defense

Describing correlated electrons

Lattice models:

Ab initio descriptions:

\hat{H}_{\text{Hubbard}} = -t\sum_{\langle ij\rangle,\sigma}(\hat{a}_{i\sigma}^{\dagger}\hat{a}_{j\sigma}+\hat{a}_{j\sigma}^{\dagger}\hat{a}_{i\sigma}) + U\sum_i\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}

\hat{H}_{\text{Hubbard}} = -t\sum_{\langle ij\rangle,\sigma}(\hat{a}_{i\sigma}^{\dagger}\hat{a}_{j\sigma}+\hat{a}_{j\sigma}^{\dagger}\hat{a}_{i\sigma}) + U\sum_i\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}

\hat{H}_{\textit{Ab initio}} = \sum_{pr} t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \sum_{prqs} v_{prqs}\hat{a}_{p}^{\dagger}\hat{a}_r\hat{a}_{q}^{\dagger}\hat{a}_{s}

\hat{H}_{\textit{Ab initio}} = \sum_{pr} t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \sum_{prqs} v_{prqs}\hat{a}_{p}^{\dagger}\hat{a}_r\hat{a}_{q}^{\dagger}\hat{a}_{s}

The quantum many-body problem and Monte Carlo

|\psi\rangle=

|\psi\rangle=

Number of configurations increases exponentially with system size

Quantum Monte Carlo: sample properties without storing full wave functions

Exact simulations $\rightarrow$ fermion sign problem

\hat{H}|\psi\rangle = E|\psi\rangle

\hat{H}|\psi\rangle = E|\psi\rangle

i\dfrac{d |\psi\rangle}{d t} = \hat{H}|\psi\rangle

i\dfrac{d |\psi\rangle}{d t} = \hat{H}|\psi\rangle

Variational Monte Carlo

Stochastic multireference perturbation theory

Auxiliary field QMC

Outline

Sampling and the sign problem in free projection

Reducing noise using selected CI wave functions

Phaseless constraint and trial state bias

Symmetry projected states in VMC

Auxiliary field QMC

Variational MC

Free projection AFQMC

e^{-\tau (\hat{H}-E_0)}|\psi_r\rangle = c_0|\Psi_0\rangle + c_1 e^{-\Delta E_1\tau}|\Psi_1\rangle+\dots

e^{-\tau (\hat{H}-E_0)}|\psi_r\rangle = c_0|\Psi_0\rangle + c_1 e^{-\Delta E_1\tau}|\Psi_1\rangle+\dots

|\psi_r\rangle = c_0|\Psi_0\rangle + c_1|\Psi_1\rangle +\dots

|\psi_r\rangle = c_0|\Psi_0\rangle + c_1|\Psi_1\rangle +\dots

Better $|\psi_l\rangle$ and $|\psi_r\rangle$ approximate $|\Psi_0\rangle$ , faster the convergence with $\tau$

E(\tau)=\dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle}

E(\tau)=\dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle}

Mixed energy estimator:

Numerically exact but noise in QMC sampling worsens exponentially with $\tau$

Imaginary time propagation:

Sampling in AFQMC

\hat{H} = \hat{K} + \hat{V} = t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \frac{1}{2}v_{prqs}\hat{a}_{p}^{\dagger}\hat{a}_r\hat{a}_{q}^{\dagger}\hat{a}_{s}

\hat{H} = \hat{K} + \hat{V} = t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \frac{1}{2}v_{prqs}\hat{a}_{p}^{\dagger}\hat{a}_r\hat{a}_{q}^{\dagger}\hat{a}_{s}

Exponentiating $\hat{H}$ : $[\hat{K}, \hat{V}]\neq 0$

Exponentiating $\hat{K}$ : orbital transformation

e^{t_{pr}\hat{a}_p^{\dagger}\hat{a}_r}|\phi\rangle=|\phi'\rangle

e^{t_{pr}\hat{a}_p^{\dagger}\hat{a}_r}|\phi\rangle=|\phi'\rangle

where $|\phi\rangle$ and $|\phi'\rangle$ are nonorthogonal determinants.

Exponentiating $\hat{V} = \frac{1}{2}\sum_{\gamma} \left(L^{\gamma}_{pr}\hat{a}_p^{\dagger}\hat{a}_r\right)^2$ :

e^{-\frac{\hat{L}_{\gamma}^2}{2}} = \int \frac{dx_{\gamma}}{\sqrt{2\pi}}\ e^{\frac{-x_{\gamma}^2}{2}}e^{ix_{\gamma}\hat{L}_{\gamma}}

e^{-\frac{\hat{L}_{\gamma}^2}{2}} = \int \frac{dx_{\gamma}}{\sqrt{2\pi}}\ e^{\frac{-x_{\gamma}^2}{2}}e^{ix_{\gamma}\hat{L}_{\gamma}}

$x_{\gamma}$ : auxiliary field

(Thouless, 1960)

(Stratonovich, 1957)

Zhang, Krakauer, Reichman, Rubenstein, ...

E(\tau) = \dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle} \approx \dfrac{\int\ dX p(X)\langle\psi_l|\hat{H}\hat{\mathcal{B}}(X)|\psi_r\rangle}{\int\ dX p(X)\langle\psi_l|\hat{\mathcal{B}}(X)|\psi_r\rangle}

E(\tau) = \dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle} \approx \dfrac{\int\ dX p(X)\langle\psi_l|\hat{H}\hat{\mathcal{B}}(X)|\psi_r\rangle}{\int\ dX p(X)\langle\psi_l|\hat{\mathcal{B}}(X)|\psi_r\rangle}

Sample Gaussian auxiliary fields $X$ , propagate, and measure

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle}

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle}

Coupled cluster as $|\psi_r\rangle$ : sampling Slater determinants from CCSD

|\psi_r\rangle = \exp\left(t_{ikjl}\hat{a}_i^{\dagger}\hat{a}_k\hat{a}_j^{\dagger}\hat{a}_l\right)\exp\left(t_{ik}\hat{a}_i^{\dagger}\hat{a}_k\right)|\phi_0\rangle

|\psi_r\rangle = \exp\left(t_{ikjl}\hat{a}_i^{\dagger}\hat{a}_k\hat{a}_j^{\dagger}\hat{a}_l\right)\exp\left(t_{ik}\hat{a}_i^{\dagger}\hat{a}_k\right)|\phi_0\rangle

commuting ph excitations $\rightarrow$ no Trotter error

Benzene (30e, 102o), Hilbert space dimension ~ $10^{35}$

E(\tau)=\dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle}

E(\tau)=\dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle}

The sign problem

\text{Var}\left(\dfrac{\overline{N}}{\overline{D}}\right) \approx \dfrac{\text{Var}(\overline{N})}{\overline{D}^2} + \dfrac{\overline{N}^2\text{Var}(\overline{D})}{\overline{D}^4} - 2\dfrac{\overline{N}\text{Cov}(\overline{N},\overline{D})}{\overline{D}^3}

\text{Var}\left(\dfrac{\overline{N}}{\overline{D}}\right) \approx \dfrac{\text{Var}(\overline{N})}{\overline{D}^2} + \dfrac{\overline{N}^2\text{Var}(\overline{D})}{\overline{D}^4} - 2\dfrac{\overline{N}\text{Cov}(\overline{N},\overline{D})}{\overline{D}^3}

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle} = \dfrac{\overline{N}}{\overline{D}}

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle} = \dfrac{\overline{N}}{\overline{D}}

Contour shift:

e^{-\frac{y^2}{2}} = \int_{-\infty}^{\infty} \frac{dx}{\sqrt{2\pi}}\ e^{\frac{-x^2}{2}+ixy}

e^{-\frac{y^2}{2}} = \int_{-\infty}^{\infty} \frac{dx}{\sqrt{2\pi}}\ e^{\frac{-x^2}{2}+ixy}

x\rightarrow x+iy

x\rightarrow x+iy

x_{\gamma} \rightarrow x_{\gamma} + i \sqrt{\tau}\langle\hat{L}_{\gamma}\rangle

x_{\gamma} \rightarrow x_{\gamma} + i \sqrt{\tau}\langle\hat{L}_{\gamma}\rangle

In AFQMC:

Baer, Head-Gordon, Neuhauser (1998)

Zero variance principle

If $|\psi_l\rangle$ is the exact ground state, then $N$ and $D$ are perfectly correlated, $\langle\psi_0|\hat{H}|\phi_i\rangle = E_0 \langle\psi_0|\phi_i\rangle$ , and the energy estimator has zero variance. More accurate $|\psi_l\rangle\ \rightarrow\$ higher $\text{Cov}(N, D)$ .

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle} = \dfrac{\overline{N}}{\overline{D}}

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle} = \dfrac{\overline{N}}{\overline{D}}

\text{Var}\left(\dfrac{\overline{N}}{\overline{D}}\right) \approx \dfrac{\text{Var}(\overline{N})}{\overline{D}^2} + \dfrac{\overline{N}^2\text{Var}(\overline{D})}{\overline{D}^4} - 2\dfrac{\overline{N}\text{Cov}(\overline{N},\overline{D})}{\overline{D}^3}

\text{Var}\left(\dfrac{\overline{N}}{\overline{D}}\right) \approx \dfrac{\text{Var}(\overline{N})}{\overline{D}^2} + \dfrac{\overline{N}^2\text{Var}(\overline{D})}{\overline{D}^4} - 2\dfrac{\overline{N}\text{Cov}(\overline{N},\overline{D})}{\overline{D}^3}

Selected configuration interaction: put the most important configurations in the state using particle-hole excitations and optimize

|\psi_l\rangle = \sum_i^{N_c} c_i |\psi_i\rangle

|\psi_l\rangle = \sum_i^{N_c} c_i |\psi_i\rangle

Benzene (30e, 102o)

E(\tau)=\dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle}

E(\tau)=\dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle}

Selected CI local energy algorithm

E_L[\phi]=\dfrac{\langle\psi_l|\hat{H}|\phi\rangle}{\langle\psi_l|\phi\rangle},\ \text{two-body part: }\ L^{\gamma}_{pr}L^{\gamma}_{qs}\dfrac{\langle\psi_l|\hat{a}_p^{\dagger}\hat{a}_q^{\dagger}\hat{a}_s\hat{a}_r|\phi\rangle}{\langle\psi_l|\phi\rangle}

E_L[\phi]=\dfrac{\langle\psi_l|\hat{H}|\phi\rangle}{\langle\psi_l|\phi\rangle},\ \text{two-body part: }\ L^{\gamma}_{pr}L^{\gamma}_{qs}\dfrac{\langle\psi_l|\hat{a}_p^{\dagger}\hat{a}_q^{\dagger}\hat{a}_s\hat{a}_r|\phi\rangle}{\langle\psi_l|\phi\rangle}

Generalized Wick's theorem: consider $|\psi_l\rangle = c_{ptqu}\hat{a}_t^{\dagger}\hat{a}_p\hat{a}_u^{\dagger}\hat{a}_q|\psi_0\rangle$

Benzene (30e, 102o)

$[\text{Cu}_2\text{O}_2]^{2+}$ isomerization

$\Delta E = E(\text{bis}) - E(\text{peroxo})$

Method
DFT (UBLYP)	36.0
DFT (UB3LYP)	52.9
DFT (UMPW1K)	74.0
CCSD(T)	30.6
CR-CCSD(TQ)	33.8
DMRG-CT	27.1
ph-AFQMC (NOCI)	32.1
fp-AFQMC	24.1(6)

kcal/mol

(32e, 108o)

Phaseless AFQMC

Bias depends on the trial state used

There is a trade-off between bias and variance

Phaseless constraint elminates the sign problem at the expense of a bias

|\psi_0\rangle = \sum_{\phi} w_{\phi}\dfrac{|\phi\rangle}{\langle\psi_T|\phi\rangle}

|\psi_0\rangle = \sum_{\phi} w_{\phi}\dfrac{|\phi\rangle}{\langle\psi_T|\phi\rangle}

w_{\phi(\mathbf{x})} = \Bigg\vert\dfrac{\langle\psi_T|\phi(\mathbf{x})\rangle}{\langle\psi_T|\phi\rangle}e^{\mathbf{x}.\bar{\mathbf{x}}-\bar{\mathbf{x}}^2/2}\Bigg\vert\max(0, \cos(\Delta\theta))

w_{\phi(\mathbf{x})} = \Bigg\vert\dfrac{\langle\psi_T|\phi(\mathbf{x})\rangle}{\langle\psi_T|\phi\rangle}e^{\mathbf{x}.\bar{\mathbf{x}}-\bar{\mathbf{x}}^2/2}\Bigg\vert\max(0, \cos(\Delta\theta))

$\text{H}_{50}$ (50e, 50o)

Ni

Excited states of conjugated systems

Butadiene: (22e, 142o)

Nickel porphyrin: (122e, 406o)

Method
NEVPT2	6.72	6.74
CCSD	6.31	7.08
AFQMC / sCI	6.50(5)	6.67(5)
Exact*	6.2	6.5

eV

Method
CASSCF (4e, 4o)	3.8
CCSD	2.55
AFQMC / sCI (50k)	3.0(1)
AFQMC / sCI (100k)	2.8(1)
Experiment	2.3-2.4

eV

1\ ^1B_{2u}

1\ ^1B_{2u}

1\ ^1B_{u}

1\ ^1B_{u}

2\ ^1A_{g}

2\ ^1A_{g}

Other properties

Species	Exact	CCSD	ph-AFQMC
	0.986	0.991	0.985(2)
	0.990	0.992	0.986(2)
CO	0.090	0.099	0.086(3)

Can be evaluated as derivatives of energy

Derivatives can be calculated just as efficiently as energy (automatic differentiation)

a.u.

Small molecule dipole moment calculations:

NH $_3$

H $_2$ O

Outline

Sampling and the sign problem in free projection

Reducing noise using selected CI wave functions

Phaseless constraint and trial state bias

Symmetry projected states in VMC

Auxiliary field QMC

Variational MC

Variational Monte Carlo (VMC)

E = \dfrac{\langle \psi|H|\psi\rangle }{\langle \psi|\psi\rangle}

E = \dfrac{\langle \psi|H|\psi\rangle }{\langle \psi|\psi\rangle}

Strategy:

Parametrize the wave function: $|\psi(\mathbf{p})\rangle$ , choose initial $\mathbf{p}$

Calculate energy and gradient: Markov chain Monte Carlo

\ \dfrac{\langle \psi(\mathbf{p})|H|\psi(\mathbf{p})\rangle }{\langle \psi(\mathbf{p})|\psi(\mathbf{p})\rangle} = \sum_{\mathbf{n}} \rho_{\mathbf{n}} E_L[\mathbf{n}]

\ \dfrac{\langle \psi(\mathbf{p})|H|\psi(\mathbf{p})\rangle }{\langle \psi(\mathbf{p})|\psi(\mathbf{p})\rangle} = \sum_{\mathbf{n}} \rho_{\mathbf{n}} E_L[\mathbf{n}]

Optimize: smart gradient descent to change parameters

\propto|\langle \mathbf{n}|\psi(\mathbf{p})\rangle|^2

\propto|\langle \mathbf{n}|\psi(\mathbf{p})\rangle|^2

Ground state minimizes

\text{walker}: |\mathbf{n}\rangle

\text{walker}: |\mathbf{n}\rangle

McMillan (1965)

Symmetry projection in VMC

Symmetry breaking $\rightarrow$ more variational freedom

Break the symmetry under a projector, to retain good quantum numbers

|\psi\rangle=\hat{P}\hat{\mathcal{J}}|\phi\rangle

|\psi\rangle=\hat{P}\hat{\mathcal{J}}|\phi\rangle

Symmetries: spin, number, complex conjugation, ...

Example: complex conjugation in $\text{H}_2$ near dissociation

|\text{RHF}\rangle = (a_{1\uparrow}^{\dagger}+a_{2\uparrow}^{\dagger})(a_{1\downarrow}^{\dagger}+a_{2\downarrow}^{\dagger})|0\rangle = (a_{1\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{2\downarrow}^{\dagger})|0\rangle

|\text{RHF}\rangle = (a_{1\uparrow}^{\dagger}+a_{2\uparrow}^{\dagger})(a_{1\downarrow}^{\dagger}+a_{2\downarrow}^{\dagger})|0\rangle = (a_{1\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{2\downarrow}^{\dagger})|0\rangle

|\text{cRHF}\rangle = (e^{i\pi/4}a_{1\uparrow}^{\dagger}+e^{-i\pi/4}a_{2\uparrow}^{\dagger})(e^{i\pi/4}a_{1\downarrow}^{\dagger}+e^{-i\pi/4}a_{2\downarrow}^{\dagger})|0\rangle\\ \quad = (ia_{1\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}-ia_{2\uparrow}^{\dagger}a_{2\downarrow}^{\dagger})|0\rangle

|\text{cRHF}\rangle = (e^{i\pi/4}a_{1\uparrow}^{\dagger}+e^{-i\pi/4}a_{2\uparrow}^{\dagger})(e^{i\pi/4}a_{1\downarrow}^{\dagger}+e^{-i\pi/4}a_{2\downarrow}^{\dagger})|0\rangle\\ \quad = (ia_{1\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}-ia_{2\uparrow}^{\dagger}a_{2\downarrow}^{\dagger})|0\rangle

\hat{K}|\text{cRHF}\rangle = (a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger})|0\rangle

\hat{K}|\text{cRHF}\rangle = (a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger})|0\rangle

Imada, Sorella, Neuscamman, ...

\hat{\mathcal{J}} = \exp\left(\sum_{p\sigma,q\gamma} v_{p\sigma,q\gamma}\hat{n}_{p\sigma}\hat{n}_{q\gamma}\right)

\hat{\mathcal{J}} = \exp\left(\sum_{p\sigma,q\gamma} v_{p\sigma,q\gamma}\hat{n}_{p\sigma}\hat{n}_{q\gamma}\right)

Density-density correlation function: 18 site 2D Hubbard model ( $U/t=4$ )

d (Bohr)	Exact (DMRG)	Jastrow-SzPfaffian	Jastrow-KSzPfaffian
1.6	-0.5344	-0.5327(2)	-0.5337(2)
1.8	-0.5408	-0.5389(2)	-0.5400(2)
2.5	-0.5187	-0.5167(2)	-0.5180(2)

H $_{50}$ linear chain (50e, 50o)

Hartree/particle

Quantum spin liquid in $\text{Ba}_4\text{Ir}_3\text{O}_{10}$ ?

Insulator with T-linear heat capacity
Interactions ~ 500 K but orders at 0.2 K
Not geometrically frustrated

G. Cao, et al. (2020) 1901.04125

Possible in $\text{Ba}_4\text{Ir}_3\text{O}_{10}$ : $U(1)$ QSL with $|\psi_0\rangle$ a metallic free fermion state

Gutzwiller projection with VMC:

Savary, Balents (2016)

Summary and outlook

Combining ideas from QMC and quantum chemistry is a useful approach

Scalability and parallelizability mean they can be used for studying much larger correlated systems

QMC techniques are very flexible, so can be employed in a variety of problems

Possible future directions:

AFQMC: excited states, dynamics, solids,...

VMC: neural network states, spin liquids,...

Thank you to:

The committee

Sandeep and the Sharma group

Friends and family

Funding from NSF and CU Boulder

Teachers, mentors, collaborators

Stochastic electronic structure theory