0.5setgray0

0.5setgray1 Spin-charge separation in

doped 2D frustratedquantum magnets

Didier Poilblanc

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

Spin-charge separation in doped 2D frustrated quantum magnets – p. 1

Outline

1. Disordered state in frustrated magnets

2. Lanczos methods & ARPES

3. Spin-charge separation in single holedynamics ?

4. New results for the J1-J2-J3 model

(Exotic superconductivity in VBS host)

Spin-charge separation in doped 2D frustrated quantum magnets – p. 2

Collaborations andreferences

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

On the J1-J2-J3 model: ongoing work withA. Läuchli, M. Mambrini & F. Mila

Other work: doped Shastry-Sutherland latticeP W. Leung, PRB 69, 180403 (2004)

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

Spin-charge separation in doped 2D frustrated quantum magnets – p. 3

2D Frustratedmagnets

Lattices with AF frustrating interactions

Melzi 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)Uemura et al., PRL 73 (’94)Spin-charge separation in doped 2D frustrated quantum magnets – p. 4

3D Frustratedmagnets

pyrochlores and spinels

Transition metal oxides- ZnCr2O4 spinel

- A2Ti2O7 titanatesRamirez et al., PRL 89,

067202 (2002)

-no ordering at low temperatures-spin gap formation

Spin-charge separation in doped 2D frustrated quantum magnets – p. 5

Exotic disorderedgroundstates

Low-spin (S=1/2) ⇒ strong quantum fluctuations

Schulz, Ziman & DP,"Magnetic systems with

competing interactions", p120,Ed. H.T. Diep, W.-S.(1994)

nature of disorderedphases ?

→ many studies (andcontroversies !)

Misguich & Lhuillier,"Frustrated spin systems", Ed.H.T. Diep, World-Scientific

(2003)

Spin-charge separation in doped 2D frustrated quantum magnets – p. 6

Confinement vsdeconfinement

Idea: use doping (or ARPES) to probe nature ofthe ground state

(a) (b)

”string potential” ”deconfined” spinon

Spin-charge separation in doped 2D frustrated quantum magnets – p. 7

Checkerboard lattice:a Valence Bond Solid

2D array ofcorner-sharing

tetrahedra:”2D pyrochlore”

VBS phase(plaquette)

Fouet et al., PRB(2003)

Finite spin gap ∼ 0.6J

Translation symmetry breaking

1. Esinglet(Q = (π, π)) − E0 → 0 when N → ∞2.

⟨

PlaqlPlaql′

⟩

→ finite when |l′ − l| → ∞Spin-charge separation in doped 2D frustrated quantum magnets – p. 8

Kagome: paradigmof a “spin liquid” (?)

Magnetically disorderedLeung & Elser PRB 47, 5459 (1993)

Small spin gap ∼ 0.05JLecheminant et al., PRB 56, 2521 (1997)

No symmetry breaking (neither SU(2) norlattice symmetries)

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

Spin-charge separation in doped 2D frustrated quantum magnets – p. 9

framework

(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

Spin-charge separation in doped 2D frustrated quantum magnets – p. 10

Frustrated holemotion

J → 0 limit

� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �

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

t < 0 E = −|t| E = −2|t|

t > 0 E = −2t E = −tSpin-charge separation in doped 2D frustrated quantum magnets – p. 11

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 & ARPESSpin-charge separation in doped 2D frustrated quantum magnets – p. 12

Dynamics withinLanczos ED

- 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 !

Spin-charge separation in doped 2D frustrated quantum magnets – p. 13

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"

Spin-charge separation in doped 2D frustrated quantum magnets – p. 14

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"

Spin-charge separation in doped 2D frustrated quantum magnets – p. 14

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" Spin-charge separation in doped 2D frustrated quantum magnets – p. 14

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

Spin-charge separation in doped 2D frustrated quantum magnets – p. 15

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

Spin-charge separation in doped 2D frustrated quantum magnets – p. 15

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

Spin-charge separation in doped 2D frustrated quantum magnets – p. 15

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

Spin-charge separation in doped 2D frustrated quantum magnets – p. 16

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

Spin-charge separation in doped 2D frustrated quantum magnets – p. 17

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 (like static case: seee.g. Dommange et al, PRB)

Spin-charge separation in doped 2D frustrated quantum magnets – p. 18

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 (like static case: seee.g. Dommange et al, PRB) Spin-charge separation in doped 2D frustrated quantum magnets – p. 18

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→ singlet corr. robust & no Nagaoka FSpin-charge separation in doped 2D frustrated quantum magnets – p. 19

J1-J2-J3 square latticeHeisenberg

J /J2 1

J /J13

0.5

0.5

(0,π)

q( ,π)

(q,q)

impZ <0.84

B

A

(π,π)

Classical phasediagram (Moreo etal., PRB 90)→ collinear vs spiral

Quantum case→ VBS vs spin liquid

columnar dimer: Leung & Lam, PPB 96

spin liquid: Capriotti, Scalapino & White, PRL 2004=⇒ Zimp = |

⟨

Φ1h|Φbare1h

⟩

|2 with |Φbare1h

⟩

= ci,↑|Φ0h

⟩

(t = 0)

Spin-charge separation in doped 2D frustrated quantum magnets – p. 20

J1-J2-J3 square latticeHeisenberg

J /J2 1

J /J13

0.5

0.5

(0,π)

q( ,π)

(q,q)

impZ <0.84

B

A

(π,π)

Classical phasediagram (Moreo etal., PRB 90)→ collinear vs spiral

Quantum case→ VBS vs spin liquid

columnar dimer: Leung & Lam, PPB 96

spin liquid: Capriotti, Scalapino & White, PRL 2004=⇒ Zimp = |

⟨

Φ1h|Φbare1h

⟩

|2 with |Φbare1h

⟩

= ci,↑|Φ0h

⟩

(t = 0)

Spin-charge separation in doped 2D frustrated quantum magnets – p. 20

Spin distribution⟨

Szi

⟩

at distance r = ri − rO from defecton a

√32 ×

√32 = 32-site square cluster

1 2 3 4−0.5

0

0.5

1

bare wf

ground state

1 2 3 4

<Siz >

|ri−rO| |ri−rO|

A B⟨

Szi

⟩

bare→ spin-spin

correlation in host⟨

Szi

⟩

gs→ location of

“spinon”

Typically, ξconf > ξAF

ξconf finite when N → ∞ ?Spin-charge separation in doped 2D frustrated quantum magnets – p. 21

Discussion &Conclusions

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

Spinon-holon bound-state in translationalsymmetry breaking VBS

Frustration of hopping → electron-holeasymmetry

Progress on frustrated square lattice AF→ help from dimer basis (Mambrini et al.)

Pairing mechanism based on kinetic energy(another time!)

Spin-charge separation in doped 2D frustrated quantum magnets – p. 22

Metallic 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 !Spin-charge separation in doped 2D frustrated quantum magnets – p. 23

Dynamics withinLanczos ED

- 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)

Spin-charge separation 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)

Spin-charge separation in doped 2D frustrated quantum magnets – p. 25

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 favoredSpin-charge separation in doped 2D frustrated quantum magnets – p. 26

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

Spin-charge separation in doped 2D frustrated quantum magnets – p. 27

Correlated pairhopping

- Analogy with fully frustrated TB model:interaction-induced delocalized 2-particle BS

Vidal & Douçot, PRB 65, 045102 (2002)Spin-charge separation in doped 2D frustrated quantum magnets – p. 28