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Spin & doped laddersSpin & doped ladders
Spin liquid
& superconducting properties
Spin liquid
& superconducting properties
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OutlineOutline
• Ladder geometry: some materials &
physical properties of spin ladders
• Doping: microscopic models & fieldtheory (bosonisation), numerical results,
phase diagram, magnetic properties …
• Pairing mechanism & beyond
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Ladder geometryLadder geometry
Two-leg ladders Three-leg ladders
Spin ladders: each site carries a S=1/2 (typically Cu-II atom)
Even & odd-leg ladders have different magnetic properties …
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A typical spin ladder:A typical spin ladder:
Derived from 2D high-Tc cuprate structure
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Copper oxide Mott insulator
Heisenberg AF super-exchange
Copper oxide Mott insulator
Heisenberg AF super-exchange
On-site repulsion UAF exchange J
between copper spins
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A ladder with 3D couplingsA ladder with 3D couplings
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Chaboussant et al., 1998
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A few simple theoretical
considerations
From the strong & weak coupling limits…
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A RVB-like spin liquidA RVB-like spin liquid
Resonating Valence Bond= linear superposition of short-range VB configurations
Anderson 87
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Strong coupling limitStrong coupling limit
• For isolated rungs: cost to excite a triplet• At finite magnon can propagate to gain
kinetic enegy:
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
π
E
k
2
0
continuum of 2 magnons states
one magnon branch
Exact diag. (G. Roux et al.)
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The spin gap is robust !The spin gap is robust !• For even # of chains
→ finite spin gap
• 2-leg ladder & strong
coupling:
• 2-leg ladder & weak
coupling:
QMC determination of the spin gap
Greven et al., PRL 96
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Spin correlation length vs T
Scaling behavior
Spin correlation length vs T
Scaling behavior
Greven et al. (QMC)
Weak coupling result
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Even-odd effects !!Even-odd effects !!• GS (& low-T low-
energy) propertiesdepend on the # of
legs
• For nc even: finite
gap
• For nc odd: gapless
Scaling of spin gap (DMRG)
White et al. (1994)
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Simple qualitative argumentSimple qualitative argument
(b)
(a)
“spin pairing”
Similar to 1D chain
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Spin gap revealed by
experiments
Spin gap revealed by
experiments
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Susceptibility: 2-leg vs 3-leg laddersSusceptibility: 2-leg vs 3-leg ladders
Azuma et al., PRL 94
Spin gap signature
No spin gap
I m p u r i t y P a u
l i c o n t r i b
u t i o n
Single magnon branchcontribution (2-leg ladder)
Troyer et al. 96
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Thermodynamic propertiesThermodynamic propertiesExemple of Cu(HpN)Cl (2-leg)
Specific heat vs TComparison with full diagonalizationMagnetization curve
Comparison with LanczosHayward et al., 1996
Calemczuk et al., 1998
Clear signatures of spin gap !
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The “telephone number” compound:
A superconducting ladder
The “telephone number” compound:
A superconducting ladder
P
l a n e s o f
c h a i n s
P l a n
e s o f 2 - l e g
l a d d e r s
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Doping the ladders:
Zhang-Rice singlets
Doping the ladders:
Zhang-Rice singlets
Basic model for lightly doped
system:
t-J Hamiltonian(or large-U Hubbard model)
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Conformal Field theoryConformal Field theory
Prediction for long wavelength (low-energy) physics
CnSm phases (Balents & Fisher)
- n =0,1 or 2 charge modes
- m=0,1 or 2 spin modes
Insulating (Mott) spin gap phase: C0S0 phase
If spin gap robust under doping: C1S0 phase:
- Spin correlations:
- Charge correlations:
- Superconducting correlations:
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Weak coupling approachWeak coupling approach
• Anisotropic Hubbardladder & small U
• Weak coupling RG:
C1S0 phase stable
Balents & Fisher 96
H a l f -
f i l l i n g
( I n s u l a t i n g
s p i n
l i q
u i d )
Single chain physics
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Strong coupling argumentStrong coupling argument
Separated holes
unpaired spins
Rung hole pair
Spin gap survises
Balance between kinetic & magnetic energies
at large rung coupling:
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Numerical results (t-J)Numerical results (t-J)Correlations vs distance (DMRG) Phase diagram (ED)
Hayward PRL 95
Dominant pairing correlations
C1S1
C1S0 superconductor
Phase separation
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Tunneling density of statesTunneling density of states
-6 -4 -2 0 20
0.1
0.2
0.3
-0.4 -0.2 0 0.2 0.4
0
0.1
0.2
0.3
0.4
0.5
ΔQP
ω
ω
N( )ω
nh=1/8
J=0.4
hole weight nh
elec. weight (1-nh)/2
Ω
Ω
Lower Hubbard band
Superconducting gap
Magnon resonance peak
Poilblanc et al. 2004
(Lanczos)
Low energy triplet excitation of energy
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Getting more insight into thenature of the (spin-fluctuation
driven) pairing interaction ?
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Unconventional d-wave pairingUnconventional d-wave pairing
have opposite signs
Typical of d-wave
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Superconducting Green function(Lanczos)
Superconducting Green function(Lanczos)
Generalize calculation of dynamical correlation
to grand-canonical ensemble
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Dynamics of d-wave pairingDynamics of d-wave pairing
-1 -0,5 0 0,5 1 -1 -0,5 0 0,5 1
2π/3
qy=0
5π/6
π
ω
I m
F ( q ,
)
2π/3
π/2
π/3
π/2
ω
qy= π(a) (b)
π/3
π/6
0
ω
Poilblanc, Scalapino, Capponi, PRL 2003
Change of sign ! => d-wave symmetry
Deviations from BCS
due to retardation
and
opposite signs !
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Summary/conclusionSummary/conclusion• Ladders exhibit fascinating behaviors
observable experimentally (except fewexperiments on doped ladders, yet)
• Simple system to investigate non-
conventional superconductivity
• Not addressed here: role of disorder,
magnetic field induced QCP, role of spinanisotropy, …