0.5setgray0
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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 2
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)
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 3
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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 6
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)
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 7
Confinement vsdeconfinement
See Sachdev
(a) (b)
”string potential” ”deconfined” spinon
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 8
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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 9
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 ??
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 10
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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 11
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)
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 12
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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 14
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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 14
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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 17
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)
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 20
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)
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 21
3D Frustratedmagnets
pyrochlores and spinels
Transition metal oxides- ZnCr2O4 spinel
- A2Ti2O7 titanatesRamirez et al., PRL 89,
067202 (2002)
-no ordering down to low temperatures
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 22
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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 23
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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 23
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)
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 25
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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 26
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)
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 28
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 !
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 29
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(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(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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 31
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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 31
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
Some exotic phenomena in doped 2D frustrated quantum magnets – p. 31