2J per bondEsinglet =

1

2

S(S + 1)J per bond

Higher dimensions = singlet lessfavorable.

Geometrical frustration: cant make all

bonds happy at the same time.= Classical order is weakened.

vs.


Quantum Spin Liquid
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1D chain:


1

2

S(S + 1)J per bond




vs.vs.


Quantum Spin Liquid
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1D chain:


1

2

S(S + 1)J per bond




vs.vs.

?


Quantum Spin Liquid
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1D chain:


1

2

S(S + 1)J per bond




vs.vs.

?

vs.


Quantum Spin Liquid
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Possible material realizations of quantum spin liquids

-(ET)2XTriangular lattice

Na4Ir3O8Hyperkagome lattice

ZnCu3(OH)6Cl2Kagome lattice


The Spin-1/2 Kagome Lattice and the U(1) DSL State
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Outline



3 Raman Scattering


5 Hole Doping


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The nearest-neighbor antiferromagnetic Heisenberg modelis highly frustrated on the kagome lattice.

Kagome lattice = 2D lattice ofcorner-sharing triangles.

Kagome lattice is frustrated: nosimple Neel order.

Classical ground states:i Si = 0 .

# classical ground states eN.[Chalker et al., PRL 68, 855 (1992)].


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Experimental realization ofS = 1/2 kagome:herbertsmithite ZnCu3(OH)6Cl2.

Herbertsmithite: layered structure with Cuforming an AF kagome lattice.

Caveats: Zn impurities and DzyaloshinskiiMoriya interactions.

Experimental results [e.g., Helton et al., PRL 98,107204 (2007); Bert et al., JP:CS 145, 012004 (2009)]:

Neutron scattering: no magnetic order downto 1.8 K.

SR: no spin freezing down to 50 K.

Heat capacity: vanishes as a power law asT 0. Spin susceptibility: diverges as T 0. NMR shift: power law as T 0.


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Leading theoretical proposals: the 36-site VBS and theU(1) Dirac spin liquid

The 36-site VBS pattern is found in seriesexpansion [Singh and Huse, PRB 76, 180407 (2007)]and entanglement renormalization [Evenbly andVidal, arXiv:0904.3383].

From VMC, the U(1) Dirac spin liquid (DSL)state has the lowest energy among various SLstates, and is stable against small VBSperturbations [e.g., Ran et al., PRL 98, 117205 (2007)].

Exact diagonalization: initially found small( J/20) spin gap; now leaning towards agapless proposal [Waldtmann et al., EPJB 2, 501(1998); arXiv:0907.4164].

vs.

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0


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U(1) DSL state: mean-field ansatz

HMF = i

f

i

(ii

0 F)fi

3J

8

ij,

(eiijf

i

fj+h.c.)

+

ihi (i

i0 B)hi t

ij

(eiijhi hj + h.c.)

Neglecting fluctuation of, spinonsand holons are decoupled.

Mean-field ansatz for SL state can bespecified by pattern of flux.

U(1) Dirac spin liquid state: flux perand 0 flux per . flux = unit cell doubled in band

structures.

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Enlarged Cell; Original Cell


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U(1) DSL state: mean-field ansatz

HMF = i

f

i

(ii

0 F)fi

3J

8

ij,

(eiijf

i

fj+h.c.)

+

ihi (i

i0 B)hi t

ij

(eiijhi hj + h.c.)

Neglecting fluctuation of, spinonsand holons are decoupled.

Mean-field ansatz for SL state can bespecified by pattern of flux.

U(1) Dirac spin liquid state: flux perand 0 flux per . flux = unit cell doubled in band

structures.teff = +t ; teff = t


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Properties of the U(1) DSL state: band structure

Real space

teff = +t ; teff = t

k-space


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Real space


k-space



( )
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Real space


k-space



f ( ) S
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Properties of the U(1) DSL state: thermodynamics andcorrelation

At low energy, the U(1) DSL state isdescribed by QED3.

i.e., gauge field coupled to Diracfermions in (2+1)-D.

Thermodynamics of the U(1) DSL stateis dominated by the spinon Fermi surface. Zero-field spin susceptibility: (T) T. Heat capacity: CV(T) T2.

U(1) DSL state is quantum criticalmany correlations decay algebraically[Hermele et al., PRB 77, 224413 (2008)].

Emergent SU(4) symmetry among Diracnodes = different correlations canhave the same scaling dimension.



R h i i d i i f h i l
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Research motivation: deriving further experimentalconsequences of the U(1) DSL state

All experiments point to a state without magnetic order. But moredata is needed to tell if it is a VBS state or a SL state, and whichVBS/SL state it is.

Without concrete theory, the experimental data are hard to interpret. Without concrete theory, unbiased theoretical calculations are difficult. The U(1) Dirac spin-liquid state is a theoretically interesting exotic

state of matter.

Thus, our approach: assume the DSL state and considerfurther experimental consequences.


Raman Scattering

O li
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Outline



3 Raman Scattering


5 Hole Doping


Raman Scattering

R tt i i M tt H bb d t th
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Raman scattering in Mott-Hubbard system: theShastryShraiman formulation

Raman scattering = inelastic scattering of photon. Good for studying excitations of the system. Probe only excitations with q 0.

We are concerned with a half-filled Hubbard system, in the regimewhere

|i

f

| U and i

U.

Both initial and final states are spin states.= T-matrix can be written in terms of spin operators.

Intermediate states are dominated by the sector where

i nini = 1. The T-matrix can be organized as an expansion in t/(U i)

[Shastry & Shraiman, IJMPB 5, 365 (1991)].

T(2) t2UiSi Sj + . . .


Raman Scattering

S i hi lit t i th Sh t Sh i f l ti
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Spin-chirality terms in the ShastryShraiman formulation

Because of holon-doublon symmetry, there is no t3 order contribution.

i

iiiii

i

iiiii+ = 0

For the square and triangular lattice, there is no t4 order contribution

to spin-chirality because of a non-trivial cancellation between 3-siteand 4-site pathways.

i

ii

iii

iv

i

ii

iii

iv

i

iiiii

iv

i

iiiii

iv+ + +

= 0

But such cancellation is absent in the kagome lattice.Wing-Ho Ko (MIT) Novel states in spin-1/2 kagome July 7, 2010 25 / 41

Raman Scattering

S i hi lit t s i th Sh st Sh i f l ti
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Spin-chirality terms in the ShastryShraiman formulation

Because of holon-doublon symmetry, there is no t3 order contribution.

i

iiiii

i

iiiii+ = 0

For the square and triangular lattice, there is no t4 order contribution

to spin-chirality because of a non-trivial cancellation between 3-siteand 4-site pathways.

i

ii

iii

iv

i

ii

iii

iv+ + +

t4(Ui)3

Si Sj Sk + . . . But such cancellation is absent in the kagome lattice.


Raman Scattering

Raman T matrix for the kagome geometry
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Raman T-matrix for the kagome geometry

For kagome geometry, the Raman T-matrix decomposes into 3 irreps:

T = T(A1g)(xx+y y)+T(E(1)g )(xxy y)+T(E(2)g )(x y+y x)+T(A2g)(xyy x)

To lowest order in inelastic terms,

T(E(1)g ) =4t2

i

U14

ij

,/ +

ij

,

\2

ij

,

Si

Sj

T(E(2)g ) =4t2

iU3

4

ij,/

ij,\

Si Sj

T(A1g) =t4

(iU)3

2

ij +

ijSi Sj

T(A2g) =2

3it4

(iU)3

R

3 + 3 + +

+ + + +

(i j

k= Si Sj Sk, etc.)


Raman Scattering

Raman signals: spinon antispinon pairs
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Raman signals: spinon-antispinon pairs

Si Sj ff ff and Si (Sj Sk) ff ff ff

= contributions from spinon-antispinon pairs,e.g.,

|f2pairs ff ff|i, f2pairs|ff ff|i = 0|f1pairs ff|i, f1pairs|ff ff|i = 0

This produces a continuum of signal:I() =

N

fNpairs|T|i2 DOS().

At low energy, one-pair states dominates.

For Dirac node, DOS1pair E, and matrix elementis suppressed in Eg and A1g, but not in A2g.

= Spinon-antispinon pairs contributeIA2g() Eand IEg/A1g() E3 at low energy.

E

3

IEg, IA1g

DOS,IA2g


Raman Scattering

Raman signals: emergent gauge mode
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Raman signals: emergent gauge mode

However, an additional collective excitation isavailable in A2g:

Si Sj Sk ff ff ff i3 exp(i d) + h.c.

3

b d2x

= I(gauge)A2g bb + . . . q2 + . . .[Recall that fi fj exp(iij) ]

In our case (QED3 with Dirac fermions), turnsout that 1/ when q 0[Ioffe & Larkin, PRB 39, 8988 (1989)].

Analogy: plasmon mode vs. particle-holecontinuum in normal metal.

b

1/


Raman Scattering

Raman signals: summary
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Raman signals: summary

A1g,

Eg

Energy shift

Intensity

3

A2g

Energy shift

Intensity

???

1/


Raman Scattering

Experimental comparisons: some qualitative agreements
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Experimental comparisons: some qualitative agreements

Wulferding et al. [arXiv:1005.4831] have obtained Raman signal onherbertsmithite.

At low T, data shows a broad background with a near-linear piece atlow-energy.

Roughly agree with the theoretical picture presented previously.


External Magnetic Field

Outline
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Outline



3 Raman Scattering


5 Hole Doping



External magnetic field and the formation of Landau levels
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External magnetic field and the formation of Landau levels

In Mott systems, B-field causes only Zeemansplitting. This induces spinon and antispinon pockets

near the Dirac node.

However, with the emergent gauge field ,Landau levels can form spontaneously.

From VMC calculations,

ePrjFP 0.33(2)B3/2 + 0.00(4)B2

ePrj

LL 0.223(6)B3/2

+ 0.03(1)B2

eLLMF < eFPMF to leading order in n

= Landau level state is favored.

B

b



Sz density fluctuation as gapless mode
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b is an emergent gauge field.

= Its strength can fluctuate in space. The fluctuation of b is tied to thefluctuation of Sz density.

In long-wavelength limit, energy cost ofb fluctuation

0.

= The system has a gapless mode! Derivative expansion = linear

dispersion.

Other density fluctuations and

quasiparticle excitations are gapped.= Gapless mode is unique. Mathematical description given by

ChernSimons theory.

b



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0.


dispersion.



ChernSimons theory.

b

vs.

b



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Sz y g p




0.


dispersion.



ChernSimons theory.

b

vs.



Gapless mode and XY-ordering
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p g

Recap: we found a single linearly-dispersing

gapless mode, which looks like... A Goldstone boson! And indeed it is.

Corresponding to this Goldstone mode is aspontaneously broken XY order.

Analogy: Superfluid: =

ei, [, ] = i, gapless

fluctuation = ordered (SF) phase; XY model: S+ = ei, [Sz, ] = i, gapless

Sz fluctuation = XY ordered phase. VMC found the q = 0 order. S+ in XY model V in QED3.

= In-plane magnetization M B.(V monopole operator, its scaling dimension)


Hole Doping

Outline
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3 Raman Scattering


5 Hole Doping


Hole Doping

Recap: band structure
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p

Real space


k-space


Hole Doping

Hole doping and the formation of Landau levels
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Doping can in principle be achieved bysubstituting Cl with S.

In the slave-boson picture, hole dopingintroduces holons and antispinons.

As before, an emergent b field can open upLandau levels in the spinon and holon bands.

At mean-field, Espinon B3/2 whileEholon B2.= LL state favored.

At mean-field, b optimal when antispinonsform = 1 LL state.= Holons form = 1/2 Laughlin state.

dope

b

f, f h

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Hole Doping

Quasiparticle statisticsintuitions


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Quasiparticles are bound states of semionsin holon sectors and/or fermions in spinon

sector. For finite energy, bound states must be

neutral w.r.t. the gapless mode.

There are two types of elementaryquasiparticles:

1 semion-antisemion pair in holon sector;2 spinon-holon pair

( Bogoliubov q.p. in conventional SC).

All other quasiparticles can be built fromthese elementary ones.

Statistics can be derived by treatingdifferent species as uncorrelated.

Semions from holon sector = existenceof semionic (mutual) statistics.

1h1

h4


Hole Doping

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1h1

h4

2f

h4


Hole Doping

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1h1

h4

2f

h4

1

2

= 2


Hole Doping

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1h1

h4

1h1

h4

1

1

= + = 2


Hole Doping

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1h1

h4

1h1

h2

1

1

=


Hole Doping

Crystal momenta of quasiparticleprojective symmetry
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group study

For spinon-holon pair, k is well-defined onthe original Brillouin zone.

These can be recovered using Projective

symmetry group (PSG).

In contrast, the semion is fractionalizedfrom a holon.=

semion-antisemion pair may not have

well-defined k.

+ +

+

+ + +

=

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Conclusions

Conclusions


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The U(1) Dirac spin-liquid state possess many unusual properties andmay be experimentally realized in herbertsmithite.

The Raman signal of the DSL state has a broad background(contributed by spinon-antispinon continuum) and a 1/ singularity(contributed by collective [gauge] excitations).

Under external magnetic field, the DSL state forms Landau levels,which corresponds to a XY symmetry broken state with Goldstoneboson corresponding to Sz density fluctuation.

When the DSL state is doped, an analogous mechanism give rise toan AndersonHiggs scenario and hence superconductivity.

But minimal vortices carry hc/4e flux and the system contains exotic

quasiparticle having semionic mutual statistics.

Reference: Ran, Ko, Lee, & Wen, PRL 102, 047205 (2009)Ko, Lee, & Wen, PRB 79, 214502 (2009)Ko, Liu, Ng, & Lee, PRB 81, 024414 (2010)

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wing-ho ko- novel magnetic and superconducting states in spin-1/2 kagome lattice and its doped...

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