[Paper Review] Wavefunction matching for solving quantum many-body problems

Serdar Elhatisari, Lukas Bovermann+16 more|arXiv (Cornell University)|Oct 31, 2022

Nuclear physics research studies1 references4 citations

TL;DR

This paper introduces wave function matching, a novel method to overcome the sign problem in quantum Monte Carlo simulations of quantum many-body systems by transforming realistic Hamiltonians into equivalent forms that match the short-distance wave functions of simpler, sign-problem-free Hamiltonians. The approach enables accurate ab initio calculations of light and medium-mass nuclei, neutron matter, and nuclear matter using chiral effective field theory interactions, achieving good agreement with empirical data and offering new insights into nuclear saturation and binding energies.

ABSTRACT

Ab initio calculations play an essential role in our fundamental understanding of quantum many-body systems across many subfields, from strongly correlated fermions to quantum chemistry and from atomic and molecular systems to nuclear physics. One of the primary challenges is to perform accurate calculations for systems where the interactions may be complicated and difficult for the chosen computational method to handle. Here we address the problem by introducing a new approach called wavefunction matching. Wavefunction matching transforms the interaction between particles so that the wavefunctions up to some finite range match that of an easily computable interaction. This allows for calculations of systems that would otherwise be impossible due to problems such as Monte Carlo sign cancellations. We apply the method to lattice Monte Carlo simulations of light nuclei, medium-mass nuclei, neutron matter, and nuclear matter. We use high-fidelity chiral effective field theory interactions and find good agreement with empirical data. These results are accompanied by new insights on the nuclear interactions that may help to resolve long-standing challenges in accurately reproducing nuclear binding energies, charge radii, and nuclear matter saturation in ab initio calculations.

Motivation & Objective

• To address the severe sign problem in quantum Monte Carlo simulations of realistic nuclear Hamiltonians derived from chiral effective field theory.
• To develop a method that preserves physical observables while enabling efficient computation by matching short-distance wave functions to those of a simpler, sign-stable Hamiltonian.
• To enable ab initio calculations of light and medium-mass nuclei, neutron matter, and nuclear matter with high-fidelity chiral interactions.
• To provide new insights into long-standing challenges in nuclear many-body theory, such as nuclear binding energy and saturation point reproduction.
• To offer a general framework applicable beyond quantum Monte Carlo, compatible with various computational schemes in ab initio nuclear physics.

Proposed method

• Wave function matching applies a unitary transformation $ U $ to a realistic high-fidelity Hamiltonian $ H $, producing a new Hamiltonian $ H' = U^⁺ H U $, which is close to a simple, sign-stable Hamiltonian $ H^S $.
• The unitary transformation $ U $ is active only for inter-particle distances $ r < R $, ensuring that the ground state wave function $ \psi'_0(r) $ of $ H' $ matches $ \psi^S_0(r) $ of $ H^S $ in this region.
• For $ r > R $, the wave function $ \psi'_0(r) $ remains equal to the original $ \psi_0(r) $, preserving long-range physics and observables.
• The method enables a rapidly converging perturbative expansion in powers of $ H' - H^S $, facilitating efficient Monte Carlo sampling.
• The approach is implemented within the auxiliary field quantum Monte Carlo (AFMC) framework using lattice-regularized chiral effective field theory Hamiltonians at N3LO.
• The transformation avoids the need for pre-diagonalization techniques like the similarity renormalization group (SRG), while maintaining compatibility with standard QMC algorithms.

Experimental results

Research questions

• RQ1 Can wave function matching effectively mitigate the sign problem in ab initio quantum Monte Carlo simulations of nuclear many-body systems?
• RQ2 To what extent can wave function matching preserve physical observables while transforming a complex, sign-problematic Hamiltonian into a computationally tractable form?
• RQ3 How accurately can wave function matching reproduce nuclear binding energies, charge radii, and nuclear matter saturation using chiral effective field theory interactions?
• RQ4 What insights into nuclear interactions can be gained from the wave function matching transformation, particularly regarding short-range correlations and saturation mechanisms?
• RQ5 Can this method be generalized to other ab initio frameworks beyond quantum Monte Carlo?

Key findings

• Wave function matching successfully enables sign-problem-free quantum Monte Carlo simulations of light and medium-mass nuclei using chiral two- and three-nucleon interactions at N3LO.
• The method reproduces empirical binding energies of $^3$H and $^4$He with high accuracy, showing good agreement with the Tjon band correlation and experimental data.
• Nuclear matter saturation properties, including energy per particle and incompressibility, are well reproduced, providing new insights into the role of three-nucleon forces.
• The approach achieves convergence with a rapidly converging expansion in $ H' - H^S $, demonstrating computational efficiency and stability.
• The wave function matching transformation reveals that short-distance correlations are crucial for nuclear saturation, and that matching to a simple Hamiltonian preserves essential physics.
• The method avoids the need for symmetry-preserving approximations or pre-diagonalization, offering a more direct and flexible alternative to existing sign-problem mitigation techniques.