Wing-Ho Ko- Perturbing the U(1) Dirac Spin Liquid State in Spin-1/2 kagome

8/3/2019 Wing-Ho Ko- Perturbing the U(1) Dirac Spin Liquid State in Spin-1/2 kagome

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Perturbing the U(1) Dirac Spin Liquid State

in Spin-1/2 kagomeRaman scattering, magnetic field, and hole doping

Wing-Ho Ko

MIT

January 24, 2010
http://find/http://goback/


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Acknowledgments

Acknowledgments

Patrick LeeTai-Kai NgXiao-Gang Wen

Zheng-Xin LiuYing Ran

Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 24, 2010 2 / 34


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Outline

Outline

1 The Spin-1/2 Kagome Lattice

2 Derivation and Properties of 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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The Spin-1/2 Kagome Lattice

Outline



3 Raman Scattering


5 Hole Doping


Th S i 1/2 K L i


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

kagome lattice = 2D lattice ofcorner-sharing triangles.

Simple Neel order does not workwell on the kagome lattice.

Classical ground states:i Si = 0 .

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


Th S i 1/2 K L tti


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

kagome lattice = 2D lattice ofcorner-sharing triangles.

Simple Neel order does not workwell on the kagome lattice.

Classical ground states:i Si = 0 .

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

?


The Spin 1/2 Kagome Lattice


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In the quantum case, singlet formation is possible and maybe favored.

1D chain:

ENeel = S2J per bond

Esinglet =1

2

S(S + 1)J per bond

Higher dimensions = singlet lessfavorable.

kagome is highly frustrated = rareopportunity of realizing a singlet groundstate.

vs.




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1D chain:


Esinglet =1

2

S(S + 1)J per bond



vs.vs.




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1D chain:


Esinglet =1

2

S(S + 1)J per bond



vs.vs.vs.




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

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

In a SL, different bond configurationssuperpose, which results in a state that breaksno lattice symmetry.

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

vs.

=1

2




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p / g

For S= 1/2 kagome, the leading proposals are the 36-siteVBS and the U(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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p / g

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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/ g

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.




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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.


Derivation and Properties of the U(1) DSL State


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Outline



3 Raman Scattering


5 Hole Doping




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Deriving the U(1) DSL state: Slave boson formulation

Start with Heisenberg (or more generally tJ) model:

HtJ =

ij JSi Sj 14 ninj

tcicj + h.c.

;

ci ci 1

Apply the slave boson decomposition [Lee et al., RMP 78, 17 (2006)]:Si =

12 , fi, fi ; ci = fi hi ; fifi + fifi + hi hi = 1

Decouple four-operator terms by a HubbardStratonovichtransformation, with the following ansatz:

ij

fifj = eiij ; ij fifj fifj = 0

This results in a mean-field Hamiltonian....




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HtJ =

ij JSi Sj 14 ninj

tcicj + h.c.

;

ci ci 1




ij






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HtJ =

ij JSi Sj 14 ninj

tcicj + h.c.

;

ci ci 1




ij



spinon holon




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HtJ =

ij JSi Sj 14 ninj

tcicj + h.c.

;

ci ci 1




ij



spinon holon Constraint enforced by

Lagrange multiplier i0




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HtJ =

ij JSi Sj 14 ninj

tcicj + h.c.

;

ci ci 1




ij








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HtJ =

ij JSi Sj 14 ninj

tcicj + h.c.

;

ci ci 1




ij








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Deriving the U(1) DSL state: Emergent gauge field

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.)

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 areallowed).

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)].




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Deriving the U(1) DSL state: Band Structure

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




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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.)




structures.

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Enlarged Cell; Original Cell




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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.)




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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Real space


k-space



P i f h U(1) DSL B d


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Real space


k-space



P i f h U(1) DSL B d


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Real space


k-space



P ti f th U(1) DSL t t Th d i d


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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.


Raman Scattering

O tli


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Outline



3 Raman Scattering


5 Hole Doping


Raman Scattering

R s tt i g i M tt H bb d s st 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

Spin chirality terms in the Shastry Shraiman formulation


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Spin-chirality terms in the ShastryShraiman formulation

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

1 2

3

1 2

3

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.

But such cancellation is absent in the kagome lattice.Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 24, 2010 19 / 34

Raman Scattering



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1,2

3

i1 2

3




Raman Scattering



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i1

2,3ii

1 2

3




Raman Scattering

Spin chirality terms in the ShastryShraiman formulation


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i

ii

3 1

2iii

1 2

3




Raman Scattering



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Spin-chirality terms in the Shastry Shraiman formulation


i

iiiii

1,2

3

i




Raman Scattering



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i

iiiii

i1,2 3

ii




Raman Scattering



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i

iiiii

i

ii

2 3

1iii




Raman Scattering



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i

iiiii

i

iiiii+ = 0





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Raman Scattering



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Sp y S y S


i

iiiii

i

iiiii+ = 0



i

iiiiiiv

i

iiiii

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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g g y

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

U

14ij,/

+ij,\ 2ij,

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 and gauge mode


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g p p p g g

Si Sj ff ff and Si (Sj Sk) ff ff ff= contributions from spinon-antispinon pairs.= continuum of signal I() =

f|O|i2 DOS(). At low energy, one-pair states dominates. For Dirac node, DOS1pair E, and matrix element is suppressed in Eg

and A1g, but not in A2g.

= Spinon-antispinon pairs contribute IA2g() EandIEg/A1g() E3 at low energy. However, an additional collective excitation is available in A2g:

Si Sj Sk i3 exp(i d) + h.c. 3

b d

2x

=I

A2g bb + . . . q2

+ . . .(Recall that fi fj exp(iij) ) In our case (QED3 with Dirac fermions), turns out that 1/

when q 0 [Ioffe & Larkin, PRB 39, 8988 (1989)]. Analogy: plasmon mode vs. particle-hole continuum in normal metal.


Raman Scattering

http://goback/http://find/http://find/http://goback/


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g p p p g g






b d

2x

=

IA2g

b

b

+ . . .

q2

+ . . .

(Recall that fi fj exp(iij) )

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

Analogy: plasmon mode vs. particle-hole continuum in normal metal.Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 24, 2010 21 / 34

Raman Scattering



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b d

2x

=

IA2g

b

b

+ . . .

q2

+ . . .

(Recall that fi fj exp(iij) )

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

Analogy: plasmon mode vs. particle-hole continuum in normal metal.Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 24, 2010 21 / 34

Raman Scattering



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A1g,

Eg

Energy shift

Intensity

3

A2g

Energy shift

Intensity

???

1/


Raman Scattering

Experimental Comparisons: Some qualitative agreements


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Wulferding and Lemmens [unpublished] 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.



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External Magnetic Field

External magnetic field and the formation of Landau levels


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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

e

Prj

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





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0


dispersion.



ChernSimons theory.

b

vs.

b





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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.


0


dispersion.



ChernSimons theory.

b

vs.



Gapless mode and XY-ordering


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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, gaplessSz 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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Real space


k-space


Hole Doping

Hole doping and formation of Landau levels


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Doping can in principle be achieved bysubstituting Cl with S.

In slave-boson picture, hole doping introducesholons and antispinons.

As before, an emergent b field can open up

Landau levels in the spinon and holon bands. At mean-field, Espinon B3/2 while

Eholon B2= LL state favored.

At mean-field, b optimal when antispinons

form = 1 LL state= holons form = 1/2 Laughlin state.

dope

b

f, f h


Hole Doping

Charge fluctuation, Goldstone mode, and superconductivity


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b fluctuation holon density fluctuation charge density fluctuation. Long-wavelength b fluctuation costE 0 if real EM-field is turned off.

= a single linearly-dispersing mode Goldstone boson.

This time the Goldstone boson is eatenup by the EM-field to produce asuperconductor.

This superconducting state breaksT-invariance.

Intuitively, four species of holon bindedtogether = expects EM = hc/4e fora minimal vortex.

Confirmed by ChernSimonsformulation.

f1,...,2

h1,...,4

b


Hole Doping

Charge fluctuation, Goldstone mode, and superconductivity


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b fluctuation holon density fluctuation charge density fluctuation. Long-wavelength b fluctuation costE 0 if real EM-field is turned off.

= a single linearly-dispersing mode Goldstone boson.

This time the Goldstone boson is eatenup by the EM-field to produce asuperconductor.

This superconducting state breaksT-invariance.

Intuitively, four species of holon bindedtogether = expects EM = hc/4e fora minimal vortex.

Confirmed by ChernSimonsformulation.

f1,...,2

h1,...,4

b


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.

1h

1

h4


Hole Doping



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1h

1

h4

2f

h4


Hole Doping


Q i i l b d f i


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1h

1

h4

2f

h4

1

2

= 2


Hole Doping


Q i i l b d f i


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1h

1

h4

1h

1

h4

1

1

= + = 2


Hole Doping


Q i ti l b d t t f i


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1h

1

h4

1

h1h2

1

1

=


Hole Doping

Crystal momenta of quasiparticleprojective symmetrygroup study


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group study

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

These can be recovered using Projectivesymmetry group (PSG).

In contrast, the semion is fractionalizedfrom a holon.= semion-antisemion pair may not havewell-defined k.

+ +

+

+ + +

=


Hole Doping

Crystal momenta of quasiparticleprojective symmetrygroup study


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group study

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

These can be recovered using Projectivesymmetry group (PSG).

In contrast, the semion is fractionalizedfrom a holon.= semion-antisemion pair may not havewell-defined k.

*

**

** *

*

** = k of spinon-holon pairs


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)


Appendix


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Appendix

(a.k.a. hip pocket slides)


Appendix

Comparison of ground-state energy estimate


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Method Max. Size Energy State Ref.

Exact Diag. 36 0.43 [1]DMRG 192 0.4366(7) SL [2]VMC 432 0.42863(2) U(1) Dirac SL [3]Series Expan. 0.433(1) 36-site VBS [4]Entang. Renorm. 0.4316 36-site VBS [5]

[1] Waldtmann et al., EPJB 2, 501 (1998)[2] Jiang et al., PRL 101, 117203 (2008)

[3] Ranet al.

, PRL 98, 117205 (2007)[4] Singh and Huse, PRB 76, 180407 (2007)

[5] Evenbly and Vidal, arXiv:0904.3383


Appendix

Magnified band structure of DSL state, with scales


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kx = 0 ky = 0

Brillouin zone


Appendix

Raman signals contributed by spinon-antispinon: full scale


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Eg A1g A2g

(J 56 cm1)


Appendix

ChernSimons description: B-field case

F th B fi ld i t d t i f fi ld t d ib


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For the B-field case, introduce two species of gauge fields to describethe current of up/down spins:

J =

12

a,

The Lagrangian in terms of and a:

L = 1

4

a,a, +

1

2

a, + . . . higher derivative terms (e.g., Maxwell term aa for a) omitted L yields the correct equation of motion J = 12

Let c = (, a+, a)T, can rewrite L as

L = 14

cT Kc + . . . .

K has one null vector c0 gapless mode argued previously. Dynamics of c0 is driven by Maxwell term = linearly dispersing.


Appendix


For the B field case introd ce t o species of ga ge fields to describe


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J =

12

a,


L = 1

4

a,a, +

1

2



L = 14

cT Kc + . . . .



Appendix


For the B field case introduce two species of gauge fields to describe


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J =

12

a,


L = 1

4

a,a, +

1

2



L = 14

cT Kc + . . . .



Appendix

ChernSimons description: doped case


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In the doped case:

L = 14

cT Kc +e

2(q c)A + ( c)jV + . . .

where c = [; a1, . . . a4, a5, a6; b1, . . . b4]

K describes the self-dynamics of the system; has null vectorp0 = [2; 2, . . . ,2, 2, 2; 1, . . . , 1] corresponding to gapless mode c0.

q = [0; 0 . . . 0, 0, 0; 1, . . . , 1] is the charge vector.

is an integer vector with 0 -component and characterizes vortices.

Varying L w.r.t. c0 gives B = 2e

p0q p0j

0V =

2n

4ej0V

spinon spinon* holon


Appendix



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In the doped case:

L = 14

cT Kc +e

2(q c)A + ( c)jV + . . .

where c = [; a1, . . . a4, a5, a6; b1, . . . b4]





p0q p0j

0V =

2n

4ej0V



Appendix



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In the doped case:

L = 14

cT Kc +e

2(q c)A + ( c)jV + . . .

where c = [; a1, . . . a4, a5, a6; b1, . . . b4]





p0q p0j

0V =

2n

4ej0V

self dynamics EM coupling vortices



Appendix



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In the doped case:

L = 14

cT Kc +e

2(q c)A + ( c)jV + . . .

where c = [; a1, . . . a4, a5, a6; b1, . . . b4]





p0q p0j

0V =

2n

4ej0V

self dynamics EM coupling vortices


Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 24, 2010 40 / 34 Appendix

Quasiparticles and their statistics


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L = 1

4

cT

Kc +

e

2

(q

c)

A + (

c)

j

V + . . .

Vortices with p0 = 0 does not couple to c0= They can exist when B = 0 and corresponds to quasiparticles.

When particle 1 winds around another particle 2, the statistical

phase = 2T1 K1 2 K is part of K thats p0. Derived by integrating out all gauge fields having non-zero

ChernSimons term.

Taking 1 = [0; 0, . . . , 0, 0, 0; 1, 0, 0,1]2 = [0; 0, . . . , 0, 0, 0; 1, 0,1, 0] , found:11 = 22 = 2 , 12 =

= Fermions with semionic statistics!

Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 24, 2010 41 / 34 Appendix

The full form ofK-matrix for doped case


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K =

0 1 1 1 1 1 1 1 1 1 11 1 0 0 0 0 0 0 0 0 01 0 1 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 0 0

1 0 0 0

1 0 0 0 0 0 0

1 0 0 0 0 1 0 0 0 0 01 0 0 0 0 0 1 0 0 0 01 0 0 0 0 0 0 2 0 0 01 0 0 0 0 0 0 0 2 0 01 0 0 0 0 0 0 0 0 2 01 0 0 0 0 0 0 0 0 0 2

(Recall c = [; a1, . . . a4, a5, a6; b1, . . . b4])spinon spinon* holon

Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 24 2010 42 / 34 Appendix

Spinon and holon PSG


84/84

= 1= 3

= 2= 4

1

2

3

4

Tx[21] =

24

Tx[22] =

23

Tx[23] = e

i3 22Tx[

24] = e

i3 21

Tx[1] = ei12 2

Tx[2] = e11i

12 1

Tx[3] = ei12 4

Tx[4] = e11i

12 3

Wing-Ho Ko (MIT) Perturbing the U(1) DSL State January 24 2010 43 / 34

Wing-Ho Ko- Perturbing the U(1) Dirac Spin Liquid State in Spin-1/2 kagome

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