Quantum Monte Carlo approaches for strongly correlated systems

Describing correlated electrons

Lattice models: make assumptions about interactions

Ab initio descriptions: include general interactions

Strongly correlated electrons

\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 t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \sum v_{prqs}\hat{a}_{p}^{\dagger}\hat{a}_r\hat{a}_{q}^{\dagger}\hat{a}_{s}

Quantum many-body problem and Monte Carlo

|\psi\rangle=

Hilbert space dimension scales exponentially with system size

Quantum Monte Carlo: sample properties without storing full wave functions

Exact simulations \(\rightarrow\) fermion sign problem (exponential decay of signal to noise)

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

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

\langle \hat{O}\rangle = \frac{\langle\psi|\hat{O}|\psi\rangle}{\langle\psi|\psi\rangle}

Variational Monte Carlo

Stochastic multireference perturbation theory

Auxiliary field QMC

1902.07690 (2019), 1908.04423 (2020), 2008.06477 (2020)

1909.06935 (2019), 2008.00220 (2020)

2104.06597 (2021)

Outline

Sampling and the sign problem in AFQMC

Reducing noise using selected CI wave functions

Benchmark results

Jastrow symmetry projected states in VMC

Auxiliary field QMC

Variational MC

Ongoing work

Variational Monte Carlo (VMC)

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}]

Optimize: smart gradient descent to change parameters

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

Ground state minimizes

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

McMillan (1965)

Symmetry projected mean field states

Symmetry breaking \(\rightarrow\) more variational freedom*

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

Projection in VMC by restricting random walk to the symmetry sector

* in finite systems

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

Break symmetries under a projector, to retain good quantum numbers

Mean field state: eigenstate of a quadratic Hamiltonian

Antisymmetrized geminal power (AGP)

Breaking number symmetry

|\psi\rangle = P_N\exp\left(\sum_{p,q}F_{p\uparrow,q\downarrow}a_{p\uparrow}^{\dagger}a_{q\downarrow}^{\dagger}\right)|0\rangle=\left(\sum_{p,q}F_{p\uparrow,q\downarrow}a_{p\uparrow}^{\dagger}a_{q\downarrow}^{\dagger}\right)^{N/2}|0\rangle

\( F_{p\uparrow,q\downarrow} \rightarrow\) amplitude for the bond between \( p \) and \( q \)

BCS wavefunction in real space

Resonating valence bond state if double occupations are filtered out

H\( _8 \) linear chain (8e, 8o)

restoring \( S_z \) symmetry

restoring \( S_z \) and \( K \) symmetries

Jastrow factor

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

counts site occupations and suppresses spurious ionic configurations (double occupations)

also correlates doublons and holons: important for describing insulators

d (Bohr)	Exact (DMRG)	Jastrow-KSzPfaffian	Green's function MC
1.6	-0.5344	-0.5337	-0.5342
1.8	-0.5408	-0.5400	-0.5406
2.5	-0.5187	-0.5180	-0.5185

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

U	Benchmark energy	Jastrow- KSzGHF	Green's function MC
2	-1.1962	-1.1920	-1.1939
4	-0.8620	-0.8566	-0.8598
8	-0.5237	-0.5183	-0.5221

2D Hubbard: 98 sites (half filling)

Hartree/particle

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

Projection QMC

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

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

Better \(|\psi\rangle\) approximates \(|\Psi_0\rangle\), 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}

Mixed energy estimator:

Trial states: Selected CI, Jastrow, MPS, ...

Noise in QMC sampling worsens exponentially with \(\tau\) (sign problem)

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}

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

e^{-\tau\hat{H}}\approx\left(e^{-\frac{\tau}{N}\frac{\hat{K}}{2}}e^{-\frac{\tau}{N}\hat{V}}e^{-\frac{\tau}{N}\frac{\hat{K}}{2}}\right)^N

Exponentiating \(\hat{K}\): orbital transformation

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}}

\(x_{\gamma}\): auxiliary field

Motta and Zhang (2017), 1711.02242

(Thouless, 1960)

(Stratonovich, 1957)

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}

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}

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 \frac{dx}{\sqrt{2\pi}}\ e^{\frac{-x^2}{2}+ixy}

x\rightarrow x+iy

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

In AFQMC:

Baer, Head-Gordon, Neuhauser (1998)

Selected CI trial state as \(|\psi_l\rangle\)

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}}

\((\text{H}_2\text{O})_2\), (16e, 80o)

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

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