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Introduction Model and Motivation Results Conclusions
Umklapp Scattering In Doped Two-Leg Ladders
Neil Robinson
Rudolf Peierls Centre for Theoretical Physics, University of Oxford
Quantum Correlations Students Workshop, 2nd July 2012
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Collaborators
Fabian Essler Eric Jeckelmann Alexei Tsvelik
University of Oxford ITP Hannover Brookhaven National Laboratory
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Outline
IntroductionUmklapp Scattering1D Hubbard model at half-filling
Model and MotivationExtended-Hubbard model on the two-leg ladderExperimental motivations
ResultsTheoretical approachPhysical pictureEffective low-energy theory
ConclusionsConclusion
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Umklapp Scattering: A Brief Refresher
DefinitionElectron-electron scattering with total initial and final momentumdiffering by a reciprocal lattice vector G
K1 + K2 = K3 + K4 + G
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Umklapp Processes in 1D SystemsI Two electrons close to Fermi point scatter
K1 + K2 = K3 + K4 + G ↔ 4kF = G = 2π→ only at half-filling can transfer momentum to lattice
EF-kF kF
Dp = 4 kF
-Π 0 Π
0
k
EHkL
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
1D Hubbard Model at Half-FillingSimplest model of interacting fermions
One electron per site = half-filling = Umklapp activation
Energy gap for single-particle excitations. Low-energy: spin chain
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Theoretical Model: Extended-Hubbard Model
H = −t∑`,σ
2∑j=1
a†j ,`+1,σaj ,`,σ + a†j ,`,σaj ,`+1,σ
− t⊥∑`,σ
a†1,`,σa2,`,σ + a†2,`,σa1,`,σ + U∑j ,`
nj ,`,↑nj ,`,↓
+ V⊥∑`
n1,`n2,` + V‖∑j ,`
nj ,`nj ,`+1 +∑j ,`
Wj cos(K`)nj ,`,
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Motivation
I X-ray scattering on “telephone number” compoundsSr14−xCaxCu24O41
I CDW order observed without lattice distortionI Origin: interladder long-range Coulomb interaction?I Treat in mean field → periodic electrostatic potential
I Stripe ordering in x = 1/8 doped La2−xSrxCuO4
I Carbon nanotubes with surface adsorbed noble gasesI Periodic structure on nanotube surfaceI External periodic electrostatic potential
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Theoretical Approach to Problem
Two limits for field theory
1. Weak interactions U,V ,V⊥ � t, t⊥
2. Strongly-interacting, weakly coupled chains t⊥ � t,U, t2/U
Weak interactions approach:
1. Linearize the spectrum and bosonize
2. Derive renormalization group (RG) equations
3. Numerically integrate RG equations
4. Perturbatively integrate out massive degrees of freedom
5. Effective Hamiltonian
6. Further RG and construct order parameters.
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Physical Picture
Change to band picture: cb = 1√2
(a1 + a2), cab = 1√2
(a1 − a2)
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Physical Picture
I Renormalization group flow → bonding charge sector massive.→ Like half-filled Hubbard chain
I Spin chain (bonding band) coupled to 1DEG (antibondingband)
→ Effective Kondo-Heisenberg Model
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Effective Low-Energy Theory: Kondo-Heisenberg Model
Phase Diagram:
I Low-energy theory controlled by J
I J < 0 “Weak Coupling Regime”
3-component Luttinger liquid
I J > 0 “Strong Coupling Regime”
OPDW (n) = c†ab,↑,nc†ab,↓,n+1
− c†ab,↓,nc†ab,↑,n+1
OCDW (n) =∑
d=b,ab
c†d,σ,ncd,σ,n
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Numerical Data - DMRG in Strong Coupling CDW Phase
I DMRG for the 96× 2 ladder with n = 88 electrons
I Quarter-filled bonding band. Applied periodic potential K = πamplitude W = 1
I Model parameters: t = 2t⊥ = 1, U = 4, V‖ = 0 and V⊥ = 5
I Fit parameter Kab,c ∼ 0.25
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Numerical Data - DMRG in Strong Coupling CDW PhaseAntibonding Green’s function:
0 10 20 30 40
0.001
0.01
0.1
1
n
Gab!n"
0.75 e!n#15 n!1DMRG data
Bonding Green’s function:
0 10 20 30 4010!9
10!7
10!5
0.001
0.1
n
Gb!n"
0.065e!n#2.6DMRG Data
OPDW two-point function:
1 2 5 10 20
10-7
10-5
0.001
0.1
n
ÈXOab
PH4
8LO
† abP
H48+
nL\È
3.3 n-4
DMRG data
OCDW two-point function:
1 2 5 10 20
10!5
10!4
0.001
0.01
0.1
n
!"O CDW#48$O
† CDW#48"n
$%! 0.15 n!2DMRG data
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders
Introduction Model and Motivation Results Conclusions
Conclusions
I Umklapp scattering profoundly changes ground stateproperties
I CDW formation can drive a system superconducting withFFLO-like superconductivity
I Long-range Coulomb interactions may play an interesting rolein the ground state properties of real crystal structures.
Reference:
N. J. Robinson, F. H. L. Essler, E. Jeckelmann andA. M. Tsvelik, Phys. Rev. B 85, 195103 (2012)
N. Robinson University of Oxford
Umklapp Scattering In Doped Two-Leg Ladders