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Best Slater approximation of a fermionic wave function

Alex D. Gottlieb, Norbert J. Mauser, J. M. Zhang

Published Oct 19, 2015• 0 citations
OpenAlex

Abstract

We study the best Slater approximation of an $N$-fermion wave function analytically. That is, we seek the Slater determinant (constructed out of $N$ orthonormal single-particle orbitals) wave function having largest overlap with a given $N$-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the GHZ state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the best Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the best Slater approximation of some special target wave functions.

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We study the best Slater approximation of an NNN-fermion wave function analytically. That is, we seek the Slater determinant (constructed out of NNN orthonormal single-particle orbitals) wave function having largest overlap with a given NNN-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the GHZ state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the best Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the best Slater approximation of some special target wave functions.
We study the optimal Slater-determinant approximation of an NNN-fermion wave function analytically. That is, we seek the Slater-determinant (constructed out of NNN orthonormal single-particle orbitals) wave function having largest overlap with a given NNN-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the Greenberger-Horne-Zeilinger state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the optimal Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the optimal Slater approximation of some special target wave functions.
The Hilbert space for three fermions in six orbitals, lately dubbed the "Borland-Dennis setting," is a proving ground for insights into electronic structure. Borland and Dennis discovered that, when referred to coordinate systems defined in terms of its natural orbitals, a wave function in the Borland-Dennis setting has the same structure as a 3-qubit state. By dint of the Borland-Dennis Theorem, canonical forms for 3-qubit states have analogs in the Borland-Dennis setting. One of these canonical forms is based upon "max-overlap Slater determinant approximations." Any max-overlap Slater determinant approximation of a given wave function is the leading term in a 5-term configuration interaction (CI) expansion of that wave function. Our main result is that "max-overlap CIS approximations" also lead to 5-term CI expansions, distinct from those based on max-overlap Slater determinant approximations, though of the same symmetric shape. We also prove the analog of this result for 3-qubit setting.
How many Slater determinants does an accurate variational description of interacting fermions require? Exact expansions in a finite basis need combinatorially many, and state-of-the-art fermionic neural quantum states stack growing numbers of them. We prove that, in the norms that govern variational calculations, at most two are needed, independently of the number of particles and of the target accuracy. A single universal Slater determinant-specified in advance, independent of both the system and the state-multiplied by a smooth bosonic wave function approximates any fermionic wave function in up to three spatial dimensions in the first-order Sobolev norm, which controls the variational energy. Reaching the second-order Sobolev norm-for Coulomb interactions, the domain of the Hamiltonian, which bounds the variance of the local energy at the core of variational Monte Carlo-requires at most one additional fixed determinant, and only in three dimensions. Antisymmetry therefore costs at most two universal determinants and no expressiveness: generalized Slater-Jastrow neural quantum states are variationally complete.
A new method for the simulation of ground-state properties of interacting fermions is introduced. A trial wave function, which is assumed to be a Slater determinant, is propagated to large imaginary times. The quantum many-body propagator is represented by a coherent superposition of single-particle propagators by means of a Hubbard-Stratonovich transformation. The resulting functional integral is performed by stochastic methods based on Langevin dynamics. Numerical stability is achieved by orthonormalizing the propagating single-particle orbitals entering the Slater determinant. The problem of the positiveness of the statistical weight is addressed and solved in most cases. Illustrative examples are given for the 1D and 2D Hubbard models.
We propose a simple iterative algorithm to construct the optimal multiconfiguration approximation of an NNN-fermion wave function. Namely, $M\ensuremath{\ge}N$ single-particle orbitals are sought iteratively so that the projection of the given wave function in the CMN{C}_{M}^{N}CMN​-dimensional configuration subspace is maximized. The algorithm has a monotonic convergence property and can be easily parallelized. The significance of the algorithm on the study of geometric entanglement in a multifermion system and its implication on the multiconfiguration time-dependent Hartree-Fock (MCTDHF) are discussed. The ground state and real-time dynamics of spinless fermions with nearest-neighbor interaction are studied using this algorithm, discussing several subtleties.