finite-size effects in transport data from quantum monte carlo simulations
DESCRIPTION
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 → ITP-Stuttgart. Thereza Paiva UFRJ. arXiv:1107.0230. - PowerPoint PPT PresentationTRANSCRIPT
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Finite-size effects in transport data from Quantum Monte Carlo simulations
Raimundo R. dos Santos
Universidade Federal do Rio de Janeiro
Financial support:
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Collaborators:
Rubem MondainiUFRJ
Karim BouadimOhio State → ITP-
Stuttgart
Thereza PaivaUFRJ
VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ
arXiv:1107.0230
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Outline
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• 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
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Ibach & Lüth (2003)Ashcroft & Mermin (1976)
Motivation: Strongly correlated fermions
Independent electrons in solids: periodic crystalline potential
a
nearly-free electrons: delocalized
a
atomic limit (tight-binding): more localized
a
dE dE
VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ
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Density of states (NFE or TB)
Ashcroft & Mermin (1976)
Insulator (eV)or
Semiconductor
(0.1 eV)
Metal
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
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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
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
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http://newscenter.lbl.gov/news-releases/2011/03/24/pseudogap/
AFM insulator
Superconductor
(Pb)
Metal!
High-Tc cupratesband theory
Insulator!
parent compound (undoped)
Superconductivity, MIT, itinerant magnetism, etc.: interactions must be included in a fundamental way
including correlations
VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ
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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: “false-positive” insulating states
VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ
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The Hubbard model (or, the “Ising model” for strongly correlated fermions)
VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ
John Hubbard (1931-1980)
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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:
band energy: favours delocalization
Coulomb repulsion: favours
localization
chemical pot’l: controls
electronic density
Special cases:
Charge and spin d.o.f.
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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)
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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
but within FSS theory, the size is as small as Lz
Lz=1
Lz=2
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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 SimulationsBlankenbecler et al., PRD (1982); Hirsch, PRB (1985); White et al., PRB (1989); Loh et al., PRB (1990); dos Santos, BJP (2003)
Use Trotter formula:
Imaginary-time interval (0,β) discretized into M = β/Δτ slices;
typically Δτ = (8U)-1/2.#sites is now Ld×M
one-body (bilinear ⇒ integrable): Ktwo-body: V
Now we can transform two-body term into a bilinear form: HS transf’n
Preparation:
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On every site (i, )
of the space-time lattice, an aux. Ising field
si ( ) is introduced
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
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...
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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
No guarantee that the “Boltzmann weight” is positive⇒ leads to the “minus-sign problem”
effective “density matrix”
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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
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
N.B.: Manipulation of Green’s functions → Ns × Ns matrices
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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
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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)
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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
Non-interacting case (U = 0)
⎯ U = 0
▲U = 2
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.
Mondaini et al., (2011)
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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
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VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ
The conductivity suffers from the same closed-shell effects as the compressibility.
Mondaini et al., (2011)
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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)
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VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ
Half filling:negative slope at high temperatures; insulator only detected at lower T
Results to order zero
ρ = 0.42:σ → 0 as T → 0 insulator? ⇒closed-shell effect!
10 x 10
ρ = 0.66:σ ↑ as T → 0 metal ⇒ ✓
Mondaini et al., (2011)
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VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ
Half filling:positive slope at high temperatures insulator detected at higher T
Results to 2nd order
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
10 x 10
Mondaini et al., (2011)
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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:
Drude weight
Incoherent response
D readily available from QMC simulations [Scalapino et al., PRB (1993)]:
Strategy: calculate a finite-temperature D and examine its behaviour as T→ 0:if D → 0 ⇒ Insulatorif D → D0 > 0 ⇒ Metal
Dagotto, RMP (1994)
4x4, T = 0Lanczos
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VI BMSP - Cuiabá, Brasil - Aug ’11 - Finite-size effects in transport data from Quantum Monte Carlo simulations - RR dos Santos - UFRJ
10 x 10
Mondaini et al., (2011)
Half filling:steady decrease as T is lowered:
⇒ insulator
Results for the Drude weight
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
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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 = 1ρ = 0.66ρ = 0.42
away from half filling: size dependence very weak, even for interacting case
(size of time slices does not influence final results)
No “hickups” at problematic density ρ=0.42 for L=10
Mondaini et al., (2011)
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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)
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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)
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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)
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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).
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32
Thanks for your attention!