finite-size effects in transport data from quantum monte carlo simulations

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 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

http://arxiv.org/abs/1107.0230

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

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


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...


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


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

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


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


The Hubbard model (or, the “Ising model” for strongly correlated fermions)


John Hubbard (1931-1980)


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.


Phase diagram: square lattice at T = 0

Mean field (Hartree-Fock) QMC

Hirsch, PRB (1985); Hirsch & Tang, PRL (1989)


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


(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:

On every site (i, )

of the space-time lattice, an aux. Ising field

si ( ) is introduced


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...


...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”


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


Sampling the HS (Ising) fields

Sampling updates (non-local!) also expressed in terms of these Green’s functions (see given refs. for details)

⇒ simplicity


The “minus-sign” problem:

dos Santos, BJP (2003)


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)


Probing MIT’s: dc-conductivity


The conductivity suffers from the same closed-shell effects as the compressibility.



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.



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 ⇒ ✓



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



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


10 x 10


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


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



A bonus: <sign> and compressibility

U = 2

U = 3

U = 4



A bonus: <sign> and compressibility



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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Thanks for your attention!

finite-size effects in transport data from quantum monte carlo simulations

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