A stochastic independent-electron approach to correlated...
Transcript of A stochastic independent-electron approach to correlated...
A stochastic independent-electron approach to correlated systems
• Why a new QMC approach?• The phaseless auxiliary-field QMC
Both electronic structure (PW or localized) and model calculationsPotential for improved accuracy and robustness
• Molecules and solids near equilibrium geometry• Bond breaking: stronger correlations• Phase separation in the Hubbard model
Shiwei Zhang College of William & Mary, USA
Outline
Collaborators:– Wissam Al-Saidi– Chia-Chen Chang– Henry Krakauer– Hendra Kwee– Wirawan Purwanto– Eric Walter
Support:– ARO, NSF, DOE petascale QMC-endstation, DOE-cmsn
Some references: (http://physics.wm.edu/~shiwei)– Zhang & Krakauer, PRL, ’03 – Al-Saidi et. al., PRB, ’06; JCP, ’06; JCP, ’06; JCP, ‘07– Suewattana et. al., PRB, ’07– Kwee et. al., PRL, ‘08– Purwanto et. al., JCP, ‘08– Chang & Zhang, arXiv:0805.4831
Recall sign problem:
1 particle -- first excited state:
Introduction: why auxiliary-field methods?
+_
In QMC, we need + and – walkers to cancel
....
2 fermions -- ground state: same problem
Fermion 1:
Fermion 2:
Sign problem – a fight against
global bosonic state
Recall sign problem:1 particle, first excited state:
Why auxiliary-field methods?
+_
Solid state or quantum chemistry?basis
Explicit --- matrix * vec
No sign problem
ψ1
ψΝ
ψ2..
…
Reduce sign problem:MnOmany free fermions:
Why auxiliary-field methods?
Solid state or quantum chemistry: basis
ψ1
ψΝ
ψ2..
ψ1ψ2
ψΝ
.
.
Matrix * matrix
Still no sign problem
(orthogonalize)
Slater det. - antisymmetric
Now turn on interaction. Can we use these as walkers ?
…
Summary: basic formalism of AF methods
To obtain ground state, use projection in imaginary-time:
Electronic Hamiltonian: (2nd quantization, given any 1-particle basis)
Hubbard-Strotonivich transf.
next
AF methods: some background• Applied in models in condensed matter, nuclear physics,
(lattice QCD), …. Scalapino, Sugar, Hirsch, White et al.; Koonin; Sorella, ….
basic idea: Monte Carlo to do sum (path integral)
• However, sign problem for “simple” interactions (Hubbard)
phase problem for realistic interaction
Fahy & Hamann; Baroni & Car; Wilson & Gyorffy; Baer et. al.; ….
• Reformulate ---
New AF QMC approachRandom walks in Slater determinant space:
H-S transformation
Schematically:
next
SZ, Carlson, Gubernatis
SZ, Krakauer
Exact so far
Sign/phase problem is due to --
“superexchange”:MnO
New approach
ψ1
ψ2
ψ1
Slater det. - antisymmetricψΝ
.
.
ψ2
ψΝ
.
.
…
Reasonable to expect it’s reduced, since
tendency for global collapse to bosonic state is removed
Controlling the phase problemSketch of approximate solution:
β
Im
Re
• Modify propagator by “gauge transformation’: phase degeneracy (use trial wf)
• Project to one overall phase:break “rotational invariance”
• subtle, but key, difference from: realhΨT|φi > 0(Fahy & Hamann; Zhang, Carlson, Gubernatis)
After:Before:
0 1 2 3 4 5 6 7 8
projection time nτ (Ry-1
)
-8.15
-8.1
-8.05
-8
-7.95
-7.9
E (
a.u.
)
free-proj (phase problem)
total energy E(nτ) --- bulk Si, 2-atom prim cellOPIUM psp Ecut=25Ry, dt=0.025
0 1 2 3 4 5 6 7 8
projection time nτ (Ry-1
)
-8.15
-8.1
-8.05
-8
-7.95
-7.9E
(a.
u.)
free-proj (phase problem)new method (phaseless)
total energy E(nτ) --- bulk Si, 2-atom prim cellOPIUM psp Ecut=25Ry, dt=0.025
0 1 2 3 4 5 6 7 8
projection time nτ (Ry-1
)
-8.15
-8.1
-8.05
-8
-7.95
-7.9
E (
a.u.
)
free-proj (phase problem)new method (phaseless)
total energy E(nτ) --- bulk Si, 2-atom prim cellOPIUM psp Ecut=25Ry, dt=0.025
0 1 2 3 4 5 6 7 8
projection time nτ (Ry-1
)
-8.15
-8.1
-8.05
-8
-7.95
-7.9
E (
a.u.
)
free-proj (phase problem)new method (phaseless)free-proj* (6K processors!)
total energy E(nτ) --- bulk Si, 2-atom prim cellOPIUM psp Ecut=25Ry, dt=0.025
Controlling the phase problemQuantify the approximation?
exact
2 3 4 5 6 7
projection time nτ (Ry-1
)
-8.058
-8.056
-8.054
-8.052
-8.05
-8.048
-8.046
E (
a.u.
)
new method (phaseless)free-proj* (6K processors!)
total energy E(nτ) --- bulk Si, 2-atom prim cellOPIUM psp Ecut=25Ry, dt=0.025
Error in total E < 1mH
0 2 4 6 8 10 12expt / exact (eV)
0
2
4
6
8
10
12
QM
C (
eV)
0 2 4 6 8 10 12expt / exact (eV)
0
2
4
6
8
10
12
QM
C (
eV)
Application: molecular binding energies
• All with single mean-field determinant as trial w.f.
• “automated” post-HF or post-DFT
- O3, H2O2, C2, F2, Be2, …- Si2, P2, S2, Cl2- As2, Br2, Sb2- TiO, MnO
3 types of calc’s:- PW +psp: - Gaussian/AE:- Gaussian/sc-ECP:
Nval up to ~ 60
0 2 4 6 8 10 12expt / exact (eV)
0
2
4
6
8
10
12
QM
C (
eV)
Molecular binding energies
• ~ 100 systems (also IP, EA, aB, ω): eq. geom., moderate correlation• Error < a few mHa (0.1 eV)• Accuracy ~ CCSD(T) (gold standard in chemistry, but N7)
• QMC: linear superposition of ‘LDA solutions’ in random fields
scaling to petascale ~ scaling LDA to O(100) processors
- O3, H2O2, C2, F2, Be2, …- Si2, P2, S2, Cl2- As2, Br2, Sb2- TiO, MnO 3 types of calc’s:
- PW +psp: - Gaussian/AE:- Gaussian/sc-ECP:
Nval up to ~ 60
1 1.5 2 2.5 3R/R
e
-199.11
-199.1
-199.09
-199.08
-199.07
-199.06
-199.05
E (
Eh)
RCCSDTQUCCSD(T)CCSD(T)QMC/UHF
1 1.5 2 2.5 3R/R
e
-199.11
-199.1
-199.09
-199.08
-199.07
-199.06
-199.05
E (
Eh)
RCCSDTQUCCSD(T)CCSD(T)
1 1.5 2 2.5 3R/R
e
-199.11
-199.1
-199.09
-199.08
-199.07
-199.06
-199.05
E (
Eh)
RCCSDTQUCCSD(T)CCSD(T)QMC/UHFQMC/UHF-sp
F2 bond breakingMimics increasing correlation effects:
Dissoc. limitEquilibrium
• UHF unbound. Nonetheless, large dependence on trial wf??
• No. Spin-contamination:- |ΨUHFi: not eigenstate of S2
- low-lying triplet in F2
• Simple fix – spin-projection:- Let |Ψ(0)i=|ΨRHFi- HS preserves spin symmetry- each walker determinant: free of contamination
“bonding” “insulating”
QMC/UHF: 20mH error!
F2 bond breaking --- larger basis
1.0 1.5 2.0 2.5 3.0
R /Re
-199.40
-199.38
-199.36
-199.34
-199.32
-199.30
-199.28
-199.26
E (
Ha)
QMC/UHF-spLDAGGA/PBEB3LYP
Potential energy curve:• LDA and GGA/PBE well-depths too deep
• B3LYP well-depth excellent, but “shoulder” too steep
• Compare with expt--- spectroscopic cnsts:
cc-pVTZ
Purwanto et. al., JCP, ‘08
Large extended systems
Cohesive energies: (eV/atom)
plane-wave + pseudopotential calculationsDMC -- Needs et al (Cambridge group)Na:
- metal: k-point integration in many-body QMC- new finite-size correction scheme Kwee, et al, PRL, ‘08
Application: Hubbard model
• Simplest model combining band structure and interaction:
• Renewed interest from experimental opportunities:- optical lattice emulators
• Long-standing questions: (conflicting numerical results)- model for high-Tcc? - phase separation? (Expt: spatial inhomog. in cuprates?)
Chang & Zhang, arXiv:0805.4831
Hubbard model: equation of state
• Constrained-path auxiliary field QMC (CPMC) is accurate.• There are kinks at closed-shell fillings => large shell effects.
4x4 square lattice, U=4.0
Exact diagonalization: Dagotto et.al. 1992CPMC: Zhang et.al., 1997
4x4, U=4t
Furukawa and Imada, 1992
(13,13)
(25,25)
Hubbard model: equation of state
Furukawa and Imada, 1992
(13,13)
(25,25)
CPMC data
e(h)=dE/dn, Maxwell constrh=1-n: doping
Hubbard model: persistent shell effects
• One signal for phase separation: does e(h) turn ?
• Shell effect persists to >40x40, leads to bias – false signal at U=0!
Non-interacting Hubbard model, LxL:
0 0.2 0.4 0.6 0.8 1
h
0
0.5
1
1.5
e(h)
481216202428323640
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
h
0
0.05
0.1
0.15
0.2
e(h)
481216202428323640
thermodynamic limit:
Twist averaged boundary conditions (TABC)
• TABC widely used in band structure methods; recently in QMC (Lin, Zhong & Ceperley)
• Hubbard:– A phase when electron goes around the lattice:
– Modifies kinetic energy:
• Breaks degeneracy in free-particle spectrum. But introduces phase problem
use the new method example of “downfolded hamilt”
Hubbard model: equation of state
• Smooth, shell effects removed• Convergence to thermodynamic limit achieved
0.5 0.6 0.7 0.8 0.9 1.0n
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
e(n)
8x8, U=08x8, U=28x8, U=48x8, U=68x8, U=812x12, U=212x12, U=416x16, U=4
Ground state energy per site
Phaseless AF QMC with TABC:
U=0
U=2
U=4
U=6U=8
Hubbard model: phase separation
• Minimum in eh(h) at doping h ~ 0.1• Appears to shift (larger) as U increases
Phaseless AF QMC with TABC:
U=0
U=2U=4
U=6
U=8
|ΨTi has no min(U=0 wf)
?
• Half-filling: antiferromagnetic (AF)(Furukawa & Imada 1991, Tang & Hirsch 1983, and many more…)
Relation: phase separation & antiferromagnetism
Calculate AF correlation as density increases ( half-filling)
Recall |ΨTi has no AF order (U=0 wf for all)
peak at if AF order
short-range correlation,indep of system size
Long-range AF correlation∝ system size
Relation: phase separation & antiferromagnetismSpin structure factor:
SummaryAF QMC : random walks in mean-field space• Potentially a method to systematically go beyond LDA
while using much of its machinery--- superposition of independent-particle calculations
• Approximate, but accuracy appears systematic
Applications & benchmarks (~100 systems)• s-, p-, and d-electron atoms, molecules, and bulk• Bond-breaking in molecules – increased correlation• Phase separation in 2-D repulsive Hubbard model:
- upon doping, two spatial regions: n=1 & n=nc (~0.9)- causes loss of long-range AF order at nc