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