High accuracy and effective models for stronglycorrelated electron systems

Hitesh J. Changlani, Huihuo Zheng, and Lucas K. WagnerDept. of Physics, University of Illinois at Urbana-Champaign

lkwagner@illinois.edu

Deriving effective models from first principlesSince we have an accurate method for calculating properties we’d like to use itto go from a big Hilbert space to a small Hilbert space, integrating out ’boring’things like short-range electron-electron correlations.

The Hamiltonians

We’d like to map from a contin-uum Hamiltonian (large Hilbert space)to discrete lattice Hamiltonian (small

Hilbert space). We do this usingreduced density matrices. We willdemonstrate this using a benzene ring,mapping from the continuum Hilbertspace onto a 6-site lattice model. Thelattice sites are the π orbitals, shownhere. The Hamiltonian in the smallspace is then

Hs = C +∑ij

tijc†i cj +

∑ijkl

Vijklc†i c†jclck

if we limit ourselves to two-body inter-actions.

A better methodWe would prefer not to solve for exact eigenvectors, since it’s hard!In the small Hilbert space, no matter the state, the expectation value of the energyis given by the expectation value of the creation/destruction operators (one- andtwo-body density matrix elements).

Es ≡ 〈H〉s = C +∑ij

tij〈c†i cj〉+∑ijkl

Vijkl〈c†i c†jclck〉

This is true for any quantum state in the small Hilbert space.So we can fit to ab-initio data by evaluating the energy expectation value Ei anddensity matrix elements 〈c†i cj〉i and 〈c†i c

†jclck〉i for a collection of non-eigenvalue

states.

E1

E2

...

EM

=

1 〈c†i cj〉1 .. 〈c†i c

†jclck〉1 ..

1 〈c†i cj〉2 .. 〈c†i c†jclck〉2 ..

1 .... .. .. ..

1 〈c†i cj〉M .. 〈c†i c†jclck〉M ..

Ctij..

Vijkl..

The procedure is:• Evaluate Ei, 〈c†i cj〉i, and 〈c†i c

†jclck〉i in ab-initio QMC

• Find the best fit by least-squares. These are the model parameters.

Comparison to experiment

4 5 6 7 8Expt Gap (eV)

4

5

6

7

8

Mod

el G

ap (e

V)

VMC paramsDMC paramsExtrap. params

The resulting model can be solved viaexact diagonalization to get excitedstate energies. The excitation energiesare in excellent agreement with experi-mental ones.

Application to an extended system

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5DMC Energy (eV) 2.783e3

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

Fitte

d En

ergy

(eV)

2.783e3

t01 =2.5(1)

∆Emax=0.83∆Erms=0.28

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5DMC Energy (eV) 2.783e3

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

Fitte

d En

ergy

(eV)

2.783e3

t01 = 3.04(3)U = 3.7(2)

∆Emax=0.40∆Erms=0.14

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5VMC Energy (eV) 2.77e3

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

Fitte

d En

ergy

(eV)

2.77e3

t01 =2.5(3)

∆Emax=0.96∆Erms=0.23

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5VMC Energy (eV) 2.77e3

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

Fitte

d En

ergy

(eV)

2.77e3

t01 =3.6(1)U =6.8(2)

∆Emax=0.18∆Erms=0.06

VMC

DMC(a) (b)

(c) (d)Tight binding HubbardWe can perform the same procedure on graphene. The value of U∗/t is in good

agreement with recent constrained-RPA values which reproduce the band struc-ture.

Implementation detailsThe results presented here were produced using QWalk, an open-source pack-age available at http://qwalk.org. Pseudopotentials were from BFD[1], andtrial wave functions were taken from GAMESS (for molecules) or CRYSTAL (forgraphene).

High accuracy for real materialsA part of this SciDAC project is to eval-uate new technologies for higher accu-racy quantum simulations. This partis focused on using quantum MonteCarlo techniques on realistic models ofmaterials to obtain predictive accuracy.The fundamental theory of condensedmatter is the Schödinger equation fornuclei and electrons:

HΨs(r1, r2, . . .) = EsΨs(r1, r2, . . .)

where

H =1

2

∑i

∇2i +

∑i<j

1

rij+

∑α,i

Zαriα

,

where for brevity we’ve excluded thenucleus-nucleus interactions. Our ob-jective is to tackle this equation as di-rectly as possible using Monte Carlotechniques to deal with the high di-mensionality of Ψ. There are twomain techniques we use. The first isa variational technique (VMC), whichapproximates the lowest eigenfunction

by minimizing the expectation value ofH with respect to a parameterization.Since we use Monte Carlo to evaluatethe necessary integrals, we have a lotof flexibility in the functional form.The second technique is diffusionMonte Carlo (DMC), which simulatesthe equation

−dΨ

dt= HΨ

stochastically. This PDE has theground state (actually any eigenstate)as its steady state distribution. Thistechnique, if implemented exactly, hasa sign problem, which we cure by us-ing a trial function (from the aboveVMC method) to approximate the val-ues of the nodes, or zeros of the wavefunction.It turns out that using relatively simpletrial nodal surfaces, very accurate re-sults can be obtained for traditionallydifficult systems.

FN-DMC

DFT(PBE)

FN

-DM

C p

erc

en

t err

or

Cohesive energy

Holes in the superconducting cuprates

AFM Block Flip FM StripeOrdering

0.00.10.20.30.40.50.6

Ene

rgy(

eV)

Charge density

Spin density

AFM Block Flip Stripe

The lowest energy is a spin-polaron that appears to have zero gap–a uniquestructure. We have obtained the metal-insulator transition in the cuprates forthe first time, provided an explanation for optical signals seen around 1eV upondoping, and provided a prediction for small peaks in X-ray and neutron experi-ments that have not previously been considered. More details in Ref [3]

Implementation and possible improvementsSince the method is based on Monte Carlo, it scales very well, up to 1,000,000threads on Mira at ALCF. The technique’s main disadvantage is the computa-tional cost. While the scaling is good at O(N3

e ), where Ne is the number of elec-trons, the prefactor is about 1,000 times larger than standard density functionaltheory calculations. The main costs are the following:• One-particle orbitals (3D function evaluation)• Updating the determinant (Matrix-vector multiplication)• Electron-electron minimum image distances• Evaluating Jastrow correlation factors (sums of many terms)

References[1] Burkatzki, M., C. Filippi, and M. Dolg. J. Chem. Phys. 126 234105 (2007), ibid 129 164115 (2008)[2] H.J. Changlani, H. Zheng and LKW arXiv:1504.03704 (submitted)[3] L.K. Wagner arXiv:1505.08091 (submitted)

AcknowledgementsWe would like to thank: DOE FG02-12ER46875 DOE INCITE SuperMatSim (mira) Many people fordiscussions and suggestions.