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

A Cluster Method for Spectral Properties of Correlated Electrons David Sénéchal Department de physique Université de Sherbrooke HPCS 2003 May 13, 2003 UNIVERSITÉ DE SHERBROOKE

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

High-temperature Superconductors The action is taking place on the CuO 2 planes UNIVERSITÉ DE SHERBROOKE

1D : Organic superconductors (TMTSF) 2 PF 6 Vertical stacks UNIVERSITÉ DE SHERBROOKE

The Hubbard Model t t ¢ • One electron orbital per atom t • Hybridation between neighbors • Screened Coulomb repulsion • Simplest model of Correlated Electrons UNIVERSITÉ DE SHERBROOKE

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) 2 Ê ˆ L ! ª 2 L L sites : ˜ Á ( L /2 ) ! L /2 ( ) ! Ë ¯ Number of states increases exponentially with size • 12 sites : 853 776 states (5.6 MB) • 16 sites : 41 409 225 states (1.3 GB) UNIVERSITÉ DE SHERBROOKE

The Hubbard Model : Hamiltonian c i s : Removes an electron of spin s ( ↑ or Ø ) at site i † : Adds an electron of spin s ( ↑ or Ø ) at site i c i s † c i s n i s = c i s : # of electrons (0 or 1) of spin s at site i t ij : Hybridation (hopping amplitude) between sites i and j U : Coulomb repulsion energy for two electrons on same site UNIVERSITÉ DE SHERBROOKE

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 UNIVERSITÉ DE SHERBROOKE

U = 0 : Band Theory Electrons are independent • They occupy plane wave states of • wavevector k and energy e ( k ) UNIVERSITÉ DE SHERBROOKE

U = 0 : Fermi Surface t > 0 t ¢ = 0 t > 0 t ¢ = –0.4 t p p k y k y 0 0 -p -p -p 0 p -p 0 p k x k x UNIVERSITÉ DE SHERBROOKE

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. UNIVERSITÉ DE SHERBROOKE

The Spectral Function A (k, w ) ground state exact eigenstate • Probability that an electron of momentum k added to, or removed from the system have energy w • Independent electrons: A ( k , w ) = d ( w-e k ) • Measurable par ARPES (angle-resolved photoemission spectroscopy) UNIVERSITÉ DE SHERBROOKE

ARPES Angle-Resolved Photoemission Spectroscopy q photon D.o.S. : A( k , w<0 ) f V i 2 r ( E , W ) G i Æ f = 2 p h k e electron z Matrix x element k material hole UNIVERSITÉ DE SHERBROOKE

Green Function & Spectral function UNIVERSITÉ DE SHERBROOKE

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 UNIVERSITÉ DE SHERBROOKE

CPT : Details Dyson’s equation Basic CPT approximation UNIVERSITÉ DE SHERBROOKE

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 • UNIVERSITÉ DE SHERBROOKE

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 UNIVERSITÉ DE SHERBROOKE

Lanczös Algorithm : Details UNIVERSITÉ DE SHERBROOKE

Continued fraction representation UNIVERSITÉ DE SHERBROOKE

Example : 1D Hubbard Model 12-site cluster, half-filling, U = 4t energy Fermi level Spin-charge separation wavevector UNIVERSITÉ DE SHERBROOKE

1D case: U Æ ∞ limit Exact result from the Bethe Ansatz solution J. Favand et al. Phys. Rev. B 55, R4859 (1997) UNIVERSITÉ DE SHERBROOKE

1D case: U Æ ∞ limit CPT 14 sites UNIVERSITÉ DE SHERBROOKE

ARPES in high-T c : NCCO Electron-doped high- T c superconductor « hot spot » (p,p) x = 0.05 x = 0.10 x = 0.15 (0,0) (p,0) Armitage et al., Phys. Rev. Lett. 88, 257001 (2002) AF zone boundary Nd 2-x Ce x CuO 4 UNIVERSITÉ DE SHERBROOKE

n = 1.111 3x3 cluster U = 3 U = 8 (0,p) (p,p) 2D Hubbard Model t’/t = -0.4, t’’/t = 0.2 Cluster Perturbation Theory UNIVERSITÉ DE (0,0) (p,0) SHERBROOKE

ARPES in high-T c : CCCO Hole-doped cuprate Ca 2-x Na x CuO 2 Cl 2 (0,p) (p,p) k y (p,0) k x G Ronning et al. , cond-mat/0301024 UNIVERSITÉ DE SHERBROOKE

n = 0.833 3x4 cluster U = 2 U = 8 (0,p) (p,p) 2D Hubbard Model t’/t = -0.4, t’’/t = 0.2 Cluster Perturbation Theory (0,0) (p,0) UNIVERSITÉ DE SHERBROOKE

2D Hubbard model U = t, 14 electrons on 3x4 cluster Imaginary part of the self-energy (scattering rate) 1 0.8 0.6 k y / p 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 k x / p UNIVERSITÉ DE SHERBROOKE

Seen this before? UNIVERSITÉ DE SHERBROOKE

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 UNIVERSITÉ DE SHERBROOKE

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) UNIVERSITÉ DE SHERBROOKE