Finite-size effects in transport data from Quantum Monte Carlo simulations Raimundo R. dos Santos Universidade Federal do Rio de Janeiro Financial support: Collaborators: Rubem Mondaini UFRJ Karim Bouadim Ohio State → ITPStuttgart Thereza Paiva UFRJ arXiv:1107.0230 VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Outline • Motivation: Strongly correlated fermions • The Hubbard model • (Determinant) Quantum MC Simulations • Probing MIT’s: compressibility • Probing MIT’s: dc-conductivity • Probing MIT’s: the Drude weight • A bonus: <sign> and compressibility • Conclusions VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Motivation: Strongly correlated fermions Independent electrons in solids: periodic crystalline potential a a a nearly-free electrons: delocalized atomic limit (tight-binding): more localized dE dE Ashcroft & Mermin (1976) Ibach & Lüth (2003) VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Density of states (NFE or TB) Insulator (eV) or Semiconductor (0.1 eV) Metal Ashcroft & Mermin (1976) Therefore, independent electron approx’n + band theory explains: metals, insulators, semiconductors... VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ However, beware of narrow bands (especially d and f): ⇒ greater tendency to localize electrons ⇒ enhances likelihood of two electrons occupying the same site ⇒ Coulomb repulsion can no longer be neglected Therefore, electrons move collectively to minimize energy: they are strongly (due to interactions) correlated Consequence: indepedent electron approx’n can fail seriously; e.g., metallic behaviour for an insulating (Mott) system VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ parent compound (undoped) band theory High-Tc cuprates Metal! including correlations Insulator! Superconductor AFM insulator Superconductivity, MIT, itinerant magnetism, etc.: interactions must be included in a fundamental way (Pb) http://newscenter.lbl.gov/news-releases/2011/03/24/pseudogap/ VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Low-dimensional systems/confined geometries: fluctuations can disrupt ordered states Add strong correlations ⇒ need unbiased methods to tackle these problems Quantum Monte Carlo simulations have proved very useful, but... ...some bottlenecks remain, some of which we address here: • finite temperatures, but usually need extrapol’ns to T=0 • “minus-sign problem” • MIT’s • lack of order parameter • inconsistent results from different probes • lattice finite size ⇔ gaps in the spectrum ⇒ bad for transport properties: “falsepositive” insulating states VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ The Hubbard model (or, the “Ising model” for strongly correlated fermions) John Hubbard (1931-1980) VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ The Hubbard model (or, the “Ising model” for strongly correlated fermions) Simplest case: s-orbitals, spin-1/2 fermions ⇒ 4 states per site: Charge and spin d.o.f. band energy: favours delocalization Coulomb repulsion: favours localization chemical pot’l: controls electronic density Special cases: VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Phase diagram: square lattice at T = 0 Mean field (Hartree-Fock) QMC Hirsch, PRB (1985); Hirsch & Tang, PRL (1989) VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Have a good testing ground to compare probes, approximations, finite-size effects, etc., in the study of MIT’s Useful in other situations with overlying structures (e.g., superlattices) The finite-size dimension Lz may include several layers: computational effort determined by the # of sites, not by Lz Lz=2 Lz=1 but within FSS theory, the size is as small as Lz VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ (Determinant) Quantum Monte Carlo Simulations Blankenbecler et al., PRD (1982); Hirsch, PRB (1985); White et al., PRB (1989); Loh et al., PRB (1990); dos Santos, BJP (2003) Preparation: one-body (bilinear ⇒ integrable): K two-body: V Use Trotter formula: Imaginary-time interval (0,β) discretized M = β/Δτ slices; typically Δτ = (8U)-1/2. #sites is now Ld×M into Now we can transform two-body term into a bilinear form: HS transf’n VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Two-body term: discrete Hubbard-Stratonovich transf’n Hirsch, PRB (1983) → continuous auxiliary-field x On every site (i, ) of the spacetime lattice, an aux. Ising field si ( ) is introduced with Now both K and V are bilinear ⇒ fermionic d.o.f. can be traced out (see given refs. for details), ⇒ remaining Ising d.o.f. are sampled by MC... VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ ...but the “Boltzmann weight” is quite distinct from the usual case: with ➔ Ld × Ld matrices; σ refers to original fermionic channels effective “density matrix” No guarantee that the “Boltzmann weight” is positive ⇒ leads to the “minus-sign problem” VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Average values e.g., single-particle Green’s functions N.B.: Manipulation of Green’s functions → Ns × Ns matrices e.g., unequal-time single-particle Green’s functions Most “measurable” quantities expressed in terms of these Green’s functions (see given refs. for details) ⇒ simplicity VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Sampling the HS (Ising) fields Sampling updates (non-local!) also expressed in terms of these Green’s functions (see given refs. for details) ⇒ simplicity VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ The “minus-sign” problem: dos Santos, BJP (2003) VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Probing MIT’s: compressibility ∴ Incompressibility (κ = 0) signals an insulating state ▲U = 2 Non-interacting case (U = 0) ⎯U=0 Mondaini et al., (2011) Conclusion: • “closed-shell” effects arise due to small lattice size; • these give rise to false insulating states; • only dismissed through a sequence of data for different lattice sizes: incompressible densities strongly dependent on size. VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Probing MIT’s: dc-conductivity VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Mondaini et al., (2011) The conductivity suffers from the same closed-shell effects as the compressibility. VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ A shortcut for calculating the dc-conductivity Randeria et al., PRL (1992); Trivedi and Randeria PRL (1995); Trivedi et al., PRB (1996) obtainable directly from QMC data (no need to invert Laplace transform!) hard to establish smallness a priori; use data for σ(ω) to assess reliability. Mondaini et al., (2011) VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Results to order zero 10 x 10 Half filling: negative slope at high temperatures; insulator only detected at lower T ρ = 0.42: σ → 0 as T → 0 ⇒ insulator? closed-shell effect! ρ = 0.66: σ ↑ as T → 0 ⇒ metal Mondaini et al., (2011) VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Results to 2nd order 10 x 10 Half filling: positive slope at high temperatures insulator detected at higher T ρ = 0.42: still σ → 0 as T → 0 ⇒ insulator closed-shell effect! (expected: present at full calc’ns) ρ = 0.66: σ ↑ as T → 0 ⇒ metal σdc(2) cannot be generically considered as a perturbation even at the lowest T’s considered Mondaini et al., (2011) VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Probing MIT’s: the Drude weight Zero-temperature limit of the frequency-dependent conductivity: 4x4, T = 0 Lanczos Drude weight Incoherent response D readily available from QMC simulations [Scalapino et al., PRB (1993)]: Dagotto, RMP (1994) Strategy: calculate a finite-temperature D and examine its behaviour as T→ 0: if D → 0 ⇒ Insulator if D → D0 > 0 ⇒ Metal VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Results for the Drude weight 10 x 10 Half filling: steady decrease as T is lowered: ⇒ insulator ρ = 0.42: D → D0 as T → 0 ⇒ metal no closed-shell effects! ρ = 0.66: D → D0 as T → 0 ⇒ metal D weakly T-dependent in metallic state ⇒ reliable extrapolations Mondaini et al., (2011) VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Size dependence of the Drude weight at finite temperatures half filling: size dependence only appreciable when interaction is on ρ = 0.42 ρ = 0.66 ρ=1 away from half filling: size dependence very weak, even for interacting case (size of time slices does not influence final results) Mondaini et al., (2011) No “hickups” at problematic density ρ=0.42 for L=10 VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ A bonus: <sign> and compressibility U=2 U=3 U=4 Mondaini et al., (2011) VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ A bonus: <sign> and compressibility Mondaini et al., (2011) VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ A bonus: <sign> and compressibility Mondaini et al., (2011) VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Conclusions • Finite-size effects in transport properties aren’t just an accuracy issue. • The Drude weight is the most reliable probe of a MIT. • Beware of closed-shell effects in other probes. • Shortcut to calculate dc-conductivity very dangerous (higher order terms very important). • Minus-sign less harmful when system is incompressible (false insulating). VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ Thanks for your attention! 32

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# DMRG study of the Periodic Anderson Model