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0.5setgray1 Some exotic phenomenain doped 2D frustrated

quantum magnetsDidier Poilblanc

Laboratoire de Physique Theorique, UMR5152-CNRS, Toulouse, France

Some exotic phenomena in doped 2D frustrated quantum magnets – p. 1

Outline

Goal: Vacancies and mobile holes in frus-trated magnets

1. Spinon-hole confinement / deconfine-ment: comparison between differenttypes of quantum disordered gs / mod-els

2. Kinetic energy-based superconductiv-ity


2D Frustratedmagnets

Lattices with AF frustrating interactionsMelzi et al., PRB 85,

1318 (2000)

J

J1

2

frustrated squarelattice (S=1/2):

Li2VOSiO4

Kagome lattice likeSrCr9−xGa3+xO19

(S=3/2)Ramirez et al., PRL 64 (’90)Broholm et al., PRL 65 (’90)


Itinerant frustratedsystems

spinel oxide LiTi2O4

Sun et al., PRB 70, 054519 (2004)

5d transition-metal pyrochlores as Cd2Re207

or KOs2O6

Hanawa et al., PRL 87, 187001 (2001)Hiroi et al., JPSJ 73, 1651 (2004)

CoO triangular layer based compoundTakada et al., Nature 422, 53 (2003)

All superconducting with Tc up to 13.7 K !Some exotic phenomena in doped 2D frustrated quantum magnets – p. 4

Quantum disorderedgroundstates

frustration + quantum fluctuations (S=1/2)

Schulz, Ziman & DP, Ed.H.T. Diep, W.-S.(1994)

Review by Misguich &Lhuillier, "Frustrated spin

systems", Ed. H.T. Diep,World-Scientific (2003)

nature of quantum disordered phase ?Some exotic phenomena in doped 2D frustrated quantum magnets – p. 5

The Valence BondSolid: Checkerboard

corner-sharingtetrahedra:

”2D pyrochlore”

VBS phase(plaquette)

Fouet et al., PRB (2003)

Finite spin gap ∼ 0.6J

Spontaneous translation symmetry breaking


The spin liquid:Kagomé lattice ?

Magnetically disorderedLeung & Elser PRB 47, 5459 (1993)

No symmetry breaking (neither SU(2) norlattice)

Large number of low energy singletsWaldtmann et al., EPJB 2, 501 (1998)Mila et al., PRL 81, 2356 (1998)


Confinement vsdeconfinement

See Sachdev

(a) (b)

”string potential” ”deconfined” spinon


Spin density arounda vacancy

⟨

Szi

⟩

at distance r = ri − rO from defect

0 1 2 3 4-0.2

0

0.2

0.4

0.6

0 1 2 3 4

-0.2

0

0.2

0.4bare wfground state

|ri-rO|

<Siz >

|ri-rO|

checkerboard

Kagome

holehole

⟨

Szi

⟩

bare→ spin-spin

correlation in host⟨

Szi

⟩

gs→ “spinon”

wavefunction

Kagomé: deconfinedCheckerboard: strongly confined

DP, A. Läuchli, M. Mambrini & F. Mila, cond-mat/0506810


Quasiparticle weight

Overlap (squared) Z = |⟨

Φgs|Φbare

⟩

|2zero or finite ?

ZKagome = 0Zcheckerboard ' 1

Dynamic hole (finite t) −→ Zk

A(k, ω) = Zkδ(ω − ωk) + Ainc ??


Hole dynamics:t-J model

(c)

(a) (b)

(d)

Γ

M

Γ

MΣ

H = −t∑

〈i,j〉,σ

P(

c†i,σcj,σ + h.c.

)

P + J∑

〈i,j〉

Si · Sj −1

4ninj


Hole dynamicsin the VB Solid

-4 -2 0 2 4 6

Γ

Σ

M

ω/t

M

A(k

, )

[a.u

.]ω

x 1.5

x 5

x 1.5

Σ

t>0 Γ

0.2%

6.7%

7.1%

13.5% 4.2%

0.04% / 2%

2.9%

1.3%

Checkerboard

-2 0 2 4 6

t<0x 5

ω/|t|

x 1.5

x 1.5

x 1.5

x 1.5 Smallquasi-particle

peaks:holon-spinonboundstate

A. Läuchli & DP, PRL 92, 236404 (2004)


Single hole dopedin a spin liquid

-4 -2 0 2 4 6 -2 0 2 4 6ω/t ω/|t|

A(k

,ω) [

a.u.

]

t>0 t<0Γ Γ

ΜΜ

x 0.5 x 0.5

kagome

IncoherentspectrumWeights

distributed onmany poleseven at low

energies

A. Läuchli & DP, PRL 92, 236404 (2004)Some exotic phenomena in doped 2D frustrated quantum magnets – p. 13

J1-J2-J3 square latticeHeisenberg

B

A

0.5

J3 /J1

J2 /J10.5

Z < 0.84

(q,q)

(q,π)

(π,π) (0,π)

0.63

(0.14,0.26)

(0.502,0.097)

0.33

0.46 0.56

Classical phasediagram (Chubukovet al., PRB 90)→ collinear vs spiral

Quantum case→ VBS vs spin liquid ?

Dimer basis computations: Mambrini et al. −→ VBS

In agreement with Leung & Lam, PPB 96


Spin density arounda vacancy

⟨

Szi

⟩

at distance r = ri − rO from defect

0 1 2 3 4

-0.1

0

0.1

0.2

bare wf

ground state

0 1 2 3 4

-0.1

0

0.1

0.2

<Siz >

|ri-rO| |ri-rO|

A B

holehole

ξ = 0.7ξ = 0.5

ξ = 5.5ξ = 2.2

AFAF

conf

conf

ξconf � ξAF

DP, A. Läuchli, M. Mambrini & F. Mila, cond-mat/0506810Compatible with "deconfined critical point" scenario (field

theory by Senthil et al.)Some exotic phenomena in doped 2D frustrated quantum magnets – p. 15

Interference effectsJ → 0 limit

� � � � � � � � � � ��

� � � � � � � � � � �� singlet triplet

t < 0 E = −|t| E = −2|t|

t > 0 E = −2t E = −tSome exotic phenomena in doped 2D frustrated quantum magnets – p. 16

Hole localisation inthe VBS host

-4 -2 0 2 4 6

(b) t>0

ω/|t|

A(k

=kX,

) [a

.u.]

ω

J/|t|=0.3, 0.4 & 0.6 (a) t<0

0

0,05

0,1

Zk(c)

(d)

k=kX=( /2, /2)π π

t<0 t>0

0 0,2 0,4 0,6 0,80

0,2

0,4

J/|t|

W

/|t|

ΓΜ

square lattice

Electron-hole asymmetry

For t > 0: destructive interference effects→ single hole almost localized


Correlated pairhopping

- s-wave and d-wave symmetries favored- Pair size ∼ 3 lattice spacings

DP, PRL 93, 197204 (2004)Some exotic phenomena in doped 2D frustrated quantum magnets – p. 18

Pairing energy

Binding on 4×4 &√

32×√

32-site clusters∆binding = E2holes + EHeis − 2E1hole

0 0,1 0,2 0,3 0,4 0,5 0,6

-0,4

-0,2

0

square lattice d-wave t<0 d-wave t>0 s-wave t>0 s-wave t<0 dxy t>0

g-wave t>0

J

Ene

rgy h-h continuum

4x44x4

EBkin<0

EBmag>0

Feynman-Hellmann:magnetic energy:

EmagB = J dEB

dJ

kinetic energy:Ekin

B = EB − EmagB < 0

⇒ gain !!

s-wave and d-wave symmetries favoredSome exotic phenomena in doped 2D frustrated quantum magnets – p. 19

Summary andoutlook

Correlate impurity problem (NMR, S-STM)with ARPES

Very heavy quasiparticle in translationalsymmetry breaking VBS

Spin-charge separation in a spin-liquid→ Generic ? Finite density of holes ?

Pairing mechanism based on kinetic energy

Perspective: doped quantum dimer model(Roksar-Kivelson)


Collaborations andreferences

Confinement around a vacancyDP, A. Läuchli, M. Mambrini & F. Mila,cond-mat/0506810

Single hole dynamicsA. Läuchli & DP, PRL 92, 236404 (2004)

Pairing in VBS hostDP, PRL 93, 197204 (2004)


3D Frustratedmagnets

pyrochlores and spinels

Transition metal oxides- ZnCr2O4 spinel

- A2Ti2O7 titanatesRamirez et al., PRL 89,

067202 (2002)

-no ordering down to low temperatures


spin & charge repel !(a)

(c) (d)

(b)

t>0 t<0

⇐ Hole-spincorrelations

Dimer correlationsin

"Holon" wavefct

Spin & charge separation: holon benefits from large dimercorrelations in neighboring triangle


Extended-sizespinon-holon BS

Single hole spectral fct (J1J2J3 model)

0 0,2 0,4 0,6 0,8 1-1,6

-1,2

-0,8

-0,4

Hol

e en

ergy

-2 0 2

J3/J1=0.5J3/J1=0.7J3/J1=1

ω/t

J1/t=0.4

J3/J1

k=(π,0)

10

J 3/t=0.

4

A(k

, )

[a

.u.]

ω

(a) (b)

Reduced Z-factor (ARPES)m

extended-range spinon wf (NMR, spin-STM)Some exotic phenomena in doped 2D frustrated quantum magnets – p. 24

Non-magnetic dopantin spin-Peierls chain

ExperimentDoping in CuGeO3: Cu2+ → Zn2+ or Mg2+

Hase et al., PRL 71, 4059 (1993)

Theory (numerics)Augier et al., PRB 60, 1075 (1999)


Hole-holecorrelations

0 1 2 3 4

0.02

0.03

0.04

0.05 t>0 (s-wave) & J=0.3 t<0 (d-wave) & J=0.3 J=0.6

0 0.2 0.4 0.6

2.3

2.4

2.5

2.6 t>0 (s-wave) t<0 (d-wave)

Chh(r)

|r| J/|t|

Rhh

(a) (b)

- No hole-hole repulsion for t > 0- Pair size ∼ 3 lattice spacings


Single particleGreen-function

"time-ordered" Green’s function → "electron"(ω < 0) and "hole" (ω > 0) parts

G(q, ω) = 〈Ne|c†−q,σ

1

ω − iε + H − ENe−1

cq,σ|Ne〉

+ 〈Ne|cq,σ1

ω + iε − H + ENe+1

c†−q,σ|Ne〉

half-filling: Ne = N , system size

"hole" and "electron" parts related: t ⇔ −t

Spectral fct Im G(q, ω) → IPES & ARPESSome exotic phenomena in doped 2D frustrated quantum magnets – p. 27

Dynamics withLanczos ED (I)

- A† is applied to GS:

|Φ1〉 =1

(〈Ψ0|AA†|Ψ0〉)1/2A†|Ψ0〉

⇒ C(z) = 〈Ψ0|AA†|Ψ0〉〈Φ1|(z′ − H)−1|Φ1〉- Lanczos procedure:

z′ − H =

z′ − e1 −b2 . . . 0

−b2. . . . . . ...

... . . . . . . −bM

0 . . . −bM z′ − eM

(1)


Dynamics withLanczos ED (II)

- Continued-fraction: z = ω + iε, A = cq,σ or A = c†−q,σ

G(z) =〈Ψ0|AA†|Ψ0〉

z + E0 − e1 −b22

z + E0 − e2 −b23

z + E0 − e3 − . . .

- Physical meaning:I(ω) =

∑

m |〈Ψm|A†|Ψ0〉|2δ(ω − Em + E0)

1. poles and weights → dynamics of A†

2. symmetry of A† → well defined quantum numbers & selection rules

3. ! calculation of eigen-states/vectors not required !


Static hole(checkerboard)

- Switch off t first ⇒ already some insight !

0 1 2 3 4 5ω/J

0

0.5

1

1.5

2

2.5

3

Ast

atic

(ω)

weight=90.1%

OverlapZ = |

⟨

Φ1h|ci,↑|Φ0h

⟩

|2finite

on checkerboardlattice

Similarity with 1D spin-Peierls g.s.⇒ confining potential between "holon" and"spinon"


Static hole(checkerboard)

- Switch off t first ⇒ already some insight !

0 1 2 3 4 5ω/J

0

0.5

1

1.5

2

2.5

3

Ast

atic

(ω)

weight=90.1%

OverlapZ = |

⟨

Φ1h|ci,↑|Φ0h

⟩

|2finite

on checkerboardlattice

Similarity with 1D spin-Peierls g.s.⇒ confining potential between "holon" and"spinon" Some exotic phenomena in doped 2D frustrated quantum magnets – p. 30

Static hole (Kagome)- Static correlations: Dommange et al., PRB 68,224416 (2003)- Dynamic correlations: Z = |

⟨

Φ1h|ci,↑|Φ0h

⟩

|2 ' 0

0 1 2 3 4 5ω/J

0

0.1

0.2

0.3

0.4

0.5

Ast

atic

(ω)

Incoherent spectrumWeights distributedon many poles even

at low energies


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