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