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!
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