A Cluster Method for Spectral Properties of Correlated Electrons - - PowerPoint PPT Presentation

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A Cluster Method for Spectral Properties

• f

Correlated Electrons

David Sénéchal Department de physique Université de Sherbrooke HPCS 2003 May 13, 2003

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Outline

• The Hubbard Model of Correlated Electrons
• The spectral function
• Cluster Perturbation Theory
• « Exact » diagonalization
• Some Results

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High-temperature Superconductors

The action is taking place

• n the CuO2 planes

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1D : Organic superconductors

(TMTSF)2PF6

Vertical stacks

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The Hubbard Model

• One electron orbital per atom
• Hybridation between neighbors
• Screened Coulomb repulsion
• Simplest model of Correlated

Electrons

t t¢ t

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The Hubbard Model : quantum states

Example : half-filled case Binary representation of states : (up spins; down spins) 1 site : (1;0) , (0;1) 2 sites : (10;01) , (01;10) , (10;10) , (01;01) L sites : Number of states increases exponentially with size

• 12 sites : 853 776 states (5.6 MB)
• 16 sites : 41 409 225 states (1.3 GB)

L! L /2

( )! L /2 ( )!

Ê Ë Á ˆ ¯ ˜

2

ª 2L

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The Hubbard Model : Hamiltonian

cis : Removes an electron of spin s (↑ or Ø) at site i

: Adds an electron of spin s (↑ or Ø) at site i

cis

†

nis = cis

† cis : # of electrons (0 or 1) of spin s at site i

tij : Hybridation (hopping amplitude) between sites i and j U : Coulomb repulsion energy for two electrons on same site

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Hamiltonian : example

2 site case at half-filling, total spin zero, NN hopping t In this basis : potential energy is diagonal, kinetic energy is off-diagonal

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U = 0 : Band Theory

• Electrons are independent
• They occupy plane wave states of

wavevector k and energy e(k)

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U = 0 : Fermi Surface

• p

p

• p

p

kx ky

• p

p

• p

p

kx ky

t > 0 t¢ = 0 t > 0 t¢ = –0.4 t

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U ≠ 0 : Correlated electrons

methods needed…

Analytical

• Perturbation theory and self-consistent variations
• Variational
• Renormalization Group
• Bosonization, conformal field theory
• Etc.

Numerical

• Quantum Monte Carlo (like in lattice QCD)
• Exact diagonalizations
• Density-Matrix renormalization group
• Dynamical Mean Field Theory (DMFT)
• Cluster methods: DCA, CDMFT, CPT…
• Etc.

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The Spectral Function A(k,w)

• Probability that an electron of momentum k added to, or

removed from the system have energy w

• Independent electrons: A(k,w) = d(w-ek)
• Measurable par ARPES (angle-resolved photoemission

spectroscopy)

ground state exact eigenstate

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ARPES

Angle-Resolved Photoemission Spectroscopy

material hole

k ke q

z x photon electron

GiÆ f = 2p

h

f V i 2r(E,W)

Matrix element D.o.S. : A(k,w<0)

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Green Function & Spectral function

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Cluster Perturbation Theory

• Numerical, exact

diagonalization on a small cluster (open BCs)

• Extension to the whole

lattice by Dyson’s equation

• Allows a continuum of

wavevectors

• Short-range effects well

rendered

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CPT : Details

Dyson’s equation Basic CPT approximation

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Exact diagonalization

• We need to calculate the Green function for a

finite cluster, with open boundary conditions

• This means:
• Calculating the ground state of H
• Adding or removing an electron from it
• Calculating

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Lanczös Algorithm

• Iterative algorithm to find the extreme

eignenvalues and eigenvectors of a very large matrix

• Works by multiply-add only
• Requires 3 vectors in memory + way of applying

matrix (stored in sparse form or otherwise)

• Used also to build a reduced-size, approximate

form of H for inversion

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Lanczös Algorithm : Details

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Continued fraction representation

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Example : 1D Hubbard Model

12-site cluster, half-filling, U = 4t

Spin-charge separation

wavevector energy

Fermi level

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1D case: U Æ ∞ limit

Exact result from the Bethe Ansatz solution

• J. Favand et al. Phys. Rev. B55, R4859 (1997)

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1D case: U Æ ∞ limit

CPT 14 sites

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ARPES in high-Tc : NCCO

Electron-doped high-Tc superconductor

(0,0) (p,0) (p,p) x = 0.05 x = 0.10 x = 0.15

Armitage et al., Phys. Rev. Lett. 88, 257001 (2002) Nd2-xCexCuO4 AF zone boundary « hot spot »

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U = 3 U = 8

(0,p) (p,p) (0,0) (p,0) n = 1.111 3x3 cluster

2D Hubbard Model t’/t = -0.4, t’’/t = 0.2 Cluster Perturbation Theory

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ARPES in high-Tc : CCCO

Hole-doped cuprate Ca2-xNaxCuO2Cl2

Ronning et al. , cond-mat/0301024

kx ky

G

(p,0) (p,p) (0,p)

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2D Hubbard Model t’/t = -0.4, t’’/t = 0.2 Cluster Perturbation Theory

U = 2 U = 8

(0,p) (p,p) (0,0) (p,0) n = 0.833 3x4 cluster

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2D Hubbard model

U = t, 14 electrons on 3x4 cluster Imaginary part of the self-energy (scattering rate)

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

kx/p ky/p

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Seen this before?

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Conclusions

• CPT is an efficient way of extending the usual

Exact Diagonalization results to an infinite lattice

• A(k,w) may be calculated for a continuum of k
• Comparison with ARPES results indicates that

the one-band Hubbard model captures the main features of high-Tc superconductors

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The End

Thanks to : NSERC (Canada) FQRNT (Québec) RQCHP

• D. Pérez, D. Plouffe, A.-M. Tremblay
• X. Barnabé-Thériault, M. Bozzo-Rey

Centre de Calcul Scientifique (CCS)